NASA Technical Memorandum 105834 

ICOMP-92-12; CMOIT-92-D8 

Center for Modeling of Turbulence and 

Transition (CMOTT) 

Research Briefs-1992 

September 1992 

NI\S/\ 

NASA Technical Memorandum 105834 

ICOMP-92-12; CMOIT-92-D8 

Center for Modeling of Turbulence and 

Transition (CMOTT) 

Research Briefs-1992 

September 1992 

NI\S/\ 
NASA Technical Memorandum 105834 

ICOMP -92-12; CMOIT -92-08 

Center for Modeling of Turbulence and 

Transition (CMOTT) 

Research Briefs-1992 

Compiled and Edited by 

William W. Liou 

Approved by 

T.-H Shih 

CMOIT Technical Leader 

and 

Louis A. Povinelli 

ICOMP Director 

September 1992 

NI\5/\ 

NASA Technical Memorandum 105834 

ICOMP -92-12; CMOIT -92-08 

Center for Modeling of Turbulence and 

Transition (CMOTT) 

Research Briefs-1992 

Compiled and Edited by 

William W. Liou 

Approved by 

T.-H Shih 

CMOIT Technical Leader 

and 

Louis A. Povinelli 

ICOMP Director 

September 1992 

NI\5/\ 
[------ - -

NASA Technical Memorandum 105834 

ICOMP -92-12; CMOTT -92-08 

Center for Modeling of Turbulence and 

Transition (CMOTT) 

Research Briefs-1992 

Compiled and Edited by 

William W. Liou 

Approved by 

T.-H Shih 

CMOTT Technical Leader 

and 

Louis A. Povinelli 

ICOMP Director 

September 1992 

NI\5/\ 

Preceding Page Blank 

-.  . . - --

[------ - -

NASA Technical Memorandum 105834 

ICOMP -92-12; CMOTT -92-08 

Center for Modeling of Turbulence and 

Transition (CMOTT) 

Research Briefs-1992 

Compiled and Edited by 

William W. Liou 

Approved by 

T.-H Shih 

CMOTT Technical Leader 

and 

Louis A. Povinelli 

ICOMP Director 

September 1992 

NI\5/\ 

Preceding Page Blank 

-.  . . - --
Preceding Page Blank 

v 
C ente r f or Mode ling o f Turbul en ce and Transition 

R e s earch Brie fs -1991 

Table of Contents 

Page 

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vll 

Lumley's Energy Cascade Dissipation Rate Model for Boundary-Free 

Turbulent Shear Flows 

Beverly Duncan 1 

PDF Turbulence Modeling and DNS 

Andrew T. Hsu . . . . . . . . . . . . . .. . . . . . .. . . . . 17 

Modeling of Turbulen ce and Transition 

Tsan-Hsing Shih . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 

Turbulence Mode ling and Experiments 

Aamir Shabbir ..... . . . .. .. . . . .... .... .. 51 

Modeling of Turbulent Shear Flows 

William W. Liou . ...... . . . .... ... .. . . . .. .. 67 

Modeling of Near Wall Turbulence and Modeling of Bypass Transition 

Zhigang Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 

Two Equation Modelling and the Pseudo Compressibility Effects 

Christopher J. Steffen, Jr. .... .. . . . .. . . . . . ..... . 95 

Renormalization Group Methods for the Reynolds Stress 

Transport Equations 

Robert Rubinstein . . . . . . . . . . . . . . . . . . 

Bypass Transition in Compressible Boundary Layers 

Jacobus J. Van der Vegt . ..... . ....... . 

Runge-Kutta Met hods Combined with Compact Difference Schemes 

for the Unsteady Euler Equations 

101 

115 

Sheng-Tao Yu . . . . . . . . . . . . . . . . . . . . . . . . . .. 127 

Appendix A : Organization - 1991 163 

Appendix B : CMOTT Biweeky Seminars/Technical Meeting 167 

Appendix C : List of M ember's Publications 187 

VI 

C ente r f or Mode ling o f Turbul en ce and Transition 

R e s earch Brie fs -1991 

Table of Contents 

Page 

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vll 

Lumley's Energy Cascade Dissipation Rate Model for Boundary-Free 

Turbulent Shear Flows 

Beverly Duncan 1 

PDF Turbulence Modeling and DNS 

Andrew T. Hsu . . . . . . . . . . . . . .. . . . . . .. . . . . 17 

Modeling of Turbulen ce and Transition 

Tsan-Hsing Shih . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 

Turbulence Mode ling and Experiments 

Aamir Shabbir ..... . . . .. .. . . . .... .... .. 51 

Modeling of Turbulent Shear Flows 

William W. Liou . ...... . . . .... ... .. . . . .. .. 67 

Modeling of Near Wall Turbulence and Modeling of Bypass Transition 

Zhigang Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 

Two Equation Modelling and the Pseudo Compressibility Effects 

Christopher J. Steffen, Jr. .... .. . . . .. . . . . . ..... . 95 

Renormalization Group Methods for the Reynolds Stress 

Transport Equations 

Robert Rubinstein . . . . . . . . . . . . . . . . . . 

Bypass Transition in Compressible Boundary Layers 

Jacobus J. Van der Vegt . ..... . ....... . 

Runge-Kutta Met hods Combined with Compact Difference Schemes 

for the Unsteady Euler Equations 

101 

115 

Sheng-Tao Yu . . . . . . . . . . . . . . . . . . . . . . . . . .. 127 

Appendix A : Organization - 1991 163 

Appendix B : CMOTT Biweeky Seminars/Technical Meeting 167 

Appendix C : List of M ember's Publications 187 

VI 
L_ 

Center lor Modeling of Turbulence and Transition 

Research Briefs· 1991 

Preface 

This research brief contains the progress reports of the Research Staff of the 

Center for Modeling of Turbulence and Transition (CMOTT) from May 1991 to May 

1992. It is intended as an annual report to the Institute for Computational Mechanics 

in Propulsion and NASA Lewis Research Center. A separate report entitled, "Workshop on Engineering Turbulence Modeling," covering some of the 1991 CMOTT 

Summer research activities was released earlier this year. 

The main objective of the CMOTT is to develop, validate and implement the 

turbulence and transition models for practical engineering flows. The flows of interest 

are three-dimensional, incompressible and compressible flows with chemical reaction. 

During this period, the research covers two-equation (e.g., IC-g) and algebraic Reynoldsstress models, second moment closure models, probability density function (pdf) 

models, Renormalization Group Theory (RNG), Large Eddy Simulation (LES) and 

Direct Numerical Simulation (DNS). Last year was CMOTT's second year in 

operation. CMOTT now has eleven members from ICOMP, NASA LeRC, and 

Sverdrup Technology Inc., working on various aspects of turbulence and transition 

modeling in collaboration with NASA-Lewis scientists and Case Western Reserve 

University faculty members. The CMOTT members have been continuously and 

actively involved in the international and national turbulence research activities. A 

biweekly CMOTT seminar series has been conducted with speakers invited from within 

and without of the NASA Lewis Research Center, including foreign speakers. The 

current CMOTT roster and its organization are listed in Appendix A. Listed in 

Appendix B are the abstracts and the scientific and technical issues discussed in 

biweekly CMOTT seminars. Appendix C gives a list of references which are the papers 

contributed by CMOTT members in the last two years. 

Tsan-Hsing-Shih 

Vll 

L_ 

Center lor Modeling of Turbulence and Transition 

Research Briefs· 1991 

Preface 

This research brief contains the progress reports of the Research Staff of the 

Center for Modeling of Turbulence and Transition (CMOTT) from May 1991 to May 

1992. It is intended as an annual report to the Institute for Computational Mechanics 

in Propulsion and NASA Lewis Research Center. A separate report entitled, "Workshop on Engineering Turbulence Modeling," covering some of the 1991 CMOTT 

Summer research activities was released earlier this year. 

The main objective of the CMOTT is to develop, validate and implement the 

turbulence and transition models for practical engineering flows. The flows of interest 

are three-dimensional, incompressible and compressible flows with chemical reaction. 

During this period, the research covers two-equation (e.g., IC-g) and algebraic Reynoldsstress models, second moment closure models, probability density function (pdf) 

models, Renormalization Group Theory (RNG), Large Eddy Simulation (LES) and 

Direct Numerical Simulation (DNS). Last year was CMOTT's second year in 

operation. CMOTT now has eleven members from ICOMP, NASA LeRC, and 

Sverdrup Technology Inc., working on various aspects of turbulence and transition 

modeling in collaboration with NASA-Lewis scientists and Case Western Reserve 

University faculty members. The CMOTT members have been continuously and 

actively involved in the international and national turbulence research activities. A 

biweekly CMOTT seminar series has been conducted with speakers invited from within 

and without of the NASA Lewis Research Center, including foreign speakers. The 

current CMOTT roster and its organization are listed in Appendix A. Listed in 

Appendix B are the abstracts and the scientific and technical issues discussed in 

biweekly CMOTT seminars. Appendix C gives a list of references which are the papers 

contributed by CMOTT members in the last two years. 

Tsan-Hsing-Shih 

Vll 
Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Lumley's Energy Cascade Dissipation Rate 

Model for Boundary-Free Turbulent Shear Flows 

B. S. Duncan 

1. Motivation and Objective 

True dissipation occurs mainly at the highest wavenumbers where the eddy sizes 

are comparatively small. These high wave numbers receive their energy through 

the spectral cascade of energy starting with the largest eddies spilling energy into 

the smaller eddies, passing through each wavenumber until it is dissipated at the 

microscopic scale. However, a small percentage of the energy does not spill continuously through the cascade but is instantly passed to the higher wave numbersl . 

Consequently, the smallest eddies receive a certain amount of energy almost immediately. As the spectral energy cascade continues, the highest wave number needs 

a certain time to receive all the energy which has been transferred from the largest 

eddies. As such, there is a time delay, of the order T, between the generation of 

energy by the largest eddies and the eventual dissipation of this energy. 

For equilibrium turbulence at high Reynolds numbers, there is a wide range where 

energy is neither produced by the large eddies nor dissipated by viscosity, but is 

conserved and passed from wavenumber to higher wavenumber. The rate at which 

energy cascades from one wavenumber to another is proportional to the energy 

contained within that wave number. This rate is constant and has been used in 

the past as a dissipation rate of turbulent kinetic energy. However, this is true 

only in steady, equilibrium turbulence. Most dissipation models contend that the 

production of dissipation is proportional the production of energy and that the 

destruction of dissipation is proportional to the destruction of energy. In essence, 

these models state that the change in the dissipation rate is proportional to the 

change in the kinetic energy. This assumption is obviously incorrect for the case 

where there is no production of turbulent energy, yet energy continues to cascade · 

from large to small eddies. If the time lag between the onset on the energy cascade 

to the destruction of energy at the microscale can be modeled, then there will be a 

better representation of the dissipation process. Development of an energy cascade 

time scale equation will be discussed in Section 2.2. 

2. Work Accomplished 

2.1 Mean Flow Equations 

For incompressible flow, the equations for continuity of mass and axial momentum 

are written as 

and 

1 

Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Lumley's Energy Cascade Dissipation Rate 

Model for Boundary-Free Turbulent Shear Flows 

B. S. Duncan 

1. Motivation and Objective 

True dissipation occurs mainly at the highest wavenumbers where the eddy sizes 

are comparatively small. These high wave numbers receive their energy through 

the spectral cascade of energy starting with the largest eddies spilling energy into 

the smaller eddies, passing through each wavenumber until it is dissipated at the 

microscopic scale. However, a small percentage of the energy does not spill continuously through the cascade but is instantly passed to the higher wave numbersl . 

Consequently, the smallest eddies receive a certain amount of energy almost immediately. As the spectral energy cascade continues, the highest wave number needs 

a certain time to receive all the energy which has been transferred from the largest 

eddies. As such, there is a time delay, of the order T, between the generation of 

energy by the largest eddies and the eventual dissipation of this energy. 

For equilibrium turbulence at high Reynolds numbers, there is a wide range where 

energy is neither produced by the large eddies nor dissipated by viscosity, but is 

conserved and passed from wavenumber to higher wavenumber. The rate at which 

energy cascades from one wavenumber to another is proportional to the energy 

contained within that wave number. This rate is constant and has been used in 

the past as a dissipation rate of turbulent kinetic energy. However, this is true 

only in steady, equilibrium turbulence. Most dissipation models contend that the 

production of dissipation is proportional the production of energy and that the 

destruction of dissipation is proportional to the destruction of energy. In essence, 

these models state that the change in the dissipation rate is proportional to the 

change in the kinetic energy. This assumption is obviously incorrect for the case 

where there is no production of turbulent energy, yet energy continues to cascade · 

from large to small eddies. If the time lag between the onset on the energy cascade 

to the destruction of energy at the microscale can be modeled, then there will be a 

better representation of the dissipation process. Development of an energy cascade 

time scale equation will be discussed in Section 2.2. 

2. Work Accomplished 

2.1 Mean Flow Equations 

For incompressible flow, the equations for continuity of mass and axial momentum 

are written as 

and 

1 
B. S. Duncan 

where, UiUj is the turbulent Reynolds stress tensor. Using the eddy viscosity 

concept2 , the Reynolds stress may be related to the mean strain rate and a turbulent 

eddy viscosity, 

-U 'U' = lit (aUi + aUi) - ~kb .. t J ax . ax . 3 tJ' 

J J 

(1) 

The turbulent viscosity given in the above equation can be interpreted as a measure 

of the turbulent kinetic energy existing in the flow times a local length scale, 

1 

lit = Ck2l 

where C is an arbitrary constant. The definition used for this length scale is the primary discriminating factor between turbulence models and determines the number 

of equations which need to be solved. The length scale can be written in terms of 

the turbulent kinetic energy,k, and its dissipation rate,€, 

k~ l -- - . € 

Now, the momentum equation can be written as 

where, 

(2) 

and cp. is a constant. The k and € are determined by solving transport equations 

for the turbulent kinetic energy and the dissipation. These equations and the modification to the "standard" dissipation equation will be discussed in Section 2.2. 

2.2 Turbulence Equations 

Taking the N avier-Stokes equations and multiplying by Ui = Ui + Ui and then 

taking the time average yields an equation that contains both the energy equation 

for the mean velocity, Ui , and a mean energy equation for the fluctuating component 

of velocity,ui. Eliminating the energy equation for the mean velocity yields the 

turbulent energy budget in the absence of a pressure gradient field3 , 

Dk = _~ (~U U' - 2I1u 's' .) - U·U ·S ·· - 211-s ' ·s·· Dt ax . 2 t t J t tJ t J tJ tJ tJ' 

J 

(3) 

The last term in equation (3) is the rate at which viscous forces perform deformation 

work to the fluctuating strain rate, defined as 

. . _ ~ (aUj aUi ) 

StJ - 2 a + a . Xi Xj 

2 

B. S. Duncan 

where, UiUj is the turbulent Reynolds stress tensor. Using the eddy viscosity 

concept2 , the Reynolds stress may be related to the mean strain rate and a turbulent 

eddy viscosity, 

-U 'U' = lit (aUi + aUi) - ~kb .. t J ax . ax . 3 tJ' 

J J 

(1) 

The turbulent viscosity given in the above equation can be interpreted as a measure 

of the turbulent kinetic energy existing in the flow times a local length scale, 

1 

lit = Ck2l 

where C is an arbitrary constant. The definition used for this length scale is the primary discriminating factor between turbulence models and determines the number 

of equations which need to be solved. The length scale can be written in terms of 

the turbulent kinetic energy,k, and its dissipation rate,€, 

k~ l -- - . € 

Now, the momentum equation can be written as 

where, 

(2) 

and cp. is a constant. The k and € are determined by solving transport equations 

for the turbulent kinetic energy and the dissipation. These equations and the modification to the "standard" dissipation equation will be discussed in Section 2.2. 

2.2 Turbulence Equations 

Taking the N avier-Stokes equations and multiplying by Ui = Ui + Ui and then 

taking the time average yields an equation that contains both the energy equation 

for the mean velocity, Ui , and a mean energy equation for the fluctuating component 

of velocity,ui. Eliminating the energy equation for the mean velocity yields the 

turbulent energy budget in the absence of a pressure gradient field3 , 

Dk = _~ (~U U' - 2I1u 's' .) - U·U ·S ·· - 211-s ' ·s·· Dt ax . 2 t t J t tJ t J tJ tJ tJ' 

J 

(3) 

The last term in equation (3) is the rate at which viscous forces perform deformation 

work to the fluctuating strain rate, defined as 

. . _ ~ (aUj aUi ) 

StJ - 2 a + a . Xi Xj 

2 
• 

Lumley's Energy Cascade Dissipation Rate Model 

This term is the true viscous dissipation rate which drains kinetic energy from the 

system, € = 2VSijSij. This dissipation rate is important to the overall turbulence 

structure. The next to the last term in equation (3) serves to transport kinetic 

energy between the mean flow and the turbulence. It is generally referred to as the 

kinetic energy production term. Writing the mean st.rain, Sij, as 

the production term for incompressible flow is 'defined as 

(4) 

The last set of terms to be modeled represents the energy redistribution by viscous 

forces and are contained within the parenthesis on the right hand side of equation 

(3). From the definition of the fluctuating strain rate, 

. ok 

2VUiSij = vaxJ ' 

and using a mean gradient hypothesis, the triple correlation can be rewritten as 

Vt ok 

UiUiUj = ---

Uk aXj 

where Uk is a constant of the order one. Combining all the$e modeling assumptions 

yields the high Reynolds number form of the turbulent kinetic equation, 

Dk = ~ ((v+!:!) ~) + lit (aUi + aUj ) aUi _ EDt aXj Uk aXj aXj aXi aXj 

An equation of the following form has been widely used for modeling of the 

dissipation, . 

~: = a~j ((II + ;:) :X€j) + P€ - ~ ~ . 

The production term, Pf. has been assumed to be proportional to the turbulent 

kinetic energy production term, equation (4). This approximation places a direct correlation between the production of turbulent energy and the production of 

dissipation. Although this may be true at equilibrium, where by definition the dissipation equals the turbulent energy production, there are some deficiencies in this 

statement for the more general case. According to Lumley!, the modeled dissipation 

rate equation should be written as, 

(5) 

3 

- , 

• 

Lumley's Energy Cascade Dissipation Rate Model 

This term is the true viscous dissipation rate which drains kinetic energy from the 

system, € = 2VSijSij. This dissipation rate is important to the overall turbulence 

structure. The next to the last term in equation (3) serves to transport kinetic 

energy between the mean flow and the turbulence. It is generally referred to as the 

kinetic energy production term. Writing the mean st.rain, Sij, as 

the production term for incompressible flow is 'defined as 

(4) 

The last set of terms to be modeled represents the energy redistribution by viscous 

forces and are contained within the parenthesis on the right hand side of equation 

(3). From the definition of the fluctuating strain rate, 

. ok 

2VUiSij = vaxJ ' 

and using a mean gradient hypothesis, the triple correlation can be rewritten as 

Vt ok 

UiUiUj = ---

Uk aXj 

where Uk is a constant of the order one. Combining all the$e modeling assumptions 

yields the high Reynolds number form of the turbulent kinetic equation, 

Dk = ~ ((v+!:!) ~) + lit (aUi + aUj ) aUi _ EDt aXj Uk aXj aXj aXi aXj 

An equation of the following form has been widely used for modeling of the 

dissipation, . 

~: = a~j ((II + ;:) :X€j) + P€ - ~ ~ . 

The production term, Pf. has been assumed to be proportional to the turbulent 

kinetic energy production term, equation (4). This approximation places a direct correlation between the production of turbulent energy and the production of 

dissipation. Although this may be true at equilibrium, where by definition the dissipation equals the turbulent energy production, there are some deficiencies in this 

statement for the more general case. According to Lumley!, the modeled dissipation 

rate equation should be written as, 

(5) 

3 

- , 
B. S. Duncan 

The production term, p~, now represents the rate at which energy arrives at the dissipative wavenumbers through the energy cascade starting with low wavenumbers. 

In eq~ation (5), 1/ S represents the characteristic time for energy to be transported 

to the dissipative wave numbers. 

In the following equation, the term on the left hand side and the first term on the 

right represent the transport of the inverse time scale by advection and diffusion, 

respectively. The remaining terms are the production from the mean straining 

forces within the large scale mean flow structures and the dissipative losses at the 

microscale, 

The production and loss terms have been scaled by a characteristic time, T, which 

in this study has been taken to be a weighted function of the time scale for the large 

eddies, T, 

T 

T = 2GB ' 

Since most energy enters the cascade at scales comparable to the integral scale, l, 

the characteristic time scale of the large eddies dictates the rate of energy transfer 

and is defined as T = ~ for nearly isotropic turbulence. Using the approximation 

for the integral length scale as, 1 = ~3 , the time scale becomes 

k 

T = 2-

E 

which is valid even in nonequilibrium flows. Letting GB be a function of mean and 

turbulent quantities, the inverse time scale equation can be written as 

where 

GB = 1 + 9kJSi

jSi

j . 

E 

Determination of the modeling constants is discussed in the next section. 

2.3 Model Constants 

In this paper, only high Reynolds number turbulence has been considered. The 

model constants are examined for the grid turbulence case where the mean velocity 

gradients and strain rates vanish. For this case, turbulent intensity decays with the 

following proportionality, 

(~) ~ (t:) -n, where, t = ;~ 

4 

I 

) 

J 

B. S. Duncan 

The production term, p~, now represents the rate at which energy arrives at the dissipative wavenumbers through the energy cascade starting with low wavenumbers. 

In eq~ation (5), 1/ S represents the characteristic time for energy to be transported 

to the dissipative wave numbers. 

In the following equation, the term on the left hand side and the first term on the 

right represent the transport of the inverse time scale by advection and diffusion, 

respectively. The remaining terms are the production from the mean straining 

forces within the large scale mean flow structures and the dissipative losses at the 

microscale, 

The production and loss terms have been scaled by a characteristic time, T, which 

in this study has been taken to be a weighted function of the time scale for the large 

eddies, T, 

T 

T = 2GB ' 

Since most energy enters the cascade at scales comparable to the integral scale, l, 

the characteristic time scale of the large eddies dictates the rate of energy transfer 

and is defined as T = ~ for nearly isotropic turbulence. Using the approximation 

for the integral length scale as, 1 = ~3 , the time scale becomes 

k 

T = 2-

E 

which is valid even in nonequilibrium flows. Letting GB be a function of mean and 

turbulent quantities, the inverse time scale equation can be written as 

where 

GB = 1 + 9kJSi

jSi

j . 

E 

Determination of the modeling constants is discussed in the next section. 

2.3 Model Constants 

In this paper, only high Reynolds number turbulence has been considered. The 

model constants are examined for the grid turbulence case where the mean velocity 

gradients and strain rates vanish. For this case, turbulent intensity decays with the 

following proportionality, 

(~) ~ (t:) -n, where, t = ;~ 

4 

I 

) 

J 
Lumley's Energy Cascade Dissipation Rate Model 

so that the dissipation rate must decay as c:) ex c:) -(n+l) 

Here, n is the decay rate and is typically in the range 1.1 ::s: n ::s: 1.3. Substituting 

these relationships into the inverse time scale equation yields 

.!dS = - nCB dt. 

S tCB 

After integration, this equation is 

where a is a constant of integration. For there to be an inverse scaling relationship 

between Sand t, the exponent in the above equation must reduce to -1 for the grid 

turbulence case. Since the coefficient CB reduces to 1.0 in grid turbulence, then 

CB = n. 

In this study, n has been set to 1.1 and held constant4, which sets the value of CB. 

Now using the fact that, initially, 

3 

Soto = M 

2v2 

the coefficients CA and CD are related as follows 

1 3 CA n + 1 

-=--+--. 

CD 2/2 CD n 

The coefficient cJ.L has been evaluated for equilibrium turbulence where, where 

Pk = E. Rewriting equation (2) as 

litE 

CJ.L = k2 

and using equations (1) and (4), this coefficients is 

- -2 U i U ' 

cJ.L = T =0.09. 

After the above analysis, the only adjustable parameters are .£A. and the U coeffi- CD 

cients, which should be of the order one. The complete set of optimized coefficients 

is given in Table 1. 

Uk = 0.9 

U € = 1.1 

Us = 1.0 

cJ.L = 0.09 

.£A. = 1.13 CD 

n = 1.1 

CB = n 

CB = 1 + 9kVS:;S:; € 

_1 _ _ 3_ .£A. + !ttl CD - 2V2 CD n 

Table 1. k- E-S Model Coefficients 

5 

---------- ---

Lumley's Energy Cascade Dissipation Rate Model 

so that the dissipation rate must decay as c:) ex c:) -(n+l) 

Here, n is the decay rate and is typically in the range 1.1 ::s: n ::s: 1.3. Substituting 

these relationships into the inverse time scale equation yields 

.!dS = - nCB dt. 

S tCB 

After integration, this equation is 

where a is a constant of integration. For there to be an inverse scaling relationship 

between Sand t, the exponent in the above equation must reduce to -1 for the grid 

turbulence case. Since the coefficient CB reduces to 1.0 in grid turbulence, then 

CB = n. 

In this study, n has been set to 1.1 and held constant4, which sets the value of CB. 

Now using the fact that, initially, 

3 

Soto = M 

2v2 

the coefficients CA and CD are related as follows 

1 3 CA n + 1 

-=--+--. 

CD 2/2 CD n 

The coefficient cJ.L has been evaluated for equilibrium turbulence where, where 

Pk = E. Rewriting equation (2) as 

litE 

CJ.L = k2 

and using equations (1) and (4), this coefficients is 

- -2 U i U ' 

cJ.L = T =0.09. 

After the above analysis, the only adjustable parameters are .£A. and the U coeffi- CD 

cients, which should be of the order one. The complete set of optimized coefficients 

is given in Table 1. 

Uk = 0.9 

U € = 1.1 

Us = 1.0 

cJ.L = 0.09 

.£A. = 1.13 CD 

n = 1.1 

CB = n 

CB = 1 + 9kVS:;S:; € 

_1 _ _ 3_ .£A. + !ttl CD - 2V2 CD n 

Table 1. k- E-S Model Coefficients 

5 

---------- ---
B . S. Duncan 

2.4 Free Shear F lows 

The three turbulent transport equations are solved implicitly using the finite 

difference technique of Spalding5 for parabolic equation systems. An initial profile 

is imposed upon the first marching plane for the turbulent kinetic energy and the 

dissipation, and S is initially set to S, = ~. In the mixing layer and jet cases, 

the initial plane was equally divided into a uniform velocity field and still air. For 

the case of the plane wake, this initial profile came from a flat plate calculation 

computed with the same solution technique. Starting with the initial plane, the flow 

field is developed by marching the transport equations downstream and applying 

the following boundary conditions, 

Planar Jet 

Centerline (y = 0) 8k = 0 8E = 0 8y , 8 y 

S - SoldCbkUII=o+/v((v+vd¥v-) 

y= O - CbkUII=O+EA x 

Far Field (y = Ymax) k = 0 , f = 0 , S = 0 

Round Jet 

Centerline (r = 0) 

Far Field (r = rmax) 

P lanar Mixing Layer 

Centerline (y = Ymin) U2 k . = 0.015, U3 EL = 3.5 X 10-6 

lI =lIm.1-n lI=lITnin 

S - E 

y = ymin - k 

Far Field (y = Ymax) k = 0 , f = 0 , S = 0 

Planar Wake 

Centerline (y = 0) 8k - 0 8E - 0 8y - 8y -

S _ = S oldCbkUII=O+!v (Cv+vd-¥v) 

y-O CbkUII=O+EAX 

Far Field (y = Ymax) k = 0 , f = 0 , S = 0 

The solutions have been checked after several hundred steps to insure that the 

profiles have become self-similar. 

2.4.1 Planar Jet 

The three equation k-f-S model performed very well for planar jet flow when 

compared to the experimental data of Gutman and Wygnanski6 , Bradbury7 and 

Heskestad8 . Mean velocities profiles given in Figure 1 are well predicted by both the 

k-f-S model and the standard k-f model, which is reassuring since the coefficients 

for both of these models were optimized for the planar jet case. The spreading rates 

predicted by both models are within the experimental range and are given in Table 

2. 

There is a slight discrepancy between the k-E-S model and the standard k-E 

model for centerline kinetic energy as can be seen in Figure 2. However, both 

predictions are well within the range of experimental measurements, which show 

a considerable spread near the centerline of the jet. Both predictions are closely 

aligned with the data of Bradbury7. 

6 

B . S. Duncan 

2.4 Free Shear F lows 

The three turbulent transport equations are solved implicitly using the finite 

difference technique of Spalding5 for parabolic equation systems. An initial profile 

is imposed upon the first marching plane for the turbulent kinetic energy and the 

dissipation, and S is initially set to S, = ~. In the mixing layer and jet cases, 

the initial plane was equally divided into a uniform velocity field and still air. For 

the case of the plane wake, this initial profile came from a flat plate calculation 

computed with the same solution technique. Starting with the initial plane, the flow 

field is developed by marching the transport equations downstream and applying 

the following boundary conditions, 

Planar Jet 

Centerline (y = 0) 8k = 0 8E = 0 8y , 8 y 

S - SoldCbkUII=o+/v((v+vd¥v-) 

y= O - CbkUII=O+EA x 

Far Field (y = Ymax) k = 0 , f = 0 , S = 0 

Round Jet 

Centerline (r = 0) 

Far Field (r = rmax) 

P lanar Mixing Layer 

Centerline (y = Ymin) U2 k . = 0.015, U3 EL = 3.5 X 10-6 

lI =lIm.1-n lI=lITnin 

S - E 

y = ymin - k 

Far Field (y = Ymax) k = 0 , f = 0 , S = 0 

Planar Wake 

Centerline (y = 0) 8k - 0 8E - 0 8y - 8y -

S _ = S oldCbkUII=O+!v (Cv+vd-¥v) 

y-O CbkUII=O+EAX 

Far Field (y = Ymax) k = 0 , f = 0 , S = 0 

The solutions have been checked after several hundred steps to insure that the 

profiles have become self-similar. 

2.4.1 Planar Jet 

The three equation k-f-S model performed very well for planar jet flow when 

compared to the experimental data of Gutman and Wygnanski6 , Bradbury7 and 

Heskestad8 . Mean velocities profiles given in Figure 1 are well predicted by both the 

k-f-S model and the standard k-f model, which is reassuring since the coefficients 

for both of these models were optimized for the planar jet case. The spreading rates 

predicted by both models are within the experimental range and are given in Table 

2. 

There is a slight discrepancy between the k-E-S model and the standard k-E 

model for centerline kinetic energy as can be seen in Figure 2. However, both 

predictions are well within the range of experimental measurements, which show 

a considerable spread near the centerline of the jet. Both predictions are closely 

aligned with the data of Bradbury7. 

6 
Lumley)s Energy Cascade Dissipation Rate Model 

Shear stress is also well predicted (see Figure 3) where the standard k - E model 

over-predicts the shear stress at the edge of the jet in comparison to the data of 

Gutman and Wynanski6 and Bradbury7. The k- E-S model predicts a narrower 

profile, also indicated by the smaller spreading rate, which agrees with Bradbury's 

measurements. Towards the centerline of the jet, both models accurately predict 

the increase in shear stress and the extrema near ~ = 0.08. 

For the next two figures, Figures 4 and 5, there is no experimental data available 

for comparison. The dissipation reaches a maximum at the location of the peak 

shear stress, Figure 4. From this extrema, the dissipation decreases toward the 

centerline and also eventually trails off to zero at the edge of the jet. Figure 5 

shows the behavior of the time scale for the planar jet. Notice that the minimum 

energy transfer time between the large scale structure to the dissipative microscale 

is in the middle of the jet (where S is a maximum since the dimension of S is t-I ) . 

2.4.2 Axisymmetric Jet 

A primary problem with the standard k-f. model is its predictive capability for 

a 2-D versus a 3-D jet, the free jet anomaly. In Figures 6 - 10, the standard k- f. 

model is unable to predict any of the turbulence quantities correctly, overpredicting 

not only the turbulent kinetic energy and the shear stress but also the mean velocity. 

The standard k-E can be "fixed" using a suggestion by Pope9 . Writing the standard 

E equation for reference, 

the correct spreading rate can be obtained by changing Cl from its standard value of 

1.45 to 1.6 (see Table 2). Clearly, by modifying a coefficient, the standard k-E model 

is capable of predicting the correct spreading rate; however, this model cannot be 

considered general. The spreading rate predicted by the k-f.-S model is very close 

to the experimentally measured rate and has been obtained with no modifications 

to the model. . 

Comparisons between the k-f.-S model and the experimental data10,1l,12, Figures 

6 - 10, indicates that the new model accurately predi~ts the mean velocity profiles 

and the shearing stress across the jet. Less accurately predicted by the k-f.-S model 

is the centerline turbulent kinetic energy, Figure 7, where there is considerable 

scatter in the experimental data. 

2.4.3 Planar Mixing Layer 

For the case of a planar mixing layer, neither model accurately predicts the high 

speed side of the layer (see Figures 11 - 15). The experimental data13 contains 

a much more dispersive edge compared to the computations, Figure 11. One explanation may be wind tunnel noise. Adding a boundary condition specifying a 

given noise level, 12 percent fluctuating velocity, increases the spreading on the 

high speed side of the computations and improves the comparisons with the data. 

This boundary condition is used with both models. Unfortunately, there is no way 

7 

Lumley)s Energy Cascade Dissipation Rate Model 

Shear stress is also well predicted (see Figure 3) where the standard k - E model 

over-predicts the shear stress at the edge of the jet in comparison to the data of 

Gutman and Wynanski6 and Bradbury7. The k- E-S model predicts a narrower 

profile, also indicated by the smaller spreading rate, which agrees with Bradbury's 

measurements. Towards the centerline of the jet, both models accurately predict 

the increase in shear stress and the extrema near ~ = 0.08. 

For the next two figures, Figures 4 and 5, there is no experimental data available 

for comparison. The dissipation reaches a maximum at the location of the peak 

shear stress, Figure 4. From this extrema, the dissipation decreases toward the 

centerline and also eventually trails off to zero at the edge of the jet. Figure 5 

shows the behavior of the time scale for the planar jet. Notice that the minimum 

energy transfer time between the large scale structure to the dissipative microscale 

is in the middle of the jet (where S is a maximum since the dimension of S is t-I ) . 

2.4.2 Axisymmetric Jet 

A primary problem with the standard k-f. model is its predictive capability for 

a 2-D versus a 3-D jet, the free jet anomaly. In Figures 6 - 10, the standard k- f. 

model is unable to predict any of the turbulence quantities correctly, overpredicting 

not only the turbulent kinetic energy and the shear stress but also the mean velocity. 

The standard k-E can be "fixed" using a suggestion by Pope9 . Writing the standard 

E equation for reference, 

the correct spreading rate can be obtained by changing Cl from its standard value of 

1.45 to 1.6 (see Table 2). Clearly, by modifying a coefficient, the standard k-E model 

is capable of predicting the correct spreading rate; however, this model cannot be 

considered general. The spreading rate predicted by the k-f.-S model is very close 

to the experimentally measured rate and has been obtained with no modifications 

to the model. . 

Comparisons between the k-f.-S model and the experimental data10,1l,12, Figures 

6 - 10, indicates that the new model accurately predi~ts the mean velocity profiles 

and the shearing stress across the jet. Less accurately predicted by the k-f.-S model 

is the centerline turbulent kinetic energy, Figure 7, where there is considerable 

scatter in the experimental data. 

2.4.3 Planar Mixing Layer 

For the case of a planar mixing layer, neither model accurately predicts the high 

speed side of the layer (see Figures 11 - 15). The experimental data13 contains 

a much more dispersive edge compared to the computations, Figure 11. One explanation may be wind tunnel noise. Adding a boundary condition specifying a 

given noise level, 12 percent fluctuating velocity, increases the spreading on the 

high speed side of the computations and improves the comparisons with the data. 

This boundary condition is used with both models. Unfortunately, there is no way 

7 
B. S. Duncan 

to determine what the true noise level was in the experiments or whether its effect 

has been properly accounted for computationally. 

An analytical solution for the mean velocity profile has been described by Schlichtingl4. 

This analyti<: solution lies between the experimental data and the two computational 

models for the case when a = 9.5 (a is a "tweeking" constant used to tune the analytic solution to match the experimental data. Typical values of a are between 913 

and 1115 .) 

Both the k-€-S and the standard k-€ model predict the peak value of tu·rbulent 

kinetic energy quite well (Figure 12). However, this peak is shifted toward the low 

speed side of the mixing layer compared to the experimental peak. PateP6 indicated that the data of Wygnanski and Fielder13 had shifted towards the low speed 

side, which is opposite from the trend noticed in this study. Since the discrepancy 

between the data and the computations lies on the high speed side of the mixing 

layer, the problem could be a result of wind tunnel noise as previously mentioned. 

In the the plot of Reynolds stresses, Figure 13, both models over predict the shear 

stress by nearly 30 percent and consequently over-predict the spreading rate given 

in Table 2. 

The k-;-€-S model and the standard k-€ model predict almost identical levels of 

dissipation (Figure 14). Since a turbulent kinetic energy level is specified at the high 

speed boundary, the dissipation is also specified on this boundary. A constant value 

of uiL = 6.5 X 10-6 gave the best fit to the data. Figure 15 shows the inverse time 

scal;distribution through the mixing layer. The minimum energy transfer time, 

where S is a maximum, corresponds to the location of the maximum shear stress. 

At the high speed edge of the mixing layer, the time scale is specified to be S = . 

2.4.4 Planar Wake 

For the ·case of the planar wake, the mean velocity profile is correctly predicted 

towards the centerline of the wake, but drops off too quickly at the wake edges (see 

Figure 16). Although the mean velocity shows the correct shape in Figure 16, the 

centerline velocity is high compared to the analytical solution described in Reference . dy~ 

[14] (see Figure 17). As a result, the spreading rates, (u:;'':!:Uo cd-), predicted by 

the turbulence models are 6 percent too low for the k-€- S model and 3 percent too 

low for the k-€ model as compared to the analytical solution. In other words, the 

turbulence models are predicting a much more compact wake with a smaller velocity 

defect than is seen in the experimental data or the analytical fit . This trend is also 

seen in the turbulent kinetic energy profiles in Figure 18 where the k-€- S model can 

predict the correct maximum intensity but drops off too quickly towards the edges. 

Also, although the computed peak value for shear stress is correct the width of the 

curve is too narrow, with the location of the extrema lying too near the centerline 

(see Figure 19). 

The dissipation levels computed by the models apparently reduce the turbulent 

intensity of the flow too much near the edges of the wake, which prevents the wake 

from spreading. There is unfortunately no experimental data to compare with the 

dissipation curves in Figure 20. Interestingly, the location of the minimum energy 

8 

B. S. Duncan 

to determine what the true noise level was in the experiments or whether its effect 

has been properly accounted for computationally. 

An analytical solution for the mean velocity profile has been described by Schlichtingl4. 

This analyti<: solution lies between the experimental data and the two computational 

models for the case when a = 9.5 (a is a "tweeking" constant used to tune the analytic solution to match the experimental data. Typical values of a are between 913 

and 1115 .) 

Both the k-€-S and the standard k-€ model predict the peak value of tu·rbulent 

kinetic energy quite well (Figure 12). However, this peak is shifted toward the low 

speed side of the mixing layer compared to the experimental peak. PateP6 indicated that the data of Wygnanski and Fielder13 had shifted towards the low speed 

side, which is opposite from the trend noticed in this study. Since the discrepancy 

between the data and the computations lies on the high speed side of the mixing 

layer, the problem could be a result of wind tunnel noise as previously mentioned. 

In the the plot of Reynolds stresses, Figure 13, both models over predict the shear 

stress by nearly 30 percent and consequently over-predict the spreading rate given 

in Table 2. 

The k-;-€-S model and the standard k-€ model predict almost identical levels of 

dissipation (Figure 14). Since a turbulent kinetic energy level is specified at the high 

speed boundary, the dissipation is also specified on this boundary. A constant value 

of uiL = 6.5 X 10-6 gave the best fit to the data. Figure 15 shows the inverse time 

scal;distribution through the mixing layer. The minimum energy transfer time, 

where S is a maximum, corresponds to the location of the maximum shear stress. 

At the high speed edge of the mixing layer, the time scale is specified to be S = . 

2.4.4 Planar Wake 

For the ·case of the planar wake, the mean velocity profile is correctly predicted 

towards the centerline of the wake, but drops off too quickly at the wake edges (see 

Figure 16). Although the mean velocity shows the correct shape in Figure 16, the 

centerline velocity is high compared to the analytical solution described in Reference . dy~ 

[14] (see Figure 17). As a result, the spreading rates, (u:;'':!:Uo cd-), predicted by 

the turbulence models are 6 percent too low for the k-€- S model and 3 percent too 

low for the k-€ model as compared to the analytical solution. In other words, the 

turbulence models are predicting a much more compact wake with a smaller velocity 

defect than is seen in the experimental data or the analytical fit . This trend is also 

seen in the turbulent kinetic energy profiles in Figure 18 where the k-€- S model can 

predict the correct maximum intensity but drops off too quickly towards the edges. 

Also, although the computed peak value for shear stress is correct the width of the 

curve is too narrow, with the location of the extrema lying too near the centerline 

(see Figure 19). 

The dissipation levels computed by the models apparently reduce the turbulent 

intensity of the flow too much near the edges of the wake, which prevents the wake 

from spreading. There is unfortunately no experimental data to compare with the 

dissipation curves in Figure 20. Interestingly, the location of the minimum energy 

8 
Lumley's Energy Cascade Dissipation Rate Model 

transfer time in Figure 21 is towards the outer edges of the wake where the mean 

flow should exert its greatest influence. 

Experiment k-f.-S Model k-f. Model 

Planar Jet 0.11-0.12 

Round Jet 0.085-0.095 

Planar Mixing Layer 0.16 

Planar Wake .10114 

0.114 

0.095 

0.186 

0.095 

0.114 

0.126, C1 = 1.45 

0.0928, C1 = 1.6 

0.186 

0.098 

Table 2. Spreading Rate Comparisons for Free Shear Flow 

3. Future Work 

Future plans include: 

,i) Extending this model to wall bounded flow. Specifically, a low Reynolds number 

version of this model will be developed for near wall turbulence. 

ii) Continuing to run this model for more test cases over a variety of flow fields. 

4. References 

1 Lumley, J. L., "Some comments on turbulence", Physics of Fluids 4 (1992). 

2 Boussinesq, J., "Essai sur la theorie des eaux courantes", Mem. pres. Acad. Sci. 

XXIII (46) (1877). 

3 Tennekes, H. & Lumley, J. L., A First Course in Turbulence The MIT Press 

(1972). 

4 Launder, B. E., " Turbulence transport models for numerical computation of complex turbulent flows" ,Measurements and Predictions of Complex Turbulent Flows 

Von Karman Institute for Fluid Dynamics, Rhode Saint Genese, Belgium (1980). 

5 Spalding,D. B., GENMIX: A General Computer Program for Two-Dimensional 

Parabolic Phenomena Pergamon Press (1977). 

6 Gutmark, E. & Wygnanski, I., "The Planar Turbulent Jet", Journal of Fluid 

Mechanics 73, 465-495 (1976). 

7 Bradbury, 1. J. S., "The Stucture of the Self-Preserving Jet" Journal of Fluid 

Mechanics 23 31-64 (1965). 

8 Heskestad, G., "Hot-Wire Measurements in a Plane Turbulent Jet" Journal of 

Applied Mechanics (1965). 

9 Pope, S. B., "An Explanation of the Turbulent Round-JetjPlane-Jet Anomaly" 

AIAA Journal 16 (1978). 

10 Wygnanski, 1. & Fiedler, H. E., "Some measurements in the self-preserving jet" 

Journal of Fluid Mechanics 38 577-612. 

11 Rodi, W., "A New Method of Analyzing hot-wire Signals in Highly Turbulent 

Flow and Its Evaluation in Round Jet" Disa Information No. 17 (1975). 

9 

Lumley's Energy Cascade Dissipation Rate Model 

transfer time in Figure 21 is towards the outer edges of the wake where the mean 

flow should exert its greatest influence. 

Experiment k-f.-S Model k-f. Model 

Planar Jet 0.11-0.12 

Round Jet 0.085-0.095 

Planar Mixing Layer 0.16 

Planar Wake .10114 

0.114 

0.095 

0.186 

0.095 

0.114 

0.126, C1 = 1.45 

0.0928, C1 = 1.6 

0.186 

0.098 

Table 2. Spreading Rate Comparisons for Free Shear Flow 

3. Future Work 

Future plans include: 

,i) Extending this model to wall bounded flow. Specifically, a low Reynolds number 

version of this model will be developed for near wall turbulence. 

ii) Continuing to run this model for more test cases over a variety of flow fields. 

4. References 

1 Lumley, J. L., "Some comments on turbulence", Physics of Fluids 4 (1992). 

2 Boussinesq, J., "Essai sur la theorie des eaux courantes", Mem. pres. Acad. Sci. 

XXIII (46) (1877). 

3 Tennekes, H. & Lumley, J. L., A First Course in Turbulence The MIT Press 

(1972). 

4 Launder, B. E., " Turbulence transport models for numerical computation of complex turbulent flows" ,Measurements and Predictions of Complex Turbulent Flows 

Von Karman Institute for Fluid Dynamics, Rhode Saint Genese, Belgium (1980). 

5 Spalding,D. B., GENMIX: A General Computer Program for Two-Dimensional 

Parabolic Phenomena Pergamon Press (1977). 

6 Gutmark, E. & Wygnanski, I., "The Planar Turbulent Jet", Journal of Fluid 

Mechanics 73, 465-495 (1976). 

7 Bradbury, 1. J. S., "The Stucture of the Self-Preserving Jet" Journal of Fluid 

Mechanics 23 31-64 (1965). 

8 Heskestad, G., "Hot-Wire Measurements in a Plane Turbulent Jet" Journal of 

Applied Mechanics (1965). 

9 Pope, S. B., "An Explanation of the Turbulent Round-JetjPlane-Jet Anomaly" 

AIAA Journal 16 (1978). 

10 Wygnanski, 1. & Fiedler, H. E., "Some measurements in the self-preserving jet" 

Journal of Fluid Mechanics 38 577-612. 

11 Rodi, W., "A New Method of Analyzing hot-wire Signals in Highly Turbulent 

Flow and Its Evaluation in Round Jet" Disa Information No. 17 (1975). 

9 
B. S. Duncan 

12 Abbiss, J. B. & Bradbury, L. J. S. & Wright, M. P., "Measurements in an 

Axisymmetric Jet Using a Photon Correlator" Proceeding of LDA Symposium 

Copenhagen (1975). 

13 Wygnanski, 1. & Fiedler, H. E., "The two-dimensional mixing region", Journal 

of Fluid Mechanics 41 327- 361 (1970). 

14 Schlichting, H.,Bounday-Layer Theory McGraw-Hill (1951). 

15 Liepmann, H. W. & Laufer, J., "Investigations of Free Turbulent Mixing" NACA 

Technical Note 1257 (1947). 

16 Patel, R, "An Experimental Study of a Plane Mixing Layer", AIAA Journal 11 

(1) (1973). 

17 Chevray, R. & Kovasznay, 1. S. G., "Turbulence Measurements in the Wake of a 

Thin Flat Plate" AIAA Journal 7 (1969). 

10 

! 

~_I 

B. S. Duncan 

12 Abbiss, J. B. & Bradbury, L. J. S. & Wright, M. P., "Measurements in an 

Axisymmetric Jet Using a Photon Correlator" Proceeding of LDA Symposium 

Copenhagen (1975). 

13 Wygnanski, 1. & Fiedler, H. E., "The two-dimensional mixing region", Journal 

of Fluid Mechanics 41 327- 361 (1970). 

14 Schlichting, H.,Bounday-Layer Theory McGraw-Hill (1951). 

15 Liepmann, H. W. & Laufer, J., "Investigations of Free Turbulent Mixing" NACA 

Technical Note 1257 (1947). 

16 Patel, R, "An Experimental Study of a Plane Mixing Layer", AIAA Journal 11 

(1) (1973). 

17 Chevray, R. & Kovasznay, 1. S. G., "Turbulence Measurements in the Wake of a 

Thin Flat Plate" AIAA Journal 7 (1969). 

10 

! 

~_I 
)( 

o 

E 

~ 

0.8 

0.6 

0.4 

0.2 

0.0 

0.00 0.05 

A Gutman Ie WYllnanski o Bradbury () Heskestad __ k-c-S lo4od,,1 ----- k-c 104od,,1 

0. 10 0.15 

YjX 

0.20 0.25 

2.5 

2.0 

N 

£ 1.5 o 

E 

::::l 

8 

~ 1.0 

'" 

O. 

0.00 

A Gutman & Wygnanski 

_____ 

~~-t.s1cod"l k-c Model 

0.05 0.10 0.15 0.20 0.25 

Y/X 

Figure 1: Mean velocity profile for a turbulent, planar 

jet. Umar: centerline velocity. 

Figure 3: Shear stress profile for a planar jet. Umar: 

centerline velocity. 

O. 

0.06 

~ 0.05 

• " o 

~ 0.04 

:;:c.. 

O.OJ 

0.02 

0.Q1 

0.00 

0.00 0.05 0.10 

yjX 

A Gutman &c Wygnonaki o Brodbury <> Heskestad 

--k-c-S lo4odel _____ k-e Model 

. 

~ 

\<? \\~ 

"\.0 

\,.~" " 

"' ..... 

1.40[-04 

1.20£-04 

1.00£-04 

': 

" 

~ 8.00£-05 ::::> 

J 

~ 6.00£-05 

2.00£-05 

0.00£+00 

0.15 0.20 0.25 0.00 0.05 0.10 

y/X 

--k-c-S lo4odel _____ k-c Model 

0.15 

. . , 

. , 

"""" 

0.20 

-, "-

0.25 

Figure 2: Turbulent kinetic energy profile for a planar 

jet. Umar: centerline velocity. 

Figure 4: Dissipation rate profile for a planar jet. Umar: 

centerline velocity. 

11 

)( 

o 

E 

~ 

0.8 

0.6 

0.4 

0.2 

0.0 

0.00 0.05 

A Gutman Ie WYllnanski o Bradbury () Heskestad __ k-c-S lo4od,,1 ----- k-c 104od,,1 

0. 10 0.15 

YjX 

0.20 0.25 

2.5 

2.0 

N 

£ 1.5 o 

E 

::::l 

8 

~ 1.0 

'" 

O. 

0.00 

A Gutman & Wygnanski 

_____ 

~~-t.s1cod"l k-c Model 

0.05 0.10 0.15 0.20 0.25 

Y/X 

Figure 1: Mean velocity profile for a turbulent, planar 

jet. Umar: centerline velocity. 

Figure 3: Shear stress profile for a planar jet. Umar: 

centerline velocity. 

O. 

0.06 

~ 0.05 

• " o 

~ 0.04 

:;:c.. 

O.OJ 

0.02 

0.Q1 

0.00 

0.00 0.05 0.10 

yjX 

A Gutman &c Wygnonaki o Brodbury <> Heskestad 

--k-c-S lo4odel _____ k-e Model 

. 

~ 

\<? \\~ 

"\.0 

\,.~" " 

"' ..... 

1.40[-04 

1.20£-04 

1.00£-04 

': 

" 

~ 8.00£-05 ::::> 

J 

~ 6.00£-05 

2.00£-05 

0.00£+00 

0.15 0.20 0.25 0.00 0.05 0.10 

y/X 

--k-c-S lo4odel _____ k-c Model 

0.15 

. . , 

. , 

"""" 

0.20 

-, "-

0.25 

Figure 2: Turbulent kinetic energy profile for a planar 

jet. Umar: centerline velocity. 

Figure 4: Dissipation rate profile for a planar jet. Umar: 

centerline velocity. 

11 
2.5 

--k-c-S Model 

.. 

~ . 

(I) 

)( 

o 

E 

~ => 

0.0 L-__ ~ ____ ~ ____ ~ __ ~~--~ 

0.00 0.05 0.1 0 0.15 0.20 0.25 

0.8 

0.6 

0.4 

0.2 

Y/X 

Figure 5: Time scale profile for a planar jet 

'" WvQnonskl & Fielder o Rodi 

k-c-S Model 

==k-c Model 

0.0 

L-----~------~----~-----0~20~--~0~~5 0.00 0.05 0.10 0.15 . 

y/x 

Figure 6: Mean velocity profile for a turbulent, axisymmetric jet. Umor : centerline velocity. 

12 

0.08 

N 

t 

~ 0.06 

E 

=> 

~ 

0.04 

0.02 

0.00 

A 

000 

OC> 

o 

~ 

... Wygronski & Fielder o Rodl 

k-c-S Model == k-c Model 

\""" 

"""""" 

'" 

" ........... 

L-__ ~ ____ ~ ____ ~ __ ~~ __ ~ 

0.00 0.05 0.10 0.15 0.20 0.25 

y/x 

Figure 7: Turbulent kinetic energy profile for an a.xisymmetric jet. Umax : centerline velocity. 

2.5 

2.0 

N 

~ 1.5 o 

E 

::::> 

"-o 

o j 1.0 

0.0 

/ __ ---__ " ~ ~ronski &< Fielder 

" .... 0 Abbiss 

,/ ',--k-c-S ,",odel 

, 

, 

, 

, 

, 

, 

, 

, 

, 

· · 

. 

. · · 

L-__ ~ ____ ~ ____ ~ __ ~=-__ ~ 

0.00 0.05 0.10 0.15 0.20 0.25 

y/x 

Figure 8: Shear stress profile for an axisymmetric jet. 

Umar : centerline velocity. 

2.5 

--k-c-S Model 

.. 

~ . 

(I) 

)( 

o 

E 

~ => 

0.0 L-__ ~ ____ ~ ____ ~ __ ~~--~ 

0.00 0.05 0.1 0 0.15 0.20 0.25 

0.8 

0.6 

0.4 

0.2 

Y/X 

Figure 5: Time scale profile for a planar jet 

'" WvQnonskl & Fielder o Rodi 

k-c-S Model 

==k-c Model 

0.0 

L-----~------~----~-----0~20~--~0~~5 0.00 0.05 0.10 0.15 . 

y/x 

Figure 6: Mean velocity profile for a turbulent, axisymmetric jet. Umor : centerline velocity. 

12 

0.08 

N 

t 

~ 0.06 

E 

=> 

~ 

0.04 

0.02 

0.00 

A 

000 

OC> 

o 

~ 

... Wygronski & Fielder o Rodl 

k-c-S Model == k-c Model 

\""" 

"""""" 

'" 

" ........... 

L-__ ~ ____ ~ ____ ~ __ ~~ __ ~ 

0.00 0.05 0.10 0.15 0.20 0.25 

y/x 

Figure 7: Turbulent kinetic energy profile for an a.xisymmetric jet. Umax : centerline velocity. 

2.5 

2.0 

N 

~ 1.5 o 

E 

::::> 

"-o 

o j 1.0 

0.0 

/ __ ---__ " ~ ~ronski &< Fielder 

" .... 0 Abbiss 

,/ ',--k-c-S ,",odel 

, 

, 

, 

, 

, 

, 

, 

, 

, 

· · 

. 

. · · 

L-__ ~ ____ ~ ____ ~ __ ~=-__ ~ 

0.00 0.05 0.10 0.15 0.20 0.25 

y/x 

Figure 8: Shear stress profile for an axisymmetric jet. 

Umar : centerline velocity. 
1

.., 

. . 

" o 

E 

:::l 

2.00E-04 

~ 1.00E-04 

5.00E-05 

... ... ... ... , 

, 

, 

, 

, 

, 

, 

, 

-- k-~-S Model ----- k-c t.4odel 

, 

, 

, 

, 

, 

, 

, 

, 

, 

\ 

, 

\ 

\ 

, 

, 

" , 

, 

" " " " " ' ............ 

" o 

E 

~ 

0.8 

0.6 

0.4 

0.2 

0.0 

- - - - - - -

II. wygnonski &< Fielder _ k-c-S Model 

k-c Model . . . . Analytical 

O.OOE+OO 

0.00 0.05 0 .1 0 0.15 0.20 0.25 -0.15 -0.10 -0.05 0 .00 0.05 0.1 0 0.15 

y/x 

Figure 9:. Dissipation rate profile for an axisymmetric 

jet. Umar: centerline velocity. 

2.0 

~ • 1.0 (/l 

0.5 

0.0 

0.00 

--k-c-S Model 

0.05 0.10 O.lS 0.20 0.25 

y/x 

Figure 10: Time scale profile for an axisymmetric jet 

13 

(y-y~/x 

Figure 11: Mean velocity profile for a planar mlxmg 

layer. Umar: cent.erline velocit.y. - analytical solut.ion 

0 .035 

0.030 A 

0.025 

A II. WYgnonski &< Fielder - k-c-S Model 

k-c Model 

': 0.020 

" 

E "

::::> 

~0 015 

A 

0.010 

0.005 

o.oooJ.---~ 

-0.15 -0.10 -0.05 0.00 0.05 

(Y-y,r)/x, 

0.10 0.15 

Figure 12: Turbulent kinetic energy profile for a planar 

mixing layer. Umar: centerline velocity. 

1

- --

.., 

. . 

" o 

E 

:::l 

2.00E-04 

~ 1.00E-04 

5.00E-05 

... ... ... ... , 

, 

, 

, 

, 

, 

, 

, 

-- k-~-S Model ----- k-c t.4odel 

, 

, 

, 

, 

, 

, 

, 

, 

, 

\ 

, 

\ 

\ 

, 

, 

" , 

, 

" " " " " ' ............ 

" o 

E 

~ 

0.8 

0.6 

0.4 

0.2 

0.0 

- - - - - - -

II. wygnonski &< Fielder _ k-c-S Model 

k-c Model . . . . Analytical 

O.OOE+OO 

0.00 0.05 0 .1 0 0.15 0.20 0.25 -0.15 -0.10 -0.05 0 .00 0.05 0.1 0 0.15 

y/x 

Figure 9:. Dissipation rate profile for an axisymmetric 

jet. Umar: centerline velocity. 

2.0 

~ • 1.0 (/l 

0.5 

0.0 

0.00 

--k-c-S Model 

0.05 0.10 O.lS 0.20 0.25 

y/x 

Figure 10: Time scale profile for an axisymmetric jet 

13 

(y-y~/x 

Figure 11: Mean velocity profile for a planar mlxmg 

layer. Umar: cent.erline velocit.y. - analytical solut.ion 

0 .035 

0.030 A 

0.025 

A II. WYgnonski &< Fielder - k-c-S Model 

k-c Model 

': 0.020 

" 

E "

::::> 

~0 015 

A 

0.010 

0.005 

o.oooJ.---~ 

-0.15 -0.10 -0.05 0.00 0.05 

(Y-y,r)/x, 

0.10 0.15 

Figure 12: Turbulent kinetic energy profile for a planar 

mixing layer. Umar: centerline velocity. 

- --
1.2 

1.0 

0.6 ", 

£ 

o 

E 

~0.6 

8 

~ 0.4 

0.2 

0.0 ~-----" 

,--..... 

... Wyononski '" Fielder 

--i<':'i-S loAodet 

----k-c loAodet 

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 

(y-y~/x 

Figure 13: Shear stress profile for a planar mixing layer. 

Umar: centerline velocity. 

5.0E-05 --i<-c-S loAodel 

----k-c loAodet 

4.0E-05 

': J.OE-05 )( 

o 

E 

::J 

J 

t 2.0E-05 

1.0E-05 

0.0£+00 ~--_--/ 

-0.15 -0.10 -0.05 0.00 0.05 0.1 0 0.15 

(y-y.rJ/x 

Figure 14: Dissipation rate profile for a planar mixing 

layer. U mar: centerline velocity. 

14 

.. 

~ 

~ 

0.6 

0.6 

0.4 

0.2 

0.0 1--------' 

-0.2 -0.1 

- k-c- S loAodel 

0.0 

(y-y~/x 

0.1 0.2 

Figure 15: Time scale profile for a planar mi..xing layer 

1. 

0.6 

1 

! 

-::J 0.6 I 

I ::J 

::::: 

S- 0.4 I 

I 

2.. 

0.2 

0.0 

... Chevroy '" Kovasynoy _ k-c-S" Model 

---- k-c loAodel 

0.0 0.5 1.0 1.5 2.0 2.5 J.O 

y/y .. 

Figure 16: Mean velocity profile for a planar wake 

behind a flat plate. Uccntcr: centerline velocity, U 00: 

freest ream velocity. - analytical solution 

--- ------

1.2 

1.0 

0.6 ", 

£ 

o 

E 

~0.6 

8 

~ 0.4 

0.2 

0.0 ~-----" 

,--..... 

... Wyononski '" Fielder 

--i<':'i-S loAodet 

----k-c loAodet 

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 

(y-y~/x 

Figure 13: Shear stress profile for a planar mixing layer. 

Umar: centerline velocity. 

5.0E-05 --i<-c-S loAodel 

----k-c loAodet 

4.0E-05 

': J.OE-05 )( 

o 

E 

::J 

J 

t 2.0E-05 

1.0E-05 

0.0£+00 ~--_--/ 

-0.15 -0.10 -0.05 0.00 0.05 0.1 0 0.15 

(y-y.rJ/x 

Figure 14: Dissipation rate profile for a planar mixing 

layer. U mar: centerline velocity. 

14 

.. 

~ 

~ 

0.6 

0.6 

0.4 

0.2 

0.0 1--------' 

-0.2 -0.1 

- k-c- S loAodel 

0.0 

(y-y~/x 

0.1 0.2 

Figure 15: Time scale profile for a planar mi..xing layer 

1. 

0.6 

1 

! 

-::J 0.6 I 

I ::J 

::::: 

S- 0.4 I 

I 

2.. 

0.2 

0.0 

... Chevroy '" Kovasynoy _ k-c-S" Model 

---- k-c loAodel 

0.0 0.5 1.0 1.5 2.0 2.5 J.O 

y/y .. 

Figure 16: Mean velocity profile for a planar wake 

behind a flat plate. Uccntcr: centerline velocity, U 00: 

freest ream velocity. - analytical solution 

--- ------
0 .25 

0.20 

• 

~ 

S" 

I 

d O.15 

0.10 

\ 

\ 

\ 

,<,"olyticol 

\ 

"-

"-

"-

"- ......... ......... .......... .......... 

---- ---- ---- -

.05 ~~--~--~----~--~--~----~--~--~-

200 400 600 800 

X/L 

0 .035 

0.030 

I') 0 .025 . 

10.020 ::l 

I 

=1 0.015 

~ 

O. 

0.0 0.5 1.0 1.5 

y/Y.a 

I>. Chevroy cIc Kovosynoy - k-c-S Model ---- k- c Model 

2.0 2.5 3.0 

Figure 17: Axial mean centerline velocity for a planar 

wake behind a flat plate. U 00: freestream veloci I.y. 

Figure 19: Shear stress profile for a planar wake behind a flat plate. U cent er : centerline velocity, U oo : 

freestream velocity. 

6 C~~ k Kovo~oy 

0.12 - k-c- Model ---- k-c Model 

A 

0.10 A 

N • i O

'

oa 

::l 

I 0.06 

• 

~ A ~ 0.04 A 

0.02 

0.00 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

Y/Y .. 

Figure 18: Turbulent kinetic energy profile for a planar 

wake behind a flat plate. Ucenter: centerline velocity, 

Uoo : freestream velocity. 

15 

0.0018 

- k-c-S Model 

0.0016 ---- k-c Model 

0.0014 

I') 

: 0.0012 

)0.0010 

I 

.0.0008 

::l 

:5 

• 0.0006 .. 

0.0004 

0.0002 

0.0000 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

y/y .. 

Figure 20: Dissipation rate profile for a planar wake 

behind a flat plate. Ucenter : centerline velocity, U 00: 

freestream velocity. 

0 .25 

0.20 

• 

~ 

S" 

I 

d O.15 

0.10 

\ 

\ 

\ 

,<,"olyticol 

\ 

"-

"-

"-

"- ......... ......... .......... .......... 

---- ---- ---- -

.05 ~~--~--~----~--~--~----~--~--~-

200 400 600 800 

X/L 

0 .035 

0.030 

I') 0 .025 . 

10.020 ::l 

I 

=1 0.015 

~ 

O. 

0.0 0.5 1.0 1.5 

y/Y.a 

I>. Chevroy cIc Kovosynoy - k-c-S Model ---- k- c Model 

2.0 2.5 3.0 

Figure 17: Axial mean centerline velocity for a planar 

wake behind a flat plate. U 00: freestream veloci I.y. 

Figure 19: Shear stress profile for a planar wake behind a flat plate. U cent er : centerline velocity, U oo : 

freestream velocity. 

6 C~~ k Kovo~oy 

0.12 - k-c- Model ---- k-c Model 

A 

0.10 A 

N • i O

'

oa 

::l 

I 0.06 

• 

~ A ~ 0.04 A 

0.02 

0.00 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

Y/Y .. 

Figure 18: Turbulent kinetic energy profile for a planar 

wake behind a flat plate. Ucenter: centerline velocity, 

Uoo : freestream velocity. 

15 

0.0018 

- k-c-S Model 

0.0016 ---- k-c Model 

0.0014 

I') 

: 0.0012 

)0.0010 

I 

.0.0008 

::l 

:5 

• 0.0006 .. 

0.0004 

0.0002 

0.0000 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

y/y .. 

Figure 20: Dissipation rate profile for a planar wake 

behind a flat plate. Ucenter : centerline velocity, U 00: 

freestream velocity. 
3.5 

3.0 

2.5 

.. 2.0 

~ 

Ih 

1.5 

1.0 

0.0 

0.0 0.5 1.0 1.5 

Y/Y.5 

- k-c-S ~odol 

2.0 2.5 3.0 

Figure 21 : Time scale profile for a plana r wake behind 

a fl at plate 

16 

I 

I 

_1 

3.5 

3.0 

2.5 

.. 2.0 

~ 

Ih 

1.5 

1.0 

0.0 

0.0 0.5 1.0 1.5 

Y/Y.5 

- k-c-S ~odol 

2.0 2.5 3.0 

Figure 21 : Time scale profile for a plana r wake behind 

a fl at plate 

16 

I 

I 

_1 
Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

PDF Turbulence Modeling and DNS 

A. T. Hsu 

1. Motivations and Objectives 

The problem oftime discontinuity (or jump condition) in the coalescence/dispersion 

(C/D) mixing model is addressed in first part (PDF) of this work. A C/D mixing model continuous in time is introduced. With the continuous mixing model, 

the process of chemical reaction can be fully coupled with mixing. In the case 

of homogeneous turbulence decay, the new model predicts a pdf very close to a 

Gaussian distribution, with finite higher moments also close to that of a Gaussian 

distribution. Results from the continuous mixing model are compared with both 

experimental data and numerical results from conventional C /D models. 

The effect of Coriolis forces on compressible homogeneous turbulence is studied 

using direct numerical simulation (DNS). The numerical method used in this study 

is an eigth order compact difference scheme. Contrary to the conclusions reached 

by previous DNS studies on incompressible isotropic turbulence, the present results 

show that the Coriolis force increases the dissipation rate of turbulent kinetic energy, 

and that anisotropy develops as the Coriolis force increases. The Taylor-Proudman 

theory does apply since the derivatives in the direction of the rotation axis vanishes 

rapidly. A closer analysis reveals that the dissipation rate of the incompressible 

component of the turbulent kinetic energy indeed decreases with a higher rotation 

rate, consistent with incompressible flow simulations (Bardina17 ), while the dissipation rate of the Compressible part increases; the net gain is positive. Inertial waves 

are observed in the simulation results. 

2. Work Accomplished 

2.1 PDF Turbulence Model for Combustion 

In Collaboration with J.Y. Chen 

Accurate prediction of turbulent reacting flows requires the solution of an evolution equation for the probability density function (pdf) of the thermo-chemical 

variables using Monte Carlo simulation. Since the pdf equation, like most equations 

describing turbulent motion, is not closed, closure models have to be devised. For 

the pdf of scalars, the terms in the pdf equation that need modeling are molecular 

mixing and turbulent convection. The present work deals with the modeling of 

molecular mixing. 

Most of the mixing models are based on the coalescence/dispersion (C/D) model 

by Curl6 This model is known to have deficiencies, and efforts had been made to 

correct these deficiencies, for example, Janicka, et al. 1979 and Pope 1982. The 

most recent efforts have been devoted to the problem of coupling between mixing 

and chemical reaction. Chen and Kollmann4 proposed a reaction conditioned model 

17 

Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

PDF Turbulence Modeling and DNS 

A. T. Hsu 

1. Motivations and Objectives 

The problem oftime discontinuity (or jump condition) in the coalescence/dispersion 

(C/D) mixing model is addressed in first part (PDF) of this work. A C/D mixing model continuous in time is introduced. With the continuous mixing model, 

the process of chemical reaction can be fully coupled with mixing. In the case 

of homogeneous turbulence decay, the new model predicts a pdf very close to a 

Gaussian distribution, with finite higher moments also close to that of a Gaussian 

distribution. Results from the continuous mixing model are compared with both 

experimental data and numerical results from conventional C /D models. 

The effect of Coriolis forces on compressible homogeneous turbulence is studied 

using direct numerical simulation (DNS). The numerical method used in this study 

is an eigth order compact difference scheme. Contrary to the conclusions reached 

by previous DNS studies on incompressible isotropic turbulence, the present results 

show that the Coriolis force increases the dissipation rate of turbulent kinetic energy, 

and that anisotropy develops as the Coriolis force increases. The Taylor-Proudman 

theory does apply since the derivatives in the direction of the rotation axis vanishes 

rapidly. A closer analysis reveals that the dissipation rate of the incompressible 

component of the turbulent kinetic energy indeed decreases with a higher rotation 

rate, consistent with incompressible flow simulations (Bardina17 ), while the dissipation rate of the Compressible part increases; the net gain is positive. Inertial waves 

are observed in the simulation results. 

2. Work Accomplished 

2.1 PDF Turbulence Model for Combustion 

In Collaboration with J.Y. Chen 

Accurate prediction of turbulent reacting flows requires the solution of an evolution equation for the probability density function (pdf) of the thermo-chemical 

variables using Monte Carlo simulation. Since the pdf equation, like most equations 

describing turbulent motion, is not closed, closure models have to be devised. For 

the pdf of scalars, the terms in the pdf equation that need modeling are molecular 

mixing and turbulent convection. The present work deals with the modeling of 

molecular mixing. 

Most of the mixing models are based on the coalescence/dispersion (C/D) model 

by Curl6 This model is known to have deficiencies, and efforts had been made to 

correct these deficiencies, for example, Janicka, et al. 1979 and Pope 1982. The 

most recent efforts have been devoted to the problem of coupling between mixing 

and chemical reaction. Chen and Kollmann4 proposed a reaction conditioned model 

17 
A. T. Hsu 

that allows correct prediction of combustion in the flame-sheet regime. Norris and 

Pope10 proposed a new model based on ordered pairing that aimed at the same end. 

All the existing models suffer in one respect, namely, they are discontinuous in 

time: once a pair of particles are chosen to participate in mixing, their properties will 

jump abruptly regardless of the step size of the time integration. This phenomenon, 

clearly non-physical, could cause difficulty in coupling the processes of mixing and 

chemical reaction. In the present work, a new model that is continuous in time is 

proposed. With this new model, the processes of molecular mixing and chemical 

reaction can be fully coupled. 

In the case of homogeneous turbulence decay of a scalar, one expects a Gaussian 

distribution for the pdf, and finite values for the higher moments. Popell pointed 

out the modified Curl model could not produce the correct pdf for this problem, 

and the higher even moments from that model tend to infinity; he suggested an age . 

biased sampling process to overcome these shortcomings. The present continuous 

model, as we will show, predicts a pdf distribution very close to Gaussian for homogeneous turbulence decay, and gives finite higher moments with values close to 

that of a Gaussian distribution. 

The continuous mixing model is applied to the study of both non-reacting and 

reacting flows, and the results are compared with earlier calculations by Hsu 7 as 

well as with experimental data. 

2.1.1 Molecular Mixing Model 

The evolution equation of a single point probability density function of scalar random variables '¢1, ... , '¢n - representing the species mass fraction and temperature 

- can be written as 

N 

POtF + pvQoQF + P L 01/J; {Wi( '¢1, ... , '¢N )F} 

i=l 

N N 

-P L L O~; 1/Ji « Eiji<pk(X) = '¢k > F) 

i=l j=l 

where the terms represent the rate of time change, mean convection, chemical reaction, turbulent convection, and molecular mixing, respectively; F is the densityweighted joint pdf: 

F = pP/p, 

E is the scalar dissipation: 

Eij = D8Q<PiOQ<Pj , 

(where D is the diffusion coefficient), and < xiy > denotes the mathematical expectation of a random function x conditioned upon y. 

The left hand side of the above equation can be evaluated exactly and requires 

no modeling; the right hand side terms contain the conditional expectation of the 

18 

J 

A. T. Hsu 

that allows correct prediction of combustion in the flame-sheet regime. Norris and 

Pope10 proposed a new model based on ordered pairing that aimed at the same end. 

All the existing models suffer in one respect, namely, they are discontinuous in 

time: once a pair of particles are chosen to participate in mixing, their properties will 

jump abruptly regardless of the step size of the time integration. This phenomenon, 

clearly non-physical, could cause difficulty in coupling the processes of mixing and 

chemical reaction. In the present work, a new model that is continuous in time is 

proposed. With this new model, the processes of molecular mixing and chemical 

reaction can be fully coupled. 

In the case of homogeneous turbulence decay of a scalar, one expects a Gaussian 

distribution for the pdf, and finite values for the higher moments. Popell pointed 

out the modified Curl model could not produce the correct pdf for this problem, 

and the higher even moments from that model tend to infinity; he suggested an age . 

biased sampling process to overcome these shortcomings. The present continuous 

model, as we will show, predicts a pdf distribution very close to Gaussian for homogeneous turbulence decay, and gives finite higher moments with values close to 

that of a Gaussian distribution. 

The continuous mixing model is applied to the study of both non-reacting and 

reacting flows, and the results are compared with earlier calculations by Hsu 7 as 

well as with experimental data. 

2.1.1 Molecular Mixing Model 

The evolution equation of a single point probability density function of scalar random variables '¢1, ... , '¢n - representing the species mass fraction and temperature 

- can be written as 

N 

POtF + pvQoQF + P L 01/J; {Wi( '¢1, ... , '¢N )F} 

i=l 

N N 

-P L L O~; 1/Ji « Eiji<pk(X) = '¢k > F) 

i=l j=l 

where the terms represent the rate of time change, mean convection, chemical reaction, turbulent convection, and molecular mixing, respectively; F is the densityweighted joint pdf: 

F = pP/p, 

E is the scalar dissipation: 

Eij = D8Q<PiOQ<Pj , 

(where D is the diffusion coefficient), and < xiy > denotes the mathematical expectation of a random function x conditioned upon y. 

The left hand side of the above equation can be evaluated exactly and requires 

no modeling; the right hand side terms contain the conditional expectation of the 

18 

J 
PDF Turbulence Modeling and DNS 

velocity fluctuation and the conditional expectation of the scalar dissipation, which 

are new unknowns and require modeling. In the present work we concentrate on 

the modeling of the second term, namely, the conditional expectation of the scalar 

dissipation, referred to as molecular mixing in the following. 

The Modified Curl Model 

The simplest and most used mixing model is the modified Curl model, which 

assumes binary interaction between sample fluid particles. As described by Pope12 

in a Monte Carlo simulation, the continuous pdf is replaced by delta functions 

1 N 

P*(1/;j t) = N L o(1/; - <Pn(t)), 

n=l 

where each delta function represents one sample fluid particle of an ensemble of N 

particles. The evolution of P* entails the movement of the particles in the 1/;-space, 

or the evolution of the individual values of <Pn's. 

With the modified Curl model, the change of <Pn due to molecular mixing is 

achieved by the following binary interaction process: divide the flow domain into 

small cells, each containing N sample particles. Given a small time interval ot and 

a turbulent time scale r, select randomly Nmx pairs of particles, 

Ot 

Nmx = 0.5 CrN, 

(C = 6.0) and let a pair, say, m and n, mix as follows 

<Pn(t + ot) = A<Pm(t) + Cl - A)<PnCt) 

<Pm(t + ot) = A<Pn(t) + (1 - A)<Pm(t) 

(1) 

(2) 

where A = 0.5e, with e a random variable uniformly distributed on the int erval 

[0,1]. The remaining N - 2Nmx particles remain unchanged: 

This model does not represent the true physical process since the properties of 

the sample particles change discontinuously regardless the size of the time interval 

ot. This deficiency can be best illustrated by rearranging eq. (1) and dividing it by 

ot 

(3) 

The derivative d1;: does not exist because as ot goes to zero the right hand side of 

the equation becomes infinite since both A and the difference between <Pm(t) and 

<Pn(t) are finite. This means that there is a sudden jump in the value of the scalar 

quantities, which is typical of a Poisson process, but is non-physical in the present 

case since the flow properties of turbulence are continuous. 

19 

PDF Turbulence Modeling and DNS 

velocity fluctuation and the conditional expectation of the scalar dissipation, which 

are new unknowns and require modeling. In the present work we concentrate on 

the modeling of the second term, namely, the conditional expectation of the scalar 

dissipation, referred to as molecular mixing in the following. 

The Modified Curl Model 

The simplest and most used mixing model is the modified Curl model, which 

assumes binary interaction between sample fluid particles. As described by Pope12 

in a Monte Carlo simulation, the continuous pdf is replaced by delta functions 

1 N 

P*(1/;j t) = N L o(1/; - <Pn(t)), 

n=l 

where each delta function represents one sample fluid particle of an ensemble of N 

particles. The evolution of P* entails the movement of the particles in the 1/;-space, 

or the evolution of the individual values of <Pn's. 

With the modified Curl model, the change of <Pn due to molecular mixing is 

achieved by the following binary interaction process: divide the flow domain into 

small cells, each containing N sample particles. Given a small time interval ot and 

a turbulent time scale r, select randomly Nmx pairs of particles, 

Ot 

Nmx = 0.5 CrN, 

(C = 6.0) and let a pair, say, m and n, mix as follows 

<Pn(t + ot) = A<Pm(t) + Cl - A)<PnCt) 

<Pm(t + ot) = A<Pn(t) + (1 - A)<Pm(t) 

(1) 

(2) 

where A = 0.5e, with e a random variable uniformly distributed on the int erval 

[0,1]. The remaining N - 2Nmx particles remain unchanged: 

This model does not represent the true physical process since the properties of 

the sample particles change discontinuously regardless the size of the time interval 

ot. This deficiency can be best illustrated by rearranging eq. (1) and dividing it by 

ot 

(3) 

The derivative d1;: does not exist because as ot goes to zero the right hand side of 

the equation becomes infinite since both A and the difference between <Pm(t) and 

<Pn(t) are finite. This means that there is a sudden jump in the value of the scalar 

quantities, which is typical of a Poisson process, but is non-physical in the present 

case since the flow properties of turbulence are continuous. 

19 
A. T. Hsu 

Continuous Mixing Model 

One can see from the previous section that the modified Curl model relies on the 

parameter Nmx to control the extent of mixing. On the individual particle level, 

it assumes complete mixing once the particle is selected as one of the mixing pair, 

without considering the size of 8t. 

In order to achieve continuous mixing, we propose the following model: during a 

time interval bt, we assume that all the particles within a cell participate in mixing. 

The extent of the mixing is controlled at the individual particle leveL That is to 

say, the N particles within a given cell are randomly grouped into N /2 pairs; the 

properties of all the particles change according to eqs. (1) and (2) The extent 

of mixing now has to be controlled at the individual particle level through the 

parameter A, which is redefined as 

where C' 

bt ~ 0, 

2.0. With this new definition, eq. (3) can be written, in the limit 

The above equation states that the change of <Pn due to mixing is proportional to 

the difference between <Pm and <Pn, and inversely proportional to the turbulence 

time scale T . 

The Coupling of Mixing and Reaction 

The processes of mixing and chemical reaction are essentially decoupled when one 

uses the discontinuous C/D models. In contrast, with the above continuous model, 

coupling becomes natural since, for a given particle, mixing and chemical reaction 

can be described with a single equation: 

where Wn is the chemical source term. 

Since the continuous mixing model allows full coupling of the reaction and mixing 

processes, the C jD model with reaction zone conditioning by Chen and Kollmann4 

can be easily implemented in the present model to simulate the fast reaction in the 

flame sheet regime. Here a modified finite difference version of eq. 12 has to be 

used since Wn is infinity in case of fast reaction. 

2.1.2 Results and Discussions 

The continuous mixing model described in the previous section has been validated using both non-reacting and reacting flow test cases. The results and their 

comparisons with earlier calculations (Hsu 7) using the modified Curl model as well 

as with experimental data are presented in this section. 

20 

A. T. Hsu 

Continuous Mixing Model 

One can see from the previous section that the modified Curl model relies on the 

parameter Nmx to control the extent of mixing. On the individual particle level, 

it assumes complete mixing once the particle is selected as one of the mixing pair, 

without considering the size of 8t. 

In order to achieve continuous mixing, we propose the following model: during a 

time interval bt, we assume that all the particles within a cell participate in mixing. 

The extent of the mixing is controlled at the individual particle leveL That is to 

say, the N particles within a given cell are randomly grouped into N /2 pairs; the 

properties of all the particles change according to eqs. (1) and (2) The extent 

of mixing now has to be controlled at the individual particle level through the 

parameter A, which is redefined as 

where C' 

bt ~ 0, 

2.0. With this new definition, eq. (3) can be written, in the limit 

The above equation states that the change of <Pn due to mixing is proportional to 

the difference between <Pm and <Pn, and inversely proportional to the turbulence 

time scale T . 

The Coupling of Mixing and Reaction 

The processes of mixing and chemical reaction are essentially decoupled when one 

uses the discontinuous C/D models. In contrast, with the above continuous model, 

coupling becomes natural since, for a given particle, mixing and chemical reaction 

can be described with a single equation: 

where Wn is the chemical source term. 

Since the continuous mixing model allows full coupling of the reaction and mixing 

processes, the C jD model with reaction zone conditioning by Chen and Kollmann4 

can be easily implemented in the present model to simulate the fast reaction in the 

flame sheet regime. Here a modified finite difference version of eq. 12 has to be 

used since Wn is infinity in case of fast reaction. 

2.1.2 Results and Discussions 

The continuous mixing model described in the previous section has been validated using both non-reacting and reacting flow test cases. The results and their 

comparisons with earlier calculations (Hsu 7) using the modified Curl model as well 

as with experimental data are presented in this section. 

20 
- --- ----- - -----

PDF Turbulence Modeling and DNS 

Homogeneo us Turbu len ce Decay of a Scalar 

The case of decaying fluct uation of a passive scalar in homogeneous turbulence 

is used to test the continuous mixing model. The initial condition is 

that is, in the Monte Carlo simulation, half of the particles are ascribed the value 

1, and the other half -l. 

The pdf distribution of the normalized variable ('lj;- < ¢ > )/(J, where < ¢ > is 

the mean and (J is the standard deviation, in the homogeneous turbulence decay 

problem converges to a single curve after certain time, and the correct distribution 

should be Gaussian. Fig. 1 and 2 are the pdf distributions from the modified Curl 

model and the present model, both compared to a normal distribution. One can 

see that the pdf from the modified Curl model deviates considerable from Gaussian, 

while the result from the present model is fairly close to a Gaussian distribution. 

The evolution history of the rms and fourth and sixth moments of the scalar 

:fluctuation are calculated using both the modified Curl model and the present 

continuous model. Fig. 3 shows the results from the modified Curl model. One 

can see that although the rms from that model behaves well, the fourth and sixth 

moments grow quickly out of bound, oscillating at a level several order of magnitudes 

higher than the value of Gaussian distribution. These results are similar to what 

Popell had observed. Fig. 4 shows the results for the same set of quantities from the 

present model. The rms behaves similar to that from the modified Curl model. The 

fourth and sixth moments, on the other hand, are quite different from those of the 

previous model; they rise smoothly to the value predicted by Gaussian distribution. 

Although the values do not seem to converge, they remain finite, and are of the 

same order of magnitude as that of the Gaussian distribution. 

The above results clearly demonstrated the advantage of the present model over 

that of the modified Curl model. Popell had devised an age biased scheme that 

achieved the same end, which required an additional variable, namely the age of 

the particles, and two extra adjustable parameters. In contrast, the present model 

needs no extra work or parameters. 

Heated Turbulent Jet 

Extensive experimental results for a heated turbulent plane jet have been reported 

by many authors (Bashir, et al. 2 , Browne et al. 3 , Uberoi and Singhl3 , Jenkins9 , Antonia, et al. l ). The turbulent jet has a slightly higher temperature than the ambient. 

Measurements of both the the mean temperature and the rms of temperature fluctuations were given. We compared the solutions for the heated turbulent jet from 

the new model with experimental data as well as with previous solutions (Hsu 7) 

obtained using the modified Curl model. 

In the present study, a combined CFD-Monte Carlo algorithm is used. The 

mean flow field is obtained by solving the Navier-Stokes equation and a two-equation 

21 

- --- ----- - -----

PDF Turbulence Modeling and DNS 

Homogeneo us Turbu len ce Decay of a Scalar 

The case of decaying fluct uation of a passive scalar in homogeneous turbulence 

is used to test the continuous mixing model. The initial condition is 

that is, in the Monte Carlo simulation, half of the particles are ascribed the value 

1, and the other half -l. 

The pdf distribution of the normalized variable ('lj;- < ¢ > )/(J, where < ¢ > is 

the mean and (J is the standard deviation, in the homogeneous turbulence decay 

problem converges to a single curve after certain time, and the correct distribution 

should be Gaussian. Fig. 1 and 2 are the pdf distributions from the modified Curl 

model and the present model, both compared to a normal distribution. One can 

see that the pdf from the modified Curl model deviates considerable from Gaussian, 

while the result from the present model is fairly close to a Gaussian distribution. 

The evolution history of the rms and fourth and sixth moments of the scalar 

:fluctuation are calculated using both the modified Curl model and the present 

continuous model. Fig. 3 shows the results from the modified Curl model. One 

can see that although the rms from that model behaves well, the fourth and sixth 

moments grow quickly out of bound, oscillating at a level several order of magnitudes 

higher than the value of Gaussian distribution. These results are similar to what 

Popell had observed. Fig. 4 shows the results for the same set of quantities from the 

present model. The rms behaves similar to that from the modified Curl model. The 

fourth and sixth moments, on the other hand, are quite different from those of the 

previous model; they rise smoothly to the value predicted by Gaussian distribution. 

Although the values do not seem to converge, they remain finite, and are of the 

same order of magnitude as that of the Gaussian distribution. 

The above results clearly demonstrated the advantage of the present model over 

that of the modified Curl model. Popell had devised an age biased scheme that 

achieved the same end, which required an additional variable, namely the age of 

the particles, and two extra adjustable parameters. In contrast, the present model 

needs no extra work or parameters. 

Heated Turbulent Jet 

Extensive experimental results for a heated turbulent plane jet have been reported 

by many authors (Bashir, et al. 2 , Browne et al. 3 , Uberoi and Singhl3 , Jenkins9 , Antonia, et al. l ). The turbulent jet has a slightly higher temperature than the ambient. 

Measurements of both the the mean temperature and the rms of temperature fluctuations were given. We compared the solutions for the heated turbulent jet from 

the new model with experimental data as well as with previous solutions (Hsu 7) 

obtained using the modified Curl model. 

In the present study, a combined CFD-Monte Carlo algorithm is used. The 

mean flow field is obtained by solving the Navier-Stokes equation and a two-equation 

21 
A. T . Hsu 

turbulence model using a finite difference scheme. The temperature is treated as a 

conserved scalar and simulated by the pdf equation. 

Fig. 5 shows the comparisons of the the mean temperature distribution from 

the pdf Monte Carlo simulations and experimental data from various authors. The 

figure shows that both mixing models predict the mean temperature distribution 

accurately. 

The results for standard variation, or rms, of the temperature distribution are 

given in Fig. 6. Although the two solutions do not show significant difference, the 

new model seems to agree slightly better with the experimental data. The skewness 

and the flatness, Le., the third and fourth moments of the temperature fluctuation, 

are given in Figs. 7 and 8; the comparisons show that in the present case of a 

turbulent jet the statistical behavior of the new model is similar to that of the 

modified Curl model. 

Hydrogen-Fluorine Diffusion Flame 

The continuous model has been applied to the H 2-F2 diffusion flame studied 

earlier by Hsu 7 . The flow conditions are set according to an experiment performed 

by Hermanson and Dimotakis6 . The flame consists two streams. The upper stream 

contains 96% of N2 and 4% of F2 , the flow velocity is U1 = 22 m/ s; the lower stream 

contains 96% of N2 and 4% of H2 , with velocity U2 = 8.8 m/ s. The estimated 

Damkohler number ranged from 25 to 130 (Hermanson and Dimotakis6 ), and a fast 

chemistry model is deemed appropriate in the calculation. Again a modified version 

of eq. 12 is used to accommodate the fast chemistry. 

Fig. 9 shows the temperature rise due to combustion. In the figure, bT is the 

shear layer thickness determined by 1% of the temperature rise, 6.T is the actual 

temperature rise due to combustion, ( the two streams have the same temperatures 

initially,) and !:1Tadf is the adiabatic flame temperature assuming complete reaction. 

Details on the flow conditions can be found in Hermanson and Dimotakis6 . The 

agreement between numerical predictions and experimental data is fairly good, and 

a comparison of the results from the continuous model and that from the modified 

Curl model shows that both performed well for this case. 

Combination with Reaction Zone Conditioning 

Chen and Kollmann4 developed a mixing model based on reaction zone conditioning, aimed at the coupling of reaction and mixing. We have shown in Section 2.3 

that with the present model, the processes of reaction and mixing can be fully coupled; therefore it is only natural to apply the reaction zone conditioning suggested 

by Chen and Kollmann here. 

The H 2-F2 diffusion flame problem is reformulated such that the chemical reaction is confined to a very narrow zone near stoichiometry. By applying reaction zone 

conditioning to the continuous mixing model, we were able to produce a scatter plot 

of the temperature vs. mixture fraction in which all the points reached the equilibrium temperature. This result is shown in Fig. 10. The mixture fraction here is 

defined as the molar concentration of fuel divided by the total molar concentration, 

and stoichiometry is located at f = 0.5. 

22 

A. T . Hsu 

turbulence model using a finite difference scheme. The temperature is treated as a 

conserved scalar and simulated by the pdf equation. 

Fig. 5 shows the comparisons of the the mean temperature distribution from 

the pdf Monte Carlo simulations and experimental data from various authors. The 

figure shows that both mixing models predict the mean temperature distribution 

accurately. 

The results for standard variation, or rms, of the temperature distribution are 

given in Fig. 6. Although the two solutions do not show significant difference, the 

new model seems to agree slightly better with the experimental data. The skewness 

and the flatness, Le., the third and fourth moments of the temperature fluctuation, 

are given in Figs. 7 and 8; the comparisons show that in the present case of a 

turbulent jet the statistical behavior of the new model is similar to that of the 

modified Curl model. 

Hydrogen-Fluorine Diffusion Flame 

The continuous model has been applied to the H 2-F2 diffusion flame studied 

earlier by Hsu 7 . The flow conditions are set according to an experiment performed 

by Hermanson and Dimotakis6 . The flame consists two streams. The upper stream 

contains 96% of N2 and 4% of F2 , the flow velocity is U1 = 22 m/ s; the lower stream 

contains 96% of N2 and 4% of H2 , with velocity U2 = 8.8 m/ s. The estimated 

Damkohler number ranged from 25 to 130 (Hermanson and Dimotakis6 ), and a fast 

chemistry model is deemed appropriate in the calculation. Again a modified version 

of eq. 12 is used to accommodate the fast chemistry. 

Fig. 9 shows the temperature rise due to combustion. In the figure, bT is the 

shear layer thickness determined by 1% of the temperature rise, 6.T is the actual 

temperature rise due to combustion, ( the two streams have the same temperatures 

initially,) and !:1Tadf is the adiabatic flame temperature assuming complete reaction. 

Details on the flow conditions can be found in Hermanson and Dimotakis6 . The 

agreement between numerical predictions and experimental data is fairly good, and 

a comparison of the results from the continuous model and that from the modified 

Curl model shows that both performed well for this case. 

Combination with Reaction Zone Conditioning 

Chen and Kollmann4 developed a mixing model based on reaction zone conditioning, aimed at the coupling of reaction and mixing. We have shown in Section 2.3 

that with the present model, the processes of reaction and mixing can be fully coupled; therefore it is only natural to apply the reaction zone conditioning suggested 

by Chen and Kollmann here. 

The H 2-F2 diffusion flame problem is reformulated such that the chemical reaction is confined to a very narrow zone near stoichiometry. By applying reaction zone 

conditioning to the continuous mixing model, we were able to produce a scatter plot 

of the temperature vs. mixture fraction in which all the points reached the equilibrium temperature. This result is shown in Fig. 10. The mixture fraction here is 

defined as the molar concentration of fuel divided by the total molar concentration, 

and stoichiometry is located at f = 0.5. 

22 
 

 

L 

PDF Turbulence Modeling and DNS 

2.1.3 Conclusions 

A turbulence mixing model that is continuous in time has been introduced. The 

deficiency of non-physical jump condition in the mixing process is removed in the 

new model. It has been shown that the new model is superior to the existing 

modified Curl model (Janicka, et al.B) in that it can predict a Gaussian distribution 

and finite higher moments in the case of homogeneous turbulence decay; it has 

accomplished what the age biased sampling scheme (Popell) is designed for, without 

the extra parameters required by that scheme. The numerical results from the 

present model compare well with experimental data. 

2.2 DNS of Homogeneous Compressible Turbulence in a Rotating frame 

Rotation is an important factor in many flow phenomena in nature and in engineering. Problems that are strongly affected by rotation include flows in turbomachinery, large scale motions in the atmosphere and oceans, and galactic motions. 

Turbulence is important in all the above examples. In order to model turbulence 

in a rotating frame, we need a better understanding on the effects of rotation on 

turbulence. 

Several experiments had been carried out by various researchers14 - 16 . However, 

the conclusions are inconsistent. Some results show that rotation increases the 

turbulence dissipation rate, others suggest the opposite. 

Numerical studies of turbulence in a rotating frame are few. In fact the only 

two are incompressible flow studies, by Bardina, et al. I7 and by Speziale et al. IB , 

using the same computer program. Studies other than the present one dealing with 

compressible flows in a rotating frame are not known to the authors. 

Bardina et al. 17 performed both large eddy and full simulations of incompressible isotropic turbulence. Their results show that, in the case of incompressible 

turbulence, the dissipation rate decreases with increasing rotation rate, and that 

anisotropy does not develop as a result of the Coriolis force. Speziale et al. confirmed these results using a smaller Rossby number, i.e. a faster rotation. 

In general, compressible turbulence in a rotating frame is not homogenous or 

isotropic because of the existence of centrifugal force and density fluctuations. However, when the rotation rate is low, or when the solution domain is very close to the 

rotation axis, the centrifugal force can be neglected and the flow is approximately 

homogeneous. In the present study, our goal is to identify the effects of the Coriolis force, as opposed to the effects of the centrifugal force or the combined effects 

of the centrifugal and Coriolis forces. Therefore, the centrifugal force is dropped 

from the governing equations, regardless of whether or not it is negligible. A future 

study will concentrate on the non-homogenous effects of the centrifugal force and 

the combined effects of the two forces. 

The numerical results presented in the following show that although the Coriolis 

force does not appear explicitly in the turbulence kinetic energy equation, its effect 

can be significant. Anisotropy does develop as a result of the Coriolis force. The 

dissipation rate of the compressible and incompressible modes of the turbulent kinetic energy behave differently under Coriolis forces, and the combined effect is a 

23 

 

 

L 

PDF Turbulence Modeling and DNS 

2.1.3 Conclusions 

A turbulence mixing model that is continuous in time has been introduced. The 

deficiency of non-physical jump condition in the mixing process is removed in the 

new model. It has been shown that the new model is superior to the existing 

modified Curl model (Janicka, et al.B) in that it can predict a Gaussian distribution 

and finite higher moments in the case of homogeneous turbulence decay; it has 

accomplished what the age biased sampling scheme (Popell) is designed for, without 

the extra parameters required by that scheme. The numerical results from the 

present model compare well with experimental data. 

2.2 DNS of Homogeneous Compressible Turbulence in a Rotating frame 

Rotation is an important factor in many flow phenomena in nature and in engineering. Problems that are strongly affected by rotation include flows in turbomachinery, large scale motions in the atmosphere and oceans, and galactic motions. 

Turbulence is important in all the above examples. In order to model turbulence 

in a rotating frame, we need a better understanding on the effects of rotation on 

turbulence. 

Several experiments had been carried out by various researchers14 - 16 . However, 

the conclusions are inconsistent. Some results show that rotation increases the 

turbulence dissipation rate, others suggest the opposite. 

Numerical studies of turbulence in a rotating frame are few. In fact the only 

two are incompressible flow studies, by Bardina, et al. I7 and by Speziale et al. IB , 

using the same computer program. Studies other than the present one dealing with 

compressible flows in a rotating frame are not known to the authors. 

Bardina et al. 17 performed both large eddy and full simulations of incompressible isotropic turbulence. Their results show that, in the case of incompressible 

turbulence, the dissipation rate decreases with increasing rotation rate, and that 

anisotropy does not develop as a result of the Coriolis force. Speziale et al. confirmed these results using a smaller Rossby number, i.e. a faster rotation. 

In general, compressible turbulence in a rotating frame is not homogenous or 

isotropic because of the existence of centrifugal force and density fluctuations. However, when the rotation rate is low, or when the solution domain is very close to the 

rotation axis, the centrifugal force can be neglected and the flow is approximately 

homogeneous. In the present study, our goal is to identify the effects of the Coriolis force, as opposed to the effects of the centrifugal force or the combined effects 

of the centrifugal and Coriolis forces. Therefore, the centrifugal force is dropped 

from the governing equations, regardless of whether or not it is negligible. A future 

study will concentrate on the non-homogenous effects of the centrifugal force and 

the combined effects of the two forces. 

The numerical results presented in the following show that although the Coriolis 

force does not appear explicitly in the turbulence kinetic energy equation, its effect 

can be significant. Anisotropy does develop as a result of the Coriolis force. The 

dissipation rate of the compressible and incompressible modes of the turbulent kinetic energy behave differently under Coriolis forces, and the combined effect is a 

23 
A. T. Hsu 

increased dissipation rate. 

2.2.1 Method 

The governing equations for compressible flows in a rotating frame are 

P,t + (pUj ),j = 0 

(pUi),t + (PUjUi),j = -P,i + Tij,j - PCijkCklmf!jf!IXm - 2pCijkf!jUk 

(pe),t + (pUje),j = -PUj,j + U'i,jTij - (kT,j),j 

p=pRT 

The last two terms in the second equation represent, respectively, the effects of 

centrifugal force and the Coriolis force. The centrifugal force is a nonuniform body 

force which would induce nonuniform pressure and density distributions; its effect 

on turbulence is to destroy homogeneity. The Coriolis force, on the other hand, is 

a uniform body force, provided that the velocity field is uniform. In the case of 

incompressible flows, the centrifugal force can be included into a modified pressure 

term and thus does not appear explicitly; however, this can not be done in the case 

of compressible flows. 

In order to identify the homogeneous effect of the Coriolis force alone, and to 

compare results with incompressible flow simulations, we simply take away the 

centrifugal force in our study. This, of course, greatly simplifies the problem: With 

a homogeneous initial condition, the flow field remains homogeneous under the 

Coriolis force. The simplified problem is a hypothetical one and can not normally 

be found in nature except in very restricted situations such as the ones mentioned 

in the introduction. 

The governing equations are solved using a compact difference scheme. Lele19 has 

shown that such schemes have spectral accuracy and are well suited for DNS. The 

spatial derivatives are approximated by 8th order finite differences, and the time 

integration is performed using a three-stage Runge-Kutta time marching scheme. 

The initial condition for the rotating flow simulation is obtained from a simulation 

of isotropic decaying turbulence. The initial condition for the decaying turbulence 

is generated using a given 3D spectrum of the form 

After the decaying turbulence shows the correct skewness, the Coriolis force is 

imposed onto the flow field . 

2.2.2 Results and Discussion 

As a primary study, three cases have been calculated, with rotation in the zdirection: The nondimensional rotation rates are f! = 0, 40, 80, which correspond 

to turbulent Rossby numbers of 00 , 0.0075, and 0.00375. These Rossby numbers 

are much smaller than those considered in References 17 and 18. 

24 

- 

I 

J 

A. T. Hsu 

increased dissipation rate. 

2.2.1 Method 

The governing equations for compressible flows in a rotating frame are 

P,t + (pUj ),j = 0 

(pUi),t + (PUjUi),j = -P,i + Tij,j - PCijkCklmf!jf!IXm - 2pCijkf!jUk 

(pe),t + (pUje),j = -PUj,j + U'i,jTij - (kT,j),j 

p=pRT 

The last two terms in the second equation represent, respectively, the effects of 

centrifugal force and the Coriolis force. The centrifugal force is a nonuniform body 

force which would induce nonuniform pressure and density distributions; its effect 

on turbulence is to destroy homogeneity. The Coriolis force, on the other hand, is 

a uniform body force, provided that the velocity field is uniform. In the case of 

incompressible flows, the centrifugal force can be included into a modified pressure 

term and thus does not appear explicitly; however, this can not be done in the case 

of compressible flows. 

In order to identify the homogeneous effect of the Coriolis force alone, and to 

compare results with incompressible flow simulations, we simply take away the 

centrifugal force in our study. This, of course, greatly simplifies the problem: With 

a homogeneous initial condition, the flow field remains homogeneous under the 

Coriolis force. The simplified problem is a hypothetical one and can not normally 

be found in nature except in very restricted situations such as the ones mentioned 

in the introduction. 

The governing equations are solved using a compact difference scheme. Lele19 has 

shown that such schemes have spectral accuracy and are well suited for DNS. The 

spatial derivatives are approximated by 8th order finite differences, and the time 

integration is performed using a three-stage Runge-Kutta time marching scheme. 

The initial condition for the rotating flow simulation is obtained from a simulation 

of isotropic decaying turbulence. The initial condition for the decaying turbulence 

is generated using a given 3D spectrum of the form 

After the decaying turbulence shows the correct skewness, the Coriolis force is 

imposed onto the flow field . 

2.2.2 Results and Discussion 

As a primary study, three cases have been calculated, with rotation in the zdirection: The nondimensional rotation rates are f! = 0, 40, 80, which correspond 

to turbulent Rossby numbers of 00 , 0.0075, and 0.00375. These Rossby numbers 

are much smaller than those considered in References 17 and 18. 

24 

- 

I 

J 
PDF Turbulence Modeling and DNS 

Figures 11 and 12 show the development of a ID two-point correlation and a ID 

spectrum in the turbulent flow field, where square symbols are initial conditions and 

lines represent later times. Figure 13 shows a comparison of the energy spectrum 

obtained using two different grids, one double the size of the other. The agreement 

affirms sufficient resolution. 

Figures 14-19 show the time history of various statistics of the three cases, where 

time is nondimensionalized using the initial eddy turn over time. 

Figure 14 is the ratio of the turbulent kinetic energy, where q5 is the initial 

turbulent kinetic energy and q2 the value at later times. It is immediately clear 

that the Coriolis force causes a faster decay of the turbulence, a conclusion that is 

contrary to what was observed in the incompressible flow simulatjons by previous 

researchers17,18. Nevertheless, as we shall see in the following, the present results 

are in fact consistent with those of Refs 4 and 5. 

To prove that the present results are consistent with the previous incompressible 

simulations, we look at the ratio of the means of the divergence squared and vorticity 

squared, which are proportional to, respectively, the compressible and incompressible turbulent kinetic energy. Figures 15 and 16 show that while the dissipation 

rate of the compressible part of the turbulent kinetic energy increases with larger 

Coriolis forces, the dissipation rate of the incompressible part decreases. The combined effect, as we have seen from Figure 14, is an increase in the total dissipation 

rate. Since the simulations of References 4 and 5 are for incompressible flows, it is 

not surprising that only a decrease in dissipation rate was observed. 

In both References 17 and 18, no anisotropy was observed. Bardina et al. 17 

speculated that a reorganization to two-dimensional flows could occur at higher 

rotation rates, while Speziale et al. suggested that Taylor-Proudman reorganization 

would not occur in a rapidly rotating isotropic turbulence. The present results show 

definite signs of anisotropy as soon as the Coriolis force is turned on. The anisotropy 

can be clearly observed from the time history of the normal components of the 

Reynolds stress, given in Figures 17, 18, and 19 for the three rotation rates. With 

the Coriolis force, the x- and y-components of the normal Reynolds stress decrease 

much faster than the z-component. In fact, it seems that the increased dissipation 

is primarily in the x- and y-directions, and the z-component of the turbulent kinetic 

energy is not affected much. 

The Taylor-Proudman theory stipulates that under strong rotation, with the 

rotation axis parallel to, say, the z axis, then for any flow variable u, we have 

(8u/8z) = O. Figures 20 and 21 are the contours of the x-component of velocity 

plotted on three planes. Without the Coriolis force (Figure 20), the flow structure remains isotropic. With the Coriolis force, columns soon appear in the flow 

field, suggesting that a Taylor-Proudman reorganization has occurred. Using larger 

Rossby numbers such as those used in References 17 and 18, the results show no 

appreciable anisotropy, hence there appears to be no contradiction. 

Finally, inertial waves can be observed from the time history of various statistics 

presented in Figures 14-19. The frequency is approximately nj7r, the theoretical 

intrinsic frequency for rotating flows. 

25 

PDF Turbulence Modeling and DNS 

Figures 11 and 12 show the development of a ID two-point correlation and a ID 

spectrum in the turbulent flow field, where square symbols are initial conditions and 

lines represent later times. Figure 13 shows a comparison of the energy spectrum 

obtained using two different grids, one double the size of the other. The agreement 

affirms sufficient resolution. 

Figures 14-19 show the time history of various statistics of the three cases, where 

time is nondimensionalized using the initial eddy turn over time. 

Figure 14 is the ratio of the turbulent kinetic energy, where q5 is the initial 

turbulent kinetic energy and q2 the value at later times. It is immediately clear 

that the Coriolis force causes a faster decay of the turbulence, a conclusion that is 

contrary to what was observed in the incompressible flow simulatjons by previous 

researchers17,18. Nevertheless, as we shall see in the following, the present results 

are in fact consistent with those of Refs 4 and 5. 

To prove that the present results are consistent with the previous incompressible 

simulations, we look at the ratio of the means of the divergence squared and vorticity 

squared, which are proportional to, respectively, the compressible and incompressible turbulent kinetic energy. Figures 15 and 16 show that while the dissipation 

rate of the compressible part of the turbulent kinetic energy increases with larger 

Coriolis forces, the dissipation rate of the incompressible part decreases. The combined effect, as we have seen from Figure 14, is an increase in the total dissipation 

rate. Since the simulations of References 4 and 5 are for incompressible flows, it is 

not surprising that only a decrease in dissipation rate was observed. 

In both References 17 and 18, no anisotropy was observed. Bardina et al. 17 

speculated that a reorganization to two-dimensional flows could occur at higher 

rotation rates, while Speziale et al. suggested that Taylor-Proudman reorganization 

would not occur in a rapidly rotating isotropic turbulence. The present results show 

definite signs of anisotropy as soon as the Coriolis force is turned on. The anisotropy 

can be clearly observed from the time history of the normal components of the 

Reynolds stress, given in Figures 17, 18, and 19 for the three rotation rates. With 

the Coriolis force, the x- and y-components of the normal Reynolds stress decrease 

much faster than the z-component. In fact, it seems that the increased dissipation 

is primarily in the x- and y-directions, and the z-component of the turbulent kinetic 

energy is not affected much. 

The Taylor-Proudman theory stipulates that under strong rotation, with the 

rotation axis parallel to, say, the z axis, then for any flow variable u, we have 

(8u/8z) = O. Figures 20 and 21 are the contours of the x-component of velocity 

plotted on three planes. Without the Coriolis force (Figure 20), the flow structure remains isotropic. With the Coriolis force, columns soon appear in the flow 

field, suggesting that a Taylor-Proudman reorganization has occurred. Using larger 

Rossby numbers such as those used in References 17 and 18, the results show no 

appreciable anisotropy, hence there appears to be no contradiction. 

Finally, inertial waves can be observed from the time history of various statistics 

presented in Figures 14-19. The frequency is approximately nj7r, the theoretical 

intrinsic frequency for rotating flows. 

25 
A . T. Hsu 

3. Future Work 

3.1 PDF 

The pdf model is being implemented into an existing compressible flow solver: 

the RPLUS2D code. Work on implementing pdf to RPLUS3D will follow. 

3.2 DNS 

To understand the effect of compressibility, incompressible homogeneous turbulence in a rotating frame will be studied. 

4. References 

1 Antonia, R.A., Browne, L.W.B., Chambers, A.J., and Rajagopalan, S., "Budget 

of the Temperature Variance in a Turbulent Plane Jet." Int. J. Heat Mass 

Transfer,26-1, 41-48, (1983). 

2 Ashir, J. and Uberoi, M.S. , "Experiments on Turbulent Structure and Heat Transfer in a Two Dimensional Jet." Physics of FLuids, 18-4, 405-410, (1975). 

3 Browne, L.W.B., Antonia, R.A., and Chembers, A.J. , "The Interaction Region 

of a Turbulent Plane Jet." J. Fluid Mech., 149, 355-373, (1984). 

4 Chen, J.-Y. and Kollmann, W., "Mixing Models for Turbulent Flows with Exothermic Reactions." Turbulent Shear Flows, Springer-Verlag, 7,277, (1991). 

5 Curl, R.L., "Dispersed Phase Mixing, I, Theory and Effects of Simple Reactors." 

A.LCh.E.J. 9, 175, (1963). 

6 Hermanson, J.C. and Dimotakis, P.E., "Effects of Heat Release in a Turbulent, 

Reacting Shear Layer." J. Fluid Mech. 199, 333-375, (1989). 

7 Hsu, A.T., "The Study of Hydrogen Diffusion Flames Using PDF Turbulence 

Models." AIAA Paper 91-1780, (1991). 

8 Janicka, J., Kolbe, W. , and Kollmann, W., "Closure of the Transport Equation 

for the Probability Density Function Scalar Field. " J Non-Equilib. Thermodyn. 

4, 47, (1979). 

9 Jenkins, P.E. and Goldschmidt, V.W., "Mean Temperature and Velocity in a 

Plane Turbulent Jet," ASME J. Fluids Eng., 95, 581-584, (1973). 

10 Norris, A.T. and Pope, S.B., "Turbulent Mixing Model Based on Ordered Pairing." Combust. and Flame, 83, 27-42, (1991). 

11 Pope, S.B., "An Improved Turbulence Mixing Model." Combust. Sci. Tech., 28, 

131, (1982). 

12 Pope, S.B. , "PDF Methods for Turbulent Reactive Flows." Prg. Energy Combust. Sci., I I, 119-192, (1985). 

13 Uberoi, M.S. and Singh, P.L, "Turbulent Mixing in a Two-Dimensional jet," 

Physics of Fluids, 18-7, 764-769, (1975). 

14 Traugott, S.C., "Influence of Solid-body Rotation on Screen-produced Turbulence." NACA TN 4135, (1958). 

15 Ibbetson, A. and Tritton, D.J., " Experiments on Turbulence in a Rotating 

Fluid." J. Fluid Mech., 56, 639-672, (1975). 

26 

A . T. Hsu 

3. Future Work 

3.1 PDF 

The pdf model is being implemented into an existing compressible flow solver: 

the RPLUS2D code. Work on implementing pdf to RPLUS3D will follow. 

3.2 DNS 

To understand the effect of compressibility, incompressible homogeneous turbulence in a rotating frame will be studied. 

4. References 

1 Antonia, R.A., Browne, L.W.B., Chambers, A.J., and Rajagopalan, S., "Budget 

of the Temperature Variance in a Turbulent Plane Jet." Int. J. Heat Mass 

Transfer,26-1, 41-48, (1983). 

2 Ashir, J. and Uberoi, M.S. , "Experiments on Turbulent Structure and Heat Transfer in a Two Dimensional Jet." Physics of FLuids, 18-4, 405-410, (1975). 

3 Browne, L.W.B., Antonia, R.A., and Chembers, A.J. , "The Interaction Region 

of a Turbulent Plane Jet." J. Fluid Mech., 149, 355-373, (1984). 

4 Chen, J.-Y. and Kollmann, W., "Mixing Models for Turbulent Flows with Exothermic Reactions." Turbulent Shear Flows, Springer-Verlag, 7,277, (1991). 

5 Curl, R.L., "Dispersed Phase Mixing, I, Theory and Effects of Simple Reactors." 

A.LCh.E.J. 9, 175, (1963). 

6 Hermanson, J.C. and Dimotakis, P.E., "Effects of Heat Release in a Turbulent, 

Reacting Shear Layer." J. Fluid Mech. 199, 333-375, (1989). 

7 Hsu, A.T., "The Study of Hydrogen Diffusion Flames Using PDF Turbulence 

Models." AIAA Paper 91-1780, (1991). 

8 Janicka, J., Kolbe, W. , and Kollmann, W., "Closure of the Transport Equation 

for the Probability Density Function Scalar Field. " J Non-Equilib. Thermodyn. 

4, 47, (1979). 

9 Jenkins, P.E. and Goldschmidt, V.W., "Mean Temperature and Velocity in a 

Plane Turbulent Jet," ASME J. Fluids Eng., 95, 581-584, (1973). 

10 Norris, A.T. and Pope, S.B., "Turbulent Mixing Model Based on Ordered Pairing." Combust. and Flame, 83, 27-42, (1991). 

11 Pope, S.B., "An Improved Turbulence Mixing Model." Combust. Sci. Tech., 28, 

131, (1982). 

12 Pope, S.B. , "PDF Methods for Turbulent Reactive Flows." Prg. Energy Combust. Sci., I I, 119-192, (1985). 

13 Uberoi, M.S. and Singh, P.L, "Turbulent Mixing in a Two-Dimensional jet," 

Physics of Fluids, 18-7, 764-769, (1975). 

14 Traugott, S.C., "Influence of Solid-body Rotation on Screen-produced Turbulence." NACA TN 4135, (1958). 

15 Ibbetson, A. and Tritton, D.J., " Experiments on Turbulence in a Rotating 

Fluid." J. Fluid Mech., 56, 639-672, (1975). 

26 
PDF Turbulence Modeling and DNS 

16 Wigeland, R.A. and Nagib, H.M. "Grid-generated Turbulence With and Without 

Rotation About the Streamwise Direction." IIT Fluids and Heat Transfer Rep. 

R78-1, Illinois Inst. of Tech., Chicago, Illinois, (1978). 

17 Bardina, J., Ferziger, J.H. and Rogallo, R.S. "Effect of Rotation on Isotropic 

Turbulence: Computation and Modeling," J. Fluid Mech., 154, 321-336, (1985). 

18 Speziale, C.G., Mansour, N.N. and Rogallo, R.S., "The Decay of Isotropic Turbulence in a Rapidly Rotating Frame," Center for Turbulence Research, Proceedings of the Summer Program, (1987). 

19 Lele, S.K., "Compact Finite Difference Schemes with Spectral Like Resolution." 

Submitted to J. Compo Physics. 

27 

PDF Turbulence Modeling and DNS 

16 Wigeland, R.A. and Nagib, H.M. "Grid-generated Turbulence With and Without 

Rotation About the Streamwise Direction." IIT Fluids and Heat Transfer Rep. 

R78-1, Illinois Inst. of Tech., Chicago, Illinois, (1978). 

17 Bardina, J., Ferziger, J.H. and Rogallo, R.S. "Effect of Rotation on Isotropic 

Turbulence: Computation and Modeling," J. Fluid Mech., 154, 321-336, (1985). 

18 Speziale, C.G., Mansour, N.N. and Rogallo, R.S., "The Decay of Isotropic Turbulence in a Rapidly Rotating Frame," Center for Turbulence Research, Proceedings of the Summer Program, (1987). 

19 Lele, S.K., "Compact Finite Difference Schemes with Spectral Like Resolution." 

Submitted to J. Compo Physics. 

27 
0.8,----------------, 

0. 7 

0.6 

' 0 .5 

~ 0 4 R. . 

0.3 

0.2 

0.1 

o. 0 -=~----'---__ """'_____ __ -==_i 

-3 - 2 -1 0 1 2 3 

{1/; - <cp> }/ a 

Fi~ 1. Arymptotic pdf distribution for & acalac in homogeneC>\U turimlence. - modified. Curl model; - - - Gauaian. 

-.3 -2 - 1 

(1/; 

J 1 

<cp»/a 

2 3 

Fi~ 2. Arymptotic pdf distr:imtion for & .calar in homogeneo\U turiluience. - present model; - - - G&Uui.&n.. 

28 

1.0 ,....-----,....-1 ---------, 

! I ; I i ~ /I ~ I I 

i · : i II I! ,1\ : I 

! i ~ : " ,\ ;; i I 

/I I,,'t:! ' I I: \ ,", \ ' ; \ I

\ 

\ 1\ . . ' / I I ';11 , .\ " v'\ I' I ' t' t II \ I I I 

J \ ;,,'. V, :/ " " 

y"/ "-,c, f) 

O.OL--C~~----------~ 

0.2 

0.8 

o 4 8 12 16 20 

Tim.e 

Figure 3. Evolution of moment. from the modified Curl 

model - standard deviation, - - - 0.01 x fxxuth central 

momcnt, - - 0.0001 x sixth central IDOmcnt, 0 0..01 x fourth 

momcnt for Gaussian diatribution, 0 0.0001 x aixth moment 

for Gauai&n distribution.. 

/) 

~ 

,... 

,... 

---

"< 

1.0,----------------, 

0.8 

0.6 

0.4 

0.2 

0. 0 

0 

\ 

\ 

\ 

I 

\ 

\ - - -- - - - p 

\ - - - - -\ - -- - --- - - '--::p -~---~ . 

4 8 12 16 2 0 

Tim.e 

Figure 4. Evolution of momenta from the pl'CM:D& model -

.tandard deYi&tion, - - - 0.1 x fourth centrallDOlDClt, - -

0.01 x sixth a:ntral moment, 0 0.1 X fourth JIlOQlent for 

G&DAian diatribution, a 0.01 x aixth moment for Gaussian 

distribution. 

0.8,----------------, 

0. 7 

0.6 

' 0 .5 

~ 0 4 R. . 

0.3 

0.2 

0.1 

o. 0 -=~----'---__ """'_____ __ -==_i 

-3 - 2 -1 0 1 2 3 

{1/; - <cp> }/ a 

Fi~ 1. Arymptotic pdf distribution for & acalac in homogeneC>\U turimlence. - modified. Curl model; - - - Gauaian. 

-.3 -2 - 1 

(1/; 

J 1 

<cp»/a 

2 3 

Fi~ 2. Arymptotic pdf distr:imtion for & .calar in homogeneo\U turiluience. - present model; - - - G&Uui.&n.. 

28 

1.0 ,....-----,....-1 ---------, 

! I ; I i ~ /I ~ I I 

i · : i II I! ,1\ : I 

! i ~ : " ,\ ;; i I 

/I I,,'t:! ' I I: \ ,", \ ' ; \ I

\ 

\ 1\ . . ' / I I ';11 , .\ " v'\ I' I ' t' t II \ I I I 

J \ ;,,'. V, :/ " " 

y"/ "-,c, f) 

O.OL--C~~----------~ 

0.2 

0.8 

o 4 8 12 16 20 

Tim.e 

Figure 3. Evolution of moment. from the modified Curl 

model - standard deviation, - - - 0.01 x fxxuth central 

momcnt, - - 0.0001 x sixth central IDOmcnt, 0 0..01 x fourth 

momcnt for Gaussian diatribution, 0 0.0001 x aixth moment 

for Gauai&n distribution.. 

/) 

~ 

,... 

,... 

---

"< 

1.0,----------------, 

0.8 

0.6 

0.4 

0.2 

0. 0 

0 

\ 

\ 

\ 

I 

\ 

\ - - -- - - - p 

\ - - - - -\ - -- - --- - - '--::p -~---~ . 

4 8 12 16 2 0 

Tim.e 

Figure 4. Evolution of momenta from the pl'CM:D& model -

.tandard deYi&tion, - - - 0.1 x fourth centrallDOlDClt, - -

0.01 x sixth a:ntral moment, 0 0.1 X fourth JIlOQlent for 

G&DAian diatribution, a 0.01 x aixth moment for Gaussian 

distribution. 
r - -

0~o~,~-----------------------------, 

0.8 

<::> 0 .6 <-. 

~ 

::- 0.4 

0.2 o 

00 , 

00" 

0 .0 '----------'-------'--==----J 

0.0 0 .5 1.0 1.5 2.0 2.5 

Y/Y.STo 

Figure 5. · Mean temperature in heated plane jet. - continuoua mixint; modd, - - - modified Curl modd, D. Browne et 

al. (1984), 0 Bashir and Uberoi (1915),0 Uberoi &lid Singh 

(1975), 0 Jenlcina and Goldschmidt (1973). 

0.40 r---------------, 

0.35 

0.30 <::> <-. 

"-.. 0.25 OIl . 

~ 0.20 

~ 0.15 v 0.10 

0.05 

• " ;p • '" 0 

, -.-.'-

• 0 

o 

o 

o 

o 

... , 

o 

o 

0.00 '------'---....:....-_-'-----.::~ __ ---J 

0.0 0 .5 1.0 1.5 2.0 2.5 

Y/Y.STO 

Figure 6. RMS of temperature variance in heated plane jet. 

- oontinu0U4 mixing modd, - - - modified Curl modd, 0 

Antonia et al. (1983),0 Ba,.,hir and Uberoi (1975), D. Uberoi 

and Singh (1975). 

20 

" 

~ 15 ;:: 

.. 

R-

~ 10 ... 

';;' 

::: 

s 

:: 

.. 

~ 0 ... 

tr. 

-5 

0.0 0.5 1.0 1.5 

Y/Y.ST, 

2.0 

, 

, 

2.5 3.0 

Figure 7. Ske1rnea. of temperature Mance in h~ted plane 

jet. - continuoWl mixins model, - •• modified Curl model 

29 

20r-----------------------------~ 

.. 18 ~ 

~ 16 

; " 14 

RIO 12 .. 

... 10 

';;' 8 ., 

:;: 6 ~ 

d 4 

;;: 2 

o~--~~--------~----~ ____ ~ __ ~ 

0. 0 0.5 1.0 1.5 

Y/lI.ST. 

2 .0 2.5 3. 0 

Figure 8. Flatness of temperature variance in heated plane 

jet. - continuous mixin~ model, ••• modified Curl model. 

1.0,-----------------, 

0.8 

:;-

d 0 . 6 

E--. 

~ :;; 0.4 

0.2 

OL-~~-~-~-~-~-~~ 

-0.8 -0.4 0.0 

y/or 

0.4 

Figure 9. Temperature rise in an H,-F, dilftmon flame. 

- rontinuoua mixing modd, - •• modified Curl modd, 0 

HermaIllIOn and Dimota.lci.s (1989). 

1.0r------------------------, 

i 

" \" 

0.8 I , 

~06 h" . 

~ ~ 0.4 / 

j 

0.2 

O.O~--~---~---~----L----~ 

0.0 0.2 0.4 0.6 0.8 1.0 

Alirture Frac~ f 

Figure 10. Joint pdf between temperature and mixture fraction on the centerline of the HrF, diffusion flame; reaction 

is restricted to a narrow zone at stoichiometry; re3ults are 

obtained by applying reaction zone conditioning (Chen and 

Kol.Jman.n, 1990) to the rontinuous mixing modd. 

---

----- -----

r - -

0~o~,~-----------------------------, 

0.8 

<::> 0 .6 <-. 

~ 

::- 0.4 

0.2 o 

00 , 

00" 

0 .0 '----------'-------'--==----J 

0.0 0 .5 1.0 1.5 2.0 2.5 

Y/Y.STo 

Figure 5. · Mean temperature in heated plane jet. - continuoua mixint; modd, - - - modified Curl modd, D. Browne et 

al. (1984), 0 Bashir and Uberoi (1915),0 Uberoi &lid Singh 

(1975), 0 Jenlcina and Goldschmidt (1973). 

0.40 r---------------, 

0.35 

0.30 <::> <-. 

"-.. 0.25 OIl . 

~ 0.20 

~ 0.15 v 0.10 

0.05 

• " ;p • '" 0 

, -.-.'-

• 0 

o 

o 

o 

o 

... , 

o 

o 

0.00 '------'---....:....-_-'-----.::~ __ ---J 

0.0 0 .5 1.0 1.5 2.0 2.5 

Y/Y.STO 

Figure 6. RMS of temperature variance in heated plane jet. 

- oontinu0U4 mixing modd, - - - modified Curl modd, 0 

Antonia et al. (1983),0 Ba,.,hir and Uberoi (1975), D. Uberoi 

and Singh (1975). 

20 

" 

~ 15 ;:: 

.. 

R-

~ 10 ... 

';;' 

::: 

s 

:: 

.. 

~ 0 ... 

tr. 

-5 

0.0 0.5 1.0 1.5 

Y/Y.ST, 

2.0 

, 

, 

2.5 3.0 

Figure 7. Ske1rnea. of temperature Mance in h~ted plane 

jet. - continuoWl mixins model, - •• modified Curl model 

29 

20r-----------------------------~ 

.. 18 ~ 

~ 16 

; " 14 

RIO 12 .. 

... 10 

';;' 8 ., 

:;: 6 ~ 

d 4 

;;: 2 

o~--~~--------~----~ ____ ~ __ ~ 

0. 0 0.5 1.0 1.5 

Y/lI.ST. 

2 .0 2.5 3. 0 

Figure 8. Flatness of temperature variance in heated plane 

jet. - continuous mixin~ model, ••• modified Curl model. 

1.0,-----------------, 

0.8 

:;-

d 0 . 6 

E--. 

~ :;; 0.4 

0.2 

OL-~~-~-~-~-~-~~ 

-0.8 -0.4 0.0 

y/or 

0.4 

Figure 9. Temperature rise in an H,-F, dilftmon flame. 

- rontinuoua mixing modd, - •• modified Curl modd, 0 

HermaIllIOn and Dimota.lci.s (1989). 

1.0r------------------------, 

i 

" \" 

0.8 I , 

~06 h" . 

~ ~ 0.4 / 

j 

0.2 

O.O~--~---~---~----L----~ 

0.0 0.2 0.4 0.6 0.8 1.0 

Alirture Frac~ f 

Figure 10. Joint pdf between temperature and mixture fraction on the centerline of the HrF, diffusion flame; reaction 

is restricted to a narrow zone at stoichiometry; re3ults are 

obtained by applying reaction zone conditioning (Chen and 

Kol.Jman.n, 1990) to the rontinuous mixing modd. 

---

----- -----
2 

1.0 

0 .8 

0 .6 

- 0 .4 

Q: 

0 .2 

0 . 0 1 

-0.2 

-0 4 

0 . 0 

Fig. 11 1 D CorqLution 

J . 

-, \ 

"-'" \ 

o· \\ 

o \'\ 

~ \" 

~\'" 

o t = C 

- - - - t = Co .DJ 

- - - T -= V ,0.' 

- - t = o.oi 

\, -- 0 \ \ " 

\ '-. "- --~ - 0 \ '- ~!! OfillHFi51J 

0' -- . -- --

~- --

0.5 1.0 .5 2 .0 2 .5 3 .0 

r 

Fig. 12 1 D Spectrum 

100 r-----------~----------------------------~ 

- 10-6 

~ 

10-tO 

100 

10- 1 

10-2 

--- 10-3 

~ 

'-

~ 10- 4 

10-5 

10-6 

10 - 7 

Fig. 

--- --- l------

I 

I 

~ I 

100 

o \ 

o \ \ ", \ 

\. . \ 

10 1 

k 

13 Energy Spectrum 

10 1 

k 

30 

2 

1.0 

0 .8 

0 .6 

- 0 .4 

Q: 

0 .2 

0 . 0 1 

-0.2 

-0 4 

0 . 0 

Fig. 11 1 D CorqLution 

J . 

-, \ 

"-'" \ 

o· \\ 

o \'\ 

~ \" 

~\'" 

o t = C 

- - - - t = Co .DJ 

- - - T -= V ,0.' 

- - t = o.oi 

\, -- 0 \ \ " 

\ '-. "- --~ - 0 \ '- ~!! OfillHFi51J 

0' -- . -- --

~- --

0.5 1.0 .5 2 .0 2 .5 3 .0 

r 

Fig. 12 1 D Spectrum 

100 r-----------~----------------------------~ 

- 10-6 

~ 

10-tO 

100 

10- 1 

10-2 

--- 10-3 

~ 

'-

~ 10- 4 

10-5 

10-6 

10 - 7 

Fig. 

--- --- l------

I 

I 

~ I 

100 

o \ 

o \ \ ", \ 

\. . \ 

10 1 

k 

13 Energy Spectrum 

10 1 

k 

30 
Fig. 11 Turbulent kinetic energy 

30 

25 

20 

10 

5 

0L-._...l-_---1 __ ....I..-_---1 __ .J.--_--.J 

0.0 0.2 0.4 0.6 

T 

Cornpresseble energey ratio. 

/\ 80 '" :::-

'-

;::> .... 

"l::! GO - v 

'""-

/\ 

'" --... 40 - -;:,.0 

'-

;::> .... 

"l::! 20 v 

0 

0. 0 0.2 0.4 0.6 

T 

Fig. 1 G Incornpresseble energey ralio. 50~-------------------------~ 

/\ 

'" 

:::- 40 '-

I-

;:l 

~ 30 - v 

'""-

/\ 

~ 20 - -;:,.0 

'-

~ 

;:l 10 - l> 

V 

n=o 

-' ,I 

0~_...l-_---1--....I..-----1--~-~ 

0. 0 0 .2 0 . 4 0.6 

T 

Fig. 17 Reynolds stresses, !1 =0., Re=20 

0 .040 r---------------, 

0 .035 

/\ § 0.030 

v 0.025 

/\ S 0.020 

v 

. 0.015 

/\ 

~ 0.010 

v 

0.005 

0.000 L-_...l.-_--1-_---l. __ .l....-_...L-_-' 

0.0 0.2 0.4 0.6 

T 

Fig.18 Reynolds stresses, 11 =40., Re =20 

0.010 r-----------------, 

0.035 

/\ 

;3 0.030 

;3 

v 0.025 

/\- '" '" ~ 0.020 " .:...WW>- v 

_ 0.015 ~ /\ "-

;:l 

v 

;:l 0.010 

0.005 

c-/ (LlU)o 

.:... Vv'"/ 

" ----- 0.000 

0 .0 0.2 0.4 0.6 

T 

Fig. 19 Reynolds slresses, 11 =80., Re=20 0.010~------------------, 

0.035 

/\ § 0.030,\ 

v 0 .025 ""\.. 

/\ 

~ 0.020 "-. '-....\. WW> v 

.0.015 

/\ 

;:! o 0.010 \ ~ ~I.\> 

0.005 ~v/ 

0.000 L--_.::l:o-_--'-___ -'--_-'-__ ----'_----' 

0 .0 

31 

Fig. 11 Turbulent kinetic energy 

30 

25 

20 

10 

5 

0L-._...l-_---1 __ ....I..-_---1 __ .J.--_--.J 

0.0 0.2 0.4 0.6 

T 

Cornpresseble energey ratio. 

/\ 80 '" :::-

'-

;::> .... 

"l::! GO - v 

'""-

/\ 

'" --... 40 - -;:,.0 

'-

;::> .... 

"l::! 20 v 

0 

0. 0 0.2 0.4 0.6 

T 

Fig. 1 G Incornpresseble energey ralio. 50~-------------------------~ 

/\ 

'" 

:::- 40 '-

I-

;:l 

~ 30 - v 

'""-

/\ 

~ 20 - -;:,.0 

'-

~ 

;:l 10 - l> 

V 

n=o 

-' ,I 

0~_...l-_---1--....I..-----1--~-~ 

0. 0 0 .2 0 . 4 0.6 

T 

Fig. 17 Reynolds stresses, !1 =0., Re=20 

0 .040 r---------------, 

0 .035 

/\ § 0.030 

v 0.025 

/\ S 0.020 

v 

. 0.015 

/\ 

~ 0.010 

v 

0.005 

0.000 L-_...l.-_--1-_---l. __ .l....-_...L-_-' 

0.0 0.2 0.4 0.6 

T 

Fig.18 Reynolds stresses, 11 =40., Re =20 

0.010 r-----------------, 

0.035 

/\ 

;3 0.030 

;3 

v 0.025 

/\- '" '" ~ 0.020 " .:...WW>- v 

_ 0.015 ~ /\ "-

;:l 

v 

;:l 0.010 

0.005 

c-/ (LlU)o 

.:... Vv'"/ 

" ----- 0.000 

0 .0 0.2 0.4 0.6 

T 

Fig. 19 Reynolds slresses, 11 =80., Re=20 0.010~------------------, 

0.035 

/\ § 0.030,\ 

v 0 .025 ""\.. 

/\ 

~ 0.020 "-. '-....\. WW> v 

.0.015 

/\ 

;:! o 0.010 \ ~ ~I.\> 

0.005 ~v/ 

0.000 L--_.::l:o-_--'-___ -'--_-'-__ ----'_----' 

0 .0 

31 
J 

Figure 20. Flaw structure without Coriolis Force, 

u-velocity. 

z 

y 

Figure 21. Flow structure with the Corio l is force, 

u-velocity. 

32 

J 

Figure 20. Flaw structure without Coriolis Force, 

u-velocity. 

z 

y 

Figure 21. Flow structure with the Corio l is force, 

u-velocity. 

32 
Center for Modeling of Turbulence and Transition 

R esearch Briefs - 1991 

Modeling of Turbulence and Transition 

Tsan-Hsing Shih 

1. Motivation and Objective 

The first objective is to evaluate current two-equation and second order closure 

turbulence models using available direct numerical simulations and experiments, 

and to identify the models which represent the state of the art in turbulence modeling. 

The second objective is to study the near-wall behavior of turbulence, and to 

develop reliable models for an engineering calculation of turbulence and transition . 

The third objective is to develop a two-scale model for compressible turbulence. 

2. Work Accomplished 

2.1 Evaluation of two-equation models (N.J. Lang and T.-H. Shihl) 

Twelve two-equation models including k - €, k - T, k - wand q - w models, 

have been evaluated using a common flow solver code (GENMIX) for wall bounded 

turbulent flows. For each model, calculations were carried out for two-dimensional, 

fully developed channel and flat plate boundary layer flows. These flows are appealing for model testing because they are simple and self-similar, yet demonstrate 

important features of wall bounded turbulent shear flows. In addition, we can 

compare these calculations with Direct Numerical Simulations (DNS). A list of the 

models which were tested are shown in the table below: 

Ch Chien2 1982 k-€ 

Sh Shih3 1991 k-f. 

LB Lam and Bremhorst4 1981 k-f. 

NH Nagano and Hishida5 1987 k-f. 

NT Nagano and TagawaO 1990 k-€ 

LS Launder and Sharma7 1974 k-f. 

JL Jones and LaunderlS 1973 k-€ 

MK Myong and Kasagill 1988 k-f. 

YS Yang and Shih10 1991 k-€ 

WIl Wilcoxll 1984 k-w 

WI2 Wilcoxl2 1991 k-w 

SAA Speziale, Abid and Anderson13 1990 k-T 

Co Coakley14 1983 q-w 

Two dimensional channel flow calculations were made at ReT = 180 and ReT = 

395. These cases were compared with DNS data of Kim et all5 . Calculations for 

the two-dimensional flat plate boundary layer flow at Reo = 1410 were compared 

with DNS data of Spalart 16 . Some flat plate boundary layer comparisons were made 

33 

-------------

Center for Modeling of Turbulence and Transition 

R esearch Briefs - 1991 

Modeling of Turbulence and Transition 

Tsan-Hsing Shih 

1. Motivation and Objective 

The first objective is to evaluate current two-equation and second order closure 

turbulence models using available direct numerical simulations and experiments, 

and to identify the models which represent the state of the art in turbulence modeling. 

The second objective is to study the near-wall behavior of turbulence, and to 

develop reliable models for an engineering calculation of turbulence and transition . 

The third objective is to develop a two-scale model for compressible turbulence. 

2. Work Accomplished 

2.1 Evaluation of two-equation models (N.J. Lang and T.-H. Shihl) 

Twelve two-equation models including k - €, k - T, k - wand q - w models, 

have been evaluated using a common flow solver code (GENMIX) for wall bounded 

turbulent flows. For each model, calculations were carried out for two-dimensional, 

fully developed channel and flat plate boundary layer flows. These flows are appealing for model testing because they are simple and self-similar, yet demonstrate 

important features of wall bounded turbulent shear flows. In addition, we can 

compare these calculations with Direct Numerical Simulations (DNS). A list of the 

models which were tested are shown in the table below: 

Ch Chien2 1982 k-€ 

Sh Shih3 1991 k-f. 

LB Lam and Bremhorst4 1981 k-f. 

NH Nagano and Hishida5 1987 k-f. 

NT Nagano and TagawaO 1990 k-€ 

LS Launder and Sharma7 1974 k-f. 

JL Jones and LaunderlS 1973 k-€ 

MK Myong and Kasagill 1988 k-f. 

YS Yang and Shih10 1991 k-€ 

WIl Wilcoxll 1984 k-w 

WI2 Wilcoxl2 1991 k-w 

SAA Speziale, Abid and Anderson13 1990 k-T 

Co Coakley14 1983 q-w 

Two dimensional channel flow calculations were made at ReT = 180 and ReT = 

395. These cases were compared with DNS data of Kim et all5 . Calculations for 

the two-dimensional flat plate boundary layer flow at Reo = 1410 were compared 

with DNS data of Spalart 16 . Some flat plate boundary layer comparisons were made 

33 

-------------
Tsan-Hsing Shih 

between the experimental data of Klebanoff17 at Reo = 7700 and solutions of the 

various models. The detailed results are reported in NASA TM 105237, 1991. 

An important criterion for two-equation model comparisons is not only how well 

the model predicts mean velocity and shear stress, but also the turbulent kinetic energy and dissipation (or specific dissipation) rate. These predictions should provide 

appropriate turbulent velocity and length scales so that the model can be applied 

to more complex flows for which a simple mixing length model often fails. Some 

researchers maintain that it is not critical whether or not the turbulent kinetic energy and the turbulent length scale are predicted with great accuracy. However, 

one may imagine that a two-equation model making unreasonable turbulent velocity and length scale predictions would be very questionable when applied to more 

general flows. A model which accurately predicts the shear stress and mean velocity does not imply that it has correctly modeled the turbulent kinetic energy and 

length scale. In fact, if the turbulent kinetic energy is incorrect, then the length 

scale must also be incorrect to compensate for the error in the turbulent kinetic 

energy. For this case, two wrongs are making a right. This warrants some caution 

when computing flows for other geometries. 

Comments on two-equation models: 

It is clear that the JL, LS, WIl and WI2 models underpredict the near wall 

turbulent kinetic energy compared to the other models. 

The standard k - E model has been proven to provide good results in the high 

Reynolds number range. It is therefore attractive for a near wall k - E turbulence 

model to approach the standard k - E model away from the wall. The LB, LS and 

YS models are the only k - E models in this study which possess this··characteristic. 

Because boundary layer and channel flows are self-similar, the solutions should 

be independent of the initial conditions. However, some of the models (SAA, Co, 

and LB) have difficulty when the initial conditions contain large gradients. The Co 

Model is particularly dependent on the initial conditions. Even slight perturbations 

of the initial conditions will yield noticeably different solutions with this model. 

JL, LS, WIl and WI2 are the only models which do not contain y+. Damping 

functions which contain y+ are not desirable because y+ is erroneous near flow 

separations and not well defined near complicated geometries. Unfortunately, these 

are the same models which poorly predict the near wall turbulent quantities. 

The Wilcox models (WIl and WI2) suffer from a numerically awkward boundary 

condition for w at the wall: 

We cannot define w at y+ = o. We have tried two ways to approximate w as y+ 

approaches o. First, we chose a large number for Wwall and, second, we used an 

asymptotic Wwall = C6V 2. Test cases showed that the solution does not converge 2Y 

as Wwall ~ 00 when y+ ~ 0 for either model. In addition, both Wilcox models 

underpredict the turbulent kinetic energy peak value for both boundary layer and 

channel flows. 

34 

Tsan-Hsing Shih 

between the experimental data of Klebanoff17 at Reo = 7700 and solutions of the 

various models. The detailed results are reported in NASA TM 105237, 1991. 

An important criterion for two-equation model comparisons is not only how well 

the model predicts mean velocity and shear stress, but also the turbulent kinetic energy and dissipation (or specific dissipation) rate. These predictions should provide 

appropriate turbulent velocity and length scales so that the model can be applied 

to more complex flows for which a simple mixing length model often fails. Some 

researchers maintain that it is not critical whether or not the turbulent kinetic energy and the turbulent length scale are predicted with great accuracy. However, 

one may imagine that a two-equation model making unreasonable turbulent velocity and length scale predictions would be very questionable when applied to more 

general flows. A model which accurately predicts the shear stress and mean velocity does not imply that it has correctly modeled the turbulent kinetic energy and 

length scale. In fact, if the turbulent kinetic energy is incorrect, then the length 

scale must also be incorrect to compensate for the error in the turbulent kinetic 

energy. For this case, two wrongs are making a right. This warrants some caution 

when computing flows for other geometries. 

Comments on two-equation models: 

It is clear that the JL, LS, WIl and WI2 models underpredict the near wall 

turbulent kinetic energy compared to the other models. 

The standard k - E model has been proven to provide good results in the high 

Reynolds number range. It is therefore attractive for a near wall k - E turbulence 

model to approach the standard k - E model away from the wall. The LB, LS and 

YS models are the only k - E models in this study which possess this··characteristic. 

Because boundary layer and channel flows are self-similar, the solutions should 

be independent of the initial conditions. However, some of the models (SAA, Co, 

and LB) have difficulty when the initial conditions contain large gradients. The Co 

Model is particularly dependent on the initial conditions. Even slight perturbations 

of the initial conditions will yield noticeably different solutions with this model. 

JL, LS, WIl and WI2 are the only models which do not contain y+. Damping 

functions which contain y+ are not desirable because y+ is erroneous near flow 

separations and not well defined near complicated geometries. Unfortunately, these 

are the same models which poorly predict the near wall turbulent quantities. 

The Wilcox models (WIl and WI2) suffer from a numerically awkward boundary 

condition for w at the wall: 

We cannot define w at y+ = o. We have tried two ways to approximate w as y+ 

approaches o. First, we chose a large number for Wwall and, second, we used an 

asymptotic Wwall = C6V 2. Test cases showed that the solution does not converge 2Y 

as Wwall ~ 00 when y+ ~ 0 for either model. In addition, both Wilcox models 

underpredict the turbulent kinetic energy peak value for both boundary layer and 

channel flows. 

34 
Tsan-Hsing Shih 

model the new unknowns T ij , IIij and Cij' At the level of the second order closure, 

these new unknowns are usually modeled with algebraic equations in terms of the 

second moments and the mean quantities (with the exception of the trace Ckk = 2c, 

which is modeled with a transport equation). 

In a general turbulent shear flow with moderate inhomogeneity, the turbulent 

diffusion terms in the second moment equations are usually smaller than the other 

terms. However, the pressure-strain rate and dissipation rate tensors are always 

among the leading terms. Therefore, the performance of modeled equations largely 

depends on the models of pressure-strain rate tensor and dissipation rate tensor. 

In this study, we only concentrate on the models of the pressure-strain rate tensor 

and the dissipation rate tensor for the velocity field: IIij - Cij, which are modeled 

as _ (1) (2) 2 

IIij - Cij - IIij + IIij - 3cbij 

where rrU), rr~~) are respectively referred to as the rapid term and the return-toisotropy term. 

Models for the rapid term II~~) 

Launder, Reece and Rodi (LRR):18 

rrD) 9C2 + 6 2 ---=- = 0.2Si j + (bikSjk + bjkSik - -bijbklSkt) 2q2 22 3 

10 - 7C2 + 22 (bikD.jk + bjkD.ik ) 

where C2 = 0.4, and 

UiUj ' 1 

bij = q2 - "3 bij , 

1 S·· - -CU· . + U· .) t] - 2 t,] ],t , 

1 n ·· - -CU· . - U· .) t] - 2 t,] ] ,t 

(2.2.1) 

This model is linear in the Reynolds-stress. It contains only one model constant C2. This model satisfies the conventional model constraints34 . It is the most general 

form at the level of linear dependence on the Reynolds-stress. However, as Lumley24 

pointed out, this model may violate realizability as the turbulence approaches a 

two-component state. 

Speziale, Sarkar and Gatski (SSG):19 

II~~) 0 .8 - C3* Ci P -= S· · --b·· 2q2 4 t] 2q2 t] 

C4 2 

+ 4 (bikSjk + bjkSik - "3bijbklSkl) (2.2.2) 

C5 + 4 (bikD.jk + bjkD.ik ) 

35 

Tsan-Hsing Shih 

model the new unknowns T ij , IIij and Cij' At the level of the second order closure, 

these new unknowns are usually modeled with algebraic equations in terms of the 

second moments and the mean quantities (with the exception of the trace Ckk = 2c, 

which is modeled with a transport equation). 

In a general turbulent shear flow with moderate inhomogeneity, the turbulent 

diffusion terms in the second moment equations are usually smaller than the other 

terms. However, the pressure-strain rate and dissipation rate tensors are always 

among the leading terms. Therefore, the performance of modeled equations largely 

depends on the models of pressure-strain rate tensor and dissipation rate tensor. 

In this study, we only concentrate on the models of the pressure-strain rate tensor 

and the dissipation rate tensor for the velocity field: IIij - Cij, which are modeled 

as _ (1) (2) 2 

IIij - Cij - IIij + IIij - 3cbij 

where rrU), rr~~) are respectively referred to as the rapid term and the return-toisotropy term. 

Models for the rapid term II~~) 

Launder, Reece and Rodi (LRR):18 

rrD) 9C2 + 6 2 ---=- = 0.2Si j + (bikSjk + bjkSik - -bijbklSkt) 2q2 22 3 

10 - 7C2 + 22 (bikD.jk + bjkD.ik ) 

where C2 = 0.4, and 

UiUj ' 1 

bij = q2 - "3 bij , 

1 S·· - -CU· . + U· .) t] - 2 t,] ],t , 

1 n ·· - -CU· . - U· .) t] - 2 t,] ] ,t 

(2.2.1) 

This model is linear in the Reynolds-stress. It contains only one model constant C2. This model satisfies the conventional model constraints34 . It is the most general 

form at the level of linear dependence on the Reynolds-stress. However, as Lumley24 

pointed out, this model may violate realizability as the turbulence approaches a 

two-component state. 

Speziale, Sarkar and Gatski (SSG):19 

II~~) 0 .8 - C3* Ci P -= S· · --b·· 2q2 4 t] 2q2 t] 

C4 2 

+ 4 (bikSjk + bjkSik - "3bijbklSkl) (2.2.2) 

C5 + 4 (bikD.jk + bjkD.ik ) 

35 
Modeling of Turbulence and Transition 

where, 

C** - C*(b b )1/2 3 - 3 ij ij , P = -UiUjUi,j 

c~ = 1.8 C; = 1.3, C4 = 1.25, C5 = 0.4 

This model is quasi-linear in the Reynolds-stress, because the coefficients in the 

first two terms are not constant, but depend on the invariant of the Reynoldsstress tensor and the production P. This model contains four model const ants 

(Ci, Ct, C4 , C5 ), therefore one may imagine that it will be difficult to correctly 

calibrate them. In addition, this model does not satisfy the normalization constraint 

which is one of the basic model constraints34 . If we impose this constraint, then 

the four coefficients will reduce to only one, and this model will reduce to the LRR 

model. Finally, like the LRR model, the SSG model may also violate realizability. 

Fu, Launder and Tselepidakis (FLT):2o 

n~~) 2 

tJ = 0.2Sij + 0.3(bikSjk + bjkSik - -8ijbklSkl) 2~ 3 

1.3( ) + """"3 biko'jk + bjko'ik 

+ 0.2(b;lSjl + b;lSil - 2bkjbliSkl - 3bijbklSkl) 

+ 0.2(b~lo'jl + b;lo'il) 

+ r[4b~n(biko'jk + bjko'ik) 

+ 12bmibnj(bmko'nk + bnkDmk)] 

where r = 0.7, btj = bikbkj . 

(2.2.3) 

This model is cubic in the Reynolds-stress. The final form selected contains 

one model constant. This model only satisfies a part of the realizability condition, 

corresponding to a two-component state of turbulence. However, when a scalar 

field is involved, this model cannot satisfy Schwarz' inequality between velocity and 

temperature. This part of realizability is sometimes called joint realizability. 

Shih and Lumley (SL):21 

where, 

n~l) 2 

tJ = 0.2Sij + 30:5 (bikSjk + bjkSik - -8ijbklSkl) 2~ 3 

1 

+ 3(2 - 70:5 )(biko'jk + bjko'id 

+ 0.2(b~lSjl + b;lSil - 2bkjbliSkl - 3bijbklSkl) 

+ 0.2(b;lo'jl + b1lo'il) 

_ 1 ( 1/2) 0:5 - - 1 + 0.8F , 10 

(2.2.4) 

This model is quasi-quadratic in the Reynolds-stress, because the model coefficient 

0:5 is a function of the invariants of the Reynolds-stress tensor. We emphasize that 

36 

Modeling of Turbulence and Transition 

where, 

C** - C*(b b )1/2 3 - 3 ij ij , P = -UiUjUi,j 

c~ = 1.8 C; = 1.3, C4 = 1.25, C5 = 0.4 

This model is quasi-linear in the Reynolds-stress, because the coefficients in the 

first two terms are not constant, but depend on the invariant of the Reynoldsstress tensor and the production P. This model contains four model const ants 

(Ci, Ct, C4 , C5 ), therefore one may imagine that it will be difficult to correctly 

calibrate them. In addition, this model does not satisfy the normalization constraint 

which is one of the basic model constraints34 . If we impose this constraint, then 

the four coefficients will reduce to only one, and this model will reduce to the LRR 

model. Finally, like the LRR model, the SSG model may also violate realizability. 

Fu, Launder and Tselepidakis (FLT):2o 

n~~) 2 

tJ = 0.2Sij + 0.3(bikSjk + bjkSik - -8ijbklSkl) 2~ 3 

1.3( ) + """"3 biko'jk + bjko'ik 

+ 0.2(b;lSjl + b;lSil - 2bkjbliSkl - 3bijbklSkl) 

+ 0.2(b~lo'jl + b;lo'il) 

+ r[4b~n(biko'jk + bjko'ik) 

+ 12bmibnj(bmko'nk + bnkDmk)] 

where r = 0.7, btj = bikbkj . 

(2.2.3) 

This model is cubic in the Reynolds-stress. The final form selected contains 

one model constant. This model only satisfies a part of the realizability condition, 

corresponding to a two-component state of turbulence. However, when a scalar 

field is involved, this model cannot satisfy Schwarz' inequality between velocity and 

temperature. This part of realizability is sometimes called joint realizability. 

Shih and Lumley (SL):21 

where, 

n~l) 2 

tJ = 0.2Sij + 30:5 (bikSjk + bjkSik - -8ijbklSkl) 2~ 3 

1 

+ 3(2 - 70:5 )(biko'jk + bjko'id 

+ 0.2(b~lSjl + b;lSil - 2bkjbliSkl - 3bijbklSkl) 

+ 0.2(b;lo'jl + b1lo'il) 

_ 1 ( 1/2) 0:5 - - 1 + 0.8F , 10 

(2.2.4) 

This model is quasi-quadratic in the Reynolds-stress, because the model coefficient 

0:5 is a function of the invariants of the Reynolds-stress tensor. We emphasize that 

36 
Tsan-Hsing Shih 

this model is obtained from a more general form of the expression than the FLT, 

and satisfies both the two component condition and Schwarz' inequality between 

the velocity and scalar fields. In addition, the final form is simpler than the model 

of FLT. 

Shih and Mansour (SM):22 

rr~~) 2 

lJ = O.2Sij + 3a5 (bikSjk + bjkSik - -DijbklSkl) 2q2 3 

1 

+ 3(2 - 7(5)(bikDjk + bjkDik ) 

+ O.2(b;lSjl + b;lSil - 2bkj bliSkl - 3bijbklSkl) 

+ 0.2(b;lo'jl + b;lo'il) 

where, fr5 = 1~ {I + 3.5[1 - (1 - F)1/4]}. 

(2.2.5) 

This model has the same form as the SL model. It was derived in a different way 

and contains a different model coefficient a5 which was calibrated from one of the 

DNS data (Rogers30 ). This model, like the SL model, fully satisfies realizability 

conditions. 

Models for the return-to-isotropy term rrU) 

Rotta: 23 

where, C = 3.0. 

(2.2.6) 

This model is linear in the Reynolds stress, and contains one model constant. It 

was widely used and adopted in the LRR model. We notice that this model will not 

allow the turbulence to reach a two-component state, because when any turbulent 

component reduces to q2/9, the model Eq.(2.2.6) will force it to grow. 

Lumley:24 

(2.2.7) 

where, ,= 0 and 

F 

(3 = 2 + 9" exp( -7.77 /.JRe){72/.JRe + 80.1In[1 + 62.4( -II + 2.3III)]} 

-2 q2 Re=- 9cv 

This model is quasi-linear in the Reynolds stress, because, is set to zero, and (3 

is a function of the invariants of Reynolds stress tensor. This model is simple, and 

satisfies realizability. 

Sarkar and Speziale (SS):25 

(2.2.8) 

37 

Tsan-Hsing Shih 

this model is obtained from a more general form of the expression than the FLT, 

and satisfies both the two component condition and Schwarz' inequality between 

the velocity and scalar fields. In addition, the final form is simpler than the model 

of FLT. 

Shih and Mansour (SM):22 

rr~~) 2 

lJ = O.2Sij + 3a5 (bikSjk + bjkSik - -DijbklSkl) 2q2 3 

1 

+ 3(2 - 7(5)(bikDjk + bjkDik ) 

+ O.2(b;lSjl + b;lSil - 2bkj bliSkl - 3bijbklSkl) 

+ 0.2(b;lo'jl + b;lo'il) 

where, fr5 = 1~ {I + 3.5[1 - (1 - F)1/4]}. 

(2.2.5) 

This model has the same form as the SL model. It was derived in a different way 

and contains a different model coefficient a5 which was calibrated from one of the 

DNS data (Rogers30 ). This model, like the SL model, fully satisfies realizability 

conditions. 

Models for the return-to-isotropy term rrU) 

Rotta: 23 

where, C = 3.0. 

(2.2.6) 

This model is linear in the Reynolds stress, and contains one model constant. It 

was widely used and adopted in the LRR model. We notice that this model will not 

allow the turbulence to reach a two-component state, because when any turbulent 

component reduces to q2/9, the model Eq.(2.2.6) will force it to grow. 

Lumley:24 

(2.2.7) 

where, ,= 0 and 

F 

(3 = 2 + 9" exp( -7.77 /.JRe){72/.JRe + 80.1In[1 + 62.4( -II + 2.3III)]} 

-2 q2 Re=- 9cv 

This model is quasi-linear in the Reynolds stress, because, is set to zero, and (3 

is a function of the invariants of Reynolds stress tensor. This model is simple, and 

satisfies realizability. 

Sarkar and Speziale (SS):25 

(2.2.8) 

37 
Modeling of Turbulence and Transition 

where C1 = 3.4. 

This is a quadratic model in the Reynolds-stress tensor. It satisfies what is 

called the weak realizability condition. Like the Rotta model Eq.(2.2.8), this model 

will not produce unphysical results. However, it will not allow the turbulence to 

approach a two-component state, which could occur in some situations,. for example, 

in near-wall turbulence. 

Haworth and Pope (HP):26 

(2.2.9) 

where C1 = 8.3, C2 = 14.8. 

Eq.(2.2.9) is a slow part of the Haworth and Pope's model for the situations 

with no mean velocity gradient. This model, like the SS model, will not produce 

unphysical results; however, it will also not allow the turbulence to approach a 

two-component state. 

Choi and Lumley (CL):27 

If III 2: 0, 

where, 

p* F1/ 2 

f3 = 2 + 1 + G X2 

p* F1/ 2 G 

1= 1 + G X2 T ~ = (III/2)1/3, 1] = (-11/3)1/2 

X = i, G = _X4 + 0.8X6 

1] 

(2.2.10.1) 

p* = exp[-9.29/ Re1

/

2

]{( ~~~~2 + 7!~7) - [296 - 16.2(X + 1)4]II} 

-2 q2 

Re = -, II = -bijbij /2 , III = bijbjkbki/3 9EV 

IfIll < 0, 

II~~ ) = Eq.(28) (2.2.10.2) 

The model coefficients in Eq.(2.2.10.1) were obtained using their comprehensive measurements of turbulence relaxing from axisymmetric expansion. Both 

Eq.(2.2.10.1) and Eq.(2.2.10.2) satisfy realizability; however, Eq.(2.2.10.1) is valid 

only for III 2: 0, because ~ is not defined when III < O. 

Craft and Launder (C&L):28 

(2.2.11) 

38 

Modeling of Turbulence and Transition 

where C1 = 3.4. 

This is a quadratic model in the Reynolds-stress tensor. It satisfies what is 

called the weak realizability condition. Like the Rotta model Eq.(2.2.8), this model 

will not produce unphysical results. However, it will not allow the turbulence to 

approach a two-component state, which could occur in some situations,. for example, 

in near-wall turbulence. 

Haworth and Pope (HP):26 

(2.2.9) 

where C1 = 8.3, C2 = 14.8. 

Eq.(2.2.9) is a slow part of the Haworth and Pope's model for the situations 

with no mean velocity gradient. This model, like the SS model, will not produce 

unphysical results; however, it will also not allow the turbulence to approach a 

two-component state. 

Choi and Lumley (CL):27 

If III 2: 0, 

where, 

p* F1/ 2 

f3 = 2 + 1 + G X2 

p* F1/ 2 G 

1= 1 + G X2 T ~ = (III/2)1/3, 1] = (-11/3)1/2 

X = i, G = _X4 + 0.8X6 

1] 

(2.2.10.1) 

p* = exp[-9.29/ Re1

/

2

]{( ~~~~2 + 7!~7) - [296 - 16.2(X + 1)4]II} 

-2 q2 

Re = -, II = -bijbij /2 , III = bijbjkbki/3 9EV 

IfIll < 0, 

II~~ ) = Eq.(28) (2.2.10.2) 

The model coefficients in Eq.(2.2.10.1) were obtained using their comprehensive measurements of turbulence relaxing from axisymmetric expansion. Both 

Eq.(2.2.10.1) and Eq.(2.2.10.2) satisfy realizability; however, Eq.(2.2.10.1) is valid 

only for III 2: 0, because ~ is not defined when III < O. 

Craft and Launder (C&L):28 

(2.2.11) 

38 
Tsan-Hsing Shih 

where, 

This model is tensorially quadratic in the Reynolds stress, and satisfies realizability. 

Yamamoto and Arakawa (YA):29 

where, 

al = 2 + P F [q (b~kr + Ibtkl S sign(btk)] 

a2 = 3 (al - 2) 

(2.2.12) 

p = -12, q = -0.65, r = 0.4, s = 0.45 

_ 9 2 3 F - 1 - 2bkk + 9 bkk 

The YA model tried to fit situations with both positive and negative b~k' It also 

meets the requirement of realizability .. 

Shih and Mansour (S&M) :22 

(2.2.13) 

where, 

Ct = (1/9) exp( -7.77 /VRe){72/.JRe + 80.11n[1 + 62.4( -II + 2.3III)]} 

-2 q2 

1 = 10 (1 - FTJ), Re = -

9cl/ 

F = 1 + 911 + 3111 

1 1 

II = -2biJ' bij, III = 3bijbjkbki 

e = 17/20, 17 = 1/20, 10 = -2 

This model matches the data of Comte-Bellot and Corrsin21 and meets the requirement that there will be no return to isotropy in the zero Reynolds number limit. 

This model also satisfies realizability. 

Concluding remarks 

We notice that the Reynolds number in all these simulations is low and therefore 

may not represent real turbulence in nature. However, the model terms concerned 

here are mainly pressure related correlations. The fluctuating pressure is not directly related to the viscosity, hence the pressure related correlation terms may 

not be directly affected by the Reynolds number, especially the rapid term. The 

return-to-isotropy term which includes the deviatoric part of the dissipation rate 

39 

Tsan-Hsing Shih 

where, 

This model is tensorially quadratic in the Reynolds stress, and satisfies realizability. 

Yamamoto and Arakawa (YA):29 

where, 

al = 2 + P F [q (b~kr + Ibtkl S sign(btk)] 

a2 = 3 (al - 2) 

(2.2.12) 

p = -12, q = -0.65, r = 0.4, s = 0.45 

_ 9 2 3 F - 1 - 2bkk + 9 bkk 

The YA model tried to fit situations with both positive and negative b~k' It also 

meets the requirement of realizability .. 

Shih and Mansour (S&M) :22 

(2.2.13) 

where, 

Ct = (1/9) exp( -7.77 /VRe){72/.JRe + 80.11n[1 + 62.4( -II + 2.3III)]} 

-2 q2 

1 = 10 (1 - FTJ), Re = -

9cl/ 

F = 1 + 911 + 3111 

1 1 

II = -2biJ' bij, III = 3bijbjkbki 

e = 17/20, 17 = 1/20, 10 = -2 

This model matches the data of Comte-Bellot and Corrsin21 and meets the requirement that there will be no return to isotropy in the zero Reynolds number limit. 

This model also satisfies realizability. 

Concluding remarks 

We notice that the Reynolds number in all these simulations is low and therefore 

may not represent real turbulence in nature. However, the model terms concerned 

here are mainly pressure related correlations. The fluctuating pressure is not directly related to the viscosity, hence the pressure related correlation terms may 

not be directly affected by the Reynolds number, especially the rapid term. The 

return-to-isotropy term which includes the deviatoric part of the dissipation rate 

39 
.. 

_._.

Modeling of Turbulence and Transition 

tensor may have some dependence on the Reynolds number. According to the above 

consideration, we think that direct comparisons with low-Reynolds DNS data are legitimate, although we should keep in mind the possible low-Reynolds number effect 

of the DNS data. 

We have directly compared five rapid models with fifteen DNS flows: four of 

Rogers et al. 'S30 shear flows, eleven of Lee et al. 'S31 irrotational strain flows (axisymmetric contraction, axisymmetric expansion and plane strain). Comparing the performance of the LRR and SSG models, which are tensorially linear in the Reynolds 

stress, we conclude that the SSG model gives very little improvement over the LRR 

model. In fact in many cases, it is worse than the LRR model. The reason is not 

very clear. However, we notice that the SSG model does not satisfy the normalization condition which may be a cause for its poor behavior. If we impose this 

constraint on the SSG model, then it will exactly reduce to the LRR model. In fact 

it can be shown that the most general form of the rapid model, which is tensorially 

linear in the Reynolds stress, is the LRR model. Therefore, in general, the treatment used in the SSG model would hardly give any improvement over the LRR 

model. A natural way to improve the model is to use a more general nonlinear form 

and more general model constraints. A typical example is the SL21 model. It starts 

with the most general form, using full realizability constraints together with the 

other conventional constraints34 . The result is a well behaved model. Indeed, from 

the direct comparisons with the DNS data, the SL21 model and its variation form 

of SM22 model give the best performance in most of the cases. As to the FLT20 

model, it is also a nonlinear model. It starts with a tensorially cubic dependance on 

the Reynolds stress with constant coefficients (in general, these coefficients should 

not be restricted to constants). In addition, the two-component conditions of turbulence have been imposed. However, the FLT model ignores Schwarz' inequality. 

Its final form contains two undetermined model constants, but one of them is set 

to zero. The performance of the FLT model, from the direct comparisons with the 

DNS data, is better overall than the linear models, but does not compare with the 

performance of the SL and SM models. So from these direct comparisons of the 

rapid models, we conclude that the SL21 model and its variation form SM22 are 

clearly the best. Having said this we notice that, as Reynolds33 pointed out, any of 

these rapid models will not show any effect of rotation on the invariants (I I, I I 1) of 

the anisotropy tensor bij . This is clearly a theoretical deficiency of the current rapid 

models. A further investigation is needed to find out how serious this deficiency 

will be in practice. 

We have directly compared eight return-to-isotropy models with thirty four DNS 

flows: four shear flows and thirty relaxation flows from axisymmetric contraction, 

axisymmetric expansion and plane strain. As was discussed earlier, all the returnto-isotropy models are variations of Eq.(2.2.7) derived by Lumley24. Therefore 

the differences in the models are due to the different choices of the model coefficients. Two linear models are due to Rotta23 and Lumley24 (which is quasi-linear 

in bij ). Lumley's model satisfies realizability, matches the data of Comte-Bellot 

40 

.. _._.

Modeling of Turbulence and Transition 

tensor may have some dependence on the Reynolds number. According to the above 

consideration, we think that direct comparisons with low-Reynolds DNS data are legitimate, although we should keep in mind the possible low-Reynolds number effect 

of the DNS data. 

We have directly compared five rapid models with fifteen DNS flows: four of 

Rogers et al. 'S30 shear flows, eleven of Lee et al. 'S31 irrotational strain flows (axisymmetric contraction, axisymmetric expansion and plane strain). Comparing the performance of the LRR and SSG models, which are tensorially linear in the Reynolds 

stress, we conclude that the SSG model gives very little improvement over the LRR 

model. In fact in many cases, it is worse than the LRR model. The reason is not 

very clear. However, we notice that the SSG model does not satisfy the normalization condition which may be a cause for its poor behavior. If we impose this 

constraint on the SSG model, then it will exactly reduce to the LRR model. In fact 

it can be shown that the most general form of the rapid model, which is tensorially 

linear in the Reynolds stress, is the LRR model. Therefore, in general, the treatment used in the SSG model would hardly give any improvement over the LRR 

model. A natural way to improve the model is to use a more general nonlinear form 

and more general model constraints. A typical example is the SL21 model. It starts 

with the most general form, using full realizability constraints together with the 

other conventional constraints34 . The result is a well behaved model. Indeed, from 

the direct comparisons with the DNS data, the SL21 model and its variation form 

of SM22 model give the best performance in most of the cases. As to the FLT20 

model, it is also a nonlinear model. It starts with a tensorially cubic dependance on 

the Reynolds stress with constant coefficients (in general, these coefficients should 

not be restricted to constants). In addition, the two-component conditions of turbulence have been imposed. However, the FLT model ignores Schwarz' inequality. 

Its final form contains two undetermined model constants, but one of them is set 

to zero. The performance of the FLT model, from the direct comparisons with the 

DNS data, is better overall than the linear models, but does not compare with the 

performance of the SL and SM models. So from these direct comparisons of the 

rapid models, we conclude that the SL21 model and its variation form SM22 are 

clearly the best. Having said this we notice that, as Reynolds33 pointed out, any of 

these rapid models will not show any effect of rotation on the invariants (I I, I I 1) of 

the anisotropy tensor bij . This is clearly a theoretical deficiency of the current rapid 

models. A further investigation is needed to find out how serious this deficiency 

will be in practice. 

We have directly compared eight return-to-isotropy models with thirty four DNS 

flows: four shear flows and thirty relaxation flows from axisymmetric contraction, 

axisymmetric expansion and plane strain. As was discussed earlier, all the returnto-isotropy models are variations of Eq.(2.2.7) derived by Lumley24. Therefore 

the differences in the models are due to the different choices of the model coefficients. Two linear models are due to Rotta23 and Lumley24 (which is quasi-linear 

in bij ). Lumley's model satisfies realizability, matches the data of Comte-Bellot 

40 
,

Tsan-Hsing Shih 

and Corrsin32 and the limit of the final period of decaying turbulence. It performs perfectly when III < O. It also compares well with the DNS data in which 

III ~ O. Rotta's model does not compare with the performance of Lumley's model. 

In fact, the nonlinear models of SS, YA, HP and C&L also do not compare with the 

performance of Lumley's model. Apparently the model coefficients chosen in these 

models are not appropriate. The CL27 model is designed for flows with I I I ~ 0 and 

is based on their experiments on relaxing turbulence. It does work better than Lumley's model when I I I ~ O. Finally, the S&M22 model is a nonlinear model; it works 

just like Lumley'S model when III < O. When III ~ 0, it shows an improvement 

over Lumley'S model according to the DNS data. So from these direct comparisons 

of the return-to-isotropy models, we conclude that the combination of Lumley'S 

model and Choi's model, that is the CL27 model, will give the best performance. 

The S&M22 model seems as good as the CL model according to these comparisons. 

Having said this, we notice that the existing return-to-isotropy models do not follow 

the relaxation flows from expansion and plane strain very well. Therefore there is 

still a need to further investigate and improve the return-to-isotropy models. 

For detailed comparison in each flow see the reference34 . 

2.3 Near-wall behavior of turbulence 

The near-wall behavior of turbulence is re-examined in a way different from that 

proposed by Hanjalic and Launder35 and followers36,37 ,38 ,3 . It is shown that at a 

certain distance from the wall, all energetic large eddies will reduce to Kolmogorov 

eddies (the smallest eddies in turbulence). All the important wall parameters, such 

as friction velocity, viscous length scale, and mean strain rate at the wall, are 

characterized by Kolmogorov microscales. According to this Kolmogorov behavior 

of near-wall turbulence, the turbulence quantities, such as turbulent kinetic energy, 

dissipation rate, etc. at the location where the large eddies become "Kolmogorov" 

eddies, can be estimated by using both direct numerical simulation (DNS) data 

and asymptotic analysis of near-wall turbulence. This information will provide 

useful boundary conditions for the turbulent transport equations. As an example, 

the concept is incorporated in the standard k-c model which is then applied to 

channel and boundary layer flows. Using appropriate boundary conditions (based 

on Kolmogorov behavior of near-wall turbulence), there is no need for any wallmodification to the k-c equations (including model constants). Results compare 

very well with the DNS and experimental data. 

Here we only list the results from this study, for the detail see NASA TM 105663. 

Model equation and boundary condition 

The K-c equations for incompressible flows can be in general modeled as: 

DK liT -Dt = [(lI + -)K·J . + lITU, ·(U· . + U· ·) - c (J"k ,t ,t t,J t ,J J ,t (2.3.1 ) 

Dc liT c -

Dt = [(lI + -)c oJ. + C1flllTU, ·(U· ° + U o .)- (J" € ,t ,t t ,J t ,J J ,t K 

• 

6 2 

- C2f2 -

K + lIlITU, °kU· Ok t,J t ,J (2.3. 2) 

41 

,

Tsan-Hsing Shih 

and Corrsin32 and the limit of the final period of decaying turbulence. It performs perfectly when III < O. It also compares well with the DNS data in which 

III ~ O. Rotta's model does not compare with the performance of Lumley's model. 

In fact, the nonlinear models of SS, YA, HP and C&L also do not compare with the 

performance of Lumley's model. Apparently the model coefficients chosen in these 

models are not appropriate. The CL27 model is designed for flows with I I I ~ 0 and 

is based on their experiments on relaxing turbulence. It does work better than Lumley's model when I I I ~ O. Finally, the S&M22 model is a nonlinear model; it works 

just like Lumley'S model when III < O. When III ~ 0, it shows an improvement 

over Lumley'S model according to the DNS data. So from these direct comparisons 

of the return-to-isotropy models, we conclude that the combination of Lumley'S 

model and Choi's model, that is the CL27 model, will give the best performance. 

The S&M22 model seems as good as the CL model according to these comparisons. 

Having said this, we notice that the existing return-to-isotropy models do not follow 

the relaxation flows from expansion and plane strain very well. Therefore there is 

still a need to further investigate and improve the return-to-isotropy models. 

For detailed comparison in each flow see the reference34 . 

2.3 Near-wall behavior of turbulence 

The near-wall behavior of turbulence is re-examined in a way different from that 

proposed by Hanjalic and Launder35 and followers36,37 ,38 ,3 . It is shown that at a 

certain distance from the wall, all energetic large eddies will reduce to Kolmogorov 

eddies (the smallest eddies in turbulence). All the important wall parameters, such 

as friction velocity, viscous length scale, and mean strain rate at the wall, are 

characterized by Kolmogorov microscales. According to this Kolmogorov behavior 

of near-wall turbulence, the turbulence quantities, such as turbulent kinetic energy, 

dissipation rate, etc. at the location where the large eddies become "Kolmogorov" 

eddies, can be estimated by using both direct numerical simulation (DNS) data 

and asymptotic analysis of near-wall turbulence. This information will provide 

useful boundary conditions for the turbulent transport equations. As an example, 

the concept is incorporated in the standard k-c model which is then applied to 

channel and boundary layer flows. Using appropriate boundary conditions (based 

on Kolmogorov behavior of near-wall turbulence), there is no need for any wallmodification to the k-c equations (including model constants). Results compare 

very well with the DNS and experimental data. 

Here we only list the results from this study, for the detail see NASA TM 105663. 

Model equation and boundary condition 

The K-c equations for incompressible flows can be in general modeled as: 

DK liT -Dt = [(lI + -)K·J . + lITU, ·(U· . + U· ·) - c (J"k ,t ,t t,J t ,J J ,t (2.3.1 ) 

Dc liT c -

Dt = [(lI + -)c oJ. + C1flllTU, ·(U· ° + U o .)- (J" € ,t ,t t ,J t ,J J ,t K 

• 

6 2 

- C2f2 -

K + lIlITU, °kU· Ok t,J t ,J (2.3. 2) 

41 
Modeling of Turbulence and Transition 

C1 = 1.44, C2 = 1.92 

R2 

h = I, h = 1 - 0.22 exp (- 3~ ), 

(Jk = I, (JE = 1.3 

(2.3.3) 

These equations are used only for the flow field outside of the turbulence limit point 

y.", where K." is non-zero. Therefore, Eq.(2.3.2) will not have singularity problems 

and will not need any near-wall modifications like other K-c models do. 2,8 

Eddy viscosity: 

where, 

fll- = [1 - exp(a1 Rk + a3 R2 + a5RDl~ 

a1 = -1.5 * 10-4 a3 = -1.0 * 10-9 a5 = -5.0 * 10-10 

Kl/2 y Rk = ---1/ 

Boundary conditions: at y." = 61/ jUr, 

u4 

c." = 0.251...-2. 1/ 

K." = 0.250u; 

(2.3.4) 

(2.3.5) 

(2.3.6) 

(2.3.7) 

In practical applications, Rer and Reoo are large numbers, hence y."j L (L is the 

length scale of a flow field) is usually very small. Therefore, as an approximation 

we may let y."j L = 0, but c." and K." must be given by Eqs.(2.3.6) and (2.3.7) 

respectively. These equations have been applied to the c.alculations of channel and 

boundary layer flows. 

Comparison of models 

To compare the present model with the DNS data and other models (e.g. Jones 

and Launder8 , and Chien2 ), we have made calculations on two channel flowS15 ,40 

and two boundary layer flows 16,17. In the present model, all the model constants 

are the same as used in the standard K-c model39 . Therefore the present model 

will also be suitable for flows away from the wall. The other two models used here 

for comparison do not have this property. Results are shown in figures 1 - 4. In 

figure 1 and figure 2, three models are compared with two DNS data for channel 

flows: one with Rer = 180, the other with Rer = 395. The profiles of mean velocity, 

Reynolds stress, turbulent kinetic energy and its dissipation rate are plotted in these 

figures. The present model is significantly better than the other two models. Figure 

3 shows the similar comparison for a turbulent boundary with ReO = 1410. The 

agreement between the present model and DNS data is excellent. Figure 4 shows 

42 

Modeling of Turbulence and Transition 

C1 = 1.44, C2 = 1.92 

R2 

h = I, h = 1 - 0.22 exp (- 3~ ), 

(Jk = I, (JE = 1.3 

(2.3.3) 

These equations are used only for the flow field outside of the turbulence limit point 

y.", where K." is non-zero. Therefore, Eq.(2.3.2) will not have singularity problems 

and will not need any near-wall modifications like other K-c models do. 2,8 

Eddy viscosity: 

where, 

fll- = [1 - exp(a1 Rk + a3 R2 + a5RDl~ 

a1 = -1.5 * 10-4 a3 = -1.0 * 10-9 a5 = -5.0 * 10-10 

Kl/2 y Rk = ---1/ 

Boundary conditions: at y." = 61/ jUr, 

u4 

c." = 0.251...-2. 1/ 

K." = 0.250u; 

(2.3.4) 

(2.3.5) 

(2.3.6) 

(2.3.7) 

In practical applications, Rer and Reoo are large numbers, hence y."j L (L is the 

length scale of a flow field) is usually very small. Therefore, as an approximation 

we may let y."j L = 0, but c." and K." must be given by Eqs.(2.3.6) and (2.3.7) 

respectively. These equations have been applied to the c.alculations of channel and 

boundary layer flows. 

Comparison of models 

To compare the present model with the DNS data and other models (e.g. Jones 

and Launder8 , and Chien2 ), we have made calculations on two channel flowS15 ,40 

and two boundary layer flows 16,17. In the present model, all the model constants 

are the same as used in the standard K-c model39 . Therefore the present model 

will also be suitable for flows away from the wall. The other two models used here 

for comparison do not have this property. Results are shown in figures 1 - 4. In 

figure 1 and figure 2, three models are compared with two DNS data for channel 

flows: one with Rer = 180, the other with Rer = 395. The profiles of mean velocity, 

Reynolds stress, turbulent kinetic energy and its dissipation rate are plotted in these 

figures. The present model is significantly better than the other two models. Figure 

3 shows the similar comparison for a turbulent boundary with ReO = 1410. The 

agreement between the present model and DNS data is excellent. Figure 4 shows 

42 
Tsan-Hsing Shih 

the results compared with Klebanoff17 and other boundary layer experiments. The 

skin friction from DNS data16 is also shown in this figure. The results of present 

model are more consistent with the DNS data than the experiments. 

It is also worthwhile to emphasize that the present model equations with the 

standard model coefficients have the simplest form among all two-equation models. 

Hence, we expect that they will have less numerical stiffness in complex turbulent 

flows. 

2.4 Modeling of transition (Z. Yang and T.-H. Shih) 

A model of intermittancy based on the shape factor is added to a two-equation 

k-€ model for prediction of boundary layer transition with a free stream turbulence. 

The detailed model and calculations are give by Yang in this year's Research Briefs. 

2.5 Modeling of compressible turbulence (W.W. Liou and T.H. Shih) 

A two-scale model is proposed based on Hanjalic-Launder's multiple-scale concept 

for compressible turbulence, in which a distinct scale created by the compressibility 

is modeled separately by considering the effects of pressure-dilatation and dissipation dilatation on large-scale energy transfer rate. The detailed model is given by 

Liou in this year's Research Briefs. 

2.6 Direct numerical simulation of compressible flows (A. Hsu 

and T.-H. Shih) 

In order to have a better understanding of the effect of compressibility on turbulence, especially the effect of the formation of eddy-shocklets on turbulence, a 

direct numerical simulation of compressible homogeneous shear turbulent flows is 

been performing. The data of all turbulence statistics are very useful for turbulence 

modeling. The detailed simulation is given by Hsu in this year's Research Briefs. 

3. Future Plans 

Development of second order closure models: pay special attention to the effects 

of inhomogeneity, non-local property, frame-rotation, compressibility, near-wall behavior in the buffer and log-layers. 

4. References 

1 Lang, H.J. and Shih, T.-H., "A critical comparison of two-equation turbulence 

models," NASA TM 105237 (1991). 

2 Chien, K.-Y., "Predictions of Channel and Boundary-Layer Flows with a LowReynolds-Number Turbulence Model," AIAA Journal, 20, 33-38, (1982). 

3 Shih, T. -H., An Improved k-€ Model for Near-Wall Turbulence and Comparison 

with Direct Numerical Simulation," NASA TM-103221, August (1990). 

4 Lam, C. K. G., and Bremhorst, K., "A Modified Form of the k-€ Model for 

Predicting Wall Turbulence," ASME Transaction, Journal of Fluids Engineering, 

103, September (1981). 

5 Nagano, Y. and Hishida, M., "Improved Form of the k-€ Model for Wall Turbulent Shear Flows," ASME Transaction, Journal of Fluids Engineering, 109, June 

(1987). 

43 

Tsan-Hsing Shih 

the results compared with Klebanoff17 and other boundary layer experiments. The 

skin friction from DNS data16 is also shown in this figure. The results of present 

model are more consistent with the DNS data than the experiments. 

It is also worthwhile to emphasize that the present model equations with the 

standard model coefficients have the simplest form among all two-equation models. 

Hence, we expect that they will have less numerical stiffness in complex turbulent 

flows. 

2.4 Modeling of transition (Z. Yang and T.-H. Shih) 

A model of intermittancy based on the shape factor is added to a two-equation 

k-€ model for prediction of boundary layer transition with a free stream turbulence. 

The detailed model and calculations are give by Yang in this year's Research Briefs. 

2.5 Modeling of compressible turbulence (W.W. Liou and T.H. Shih) 

A two-scale model is proposed based on Hanjalic-Launder's multiple-scale concept 

for compressible turbulence, in which a distinct scale created by the compressibility 

is modeled separately by considering the effects of pressure-dilatation and dissipation dilatation on large-scale energy transfer rate. The detailed model is given by 

Liou in this year's Research Briefs. 

2.6 Direct numerical simulation of compressible flows (A. Hsu 

and T.-H. Shih) 

In order to have a better understanding of the effect of compressibility on turbulence, especially the effect of the formation of eddy-shocklets on turbulence, a 

direct numerical simulation of compressible homogeneous shear turbulent flows is 

been performing. The data of all turbulence statistics are very useful for turbulence 

modeling. The detailed simulation is given by Hsu in this year's Research Briefs. 

3. Future Plans 

Development of second order closure models: pay special attention to the effects 

of inhomogeneity, non-local property, frame-rotation, compressibility, near-wall behavior in the buffer and log-layers. 

4. References 

1 Lang, H.J. and Shih, T.-H., "A critical comparison of two-equation turbulence 

models," NASA TM 105237 (1991). 

2 Chien, K.-Y., "Predictions of Channel and Boundary-Layer Flows with a LowReynolds-Number Turbulence Model," AIAA Journal, 20, 33-38, (1982). 

3 Shih, T. -H., An Improved k-€ Model for Near-Wall Turbulence and Comparison 

with Direct Numerical Simulation," NASA TM-103221, August (1990). 

4 Lam, C. K. G., and Bremhorst, K., "A Modified Form of the k-€ Model for 

Predicting Wall Turbulence," ASME Transaction, Journal of Fluids Engineering, 

103, September (1981). 

5 Nagano, Y. and Hishida, M., "Improved Form of the k-€ Model for Wall Turbulent Shear Flows," ASME Transaction, Journal of Fluids Engineering, 109, June 

(1987). 

43 
Modeling of Turbulence and Transition 

6 Nagano, Y., and Tagawa, M., "An Improved k-E Model for Boundary Layer 

Flows," ASME Transaction, Journal of Fluids Engineering, 112, March (1990). 

7 Launder, B. E. and Sharma, B. 1., "Application of the Energy-Dissipation Model 

of Thrbulence to the Calculation of a Flow near a Spinning Disk," Letters in Heat 

and Mass Transfer, 1, 131-138 (1974). 

8 Jones, W. P. and Launder, B. E., "The Calculation of Low-Reynolds Number 

Phenomena with a Two-Equation Model of Thrbulence," International Journal 

of Heat and Mass Transfer, 16, 1119-1130 (1973). 

9 Myong, H. K. and Kasagi, N., "A new Proposal for a k-E Thrbulence Model and 

its Evaluation." JSME Transaction, 54, 3003-3009 and 3512-3520 (1988). 

10 Yang, Z. and Shih, T. -H., "A k - E Modeling of Near Wall Thrbulence,' Proceedings of 4th International Symposium on Computational Fluid Dynamics, UC 

Davis (1991). 

11 Wilcox, D. C., "A Complete Model fo Turbulence Revisited, " AIAA Paper no. 

84- 0176, Reno, Nevada (1984). 

12 Wilcox, D. C., "Progress in Hypersonic Turbulence Modeling," AIAA Paper 

no.91-1785, Honolulu, HI (1991). 

13 Speziale, C. G., Abid, R and Anderson, E. C., "A Critical Evaluation of TwoEquation Models for Near Wall Turbulence," AIAA Paper no. 90-1481, Seattle, 

WA (1990). 

14 Coakley, T. J., "Thrbulence Modeling Methods for the Compressible NavierStokes Equations," AIAA Paper no. 83-1693, Danvers, Massachusetts (1983). 

15 Kim, J., Moin, P. and Moser, R, "Turbulent Statistics in fully Developed Channel Flow at Low Reynolds Number," Journal of Fluid Mechanics, 177, 133-166 

(1987). 

16 Spalart, P. R, "Direct Simulation of a Turbulent Boundary Layer up to Reo = 

1410," Journal of Fluid Mechanics, 187, 61-98 (1988). 

17 Klebanoff, P. S., "Characteristics of Turbulence in a Boundary Layer with Zero 

Pressure Gradient," NACA-TN-3178 (1954). 

18 Launder, B.E., Reece, G.J., & Rodi, W. , " Progress in the development of a 

Reynolds-stress turbulence closure," J. Fluid Mech. (1975), 68, 537-566 (1975). 

19 Speziale, C.G., Sarkar, S. and Gatski, T.B., "Modeling the pressure-strain correlation of turbulence: an invariant dynamical systems approach," J. Fluid Mech., 

227, 245-272 (1991). 

20 Fu, S., Launder, B.E. and Tselepidakis, D.P., "Accommodating the effects of high 

strain rates in modeling the pressure-strain correlation. UMIST Mech . Engng 

Dept Rep. TFD /87/5, 1987. 

21 Shih, T.-H. & Lumley, J.L., " Modeling of pressure correlation terms in Reynolds 

stress and scalar flux equations." Rep. FDA-85-3, Sibley School of Mech. & 

Aero. Engrg., Cornell University (1985). 

22 Shih, T.-H. and Mansour, N.N. , "Reynolds Stress Modeling of Homogeneous 

Turbulence and Comparison with Numerical Simulations," Center for Turbulence 

Research Proceedings of the Summer Program (1987). 

23 Rotta, J.C., Z. Phys. 129, 547 (1951). 

44 

Modeling of Turbulence and Transition 

6 Nagano, Y., and Tagawa, M., "An Improved k-E Model for Boundary Layer 

Flows," ASME Transaction, Journal of Fluids Engineering, 112, March (1990). 

7 Launder, B. E. and Sharma, B. 1., "Application of the Energy-Dissipation Model 

of Thrbulence to the Calculation of a Flow near a Spinning Disk," Letters in Heat 

and Mass Transfer, 1, 131-138 (1974). 

8 Jones, W. P. and Launder, B. E., "The Calculation of Low-Reynolds Number 

Phenomena with a Two-Equation Model of Thrbulence," International Journal 

of Heat and Mass Transfer, 16, 1119-1130 (1973). 

9 Myong, H. K. and Kasagi, N., "A new Proposal for a k-E Thrbulence Model and 

its Evaluation." JSME Transaction, 54, 3003-3009 and 3512-3520 (1988). 

10 Yang, Z. and Shih, T. -H., "A k - E Modeling of Near Wall Thrbulence,' Proceedings of 4th International Symposium on Computational Fluid Dynamics, UC 

Davis (1991). 

11 Wilcox, D. C., "A Complete Model fo Turbulence Revisited, " AIAA Paper no. 

84- 0176, Reno, Nevada (1984). 

12 Wilcox, D. C., "Progress in Hypersonic Turbulence Modeling," AIAA Paper 

no.91-1785, Honolulu, HI (1991). 

13 Speziale, C. G., Abid, R and Anderson, E. C., "A Critical Evaluation of TwoEquation Models for Near Wall Turbulence," AIAA Paper no. 90-1481, Seattle, 

WA (1990). 

14 Coakley, T. J., "Thrbulence Modeling Methods for the Compressible NavierStokes Equations," AIAA Paper no. 83-1693, Danvers, Massachusetts (1983). 

15 Kim, J., Moin, P. and Moser, R, "Turbulent Statistics in fully Developed Channel Flow at Low Reynolds Number," Journal of Fluid Mechanics, 177, 133-166 

(1987). 

16 Spalart, P. R, "Direct Simulation of a Turbulent Boundary Layer up to Reo = 

1410," Journal of Fluid Mechanics, 187, 61-98 (1988). 

17 Klebanoff, P. S., "Characteristics of Turbulence in a Boundary Layer with Zero 

Pressure Gradient," NACA-TN-3178 (1954). 

18 Launder, B.E., Reece, G.J., & Rodi, W. , " Progress in the development of a 

Reynolds-stress turbulence closure," J. Fluid Mech. (1975), 68, 537-566 (1975). 

19 Speziale, C.G., Sarkar, S. and Gatski, T.B., "Modeling the pressure-strain correlation of turbulence: an invariant dynamical systems approach," J. Fluid Mech., 

227, 245-272 (1991). 

20 Fu, S., Launder, B.E. and Tselepidakis, D.P., "Accommodating the effects of high 

strain rates in modeling the pressure-strain correlation. UMIST Mech . Engng 

Dept Rep. TFD /87/5, 1987. 

21 Shih, T.-H. & Lumley, J.L., " Modeling of pressure correlation terms in Reynolds 

stress and scalar flux equations." Rep. FDA-85-3, Sibley School of Mech. & 

Aero. Engrg., Cornell University (1985). 

22 Shih, T.-H. and Mansour, N.N. , "Reynolds Stress Modeling of Homogeneous 

Turbulence and Comparison with Numerical Simulations," Center for Turbulence 

Research Proceedings of the Summer Program (1987). 

23 Rotta, J.C., Z. Phys. 129, 547 (1951). 

44 
Tsan-Hsing Shih 

24 Lumley, J.1., "Computational modeling of turbulent flows," Adv. Appl. Mech. 

18, 123 (1978). 

25 Sarkar, S. and Speziale, C.G., "A simple nonlinear model for the return to isotropy 

in turbulence," Phys. Fluids, A2 (1), January, (1990). 

26 Haworth, D.C. and Pope, S.B., "A generalized Langevin model for turbulent 

flows," Phys. fluid 29 (2), February (1986). 

27 Choi, K.S. and Lumley, J.1., Proceedings of the IUTAM symposium, Kyoto, 

1983, 'Turbulence and Chaotic phenomena in Fluids, edited by T.Tatsumi (NorthHolland, Amsterdam), p.267 (1984). 

28 Craft, T.J. and Launder, B.E., "Computation of Impinging Flows Using SecondMoment Closure," Eighth Symposium on Turbulent Shear Flows, Technical University of Munich, September 9-11 (1991). 

29 Yamamoto, M. and Arakawa, c., "Study on the pressure-strain term in Reynolds 

stress model," Eighth Symposium on 'Turbulent Shear Flows, Technical University 

of Munich, September, 9-11 (1991). 

30 Rogers, M.M., Moin, P. & Reynolds, W.C., "The structure and modeling of the 

hydrodynamic and passive scalar fields in homogeneous turbulent shear flow," 

Dept. Mech. Engng. Rep. TF-24, Stanford University: Stanford California 

(1986). 

31 Lee, M.J. & Reynolds, W.C., " Numerical experiments on structure of homogeneous turbulence," Dept. Mech. Engng. Rep. TF-24, Stanford University: 

Stanford California (1985). 

32 Comte-Bellot, G. and Corrsin, S., " The use of a contraction to improve the 

isotropy of grid-generated turbulence," J. Fluid Mech. 25 , 657-682 (1966). 

33 Reynolds, W.C.," Effect of Rotation on Homogeneous Turbulence," Tenth Australian Fluid Mechanics Conference, December 11-15 (1989). 

34 Shih, T.-H. and Lumley, J.L., "A Critical Comparison of Second Order Closures with Direct Numerical Simulation of Homogeneous Turbulence," NSAS TM 

105351 (1991). 

35 Hanjalic, K. and Launder, B.E., "Contribution Towards a Reynolds-stress Closure 

for Low-Reynolds- Number Turbulence," J. Fluid Mech, 74, 593-619 (1976). 

36 Patel, V.C., Rodi, W. and Scheuerer, G., "Turbulence models for near-wall and 

low-Reynolds-number flows: A review," AIAA Journal, 23, 1308-1319 (1985). 

37 Mansour, N.N., Kim,J. and Moin. P., "Reynolds-Stress and Dissipation Rate· 

Budgets in a Turbulent Channel Flow," J. Fluid Mech., 194, 15-44 (1988). 

38 Lai, Y.G. and So, R.M.C., "On near-wall turbulent flow modeling," J. Fluid 

Mech. 221, 641-673 (1990). 

39 Launder, B.E. and Spalding, D.B., "Computer Methods in Applied Mechanics 

and Engineering, 3, 269 (1974). 

40 Kim, J., Personal communication, 1992. 

45 

Tsan-Hsing Shih 

24 Lumley, J.1., "Computational modeling of turbulent flows," Adv. Appl. Mech. 

18, 123 (1978). 

25 Sarkar, S. and Speziale, C.G., "A simple nonlinear model for the return to isotropy 

in turbulence," Phys. Fluids, A2 (1), January, (1990). 

26 Haworth, D.C. and Pope, S.B., "A generalized Langevin model for turbulent 

flows," Phys. fluid 29 (2), February (1986). 

27 Choi, K.S. and Lumley, J.1., Proceedings of the IUTAM symposium, Kyoto, 

1983, 'Turbulence and Chaotic phenomena in Fluids, edited by T.Tatsumi (NorthHolland, Amsterdam), p.267 (1984). 

28 Craft, T.J. and Launder, B.E., "Computation of Impinging Flows Using SecondMoment Closure," Eighth Symposium on Turbulent Shear Flows, Technical University of Munich, September 9-11 (1991). 

29 Yamamoto, M. and Arakawa, c., "Study on the pressure-strain term in Reynolds 

stress model," Eighth Symposium on 'Turbulent Shear Flows, Technical University 

of Munich, September, 9-11 (1991). 

30 Rogers, M.M., Moin, P. & Reynolds, W.C., "The structure and modeling of the 

hydrodynamic and passive scalar fields in homogeneous turbulent shear flow," 

Dept. Mech. Engng. Rep. TF-24, Stanford University: Stanford California 

(1986). 

31 Lee, M.J. & Reynolds, W.C., " Numerical experiments on structure of homogeneous turbulence," Dept. Mech. Engng. Rep. TF-24, Stanford University: 

Stanford California (1985). 

32 Comte-Bellot, G. and Corrsin, S., " The use of a contraction to improve the 

isotropy of grid-generated turbulence," J. Fluid Mech. 25 , 657-682 (1966). 

33 Reynolds, W.C.," Effect of Rotation on Homogeneous Turbulence," Tenth Australian Fluid Mechanics Conference, December 11-15 (1989). 

34 Shih, T.-H. and Lumley, J.L., "A Critical Comparison of Second Order Closures with Direct Numerical Simulation of Homogeneous Turbulence," NSAS TM 

105351 (1991). 

35 Hanjalic, K. and Launder, B.E., "Contribution Towards a Reynolds-stress Closure 

for Low-Reynolds- Number Turbulence," J. Fluid Mech, 74, 593-619 (1976). 

36 Patel, V.C., Rodi, W. and Scheuerer, G., "Turbulence models for near-wall and 

low-Reynolds-number flows: A review," AIAA Journal, 23, 1308-1319 (1985). 

37 Mansour, N.N., Kim,J. and Moin. P., "Reynolds-Stress and Dissipation Rate· 

Budgets in a Turbulent Channel Flow," J. Fluid Mech., 194, 15-44 (1988). 

38 Lai, Y.G. and So, R.M.C., "On near-wall turbulent flow modeling," J. Fluid 

Mech. 221, 641-673 (1990). 

39 Launder, B.E. and Spalding, D.B., "Computer Methods in Applied Mechanics 

and Engineering, 3, 269 (1974). 

40 Kim, J., Personal communication, 1992. 

45 
+ p 

+ 

~ 

Modeling of Turbulence and Transition 

20.0 0.8 

--_ . 

JL 

--- Chien 

15.0 .,/ 0.6 Present 

./' 

/' o DNS data 

/ 

/ -- JL + 

10.0 / - - - dhien ~ 0.4 

PreSent I 

o DNS data 

5.0 0.2 

0.0 m----'_--1...._-'--_"----'-_-L-_~--....J 0.0 8'----'_---L.._--'--_L....--'-_-L-_.JQl---....l 

0.0 

5.0 

4.0 

3.0 

2.0 

1.0 

0.0 

0.0 

50.0 

50.0 

100.0 

y+ 

150.0 200.0 0.0 

JL 

dhien 

0.30 

-- Present 

o DNS data 0.20 

0.10 

0.00 

100.0 150.0 200.0 0.0 

y+ 

50.0 

50.0 

100.0 

y+ 

JL 

dhien 

150.0 200. ~ 

-- Present 

o DNS data 

100.0 150.0 200'« 

y+ 

Figure 1. Comparison of models with DNS of 2-D channel with ReT = 

180. 

46 

+ p 

+ 

~ 

Modeling of Turbulence and Transition 

20.0 0.8 

--_ . 

JL 

--- Chien 

15.0 .,/ 0.6 Present 

./' 

/' o DNS data 

/ 

/ -- JL + 

10.0 / - - - dhien ~ 0.4 

PreSent I 

o DNS data 

5.0 0.2 

0.0 m----'_--1...._-'--_"----'-_-L-_~--....J 0.0 8'----'_---L.._--'--_L....--'-_-L-_.JQl---....l 

0.0 

5.0 

4.0 

3.0 

2.0 

1.0 

0.0 

0.0 

50.0 

50.0 

100.0 

y+ 

150.0 200.0 0.0 

JL 

dhien 

0.30 

-- Present 

o DNS data 0.20 

0.10 

0.00 

100.0 150.0 200.0 0.0 

y+ 

50.0 

50.0 

100.0 

y+ 

JL 

dhien 

150.0 200. ~ 

-- Present 

o DNS data 

100.0 150.0 200'« 

y+ 

Figure 1. Comparison of models with DNS of 2-D channel with ReT = 

180. 

46 
20.0 

0.0 

0.0 

5.0 

4.0 

3.0 

+ 

~ 

2.0 

1.0 

0.0 

0.0 

Tsan-Hsing Shih 

1.0 ----

-- IL 

- - - dhien 

-- Present 

o DNS data 

0.0 

100.0 200.0 300.0 400.0 0.0 100.0 200.0 300.0 

y+ y+ 

JL 

dhien -- JL ---

Present - - - dhien 

Present o DNS data 0.20 

o DNS data 

+ " 171 

~ 

I'>Q 

0.10 

/I 0 

0.00 

100.0 200.0 300.0 400.0 0.0 100.0 

y+ y+ 

Figure 2. Comparison of models with DNS of 2-D channel with Rer = 

395. 

47 

400.0 

200.0 

20.0 

0.0 

0.0 

5.0 

4.0 

3.0 

+ 

~ 

2.0 

1.0 

0.0 

0.0 

Tsan-Hsing Shih 

1.0 ----

-- IL 

- - - dhien 

-- Present 

o DNS data 

0.0 

100.0 200.0 300.0 400.0 0.0 100.0 200.0 300.0 

y+ y+ 

JL 

dhien -- JL ---

Present - - - dhien 

Present o DNS data 0.20 

o DNS data 

+ " 171 

~ 

I'>Q 

0.10 

/I 0 

0.00 

100.0 200.0 300.0 400.0 0.0 100.0 

y+ y+ 

Figure 2. Comparison of models with DNS of 2-D channel with Rer = 

395. 

47 

400.0 

200.0 
20.0 

+ p 

10.0 

0.0 

0.0 

5.0 

4.0 

3.0 

+ 

~ 

2.0 

1.0 

0.0 

0.0 

Modeling of Turbulence and Transition 

1.0 

----

----- -- JL 0.8 - - - dhien 

Present 

0.6 o DNS data 

-- JL + - - - dhien ~ 

Present I 

0.4 

o DNS data 

0.2 

0.0 

200.0 400.0 600.0 800.0 0.0 200.0 400.0 600.0 

y+ y+ 

JL -- .JL --- dhien - - - Chien 

Present Present o DNS data 0.20 o DNS data 

+ 

111 

P. 

r:.:l 

0.10 

0.00 

200.0 400.0 600.0 800.0 0.0 50.0 100.0 150.0 

y+ y+ 

Figure 3. Comparison of models with DNS of 2-D boundary layer flow 

with R eB = 1410. 

48 

800. ~, 

200'« 

20.0 

+ p 

10.0 

0.0 

0.0 

5.0 

4.0 

3.0 

+ 

~ 

2.0 

1.0 

0.0 

0.0 

Modeling of Turbulence and Transition 

1.0 

----

----- -- JL 0.8 - - - dhien 

Present 

0.6 o DNS data 

-- JL + - - - dhien ~ 

Present I 

0.4 

o DNS data 

0.2 

0.0 

200.0 400.0 600.0 800.0 0.0 200.0 400.0 600.0 

y+ y+ 

JL -- .JL --- dhien - - - Chien 

Present Present o DNS data 0.20 o DNS data 

+ 

111 

P. 

r:.:l 

0.10 

0.00 

200.0 400.0 600.0 800.0 0.0 50.0 100.0 150.0 

y+ y+ 

Figure 3. Comparison of models with DNS of 2-D boundary layer flow 

with R eB = 1410. 

48 

800. ~, 

200'« 
I 

I 

+ P 

30.0 

20.0 

10.0 

0.0 

10~o 

0.0 

0.0 

- --- 11 

--- dhlen 

Pre~ent 

oExp.Jq1 

Tsan-Hsing Shih 

;; 

;; I 

/ I 

~ I 

/ / 

D~ // / / 

/ 

/ 

y+ 

I 

---- 11 

dhlen 

Pre~ent 

o Exp.Jq1 

.6. Exp. ~W 

o Exp.WT 

• DN~ data 

-P 

• P 

'-' 

~ 

o.ooa 

0.006 

0.004 

0.002 

o 

\ 

0 .000 10~o 0.0 

0.010 

0.008 

0.006 \ 

, . ~ 

0.004 

0.002 

---- 11 

dhien 

Pre~ent 

o Exp.Jq1 

o 

0.5 1.0 1.5 

y/dlta 

---- 11 

dhien 

Pre~ent 

• Exp. IQ3 

---- -- ---------

1 0.000 '--_---'-__ -'--__ -l--_----'L....-_--'-__ -' 

5000.0 10000.0 0.0 1000.0 2000.0 

Re2 Re2 

Figure 4. Comparison of models with the experiments of 2-D boundary 

layer flows (Klebanoffl13] and others). 

49 

3000.0 

I 

I 

+ P 

30.0 

20.0 

10.0 

0.0 

10~o 

0.0 

0.0 

- --- 11 

--- dhlen 

Pre~ent 

oExp.Jq1 

Tsan-Hsing Shih 

;; 

;; I 

/ I 

~ I 

/ / 

D~ // / / 

/ 

/ 

y+ 

I 

---- 11 

dhlen 

Pre~ent 

o Exp.Jq1 

.6. Exp. ~W 

o Exp.WT 

• DN~ data 

-P 

• P 

'-' 

~ 

o.ooa 

0.006 

0.004 

0.002 

o 

\ 

0 .000 10~o 0.0 

0.010 

0.008 

0.006 \ 

, . ~ 

0.004 

0.002 

---- 11 

dhien 

Pre~ent 

o Exp.Jq1 

o 

0.5 1.0 1.5 

y/dlta 

---- 11 

dhien 

Pre~ent 

• Exp. IQ3 

---- -- ---------

1 0.000 '--_---'-__ -'--__ -l--_----'L....-_--'-__ -' 

5000.0 10000.0 0.0 1000.0 2000.0 

Re2 Re2 

Figure 4. Comparison of models with the experiments of 2-D boundary 

layer flows (Klebanoffl13] and others). 

49 

3000.0 
Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Turbulence Modeling and Experiments 

Aamir Shabbir 

1. Motivation and Objective 

The best way of verifying turbulence models is to do a direct comparison between 

the various terms and their modelsl ,2,3. The success of this approach depends upon 

the availability of the data for the exact correlations (both experimental and DNS). 

The other approach involves numerically solving the differential equations and then 

comparing the results with the data. The results of such a computation will depend 

upon the accuracy of all the modeled terms and constants. Because of this it is 

sometimes difficult to find the cause of a poor performance by a model. However, 

such a calculation is still meaningful in other ways as it shows how a complete 

Reynolds stress model performs. 

In this study thirteen homogeneous flows are numerically computed using the 

second order closure models. We concentrate only on those models which use a 

linear (or quasi-linear) model for the rapid term. This, therefore, includes the 

Launder ,Reece and Rodi4 (LRR) model; the isotropization of production 4 (IP) 

model; and the Speziale, Sarkar and Gatski 5 (SSG) model. The purpose of this 

study is to find out which of the three models performs better and what are their 

weaknesses, if any. 

The other work reported here deals with the experimental balnces of the second 

moment equations for a buoyant plume. Despite the tremendous amount of activity toward the second order closure modeling of turbulence, very little experimental 

information is available about the budgets of the second moment equations. Part 

of the problem stems from our inability to measure the pressure correlations. However, if everything else appearing in these equations is known from the experiment, 

pressure correlations can be obtained as the closing terms. This is the closest we 

can come to in obtaining these terms from experiment, and despite the measurement errors which might be present in such balances, the resulting information will 

be extremely useful for the turbulence modelers. The purpose of this part of the 

work reported here was to provide such balances of the Reynolds stress and heat 

flux equations for the buoyant plume. 

2.0.0 Work Accomplished 

2.1.0 Comparison of Second Order Models in Homogeneous Flows 

Before presenting the results a note about the LRR model constants used in the 

present study is in order. These constants have evolved to slightly different values 

than those orginally recommended by LRR4. The value of the Rotta constant Cl 

(in the return to isotropy term) used in the present study is 3.6 (note that due to a 

different definition of bij used here the value of Cl differs by a factor of two). The 

51 Preceding Page Blank 

Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Turbulence Modeling and Experiments 

Aamir Shabbir 

1. Motivation and Objective 

The best way of verifying turbulence models is to do a direct comparison between 

the various terms and their modelsl ,2,3. The success of this approach depends upon 

the availability of the data for the exact correlations (both experimental and DNS). 

The other approach involves numerically solving the differential equations and then 

comparing the results with the data. The results of such a computation will depend 

upon the accuracy of all the modeled terms and constants. Because of this it is 

sometimes difficult to find the cause of a poor performance by a model. However, 

such a calculation is still meaningful in other ways as it shows how a complete 

Reynolds stress model performs. 

In this study thirteen homogeneous flows are numerically computed using the 

second order closure models. We concentrate only on those models which use a 

linear (or quasi-linear) model for the rapid term. This, therefore, includes the 

Launder ,Reece and Rodi4 (LRR) model; the isotropization of production 4 (IP) 

model; and the Speziale, Sarkar and Gatski 5 (SSG) model. The purpose of this 

study is to find out which of the three models performs better and what are their 

weaknesses, if any. 

The other work reported here deals with the experimental balnces of the second 

moment equations for a buoyant plume. Despite the tremendous amount of activity toward the second order closure modeling of turbulence, very little experimental 

information is available about the budgets of the second moment equations. Part 

of the problem stems from our inability to measure the pressure correlations. However, if everything else appearing in these equations is known from the experiment, 

pressure correlations can be obtained as the closing terms. This is the closest we 

can come to in obtaining these terms from experiment, and despite the measurement errors which might be present in such balances, the resulting information will 

be extremely useful for the turbulence modelers. The purpose of this part of the 

work reported here was to provide such balances of the Reynolds stress and heat 

flux equations for the buoyant plume. 

2.0.0 Work Accomplished 

2.1.0 Comparison of Second Order Models in Homogeneous Flows 

Before presenting the results a note about the LRR model constants used in the 

present study is in order. These constants have evolved to slightly different values 

than those orginally recommended by LRR4. The value of the Rotta constant Cl 

(in the return to isotropy term) used in the present study is 3.6 (note that due to a 

different definition of bij used here the value of Cl differs by a factor of two). The 

51 Preceding Page Blank 
Aamir Shabbir 

rapid term constant Cz was assigned a value of 0.4 in the original LRR model. In 

the present study the value used for this constant is 0.55 which is slightly higher 

than the value of 0.5 recommended by Morris6. It was found out that the value of 

0.55 led to improvement in the performance of LRR model in all the flows tested 

here. (The improvements were slight for the irrotationally strained flows but 

Figure 1 compares the development of Reynolds stresses computed using these 

three models in a flow through axisymmetric contraction with the DNS data 7 . 

Here we show a typical case of 5 = 100.00 (5ko /co = 55.73, case AXM). All 

the models deviate from the DNS data. However, LRR model gives slightly better 

results than the SSG model with IP model performing the worst. 

Figures 2 and 3 show a similar comparison for flow through axisymmetric expansion for two different strain: rates. For the smaller strain rate flow (S = 0.717, S ko / Co = 

.408, ~ase EXO) SSG model reproduces the u2development quite well while both IP 

and LRR models underpredict it. For the v 2 component all the models give similar 

results. Therefore, for this low strain rate flow SSG model is better than the other 

two rhodels. For the flow with higher strain rate (5 = 7.17, 5ko / Co = 4.08, case 

EXP)~ the LRR model is in excellent agreement with the DNS data for both the 

components while both IP and SSG models show overprediction So for this flow 

LRR model works the best. 

Now we show comparisons for the distortion of turbulence by plane strain for 

four cases of differing strain ratfts. We start from the lower strain rate case. Figure 

4 compares the evolution of the three non-zero Reynolds stress components for the 

flow with strain rate 5 = 2.6 (Sko/ Co = 2.309, case PXC), For u2component all 

the models under predict the DNS data. LRR model is slightly better than the SSG 

model. IP model is the worst of the thre~For v 2 component IP model works the 

best. LRR model slightly underpredicts v 2 while SSG overpredicts it. The third 

component w2 is overpredicted by all the models with LRR model being better than 

the other two. Figure 5 shows the similar comparisons for the highest strain rate 

case (S = 25.0, S ok/co = 22.227, case PXE). All the three models underpredict the 

u2component. IP m~o!. ( is -the worst of t~e three models. LRR mode~ives slightly 

better result than t SSG model for this stress component. For v 2 component 

LRR model 'is the ~ st and SSG model is the worst of the three. For the w2 

component all the tIi'~ee models overpredict the DNS data with LRR mdoel being 

closest to the data. I From the above four plane strain flow comparisons, we note 

that the performance of all the three models deteriorates as the strain rate increases. 

However, on the overall LRR model works better than the other two models. 

Figure 6 shows the same comparison with the homogeneous shear flow experiment8 

(S = 46.8, Sko/co = 6.46). For the u2 component LRR model gives the best result 

whereas SSG and IP models overpredict it. For the v2 component also the LRR 

works the best. SSG model slightly overpredicts the data where as IP model if off 

by a larger margin. For the w 2 component both SSG and IP models reproduce 

the data very well whereas LRR model overpredicts the data. For the shear stress 

component LRR performs reasonably whereas SSG model slightly overpredicts the 

data and IP model is off the data by a higher margin. So, for this experiment, LRR 

52 

Aamir Shabbir 

rapid term constant Cz was assigned a value of 0.4 in the original LRR model. In 

the present study the value used for this constant is 0.55 which is slightly higher 

than the value of 0.5 recommended by Morris6. It was found out that the value of 

0.55 led to improvement in the performance of LRR model in all the flows tested 

here. (The improvements were slight for the irrotationally strained flows but 

Figure 1 compares the development of Reynolds stresses computed using these 

three models in a flow through axisymmetric contraction with the DNS data 7 . 

Here we show a typical case of 5 = 100.00 (5ko /co = 55.73, case AXM). All 

the models deviate from the DNS data. However, LRR model gives slightly better 

results than the SSG model with IP model performing the worst. 

Figures 2 and 3 show a similar comparison for flow through axisymmetric expansion for two different strain: rates. For the smaller strain rate flow (S = 0.717, S ko / Co = 

.408, ~ase EXO) SSG model reproduces the u2development quite well while both IP 

and LRR models underpredict it. For the v 2 component all the models give similar 

results. Therefore, for this low strain rate flow SSG model is better than the other 

two rhodels. For the flow with higher strain rate (5 = 7.17, 5ko / Co = 4.08, case 

EXP)~ the LRR model is in excellent agreement with the DNS data for both the 

components while both IP and SSG models show overprediction So for this flow 

LRR model works the best. 

Now we show comparisons for the distortion of turbulence by plane strain for 

four cases of differing strain ratfts. We start from the lower strain rate case. Figure 

4 compares the evolution of the three non-zero Reynolds stress components for the 

flow with strain rate 5 = 2.6 (Sko/ Co = 2.309, case PXC), For u2component all 

the models under predict the DNS data. LRR model is slightly better than the SSG 

model. IP model is the worst of the thre~For v 2 component IP model works the 

best. LRR model slightly underpredicts v 2 while SSG overpredicts it. The third 

component w2 is overpredicted by all the models with LRR model being better than 

the other two. Figure 5 shows the similar comparisons for the highest strain rate 

case (S = 25.0, S ok/co = 22.227, case PXE). All the three models underpredict the 

u2component. IP m~o!. ( is -the worst of t~e three models. LRR mode~ives slightly 

better result than t SSG model for this stress component. For v 2 component 

LRR model 'is the ~ st and SSG model is the worst of the three. For the w2 

component all the tIi'~ee models overpredict the DNS data with LRR mdoel being 

closest to the data. I From the above four plane strain flow comparisons, we note 

that the performance of all the three models deteriorates as the strain rate increases. 

However, on the overall LRR model works better than the other two models. 

Figure 6 shows the same comparison with the homogeneous shear flow experiment8 

(S = 46.8, Sko/co = 6.46). For the u2 component LRR model gives the best result 

whereas SSG and IP models overpredict it. For the v2 component also the LRR 

works the best. SSG model slightly overpredicts the data where as IP model if off 

by a larger margin. For the w 2 component both SSG and IP models reproduce 

the data very well whereas LRR model overpredicts the data. For the shear stress 

component LRR performs reasonably whereas SSG model slightly overpredicts the 

data and IP model is off the data by a higher margin. So, for this experiment, LRR 

52 
Turbulence M adeling and Experimetns 

model has better overall performance than the other two models. 

Last, we discuss the evolution of q2 for the case of rotating homogeneous shear 

flow. Since no experimental or DNS data is available for this flow the comparisons . 

will be made (for two cases) with the LES9. Bardina10 pointed out that in this 

case we should be careful in interpreting the comparisons for anything more than 

the trends shown by the LES. In all the cases shown here the initial conditions 

corresponded to isotropic turbulence with toj Sko = 0.296. Figure 7 shows the 

comparisons for the three cases of different Rosby numbers (= DIS). For nls = .25 

we note that all three models significantly underpredict the LES results for q2 ; SSG 

being closest to the LES data and the LRR being the furthest. Qualitatively all 

the three models reproduce the LES trends. For the case of nj S = 0.50 SSG is in 

excellent agreement with th~ES results. Both IP and LRR give identical results 

and give a smaller value of q2 than the LES. It should be pointed out that SSG 

model constants were partially calibrated against this flow. For the third case of 

Dj S = 1.0, all the three models give identicai results. Since no LES results are 

available for this case the only purpose of showing the results is to see how the 

three models compare with each other. 

2.1.2 Conclusions 

Results were shown from numerical computation of various homogeneous turbulent flows using three different turbulence models. All of these models use a linear 

(or quasi-linear) model for the rapid part of the pressure strain model. Based on 

their overall performance it is found that LRR model works better than both SSG 

and IP models. For the irrotationally flows the differences between the models and 

DNS data increased with the strain rate with LRR model performing better than 

the other two models. For the simple homogeneous shear flow LRR model better 

than the SSG model (for the DNS both performed equally good but for the experiment LRR worked better). For the homogeneous shear flows both SSG and LRR 

model showed trends similar to those shown by LES with SSG performing better 

than the LRR model. It is worth noting that SSG model has seven empirical constants as compared to two in LRR model and on the overall it still does not perform 

better than LRR model. Part of the reason for this may be due to the fact that 

the SSG model does not satisfy the normalization constraint where as LRR model 

does. (Normalization is an exact property of the pressure strain correlation; see 

references 4 and 11 for details.) As has been pointed out by Shih and Lumley3, for 

a model of the rapid pressure strain part which is linear in the anisotropic tensor 

and satisfies all of its exact properties, LRR is the most general model. 

2.2.0 Experimental Balances for the Second Moments for a Buoyant Plume 

2.2.1 Heat Flux Budgets 

The transport equation for the vertical (streamwise) heat flux can be written as 

awt awt 1 a - a - aT - aT u- + w- = - --(ruwt) - -(wwt) - uw- - w2 - aT a Z T ar a z ar a z 

53 

Turbulence M adeling and Experimetns 

model has better overall performance than the other two models. 

Last, we discuss the evolution of q2 for the case of rotating homogeneous shear 

flow. Since no experimental or DNS data is available for this flow the comparisons . 

will be made (for two cases) with the LES9. Bardina10 pointed out that in this 

case we should be careful in interpreting the comparisons for anything more than 

the trends shown by the LES. In all the cases shown here the initial conditions 

corresponded to isotropic turbulence with toj Sko = 0.296. Figure 7 shows the 

comparisons for the three cases of different Rosby numbers (= DIS). For nls = .25 

we note that all three models significantly underpredict the LES results for q2 ; SSG 

being closest to the LES data and the LRR being the furthest. Qualitatively all 

the three models reproduce the LES trends. For the case of nj S = 0.50 SSG is in 

excellent agreement with th~ES results. Both IP and LRR give identical results 

and give a smaller value of q2 than the LES. It should be pointed out that SSG 

model constants were partially calibrated against this flow. For the third case of 

Dj S = 1.0, all the three models give identicai results. Since no LES results are 

available for this case the only purpose of showing the results is to see how the 

three models compare with each other. 

2.1.2 Conclusions 

Results were shown from numerical computation of various homogeneous turbulent flows using three different turbulence models. All of these models use a linear 

(or quasi-linear) model for the rapid part of the pressure strain model. Based on 

their overall performance it is found that LRR model works better than both SSG 

and IP models. For the irrotationally flows the differences between the models and 

DNS data increased with the strain rate with LRR model performing better than 

the other two models. For the simple homogeneous shear flow LRR model better 

than the SSG model (for the DNS both performed equally good but for the experiment LRR worked better). For the homogeneous shear flows both SSG and LRR 

model showed trends similar to those shown by LES with SSG performing better 

than the LRR model. It is worth noting that SSG model has seven empirical constants as compared to two in LRR model and on the overall it still does not perform 

better than LRR model. Part of the reason for this may be due to the fact that 

the SSG model does not satisfy the normalization constraint where as LRR model 

does. (Normalization is an exact property of the pressure strain correlation; see 

references 4 and 11 for details.) As has been pointed out by Shih and Lumley3, for 

a model of the rapid pressure strain part which is linear in the anisotropic tensor 

and satisfies all of its exact properties, LRR is the most general model. 

2.2.0 Experimental Balances for the Second Moments for a Buoyant Plume 

2.2.1 Heat Flux Budgets 

The transport equation for the vertical (streamwise) heat flux can be written as 

awt awt 1 a - a - aT - aT u- + w- = - --(ruwt) - -(wwt) - uw- - w2 - aT a Z T ar a z ar a z 

53 
Aamir Shabbir 

-oW -oW - 1 op ( )-- - ut- - wt- + g(3t 2 - - t- - v + r t ·w . or a z paz ,J ,1 (1) 

N ate that the molecular term is written in local cartesian coordinates. The balance 

of this equation is shown in figure 8. Advection term is the smallest in this balance 

and, therefore, contributes least to the transport of the heat flux wt. It is clear 

that in the central core of the flow.(r/z < 0.04), the production of this heat flux is 

maintained by the mean buoyancy gradients and the turbulent buoyancy force i.e. 

the source of energy is the gravitational field. The shear production is relatively 

small in this region. Then there is an intermediate region where the production 

from mean velocity and gravitational field are of the same order. However, for 

r / z > 0.1 (which approximately corresponds to the plume half width), most of the 

production is maintained by the mean velocity and buoyancy gradients and the 

turbulent buoyancy production is only a small fraction of these two. The closing 

term in the heat flux balances is labelled as IIi and represents the sum of the pressure 

correlation and the molecular destruction terms i.e. 

IIi = ! t op _ (v + r) OUi at 

P OXi OXj OXj 

(2) 

The molecular term in (2) is thought to get weaker with increasing Reynolds and 

Peclet numbers, eventually approaching a value of zero in the limit of local (small 

scale) isotropy. This term was not measured and, therefore, its magnitude relative 

to others can not be established. However, in turbulence modeling, it is customary 

to combine this term with the pressure correlation term8 and, therefore, from that 

point of view not knowing each term separately does not reduce the usefulness of 

these budgets. Notice that the shape of this term is very similar to the shape of the 

heat flux wt and its magnitude remains large throughout the flow field. 

The equation for the radial heat flux is 

out out 1 a - a - - aT aT 

U or + W oz = - -:;: or (ruut) - oz (wut) - u

2 

or - uw oz 

- aU -aU 1 op ( )- - ut- - wt- - -t- - v + r t ·U . or a z p or ,1 ,] (3) 

The balance of this equation is shown in figure 9. Again, we note that the advection 

term is quite small as compared to the other dominant terms in the equation. Unlike 

the wt heat flux balance, the shear production is extremely small here. This is 

because the gradients of mean radial velocity are much smaller than the gradients in 

the mean vertical (streamwise) velocity. There is no turbulent buoyancy production 

in this equation and all the production is due to the mean buoyancy gradients. We 

note that the term representing sum of the pressure correlation and the molecular 

destruction makes up a substantial part of the budget and its shape is similar to 

the radial heat flux. We also note that this budget can not be divided into any 

subregions, where some phenomenon are more dominant than others, because the 

relative magnitude of each of the terms in equation (3) remains the same across the 

flow field. 

54 

Aamir Shabbir 

-oW -oW - 1 op ( )-- - ut- - wt- + g(3t 2 - - t- - v + r t ·w . or a z paz ,J ,1 (1) 

N ate that the molecular term is written in local cartesian coordinates. The balance 

of this equation is shown in figure 8. Advection term is the smallest in this balance 

and, therefore, contributes least to the transport of the heat flux wt. It is clear 

that in the central core of the flow.(r/z < 0.04), the production of this heat flux is 

maintained by the mean buoyancy gradients and the turbulent buoyancy force i.e. 

the source of energy is the gravitational field. The shear production is relatively 

small in this region. Then there is an intermediate region where the production 

from mean velocity and gravitational field are of the same order. However, for 

r / z > 0.1 (which approximately corresponds to the plume half width), most of the 

production is maintained by the mean velocity and buoyancy gradients and the 

turbulent buoyancy production is only a small fraction of these two. The closing 

term in the heat flux balances is labelled as IIi and represents the sum of the pressure 

correlation and the molecular destruction terms i.e. 

IIi = ! t op _ (v + r) OUi at 

P OXi OXj OXj 

(2) 

The molecular term in (2) is thought to get weaker with increasing Reynolds and 

Peclet numbers, eventually approaching a value of zero in the limit of local (small 

scale) isotropy. This term was not measured and, therefore, its magnitude relative 

to others can not be established. However, in turbulence modeling, it is customary 

to combine this term with the pressure correlation term8 and, therefore, from that 

point of view not knowing each term separately does not reduce the usefulness of 

these budgets. Notice that the shape of this term is very similar to the shape of the 

heat flux wt and its magnitude remains large throughout the flow field. 

The equation for the radial heat flux is 

out out 1 a - a - - aT aT 

U or + W oz = - -:;: or (ruut) - oz (wut) - u

2 

or - uw oz 

- aU -aU 1 op ( )- - ut- - wt- - -t- - v + r t ·U . or a z p or ,1 ,] (3) 

The balance of this equation is shown in figure 9. Again, we note that the advection 

term is quite small as compared to the other dominant terms in the equation. Unlike 

the wt heat flux balance, the shear production is extremely small here. This is 

because the gradients of mean radial velocity are much smaller than the gradients in 

the mean vertical (streamwise) velocity. There is no turbulent buoyancy production 

in this equation and all the production is due to the mean buoyancy gradients. We 

note that the term representing sum of the pressure correlation and the molecular 

destruction makes up a substantial part of the budget and its shape is similar to 

the radial heat flux. We also note that this budget can not be divided into any 

subregions, where some phenomenon are more dominant than others, because the 

relative magnitude of each of the terms in equation (3) remains the same across the 

flow field. 

54 
· Turbulence M adeling and Experimetns 

2.2.2 Reynolds Stress Budgets 

The transport equation for the Reynolds stress, within Bussinesq approximation, 

is 

Uk (UiUj) ,k = - [UiUjUkl,k - (UiUk Uj,k + UjUk Ui,d 

- - 1( ) + a·u·t + a. ·u·t + - U 'p . + U'p ' - 211u ' kU ' /,; iJt J iJJ t t ,J J ,t t, J, 

P 

(4) 

where the viscous diffusion term has been neglected since it will be small as compared to the turbulent diffusion. 

For reasons of convenience, turbulence modelers do not model the pressure correlation term in the form as it appears in the above equation but re-write it in a 

different form by separating it into a deviatoric and a non-deviatoric part. Two 

ways of doing this have been suggested in the literature and we will look at both of 

these before deciding which one to use in the present study. The traditional way of 

writing this term is4 

where the first term on the right hand side is the deviatoric part. The second term 

is the so called pressure diffusion term. Lumley12 (1975) has instead suggested the 

following separation 

where the term in the square brackets is the deviatoric part and the last term on 

the right hand side is the pressure diffusion term. Regardless of which separation is 

employed a correction or model has to be used for the correlation PUk. The model 

used here is due to Lumley8 is given by PUk = _q2 Uk /5. This study indicates that 

the use of this model with (5) produces so much pressure diffusion that it negates 

the velocity diffusion (i.e. due to uiujud. On this basis it was concluded to use the 

separation given by (6) in the present study. (For further details see Shabbir13). 

Therefore, using (6) the equation for the Reynolds stress can be re-written as 

where C = Cii. Note that anisotropic part of the dissipation part has been combined 

with the pressure correlation termll . The term in the curly parenthesis has a zero 

trace and will be denoted by <I>ij in the rest of the paper. It is this term whose models 

55 

· Turbulence M adeling and Experimetns 

2.2.2 Reynolds Stress Budgets 

The transport equation for the Reynolds stress, within Bussinesq approximation, 

is 

Uk (UiUj) ,k = - [UiUjUkl,k - (UiUk Uj,k + UjUk Ui,d 

- - 1( ) + a·u·t + a. ·u·t + - U 'p . + U'p ' - 211u ' kU ' /,; iJt J iJJ t t ,J J ,t t, J, 

P 

(4) 

where the viscous diffusion term has been neglected since it will be small as compared to the turbulent diffusion. 

For reasons of convenience, turbulence modelers do not model the pressure correlation term in the form as it appears in the above equation but re-write it in a 

different form by separating it into a deviatoric and a non-deviatoric part. Two 

ways of doing this have been suggested in the literature and we will look at both of 

these before deciding which one to use in the present study. The traditional way of 

writing this term is4 

where the first term on the right hand side is the deviatoric part. The second term 

is the so called pressure diffusion term. Lumley12 (1975) has instead suggested the 

following separation 

where the term in the square brackets is the deviatoric part and the last term on 

the right hand side is the pressure diffusion term. Regardless of which separation is 

employed a correction or model has to be used for the correlation PUk. The model 

used here is due to Lumley8 is given by PUk = _q2 Uk /5. This study indicates that 

the use of this model with (5) produces so much pressure diffusion that it negates 

the velocity diffusion (i.e. due to uiujud. On this basis it was concluded to use the 

separation given by (6) in the present study. (For further details see Shabbir13). 

Therefore, using (6) the equation for the Reynolds stress can be re-written as 

where C = Cii. Note that anisotropic part of the dissipation part has been combined 

with the pressure correlation termll . The term in the curly parenthesis has a zero 

trace and will be denoted by <I>ij in the rest of the paper. It is this term whose models 

55 
Aamir Shabbir 

have been proposed. Note that the above equation is exact since no approximations 

have been used so far. Now we introduce the model for the pressure diffusion term, 

as given above, and with this approximation the above equation becomes 

Uk (UiUj) ,k ~ - [UiUjUk + 1

2

5 (q2Uk)bij] k - (UiUk Uj,k + UjUk Ui,l.) + !3iUjt + !3jUit , 

2 

+ if?ij - "3cbij (8) 

Note that due to the model for the pressure diffusion term this is no longer an exact 

equation and ~ has been used to emphasize this fact. It is t his equation which will 

be balanced out with the experimental dat a and the term if? i j will be obtained as 

the closing term. It should be reminded that in addition to the measurement errors, 

any uncertainty in the approximation of the pressure diffusion will also be lumped 

into if? ij . 

The equation for the streamwise Reynolds stress w2 is given by 

aw2 [Jw2 1 a - 0 -- 2 [1 0 - 0 - ] U - , + W - ~ - --(ruw2) - - (ww2) + - __ (ruq2) + _(wq2) or az r or oz 15 r or oz 

[JW -2 oW!3- 2 - 2uw - - 2w - + 29 wi - if? - - c or oz zz 3 (9) 

The balance of this equation is shown in figure 10. Advection is the smallest of 

all the terms, Diffusion term is a gain near the center of the plume and a loss 

in the rest of the flow. Also, its magnitude near the center is comparable to the 

other dominant terms in the balance, We note that the buoyancy production is 

comparable to the production due to mean velocity gradients near the plume center 

but over the rest of the flow fi eld the shear production is much larger than the 

buoyancy production, It is also interesting to note that the buoyancy production 

and dissipation rate approximat ely balance each other. The closing term in this 

balance is if? zz and represents the sum of the pressure correlation term and the 

anisotropic part of the dissipation. This term is a loss for the u2 budget and we 

note that beyond r / z = 0.08 this term and shear production approximately balance 

each other. 

The equation for the radial component u2 is given by 

and its balance is shown i~gure 11. Obviously the advection of u2 has the same 

form as the advection of w2 . The production due to velocity gradients is a loss 

near the plume center and is a gain after about r / z = 0.04. This is because the 

radial gradient of the radial mean velocity is positive near the plume center. The 

56 

Aamir Shabbir 

have been proposed. Note that the above equation is exact since no approximations 

have been used so far. Now we introduce the model for the pressure diffusion term, 

as given above, and with this approximation the above equation becomes 

Uk (UiUj) ,k ~ - [UiUjUk + 1

2

5 (q2Uk)bij] k - (UiUk Uj,k + UjUk Ui,l.) + !3iUjt + !3jUit , 

2 

+ if?ij - "3cbij (8) 

Note that due to the model for the pressure diffusion term this is no longer an exact 

equation and ~ has been used to emphasize this fact. It is t his equation which will 

be balanced out with the experimental dat a and the term if? i j will be obtained as 

the closing term. It should be reminded that in addition to the measurement errors, 

any uncertainty in the approximation of the pressure diffusion will also be lumped 

into if? ij . 

The equation for the streamwise Reynolds stress w2 is given by 

aw2 [Jw2 1 a - 0 -- 2 [1 0 - 0 - ] U - , + W - ~ - --(ruw2) - - (ww2) + - __ (ruq2) + _(wq2) or az r or oz 15 r or oz 

[JW -2 oW!3- 2 - 2uw - - 2w - + 29 wi - if? - - c or oz zz 3 (9) 

The balance of this equation is shown in figure 10. Advection is the smallest of 

all the terms, Diffusion term is a gain near the center of the plume and a loss 

in the rest of the flow. Also, its magnitude near the center is comparable to the 

other dominant terms in the balance, We note that the buoyancy production is 

comparable to the production due to mean velocity gradients near the plume center 

but over the rest of the flow fi eld the shear production is much larger than the 

buoyancy production, It is also interesting to note that the buoyancy production 

and dissipation rate approximat ely balance each other. The closing term in this 

balance is if? zz and represents the sum of the pressure correlation term and the 

anisotropic part of the dissipation. This term is a loss for the u2 budget and we 

note that beyond r / z = 0.08 this term and shear production approximately balance 

each other. 

The equation for the radial component u2 is given by 

and its balance is shown i~gure 11. Obviously the advection of u2 has the same 

form as the advection of w2 . The production due to velocity gradients is a loss 

near the plume center and is a gain after about r / z = 0.04. This is because the 

radial gradient of the radial mean velocity is positive near the plume center. The 

56 
Turbulence Modeling and Experimetns 

mechanical production term is not large. The diffusion term is a loss over most of 

the flow field and becomes a gain toward the outer edge of the flow field . The sum 

of the pressure correlation term and the anisotropic part of the dissipation rate is 

obtained as the closing term in the budget and represents a gain for u2 . We further 

note that beyond r / z = 0.08 it approximately balances the dissipation rate. 

Finally we look at the budget for the shear stress uw as shown in figure 12. Its 

equation is given by 

&uw &uw 1 & 8 8U &U -8W U-- + W-- ~ - --(ruuw) - -(wuw) - uw- - uw- - u2 - &r 8 z r &r 8 z 8r & z 8r 

8W - ( - uw 8z + gf3ut - <Prz 11) 

Both advection and the turbulent buoyancy production are of very small magnitude 

and over most of the flow field these approximately balance each other. Neglecting 

these two terms would not cause any significant change in the shear stress balance. 

We note that the diffusion term is not negligible in this budget. The term <Prz is 

essentially balanced by the difference between the shear production and diffusion 

processes. The shape of <Prz is obviously similar to that of the shear stress and its 

peak approximately corresponds to the peak in the shear production. 

3. Future Plans 

3.1 Turbulence Modeling (with T.-H. Shih) 

(a). Compare the performance of the various non-linear second order models in 

different homogeneous flows in order to find out their strengths and weakneses. This 

will be an extension of the work presented in section 1 of this brief. 

(b). To develop and test models for turbulent diffusion terms in the Reynolds 

stress equations using Lumley's theory of third momentsll. 

3.2 DNS of Bypass Transistion (with T.-H. Shih and G. Karniadakis). 

The bypass transition is an important engineering problem due to its relevence 

to turbomachinery environment and, therefore, there is a considerable interest both 

at LeRC and at CMOTT to study this phenomenon. We are interested in carrying 

out the DNS for this problem both in order to provide a data base for the modeling 

efforts of bypass transition at CMOTT and to study its physics. For the former we 

are interesting in finding out what kind of global parameters, if any, are linked to 

the transition process. For the later we are interested in finding out, for instance, 

what is the effect of anisotropy in the free stream turbulence velocity and length 

scale on the transition process. 

These simulations will be designed after the experiments of Sohn and Reshotko14 

who studied the bypass transition over a flat plate with differing free stream turbulence intensities. The results of DNS will be compared with these experiments. 

57 

Turbulence Modeling and Experimetns 

mechanical production term is not large. The diffusion term is a loss over most of 

the flow field and becomes a gain toward the outer edge of the flow field . The sum 

of the pressure correlation term and the anisotropic part of the dissipation rate is 

obtained as the closing term in the budget and represents a gain for u2 . We further 

note that beyond r / z = 0.08 it approximately balances the dissipation rate. 

Finally we look at the budget for the shear stress uw as shown in figure 12. Its 

equation is given by 

&uw &uw 1 & 8 8U &U -8W U-- + W-- ~ - --(ruuw) - -(wuw) - uw- - uw- - u2 - &r 8 z r &r 8 z 8r & z 8r 

8W - ( - uw 8z + gf3ut - <Prz 11) 

Both advection and the turbulent buoyancy production are of very small magnitude 

and over most of the flow field these approximately balance each other. Neglecting 

these two terms would not cause any significant change in the shear stress balance. 

We note that the diffusion term is not negligible in this budget. The term <Prz is 

essentially balanced by the difference between the shear production and diffusion 

processes. The shape of <Prz is obviously similar to that of the shear stress and its 

peak approximately corresponds to the peak in the shear production. 

3. Future Plans 

3.1 Turbulence Modeling (with T.-H. Shih) 

(a). Compare the performance of the various non-linear second order models in 

different homogeneous flows in order to find out their strengths and weakneses. This 

will be an extension of the work presented in section 1 of this brief. 

(b). To develop and test models for turbulent diffusion terms in the Reynolds 

stress equations using Lumley's theory of third momentsll. 

3.2 DNS of Bypass Transistion (with T.-H. Shih and G. Karniadakis). 

The bypass transition is an important engineering problem due to its relevence 

to turbomachinery environment and, therefore, there is a considerable interest both 

at LeRC and at CMOTT to study this phenomenon. We are interested in carrying 

out the DNS for this problem both in order to provide a data base for the modeling 

efforts of bypass transition at CMOTT and to study its physics. For the former we 

are interesting in finding out what kind of global parameters, if any, are linked to 

the transition process. For the later we are interested in finding out, for instance, 

what is the effect of anisotropy in the free stream turbulence velocity and length 

scale on the transition process. 

These simulations will be designed after the experiments of Sohn and Reshotko14 

who studied the bypass transition over a flat plate with differing free stream turbulence intensities. The results of DNS will be compared with these experiments. 

57 
Aamir Shabbir 

3.2.1 Numerical Scheme. 

Currently we are exploring the possibility of using a spectral element code for its 

suitability for doing such a DNS. We are inclined to use a spectral element method 

because of its higher accuracy and its ease of local grid refinement. 

The numerical scheme used in the code involves fractional time discretization 

which results in three sets of semi discrete equations. In the first step advection term 

is handled explicitly using a third order Adams-Bashforth scheme. In the second 

step Poisson equation for pressure is solved implicitly and continuity is satisfied. In 

the third fractional step the diffusion terms are accounted implicitly by a second 

order Crank-:r~kholson method. 

In order to carry out the spatial discretization the flow domain is first decomposed into macro elements. Each of these macro elements uses a local cartesian 

mesh by employing Gauss-Labatto collocation points. Then within each macro element the flow variables are represented as tensor product of Chebychev polynomial. 

These representations of the flow variables are then substituted into the governing 

equations and discrete equations are obtained by applying the weighted residual 

technique. 

3.2.2 Test Cases to be run 

Several test cases will have to be run in order to validate the code before a full 

DNS can be carried out. First of these is to solve the laminar boundary layer flow 

over a flat plate in order to insure that the numerical method gives the Blasius 

solution. This will also help us explore the various boundary conditions which can 

be used at the top boundary and at the outflow and latter can be used for the 

mean flow during the DNS. After this has been successfully accomplished the most 

unstable mode disturbances based on the linear stability theory will be intorduced. 

This will allow comparing their growth rates (in the linear region) with the solutions 

from the linear stability theory. The third case will be that of suction and blowing 

through the flat plate and the resutls will be compared with those obtained by 

previous workers. 

4. References 

1 Shabbir, A. and D.B. Taulbee. Evaluation of Turbulence Models for Predicting 

Buoyant Flows. J. Heat Trans., 112, 945-951 (1990). 

2 Shih, T.-H., A. Shabbir and J.L. Lumley. Advances in the Modeling of Pressure 

Correlations terms in the Second Moment Equations. NASA TM. 104413 (1991). 

3 Shih,T.-H. and J.L. Lumley. A Critical Comparison of Second Order Closures 

with Direct Numerical Simulation of Homogeneous Turbulence. NASA TM 105351 

(1992). 

4 Launder, B.E., Reece and W. Rodi.Progress in the Development of Reynoldsstress Turbulence Closure. JFM, 68, 537-566 (1975). 

5 Speziale, C.G., S. Sarkar and T.B. Gatski. Modeling the Pressure-Strain Correlation of Turbulence; An Invariant Dynamical Systems Approach. JFM, 227, 

245-272, (1991). 

58 

Aamir Shabbir 

3.2.1 Numerical Scheme. 

Currently we are exploring the possibility of using a spectral element code for its 

suitability for doing such a DNS. We are inclined to use a spectral element method 

because of its higher accuracy and its ease of local grid refinement. 

The numerical scheme used in the code involves fractional time discretization 

which results in three sets of semi discrete equations. In the first step advection term 

is handled explicitly using a third order Adams-Bashforth scheme. In the second 

step Poisson equation for pressure is solved implicitly and continuity is satisfied. In 

the third fractional step the diffusion terms are accounted implicitly by a second 

order Crank-:r~kholson method. 

In order to carry out the spatial discretization the flow domain is first decomposed into macro elements. Each of these macro elements uses a local cartesian 

mesh by employing Gauss-Labatto collocation points. Then within each macro element the flow variables are represented as tensor product of Chebychev polynomial. 

These representations of the flow variables are then substituted into the governing 

equations and discrete equations are obtained by applying the weighted residual 

technique. 

3.2.2 Test Cases to be run 

Several test cases will have to be run in order to validate the code before a full 

DNS can be carried out. First of these is to solve the laminar boundary layer flow 

over a flat plate in order to insure that the numerical method gives the Blasius 

solution. This will also help us explore the various boundary conditions which can 

be used at the top boundary and at the outflow and latter can be used for the 

mean flow during the DNS. After this has been successfully accomplished the most 

unstable mode disturbances based on the linear stability theory will be intorduced. 

This will allow comparing their growth rates (in the linear region) with the solutions 

from the linear stability theory. The third case will be that of suction and blowing 

through the flat plate and the resutls will be compared with those obtained by 

previous workers. 

4. References 

1 Shabbir, A. and D.B. Taulbee. Evaluation of Turbulence Models for Predicting 

Buoyant Flows. J. Heat Trans., 112, 945-951 (1990). 

2 Shih, T.-H., A. Shabbir and J.L. Lumley. Advances in the Modeling of Pressure 

Correlations terms in the Second Moment Equations. NASA TM. 104413 (1991). 

3 Shih,T.-H. and J.L. Lumley. A Critical Comparison of Second Order Closures 

with Direct Numerical Simulation of Homogeneous Turbulence. NASA TM 105351 

(1992). 

4 Launder, B.E., Reece and W. Rodi.Progress in the Development of Reynoldsstress Turbulence Closure. JFM, 68, 537-566 (1975). 

5 Speziale, C.G., S. Sarkar and T.B. Gatski. Modeling the Pressure-Strain Correlation of Turbulence; An Invariant Dynamical Systems Approach. JFM, 227, 

245-272, (1991). 

58 
Turbulence M adeling and Experimetns 

6 Morris, J. Modeling the Pressure Redistribution Terms. Phys. Fluids, 27, 7, 

1620-1623 (1984). 

7 Lee, M. and W.C. Reynolds. Numerical Experiments on the Structure of Homogeneous Turbulence. Stanford University Report No. TF-24 (1985). 

8 Tavoularis, Sand S. Corrsin. Experiments in Nearly Homogeneous Turbulent 

Shear Flow with a Uniform Mean Temperature Gradient. Part 1. JFM, 104, 

311-347 (1981). 

9 Bardina, J., J.H. Ferziger and W.C. Reynolds Improved Turbulence Models Based 

on Large-Eddy Simulation of Homogeneous, Incompressible Turbulent Flows. 

Stanford University Technical Report TF-19 (1983). 

10 Bardina, J. Private communication (1992). 

11 Lumley, J. L. Computational Modeling of Turbulent Flows. Advances in Applied 

Mechanics, 18, 123-176 (1978). 

12 Lumley, J.L. Phys. Fluids, 18, 750 (1975). 

13 Shabbir, A. of the 8th Symposium on Turbulent Shear Flows (1991). 

14 Sollll, K.-H., and E. Reshotko Experimental Study of Boundary Layer Transition 

With Elevated Freestream Turbulence on a Heated Flat Plate. NASA CR 187068 

(1991 ). 

59 

Turbulence M adeling and Experimetns 

6 Morris, J. Modeling the Pressure Redistribution Terms. Phys. Fluids, 27, 7, 

1620-1623 (1984). 

7 Lee, M. and W.C. Reynolds. Numerical Experiments on the Structure of Homogeneous Turbulence. Stanford University Report No. TF-24 (1985). 

8 Tavoularis, Sand S. Corrsin. Experiments in Nearly Homogeneous Turbulent 

Shear Flow with a Uniform Mean Temperature Gradient. Part 1. JFM, 104, 

311-347 (1981). 

9 Bardina, J., J.H. Ferziger and W.C. Reynolds Improved Turbulence Models Based 

on Large-Eddy Simulation of Homogeneous, Incompressible Turbulent Flows. 

Stanford University Technical Report TF-19 (1983). 

10 Bardina, J. Private communication (1992). 

11 Lumley, J. L. Computational Modeling of Turbulent Flows. Advances in Applied 

Mechanics, 18, 123-176 (1978). 

12 Lumley, J.L. Phys. Fluids, 18, 750 (1975). 

13 Shabbir, A. of the 8th Symposium on Turbulent Shear Flows (1991). 

14 Sollll, K.-H., and E. Reshotko Experimental Study of Boundary Layer Transition 

With Elevated Freestream Turbulence on a Heated Flat Plate. NASA CR 187068 

(1991 ). 

59 
0.10 

0.08 . 

006J 

u 2 

0.04 

002-l0.00-1 

0.00 

0) 

0 

0.10 

0.08 

0.06 

u 2 

0.0 4 

0.02 

0.00 

0.00 

- ---

0.30 

• DNS 

-- LRI1. 0. 251 

. 

- - IP 

- - -

. SSG • 

0.20 

.. / 

- • v 2 :. -. -. -. _. -. -' -' - =.: / / 0. 15 • /, / 

------

I 

0.50 

0.50 

0.10 

• t • DNS • -- LRI1. • 0.05 . - - IF 

---·SSG 

I I I- 0.00 

1.00 1.50 2.00 0.00 0.50 1.00 1.50 

St St 

Figure L Comparison of the models for flow through axisymmetric contraction with the 

DNS data of of Lee and Reynolds (lD8S)_ 

2.0

0.1 __ ___ -+-

• DNS 

-- LRI1. 

. 

-

.

- - IP 

- - - - SSG 

---

0.08 

0.06 

') 'U0.04 

0.02 

• DNS 

-- LRI1. 

'

-

'

- IF 

- - - - SSG 

.

--.----------

1.00 1.50 2.00 

St 

0.00 0.50 1.00 

St 

1.50 

Figure 2. Comparison of the models for flow through axisymmetric expansion with the 

DNS data of of Lee and Reynolds (1985), case EXO. 

2.00 

0) 

0 

0.10 

0.08 

0.06 

u2 

0.04 

0.02 

0.00 

0.00 0.50 

1 

__ 

Figure 1. Comparison of the models for flow through axisymmetric contraction with the DNS data of of Lee and Reynolds (l[J85). 

• DNS 

-- LRll 

. 

_

. 

-

. IP 

- - -

. SSG 

---

0.08 

0.06 

') 

.(1-

0.04 

0.02 

. ' 

• DNS 

-- LRn 

.-. - . IF 

- - - . SSG 

1.00 1.50 2.00 

St 

0.00 0.50 1.00 

St 

1.50 

Figure 2. Comparison of the models for flow through axisymmetric expansion with the 

DNS data of of Lee and Reynolds (1985), case EXO. 

2.00 
0) 

I-' 

, O.JO I " "' 

0.20 

u2 

,,'" ",.'" 

0.10-

, 

, 

"' . "' 

• DNS 

-- LRTl 

. 

_

. 

-

. IP 

- - -

. SSG 

0.00 0.50 1.00 

St 

1.50 2.00 

0.20 

v 2 

0.10 -

• DNS 

-LRR 

. 

-

.-. IF 

- - -. SSG 

0 .00 0 .:0 1.00 1.50 2.00 

St 

Figure 3 .. Comparison of the models for flow through axisymmetric expansion with the 

DNS data of of Lee and Reynolds (1085), case EXP. 

, 

0.20 

0.10-

./ ./ 

.,. '" :".", 

. ./ 

./ 

, ./ 

; / . , ; ' 

./ ./ 

• DNS 

-- LRII. 

_

. 

-

. IP 

- - -

. SSG 

0.20 

0.10· 

• DNS 

-LRR 

. 

_

.

-

. IP 

- - -. SSG 

0.00 0.50 1.00 

St 

1.50 2.00 0.00 0.:0 1.00 1.50 

St 

Figure 3 .. Comparison of the models for flow through axisymmetric expansion with the 

DNS data of of Lee and Reynolds (1985), case EXP. 

2.00 
0) 

t-.:) 

0.1 O-tt~~~---+--~~-+--~~~--I--~~~T 

0.08 

0.06 

U 2 

0.04 

0.02 

- ---~---::: ... -

• D~S 

-- LRIl 

. 

-

. 

- IP 

- - -

. SSG 

0.00 I I0 .00 

0.1 0 

0.08 

0.06 

V 2 

0.04 

0.02 

0.00 

0.00 

0.50 

0.50 

1.00 

St 

1.00 

St 

1.50 

, 

, 

, 

. " 

. ' 

• D~S 

-- LRIl 

-

. 

-

. IP 

- - -

. SSG 

2.00 

1.:0 2.00 

0 .1 O-tI~~~~-+---~--4-~~~-l-~~~-+-

0.08 

0.06 

W 2 

0.04 

0.02 

• DNS 

-- LRIl 

-. - . IP 

- - -

. SSG . _. _. _. -:.:: :.-.::: ._-,;-:..::..--

• • 

0.001 

0.00 0.50 1.00 1.50 2

St 

Figure 4 . Comparison of the models for distortion by plane strain with the DNS data of 

Lee and Reynolds (1985), case PXC, S=2.6. 

Q) 

~ 

0.08 

0.06 

U 2 

0.04 

0.02 

---!'- ~--. = --~ :: .--

• O:-;S 

-- LRR 

. 

-

. 

-

. IP 

- - -

. SSG 

OO+-~~--t--~~r--~~~t-"~--l0.00 0 .50 1.00 

St 

0.10 

0.08 

0. 06 

V 2 

0 .04 

0 .02 

0 .00 

0.00 0 .50 1.00 

St 

/ 

1.50 

/ 

/ 

/ / 

, . .... 

. / 

• O:-;S 

-- LRR 

. 

-

. 

-

. IP 

- - -

. SSG 

2.00 

1.:0 2.00 

0.08 

0.06 

W 2 

0.04 

0 .02 

• ONS 

-- LRR 

.-. - . IP 

- - -

. SSG 

-' = '--:' ~ '.,: ;.,--

'

- '- ' -'-' -:": ::' -

• • 

O.OO+-~~~-r-~~-t~-~~t---~~-f.

0.00 0.50 1.00 1.50 2

St 

Figure 4. Comparison of the models for distortion by plane strain with the DNS data of 

Lee and Reynolds (1985), case PXC, S=2.6. 

-------------
0:. 

c,., 

~--

0.20 • DNS 

~-

LRR 

'-.-. IP 

~J 

- - -. SSG 

U 2 

/ 

/ 

0.10 

• 

/ 

0.05. ____ 

0.00+1 ~~~~~~~-;-~~~-r-~~~+ 

0.00 0.50 1.00 1.50 

0.20 I J 

0.15 

v 2 

0.10 

0.05 

/ I 

I • 

/ / 

/ . 

/ 

/ . // 

/ 

,',' 

/ 

. 

/ / 

/ :/ 

/ / ':/'. 

'" ~,- ' "'

."'" 

• DNS 

-- LRR 

. 

_

. 

- IP 

- - -

. SSG 

2.00 

0.00+1 ~~~-r-~~---r-~~r---~~-+ 

0.00 0.50 1.00 l.S0 2.00 

St 

0.20 

o,,~ 

• DNS 

-LRR 

'-'-' IP 

- - -. SSG 

W 2 

0.10 

0.05t---.. 

.-._. _. _. -':~ --'::"'-? • • 

oo-+I~~~--t~~~--t~~~--t~~~-+ 

0.00 0.50 1.00 

St 

1.50 2.0

Figure 5 . Comparison of the models for distortion by plane strain with the DNS data of 

Lee and Reynolds (1085), case PXE, S=25.0 

0.20 • DNS --LRR '-'- ' IP 0.15 - - -. SSG U 2 • , 0.10 , ~ 0.05. ___ ~~ oo-l--~~~_;_~~~__r_~~~__r_~~~+ 0.00 0.50 1.00 1.50 2.00 20+------+------+----+--~r_-_t , , 0.15 v 2 0.10 0.05 , , , , .' , , , . , , . I . ' " ." I I I , , I I • ",,,.' . " ~, ' , , • • DNS --LRR . _ . - IP - - - . SSG 

O.oo+-~~-r--~~--;--~~-r--

~~ 

0.00 0.50 1.00 

St 

1.50 2.00 

0.20 

• DNS --LRR 

0.15 

'

-.-. IP 

- - -. SSG 

W 2 

0.10 

0.05 It----. 

'-' - '-'-'- '::--'::'--';? 

• •• • 

oo+~~~-r~~~--t~~~---jr--'~~'---+ 

0.00 0.50 1.00 

St 

1.!:O 2.0

Figure 5 . Comparison of the models for distortion by plane strain with the DNS data of 

Lee and Reynolds (1985), case PXE, S=25.0 
0) 

..,. 

40.00 40.00 

• EXPER.!MENT • EXPERIMENT 

-- LRR -- LRR 

.

_

.

- IP .- .- . IP 

- - -. SSG 

X 104 X 104 

20.00 20.00 

".OO} __ _ .SSG 

-----"ii, 

.

-.... ... 

10001 

0.00 

6.00 

20.00-t-.-

15.00 

X 104 

10.00 

5.00 

0.00 

6.00 

... 

r 

10.00 ' 

I I I I I 0.00 

7.00 8.00 9.00 10.00 11.00 12.00 6.00 7.00 8.00 9.00 10.00 

x/H x/H 

20.00 

• EXPERlMENT • 

--LRR 

.

-

.

-

. IF 

- - -. SSG 

7.00 

-- LRR 

.

-

.-. IP 

15.00 - - -. SSG 

_ uv X 104 U2 

10.00 

5.00 

0.00 

8.00 9.00 10.00 11.00 12.00 6.00 7.00 8.00 9.00 10.00 

x/H x/H 

Figure 6 . Comparison of the models for homogeneous shear flow with the data Tavoulans 

and Corrsin (1981). 

11.00 

11.00 

Figure 6 . Comparison of the models for homogeneous shear flow with the data Tavoularis 

and Corrsin (1981). 
0) 

C1l 

,/ /,} n/s = .25 4.00-+ 

I 

I 

I _3oaJ q2 o o ,,/} ,. I • 

q; 

o I / o ' , , ,. 2.00 , ,. 

o LES 

-LRR ,oar= 00 1 - -

_. SSG _ 

0.00 I I I 

0.00 2.00 4.00 6.00 8.00 10.00 

St 

4.00 

3.00 

2.00 

0/5=0.50 o LES --LRR 

.-.-. IP 

- - _ . SSG 

I n-O 0-0 - - - - - - - - - - - - - -

0.00 2.00 4.00 6.00 8.00 10.00 

St 

5.00 

4.00{ 

n/s = 1.0 -LRR 

._.-. IP 

- - -. SSG 

_3.00 

2.00 

0.00 

0.00 2.00 4.00 6.00 8.00 

St 

Figure! 7.. Comparison of the models for the rotating hOlllogcncous shear flow with the 

large eddy simulation of Bardinn ct al. (1983). 

5.00 

4. 00 

3.00 

q2 

2.00 

1.00 

0.00 

0.00 

4.00 

3.00 

2.00 

filS = .25 

2.00 

0/5 = 0. 50 

1.00 

I 

I 

0 I I 

0 

0 

0 

4.00 6.00 

St 

I 

I 

I 

I 

I 

0 LES 

-LRR 

.-.-. IP 

- -

_. SSG 

8.00 10.00 

o LES 

-- LRR 

.- .- . IP 

- - _ . SSG 

----

----

5.00 

fl/S = 1.0 -LRR 

.

- .-. IP 

4.00 

- - -. SSG 

_

3.00 

2.00 

1.00 

0.00 

0.00 2.00 4.00 6.00 8.00 

St 

0.00 2.00 4.00 6.00 8.00 10.00 

St 

Figure! 7 .. Comparison of the models for the rotating hOlllogcncous shear flow with the 

large eddy simulation of Bardinl\ ct al. (1983). 
Aamir Shabbir 

lC.00t---_----:..L-_~ ____ ~ ____ - _ _+ 

~ o.o,~~. -~. -~. -=' -:::' -~';;-:;"~-:.=~~~~===~==;::;;;;;;;;- . _. _. _. _ .... ~ t).;rn.~tOS 

i 

· > 

~ 

~ ~ -30. 

suw 'lr 'U:SSUlr:.COU.[LA O~ ... so 

lJOlC":-:'::ul TtluS 

-'0·og~.C~0~-------------0~.~b-5----------------0~\-.----------------0+.~ . 

r/. 

Figure 8 . Budget for wt equation. 

.5.no,t-------________ --~ ________________ _L ________________ _+ 

.x> rnC'Ol'C":'":C:-; :U':C TO T().,r(ru,TUJl( Gf\AOIC :-o TS 

: 20. 00-:. 

o 

· e 

< 

> 

t 

e 

· ~ 

:i 

;;; 

,. 

0.00 0.08 

'!'t....-- .! ... ~ or pp.[S5C!t[..CO::~.!:L'no:_: ASD 

)':O:"CCCLA.' itl\~S 

0 .'16 

Figure 10; Budget for ut equation. 

l~.OO~~--------__ --~?_~~---------------L----------------~ 

~..."..:...>,----.JC'-_._UOYA~CY rItO::::-C7;O:-t 

~ 'i _ 5.0""';--------.--------

~ SC':.l cr ~i\CSH kC-CO""'tLATIOJ'f ~O 

"S:sc7ltO'IC P",IlT or OISlI"'ATIO:-f A..,. \T[ 

10.0'0+-_ ___ _ - 1---____ - -4--- - --·---+ 

SUN or , a u .su",c .cO'UlCLAT1011 ASO 

ASlsoTJ.orlC r.u.r or OISSIPATIO!f ""Tt 

~ DISSIPAT10S UTE 

-5.00+------ -- ---r------~--;--------+ 0.00 0.08 0.16 

r/. 

Figure 9 Budget for trW equation. 

".00+----- ----'--------'---------+ rkODUCTI0!< Due TO '"'CLOCrrY C1AOICl'-"TS 

"; ~"O 

. 

o 

~ > 

5U)J or I'ltSSUlt-COllllttATlO;O; A..'>D 

A1\rsoTkorlC 1AlT or DISSI'AT O~ ~'7'[ 

15 .000+0---~------------0.7C-5--------------~O~.r,~6--------------~O~.t2( 

Figur 11. Budget for trW equation. 

;;; ~"" . .,~':'" 

- I ~.og bo o.oa o. "6 0 '2'-

'I' 

Figure 12. Budget for w 2 equation. 

66 

Aamir Shabbir 

lC.00t---_----:..L-_~ ____ ~ ____ - _ _+ 

~ o.o,~~. -~. -~. -=' -:::' -~';;-:;"~-:.=~~~~===~==;::;;;;;;;;- . _. _. _. _ .... ~ t).;rn.~tOS 

i 

· > 

~ 

~ ~ -30. 

suw 'lr 'U:SSUlr:.COU.[LA O~ ... so 

lJOlC":-:'::ul TtluS 

-'0·og~.C~0~-------------0~.~b-5----------------0~\-.----------------0+.~ . 

r/. 

Figure 8 . Budget for wt equation. 

.5.no,t-------________ --~ ________________ _L ________________ _+ 

.x> rnC'Ol'C":'":C:-; :U':C TO T().,r(ru,TUJl( Gf\AOIC :-o TS 

: 20. 00-:. 

o 

· e 

< 

> 

t 

e 

· ~ 

:i 

;;; 

,. 

0.00 0.08 

'!'t....-- .! ... ~ or pp.[S5C!t[..CO::~.!:L'no:_: ASD 

)':O:"CCCLA.' itl\~S 

0 .'16 

Figure 10; Budget for ut equation. 

l~.OO~~--------__ --~?_~~---------------L----------------~ 

~..."..:...>,----.JC'-_._UOYA~CY rItO::::-C7;O:-t 

~ 'i _ 5.0""';--------.--------

~ SC':.l cr ~i\CSH kC-CO""'tLATIOJ'f ~O 

"S:sc7ltO'IC P",IlT or OISlI"'ATIO:-f A..,. \T[ 

10.0'0+-_ ___ _ - 1---____ - -4--- - --·---+ 

SUN or , a u .su",c .cO'UlCLAT1011 ASO 

ASlsoTJ.orlC r.u.r or OISSIPATIO!f ""Tt 

~ DISSIPAT10S UTE 

-5.00+------ -- ---r------~--;--------+ 0.00 0.08 0.16 

r/. 

Figure 9 Budget for trW equation. 

".00+----- ----'--------'---------+ rkODUCTI0!< Due TO '"'CLOCrrY C1AOICl'-"TS 

"; ~"O 

. 

o 

~ > 

5U)J or I'ltSSUlt-COllllttATlO;O; A..'>D 

A1\rsoTkorlC 1AlT or DISSI'AT O~ ~'7'[ 

15 .000+0---~------------0.7C-5--------------~O~.r,~6--------------~O~.t2( 

Figur 11. Budget for trW equation. 

;;; ~"" . .,~':'" 

- I ~.og bo o.oa o. "6 0 '2'-

'I' 

Figure 12. Budget for w 2 equation. 

66 
_I _ 

Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Modeling of Turbulent Shear Flows 

William W. Liou 

1. Motivation and Objective 

This report documents the current progress in the research and development of 

modeling techniques for turbulent shear flows. These include a two-scale model for 

compressible turbulent flows and a new energy transfer model. The former represents the status of our efforts to identify compressibility effects in turbulence. 

The energy transfer model refines a weakly nonlinear wave model developed earlier, which models directly the turbulent large structures. The objective of these 

activities is to develop second-order closures for compressible turbulent flows. 

2. Work Accomplished 

2.1 A Two-Scale Model for Compressible Turbulent Flows 

Numerical simulations of 2D and 3D compressible turbulence have shown tha.t 

the existence of shocklet structures and the energy transfer mechanism between 

the kinetic energy and the thermal-energy are the two important compressibility 

effects1 ,2 ,3 ,4. These compressibility effects are incorporated into a new two-scale 

model. The model is based on the proposition that the effect of compressibility in 

turbulence is mainly on the energetic large eddies in turbulent shear flows. The small 

eddies are affected only indirectly through the increased spectral energy transfer. 

The development of the model and some results of its application to compressible 

free shear layers are briefly described here. A more detailed analysis is included in 

a NASA TM5. 

Firstly, it is assumed that the shocklet structures that may occur intermittently 

in compressible turbulent flows are formed mainly by the collision of the energetic 

turbulent eddies of large scale. The small eddies, which contain much less energy, 

are less efficient in the formation of shocklet structures when they collide with 

other eddies. Thus, the eddy shocklets scale with the energy containing eddies 

and have more direct influence on the evolution of the large eddies than on the 

smaller ones. The large vortical structures are intensified as they pass through the 

shocklet. This process, in other words, enhances the vortex stretching mechanism 

and increases the spectral energy transfer. In addition to the usual route of the 

vortex stretching mechanism that has already been enhanced, the small eddies may 

be generated directly from the passage of the large vortical structures through shock 

waves These processes of enhanced energy transfer may then cause the spectrum 

to depart from equilibrium. Another mechanism that may also contribute to the 

non-equilibrium spectrum or the creation of vorticity is strongly related to the 

pressure fluctuation. It has been shown by Kida and Orszag3 and Lee et al.

2 , 

among others, that substantial vorticity is created by the baroclinic terms. The 

creation of vorticity, however, occurs mainly at the shock wave. 

67 

_I _ 

Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Modeling of Turbulent Shear Flows 

William W. Liou 

1. Motivation and Objective 

This report documents the current progress in the research and development of 

modeling techniques for turbulent shear flows. These include a two-scale model for 

compressible turbulent flows and a new energy transfer model. The former represents the status of our efforts to identify compressibility effects in turbulence. 

The energy transfer model refines a weakly nonlinear wave model developed earlier, which models directly the turbulent large structures. The objective of these 

activities is to develop second-order closures for compressible turbulent flows. 

2. Work Accomplished 

2.1 A Two-Scale Model for Compressible Turbulent Flows 

Numerical simulations of 2D and 3D compressible turbulence have shown tha.t 

the existence of shocklet structures and the energy transfer mechanism between 

the kinetic energy and the thermal-energy are the two important compressibility 

effects1 ,2 ,3 ,4. These compressibility effects are incorporated into a new two-scale 

model. The model is based on the proposition that the effect of compressibility in 

turbulence is mainly on the energetic large eddies in turbulent shear flows. The small 

eddies are affected only indirectly through the increased spectral energy transfer. 

The development of the model and some results of its application to compressible 

free shear layers are briefly described here. A more detailed analysis is included in 

a NASA TM5. 

Firstly, it is assumed that the shocklet structures that may occur intermittently 

in compressible turbulent flows are formed mainly by the collision of the energetic 

turbulent eddies of large scale. The small eddies, which contain much less energy, 

are less efficient in the formation of shocklet structures when they collide with 

other eddies. Thus, the eddy shocklets scale with the energy containing eddies 

and have more direct influence on the evolution of the large eddies than on the 

smaller ones. The large vortical structures are intensified as they pass through the 

shocklet. This process, in other words, enhances the vortex stretching mechanism 

and increases the spectral energy transfer. In addition to the usual route of the 

vortex stretching mechanism that has already been enhanced, the small eddies may 

be generated directly from the passage of the large vortical structures through shock 

waves These processes of enhanced energy transfer may then cause the spectrum 

to depart from equilibrium. Another mechanism that may also contribute to the 

non-equilibrium spectrum or the creation of vorticity is strongly related to the 

pressure fluctuation. It has been shown by Kida and Orszag3 and Lee et al.

2 , 

among others, that substantial vorticity is created by the baroclinic terms. The 

creation of vorticity, however, occurs mainly at the shock wave. 

67 
William W. Liou 

Based on the picture described above, the effect of compressibility in turbulence 

is mainly on the energy containing large eddies or the low wavenumber fluctuations. 

The large eddies respond more readily to changes in the compressible mean flow 

resulting from either high speed or combustion. The straining of the large eddies due 

to compressibility effects increases the spectral energy transfer to the small scales 

through the mechanism of vortex stretching and direct generation. The smaller 

scales, on the other hand, are only indirectly affected by compressibility. The energy 

contained in the small scales in the high wavenumber part of the energy spectrum 

is increased only as more energy is pumped in from the large eddies associated with 

the low wavenumber part of the spectrum. To model the Favre-averaged mean 

compressible turbulent quantities associated with these two distinct regimes in the 

energy spectrum we solve the modeled transport equations for the kinetic energy 

of the large eddy (kp ) and the small eddy (kt ) and the rate of energy transfer from 

the large eddy to the small eddy (cp) and the rate of energy dissipation (Ct). The 

transport equations are 

- - _Dkp _ d [(_ J.lT ) dkp] (dU)2 

p-=---- - - J.l + - - + /-LT dy Dt dy uk dy p 

_DE; _ d [(- /-LT )dE;J C E; (dU)2 p-=- - - /-L + - - + Pl-=-/-LT- Dt dy u- dy k dy ~p p 

_Dkt d [(_ /-LT) dkt] __ p-=- = - J.l + - - + pc Dt dy uk: dy p 

_DEt d [(_ /IT) dEt ] c _EtEp p-=- = - J.l + - - + tIP--=- Dt dy u;; dy kt 

pEp + P.D. (1) 

(2) 

--PCt (3) 

(4) 

P.D. and E.S. denote the effects of pressure-dilatation and eddy shocklets, respectively. The definition of the model constants can be found in the NASA TM. The 

present two-scale model for compressible turbulence is built upon a parallel model 

for incompressible flows, Duncan et al. 6 . Models for the terms responsible for the 

compressibility effects are needed to close the equations. In this analysis, we have 

adopted Sarkar's 7 model for the pressure-dilatation terms. To model the effects of 

the increased spectral energy transfer due to compressibility, a simple model has 

been constructed through dimensional reasoning. Its coefficient has a M; dependence, similar to the dilatation dissipation model proposed by Zeman8 and Sarkar 

et al. 9 . The compressibility correctio"ns that they proposed have been implemented 

successfully into k - c models, Viegas and Rubesin 10, into k - w models, Wilcoxll 

and into second-order closure models, Speziale and Sarkarl2 . 

Fig. 1 shows the variation of the vorticity thickness growth rate, dow/dx, as a 

function of convective Mach number. The vorticity thickness, Ow, is defined by 

(5) 

68 

_I 

William W. Liou 

Based on the picture described above, the effect of compressibility in turbulence 

is mainly on the energy containing large eddies or the low wavenumber fluctuations. 

The large eddies respond more readily to changes in the compressible mean flow 

resulting from either high speed or combustion. The straining of the large eddies due 

to compressibility effects increases the spectral energy transfer to the small scales 

through the mechanism of vortex stretching and direct generation. The smaller 

scales, on the other hand, are only indirectly affected by compressibility. The energy 

contained in the small scales in the high wavenumber part of the energy spectrum 

is increased only as more energy is pumped in from the large eddies associated with 

the low wavenumber part of the spectrum. To model the Favre-averaged mean 

compressible turbulent quantities associated with these two distinct regimes in the 

energy spectrum we solve the modeled transport equations for the kinetic energy 

of the large eddy (kp ) and the small eddy (kt ) and the rate of energy transfer from 

the large eddy to the small eddy (cp) and the rate of energy dissipation (Ct). The 

transport equations are 

- - _Dkp _ d [(_ J.lT ) dkp] (dU)2 

p-=---- - - J.l + - - + /-LT dy Dt dy uk dy p 

_DE; _ d [(- /-LT )dE;J C E; (dU)2 p-=- - - /-L + - - + Pl-=-/-LT- Dt dy u- dy k dy ~p p 

_Dkt d [(_ /-LT) dkt] __ p-=- = - J.l + - - + pc Dt dy uk: dy p 

_DEt d [(_ /IT) dEt ] c _EtEp p-=- = - J.l + - - + tIP--=- Dt dy u;; dy kt 

pEp + P.D. (1) 

(2) 

--PCt (3) 

(4) 

P.D. and E.S. denote the effects of pressure-dilatation and eddy shocklets, respectively. The definition of the model constants can be found in the NASA TM. The 

present two-scale model for compressible turbulence is built upon a parallel model 

for incompressible flows, Duncan et al. 6 . Models for the terms responsible for the 

compressibility effects are needed to close the equations. In this analysis, we have 

adopted Sarkar's 7 model for the pressure-dilatation terms. To model the effects of 

the increased spectral energy transfer due to compressibility, a simple model has 

been constructed through dimensional reasoning. Its coefficient has a M; dependence, similar to the dilatation dissipation model proposed by Zeman8 and Sarkar 

et al. 9 . The compressibility correctio"ns that they proposed have been implemented 

successfully into k - c models, Viegas and Rubesin 10, into k - w models, Wilcoxll 

and into second-order closure models, Speziale and Sarkarl2 . 

Fig. 1 shows the variation of the vorticity thickness growth rate, dow/dx, as a 

function of convective Mach number. The vorticity thickness, Ow, is defined by 

(5) 

68 

_I 
Modeling of Turbulent Shear Flows 

The convective Mach number is defined as the ratio of the average convective 

velocity of the dominant large scale structures, relative to the free stream, to the free 

stream speeds of sound, Papamoschou and Roshko13 . The convective Mach number 

has been shown to be an appropriate parameter to correlate experimental data and 

to identify the effects of cOII'pressibility. The vorticity thickness growth rates for 

compressible free shear flows, (dow/dx)(Mc, Us/Uf , Psi Pf), have been normalized 

by the corresponding values for incompressible flows, (dow/dx)i(O, UsIUf, Psi Pf), 

and are presented in Fig. 1. The value of (dow / dX)i is obtained by using a relation, 

Papamoschou and Roshko13 , 

~ (1 - !b.. )(1 + (b )1/2) 

( U w ) Uj Pj 

dx i r.J 1 + !b..(b)1/2 

uj Pj 

(6) 

The constant of proportionality is obtained by the present model calculations performed in the limit of Mc --t O. Measured data are denoted by open symbols in Fig. 

1. Without the compressibility corrections, the current two-scale model and the 

two-scale model developed by Kim and Chen14 (KC) predict a large reduction of 

the growth rate only at very high convective Mach numbers. With the inclusion of 

the effects of eddy shocklets and the pressure work, the current compressible twoscale model predicts a smooth reduction of the vorticity thickness growth rate as 

the convective Mach number increases. The calculated growth rate curve levels off 

at high convective Mach numbers. It should be noted that in the present analysis 

the convective Mach number of the shear layer is increased by increasing the Mach 

number of -the high speed stream. According to the definition of the convective 

Mach number, there exists a maximum convective Mach number for a plane mixing 

layer of the same fluid with matched total temperature. That is, 

lim M = 1 - r (7) 

Mr-+(XJ e (-y-1)~1-r2) )1/2 

where r = Us/Uf and I denotes the ratio of the specific heats of the working fluid. 

For a value of R=O.l, the limiting convective Mach number for a plane shear layer 

of air is about 2.0. 

Since it is the Reynolds shear stress that appears in the mean momentum equations and influences directly the development of the mean flow, it is interesting to 

see how its peak value varies as a function of Me. Note that in the current analysis, 

the Reynolds shear stress is related to the mean flow by a turbulent eddy viscosity, 

/Lt. That is, 

-pu'v' 

and 

/.1T 

8u 

/LT -

8y 

69 

(8) 

(9) 

Modeling of Turbulent Shear Flows 

The convective Mach number is defined as the ratio of the average convective 

velocity of the dominant large scale structures, relative to the free stream, to the free 

stream speeds of sound, Papamoschou and Roshko13 . The convective Mach number 

has been shown to be an appropriate parameter to correlate experimental data and 

to identify the effects of cOII'pressibility. The vorticity thickness growth rates for 

compressible free shear flows, (dow/dx)(Mc, Us/Uf , Psi Pf), have been normalized 

by the corresponding values for incompressible flows, (dow/dx)i(O, UsIUf, Psi Pf), 

and are presented in Fig. 1. The value of (dow / dX)i is obtained by using a relation, 

Papamoschou and Roshko13 , 

~ (1 - !b.. )(1 + (b )1/2) 

( U w ) Uj Pj 

dx i r.J 1 + !b..(b)1/2 

uj Pj 

(6) 

The constant of proportionality is obtained by the present model calculations performed in the limit of Mc --t O. Measured data are denoted by open symbols in Fig. 

1. Without the compressibility corrections, the current two-scale model and the 

two-scale model developed by Kim and Chen14 (KC) predict a large reduction of 

the growth rate only at very high convective Mach numbers. With the inclusion of 

the effects of eddy shocklets and the pressure work, the current compressible twoscale model predicts a smooth reduction of the vorticity thickness growth rate as 

the convective Mach number increases. The calculated growth rate curve levels off 

at high convective Mach numbers. It should be noted that in the present analysis 

the convective Mach number of the shear layer is increased by increasing the Mach 

number of -the high speed stream. According to the definition of the convective 

Mach number, there exists a maximum convective Mach number for a plane mixing 

layer of the same fluid with matched total temperature. That is, 

lim M = 1 - r (7) 

Mr-+(XJ e (-y-1)~1-r2) )1/2 

where r = Us/Uf and I denotes the ratio of the specific heats of the working fluid. 

For a value of R=O.l, the limiting convective Mach number for a plane shear layer 

of air is about 2.0. 

Since it is the Reynolds shear stress that appears in the mean momentum equations and influences directly the development of the mean flow, it is interesting to 

see how its peak value varies as a function of Me. Note that in the current analysis, 

the Reynolds shear stress is related to the mean flow by a turbulent eddy viscosity, 

/Lt. That is, 

-pu'v' 

and 

/.1T 

8u 

/LT -

8y 

69 

(8) 

(9) 
-~------

William W. Liou 

In Fig. 2, the peak Reynolds shear stresses predicted by the present compressible 

two-scale model are compared with measured data, Elliott and Samimy15. The 

predictions of the present model, without compressibility corrections and of the 

KC model are also shown for comparison. The results show that, without the 

inclusion of some forms of compressibility corrections, none of the two-scale models 

tested, including the current model and the KC model, performs satisfactorily in the 

calculations of compressible free shear layers. The present compressible two-scale 

model under-predicts the absolute value of the peak Reynolds shear stress. However, 

the trend observed in the experiment that the level of the peak Reynolds shear stress 

decreases with increasing convective Mach number is picked up consistently by the 

current compressible two-scale model. Note that the Reynolds shear stress has 

been normalized by the square of the velocity difference of the two free streams. 

The model also shows that the value of the peak Reynolds shear stress appears to be 

independent of the velocity ratio of the free streams. In fact, the predicted variation 

of the peak value of the Reynolds shear stress as a function of the convective Mach 

number is similar to the predicted variation of the normalized vorticity thickness 

growth rate as a function of the convective Mach number. This characteristic of 

the present compressible two-scale model is consistent with the observation made 

by Elliott and Samimy15. They argue that, based on an integral analysis, the 

decreasing trend of the level of the Reynolds shear stress, as the convective Mach 

number is increased, is due mainly to the decrease of momentum thickness growth 

rate. However, the two speed ratios considered here, 0.1 and 0.2, are nearly equal 

to each other. Cases with a wider range of operating conditions, such as the speed 

ratios and the working fluids, need to be examined before any conclusive statement 

can be made. 

To further validate the present compressible two-scale model, it is applied to the 

compressible free shear layer corresponding to the Case 1 in Samimy and Elliott 16 . 

In this case, a fully expanded plane shear layer of air with Me = 0.51 and r = 0.36 is 

examined. The calculated mean profile shown in Fig. 3 agrees reasonably well with 

the measurement. As described previously, there are many possible causes for the 

small difference in the outer region of the mixing zone. Fig. 4 shows the comparison 

of the computed and the measured Reynolds shear stress. The present two-scale 

model under-predicts the peak Reynolds shear stress by about 12%. The profile of 

the Reynolds shear shear stress, however, agree very well with the measurement. 

2.2 A New Energy Transfer Model for Turbulent Free Shear Flows 

The model is built upon the weakly nonlinear wave models developed by Liou 

and Morris17 . The development of the energy transfer model and some results of 

its application to an incompressible free shear layer are briefly described here. A 

more detailed analysis is included in a NASA TM18. 

The random flow properties are split into three components, 

(10) 

The fluctuation with respect to the long time-average component, Fi , is separated 

into a component representing the large-scale motion, Ii, and one representing 

70 

-~------

William W. Liou 

In Fig. 2, the peak Reynolds shear stresses predicted by the present compressible 

two-scale model are compared with measured data, Elliott and Samimy15. The 

predictions of the present model, without compressibility corrections and of the 

KC model are also shown for comparison. The results show that, without the 

inclusion of some forms of compressibility corrections, none of the two-scale models 

tested, including the current model and the KC model, performs satisfactorily in the 

calculations of compressible free shear layers. The present compressible two-scale 

model under-predicts the absolute value of the peak Reynolds shear stress. However, 

the trend observed in the experiment that the level of the peak Reynolds shear stress 

decreases with increasing convective Mach number is picked up consistently by the 

current compressible two-scale model. Note that the Reynolds shear stress has 

been normalized by the square of the velocity difference of the two free streams. 

The model also shows that the value of the peak Reynolds shear stress appears to be 

independent of the velocity ratio of the free streams. In fact, the predicted variation 

of the peak value of the Reynolds shear stress as a function of the convective Mach 

number is similar to the predicted variation of the normalized vorticity thickness 

growth rate as a function of the convective Mach number. This characteristic of 

the present compressible two-scale model is consistent with the observation made 

by Elliott and Samimy15. They argue that, based on an integral analysis, the 

decreasing trend of the level of the Reynolds shear stress, as the convective Mach 

number is increased, is due mainly to the decrease of momentum thickness growth 

rate. However, the two speed ratios considered here, 0.1 and 0.2, are nearly equal 

to each other. Cases with a wider range of operating conditions, such as the speed 

ratios and the working fluids, need to be examined before any conclusive statement 

can be made. 

To further validate the present compressible two-scale model, it is applied to the 

compressible free shear layer corresponding to the Case 1 in Samimy and Elliott 16 . 

In this case, a fully expanded plane shear layer of air with Me = 0.51 and r = 0.36 is 

examined. The calculated mean profile shown in Fig. 3 agrees reasonably well with 

the measurement. As described previously, there are many possible causes for the 

small difference in the outer region of the mixing zone. Fig. 4 shows the comparison 

of the computed and the measured Reynolds shear stress. The present two-scale 

model under-predicts the peak Reynolds shear stress by about 12%. The profile of 

the Reynolds shear shear stress, however, agree very well with the measurement. 

2.2 A New Energy Transfer Model for Turbulent Free Shear Flows 

The model is built upon the weakly nonlinear wave models developed by Liou 

and Morris17 . The development of the energy transfer model and some results of 

its application to an incompressible free shear layer are briefly described here. A 

more detailed analysis is included in a NASA TM18. 

The random flow properties are split into three components, 

(10) 

The fluctuation with respect to the long time-average component, Fi , is separated 

into a component representing the large-scale motion, Ii, and one representing 

70 
Modeling of Turbulent Shear Flows 

the residual fluctuations, f: . The long time-average of the instantaneous value 

is denoted by an overbar, 

(11) 

For the large-scale fluctuation, a separable form of solut'ion is assumed, 

{u, v, p} = A(x) [u(y),v(Y),P(y)] exp [i(ax - wt)]. (12) 

The bold face quantities denote a complex solution whose real part describes the 

physical properties of the large-scale structures. a (= ar + iai) denotes a complex 

wavenumber and w the frequency. The governing equations for the local distributions of the large structures can be reduced to the Rayleigh equation in terms of 

v, 

(13) 

The amplitude, A( x), appears as a parameter in the local calculation for the u, v, p 

and is determined separately from the large scale turbulent kinetic energy equation, 

__ oUi 

-u·u·-

t J ox ' J 

o 

(Ui < uiuj » + viscous terms 

OXj 

(14) 

where k = ~UiUi' k denotes the turbulent kinetic energy of the large-scale structure. Note that in this analysis k den£.tes the turbulent kinetic energy of large 

scale structure of a single mode. The kp defined in the first part of this report 

represents the sum of the turbulent kinetic energy of all the modes in the entire 

large-scale spectrum. <> represents a short time-average with an average interval much smaller than Tl but much larger than the characteristic time scale of the 

background small-scale fluctuation, Strange and Crighton19 . The interaction terms, 

the third expression on the right hand side of equation (14), describe the transfer 

of large-scale energy, presumably, to the small scales where energy is eventually 

dissipated by viscosity. The detailed analysis of the weakly nonlinear wave models 

and the numerical solution procedure used here can be found in Liou and Morris 17 . 

The spectral energy transfer results from the interactions between turbulent fluctuations of different scales. For the weakly nonlinear wave turbulence models, the 

energy transfer is of crucial importance in the determination of the wave amplitude and needs to be considered carefully. Very little information, experimental or 

theoretical, is available regarding the stresses, - < u~uj >. 

The weakly nonlinear analysis seeks normal mode solution of the large-scale turbulent fluctuation. Locally, the fluctuations are described by the linearized Euler 

71 

Modeling of Turbulent Shear Flows 

the residual fluctuations, f: . The long time-average of the instantaneous value 

is denoted by an overbar, 

(11) 

For the large-scale fluctuation, a separable form of solut'ion is assumed, 

{u, v, p} = A(x) [u(y),v(Y),P(y)] exp [i(ax - wt)]. (12) 

The bold face quantities denote a complex solution whose real part describes the 

physical properties of the large-scale structures. a (= ar + iai) denotes a complex 

wavenumber and w the frequency. The governing equations for the local distributions of the large structures can be reduced to the Rayleigh equation in terms of 

v, 

(13) 

The amplitude, A( x), appears as a parameter in the local calculation for the u, v, p 

and is determined separately from the large scale turbulent kinetic energy equation, 

__ oUi 

-u·u·-

t J ox ' J 

o 

(Ui < uiuj » + viscous terms 

OXj 

(14) 

where k = ~UiUi' k denotes the turbulent kinetic energy of the large-scale structure. Note that in this analysis k den£.tes the turbulent kinetic energy of large 

scale structure of a single mode. The kp defined in the first part of this report 

represents the sum of the turbulent kinetic energy of all the modes in the entire 

large-scale spectrum. <> represents a short time-average with an average interval much smaller than Tl but much larger than the characteristic time scale of the 

background small-scale fluctuation, Strange and Crighton19 . The interaction terms, 

the third expression on the right hand side of equation (14), describe the transfer 

of large-scale energy, presumably, to the small scales where energy is eventually 

dissipated by viscosity. The detailed analysis of the weakly nonlinear wave models 

and the numerical solution procedure used here can be found in Liou and Morris 17 . 

The spectral energy transfer results from the interactions between turbulent fluctuations of different scales. For the weakly nonlinear wave turbulence models, the 

energy transfer is of crucial importance in the determination of the wave amplitude and needs to be considered carefully. Very little information, experimental or 

theoretical, is available regarding the stresses, - < u~uj >. 

The weakly nonlinear analysis seeks normal mode solution of the large-scale turbulent fluctuation. Locally, the fluctuations are described by the linearized Euler 

71 
William W. Liou 

equations. On the other hand, the spatial extent of each of the modes of the largescale structures could be regarded as being determined by the wavenumber, a r . 

Therefore, the proposition here is to estimate the characteristic size of the large 

scales as the wavelength associated with the large structures, which are predicted 

by the weakly nonlinear analysis. That is, 

271" 

l = lw = (15) 

where lw denotes the wavelength. Through dimensional reasoning, the enregy transfer can be modeled by 

(16) 

This is the proposed model for the energy transfer from the large scale to the small 

scale. This estimate is in accord with the classic assumption of turbulence theory 

that dissipation " .. proceeds at a rate dictated by the inviscid inertia behavior of 

the large eddies", Tennekes and Lumley 20. Computationally, since the wavenumber 

is already a part of the solution of the equations for the large-scale fluctuation, this 

model involves no extra efforts in estimating the characteristic size of the energy 

containing large scales. This rather simple model provides a closure to the equations 

for the large-scale structure, thereby allowing render the weakly nonlinear wave 

description of the large-scale structure to be self-contained. This self-contained 

nature of the weakly nonlinear wave turbulence models may be important in the 

future applications to other turbulent free shear flows. 

The model is tested against an incompressible plane mixing layer. Since the most 

unstable mode interacts most strongly with the mean flow17 , the most amplifying 

local instability is used in the modeling of the average, overall interactions between 

the mean and the large scale motions. Therefore, in the present formulation, the 

characteristic length scale lw is determined only by the locally most unstable modes. 

Fig. 5 shows the predicted evolution of the streamwise mean velocity profiles 

with axial distance. '" is a similarity coordinate, 

Y - Yl/2 

'" = X - xo 

(17) 

where Yl/2 denotes the location where the local mean velocity is one half of the free 

stream velocity. The predicted self-similar profiles agree well with that compiled 

by Patel21 except at the low speed edge of the layer. Similar differences were also 

observed by Liou and Morris17 . They attributed this difference to the single mode 

representation of the entire large scale spectrum and the uncertainties in the measurements in this region resulting from the local large changes in the instantaneous 

flow direction. 

The streamwise evolution of the amplitude of the large-scale structures is shown 

in Fig. 6. After a region of establishment, the amplitude reaches a saturated value. 

72 

William W. Liou 

equations. On the other hand, the spatial extent of each of the modes of the largescale structures could be regarded as being determined by the wavenumber, a r . 

Therefore, the proposition here is to estimate the characteristic size of the large 

scales as the wavelength associated with the large structures, which are predicted 

by the weakly nonlinear analysis. That is, 

271" 

l = lw = (15) 

where lw denotes the wavelength. Through dimensional reasoning, the enregy transfer can be modeled by 

(16) 

This is the proposed model for the energy transfer from the large scale to the small 

scale. This estimate is in accord with the classic assumption of turbulence theory 

that dissipation " .. proceeds at a rate dictated by the inviscid inertia behavior of 

the large eddies", Tennekes and Lumley 20. Computationally, since the wavenumber 

is already a part of the solution of the equations for the large-scale fluctuation, this 

model involves no extra efforts in estimating the characteristic size of the energy 

containing large scales. This rather simple model provides a closure to the equations 

for the large-scale structure, thereby allowing render the weakly nonlinear wave 

description of the large-scale structure to be self-contained. This self-contained 

nature of the weakly nonlinear wave turbulence models may be important in the 

future applications to other turbulent free shear flows. 

The model is tested against an incompressible plane mixing layer. Since the most 

unstable mode interacts most strongly with the mean flow17 , the most amplifying 

local instability is used in the modeling of the average, overall interactions between 

the mean and the large scale motions. Therefore, in the present formulation, the 

characteristic length scale lw is determined only by the locally most unstable modes. 

Fig. 5 shows the predicted evolution of the streamwise mean velocity profiles 

with axial distance. '" is a similarity coordinate, 

Y - Yl/2 

'" = X - xo 

(17) 

where Yl/2 denotes the location where the local mean velocity is one half of the free 

stream velocity. The predicted self-similar profiles agree well with that compiled 

by Patel21 except at the low speed edge of the layer. Similar differences were also 

observed by Liou and Morris17 . They attributed this difference to the single mode 

representation of the entire large scale spectrum and the uncertainties in the measurements in this region resulting from the local large changes in the instantaneous 

flow direction. 

The streamwise evolution of the amplitude of the large-scale structures is shown 

in Fig. 6. After a region of establishment, the amplitude reaches a saturated value. 

72 
Modeling of Turbulent Shear Flows 

In this region, the rate of the production of the large-scale turbulent kinetic energy 

from the mean flow is balanced by the rate of energy transfer from the large scales 

to the small scales. Note that, for the present energy transfer model, the amplitude 

equation becomes, 

(18) 

G3 and G4 denote the normalized positive definite integrals of the production terms 

and interaction terms across the layer, respectively. The critical points ofthe nonlinear equation (18), where dA2 jdx = 0, are Al = 0 and G4(X2)A2 = G3(X2). Simple 

analyses by applying the Liapunov function method22 show that Al is an unstable 

critical point. Any small disturbances to AI, say A~ would grow exponentially. In 

fact, 

(19) 

A2 , on the other hand, is asymptotically stable. A disturbance about the A2 , say 

A~, would decay exponentially, 

(A2)',...., -~ x 2 ,...., e 2 (20) 

The saturated value of the amplitude, A2 , is an asymptotically equilibrium value. 

It indicates an asymptotically equilibrium state of the large-scale structures. The 

simple instability analyses also show that any deviation away from this equilibrium 

state would be damped out exponentially. Consequently, the saturation of the wave 

amplitude may provide an indication of the the self-similarity of the flow in terms 

of the development of the large-scale structures. 

3. Future Plans 

3.1 A Two-Scale Model for Compressible Turbulent Flows 

(1) Extend the two-scale model to wall-bounded flows. 

(2) Continue the development of second-order closure models that account explicitly for the compressibility effects identified during the development of the two-scale 

eddy-viscosity model. 

3.2 A New Energy Transfer Model for Turbulent Free Shear Flows 

(1) Apply the weakly nonlinear wave model to compressible mixing layers to 

investigate the effects of compressibility on the characteristics of the coherent largescale structures. 

4. References 

I Passot, T. and Pouquet, A., "Numerical simulation of compressible homogeneous 

flows in the turbulent regime," J. Fluid Mech., 181, 441-466 (1987). 

2 Lee, S., Lele, S. K. and Moin, P., "Direct numerical simulation and analysis of 

shock turbulence interaction," AlA A paper 91-0523. 

73 

Modeling of Turbulent Shear Flows 

In this region, the rate of the production of the large-scale turbulent kinetic energy 

from the mean flow is balanced by the rate of energy transfer from the large scales 

to the small scales. Note that, for the present energy transfer model, the amplitude 

equation becomes, 

(18) 

G3 and G4 denote the normalized positive definite integrals of the production terms 

and interaction terms across the layer, respectively. The critical points ofthe nonlinear equation (18), where dA2 jdx = 0, are Al = 0 and G4(X2)A2 = G3(X2). Simple 

analyses by applying the Liapunov function method22 show that Al is an unstable 

critical point. Any small disturbances to AI, say A~ would grow exponentially. In 

fact, 

(19) 

A2 , on the other hand, is asymptotically stable. A disturbance about the A2 , say 

A~, would decay exponentially, 

(A2)',...., -~ x 2 ,...., e 2 (20) 

The saturated value of the amplitude, A2 , is an asymptotically equilibrium value. 

It indicates an asymptotically equilibrium state of the large-scale structures. The 

simple instability analyses also show that any deviation away from this equilibrium 

state would be damped out exponentially. Consequently, the saturation of the wave 

amplitude may provide an indication of the the self-similarity of the flow in terms 

of the development of the large-scale structures. 

3. Future Plans 

3.1 A Two-Scale Model for Compressible Turbulent Flows 

(1) Extend the two-scale model to wall-bounded flows. 

(2) Continue the development of second-order closure models that account explicitly for the compressibility effects identified during the development of the two-scale 

eddy-viscosity model. 

3.2 A New Energy Transfer Model for Turbulent Free Shear Flows 

(1) Apply the weakly nonlinear wave model to compressible mixing layers to 

investigate the effects of compressibility on the characteristics of the coherent largescale structures. 

4. References 

I Passot, T. and Pouquet, A., "Numerical simulation of compressible homogeneous 

flows in the turbulent regime," J. Fluid Mech., 181, 441-466 (1987). 

2 Lee, S., Lele, S. K. and Moin, P., "Direct numerical simulation and analysis of 

shock turbulence interaction," AlA A paper 91-0523. 

73 
William W. Liou 

3 Kida, S. and Orszag, S. A., "Energy and spectral dynamics in forced compressible 

turbulence," J. Sci. Compo 5 (1990). 

4 Kida, S. and Orszag, S. A., "Enstrophy budget in decaying compressible turbulence," J. Sci. Compo (1991) 

5 Liou, W. W. and Shih, T.-H., "A two-scale model for compressible turbulent 

flows," NASA TM (to appear) (1992). 

6 Duncan, B. S., Liou, W. W. and Shih, T.-H., "A Multiple-scale turbulence model 

for incompressible flow," NASA TM (to appear) (1992). 

7 Sarkar, S., Erlebacher, G., Hussaini, M. Y. and Kreiss, H. 0., " The analysis 

and modeling of dilatational terms in compressible turbulence," NASA CR 18195 

(1989). 

8 Zeman, 0., "Dilatation dissipation: The concept and application in modeling 

compressible mixing layers," Phys. Fluids A 2 (2), 178-188 (1990). 

9 Sarkar, S., Erlebacher, G. and Hussaini, M. Y., "Compressible homogeneous 

shear: simulation and modeling," NASA CR 189611 (1992). 

10 Viegas, J. R. and Rubesin, M. W., "A comparative study of several compressibility corrections to turbulence models applied to high-speed shear layers," AIAA 

paper 91-1783. 

11 Wilcox, D. C., "Progress in hypersonic turbulence modeling," AIAA paper 91-

1785. 

12 Speziale, C. G. and Sarkar, S., "Second-order closure models for supersonic turbulent flows," AIAA paper 91-0217. 

13 Papamoschou, D. and Roshko, A., "The compressible turbulent shear layer: an 

experimental study," J. Fluid Mech. 197, 453-477 (1988). 

14 Kim, S.-W. and Chen, C.-P., "A multiple-time-scale turbulence model based on 

variable partitioning of turbulent kinetic energy spectrum," NASA CR-179222 

(1987). 

15 Elliott, G. S. and Samimy, M., "Compressibility effects in free shear layers," Phys. 

Fluids A (1991) 

16 Samimy, M. and Elliott, G. S., "Effects of compressibility on the characteristics 

of free shear layers," Phys. Fluids A 28, 439-445 (1990). 

17 Liou, W. W. and Morris, P. J., "Weakly nonlinear models for turbulent mixing 

in a plane mixing layer," to appear in Phys. Fluids (1992). 

18 Liou, W. W., "A new energy transfer model for turbulent free shear flows," NASA 

TM (to appear) (1992). 

19 Strange, P. J. R. and Crighton, D. G., "Spinning modes on axisymmetric jets, 

Part I," J. Fluid Mech. 134, 231-245 (1983). 

20 Tennekes, H. and Lumley, J. L., A First Course in Turbulence, M.LT. Press, 

1972. 

21 Patel, R. P., "An experimental study of a plane mixing layer," AIAA Journal II, 

67-71 (1973). 

74 

William W. Liou 

3 Kida, S. and Orszag, S. A., "Energy and spectral dynamics in forced compressible 

turbulence," J. Sci. Compo 5 (1990). 

4 Kida, S. and Orszag, S. A., "Enstrophy budget in decaying compressible turbulence," J. Sci. Compo (1991) 

5 Liou, W. W. and Shih, T.-H., "A two-scale model for compressible turbulent 

flows," NASA TM (to appear) (1992). 

6 Duncan, B. S., Liou, W. W. and Shih, T.-H., "A Multiple-scale turbulence model 

for incompressible flow," NASA TM (to appear) (1992). 

7 Sarkar, S., Erlebacher, G., Hussaini, M. Y. and Kreiss, H. 0., " The analysis 

and modeling of dilatational terms in compressible turbulence," NASA CR 18195 

(1989). 

8 Zeman, 0., "Dilatation dissipation: The concept and application in modeling 

compressible mixing layers," Phys. Fluids A 2 (2), 178-188 (1990). 

9 Sarkar, S., Erlebacher, G. and Hussaini, M. Y., "Compressible homogeneous 

shear: simulation and modeling," NASA CR 189611 (1992). 

10 Viegas, J. R. and Rubesin, M. W., "A comparative study of several compressibility corrections to turbulence models applied to high-speed shear layers," AIAA 

paper 91-1783. 

11 Wilcox, D. C., "Progress in hypersonic turbulence modeling," AIAA paper 91-

1785. 

12 Speziale, C. G. and Sarkar, S., "Second-order closure models for supersonic turbulent flows," AIAA paper 91-0217. 

13 Papamoschou, D. and Roshko, A., "The compressible turbulent shear layer: an 

experimental study," J. Fluid Mech. 197, 453-477 (1988). 

14 Kim, S.-W. and Chen, C.-P., "A multiple-time-scale turbulence model based on 

variable partitioning of turbulent kinetic energy spectrum," NASA CR-179222 

(1987). 

15 Elliott, G. S. and Samimy, M., "Compressibility effects in free shear layers," Phys. 

Fluids A (1991) 

16 Samimy, M. and Elliott, G. S., "Effects of compressibility on the characteristics 

of free shear layers," Phys. Fluids A 28, 439-445 (1990). 

17 Liou, W. W. and Morris, P. J., "Weakly nonlinear models for turbulent mixing 

in a plane mixing layer," to appear in Phys. Fluids (1992). 

18 Liou, W. W., "A new energy transfer model for turbulent free shear flows," NASA 

TM (to appear) (1992). 

19 Strange, P. J. R. and Crighton, D. G., "Spinning modes on axisymmetric jets, 

Part I," J. Fluid Mech. 134, 231-245 (1983). 

20 Tennekes, H. and Lumley, J. L., A First Course in Turbulence, M.LT. Press, 

1972. 

21 Patel, R. P., "An experimental study of a plane mixing layer," AIAA Journal II, 

67-71 (1973). 

74 
Modeling of Turbulent Shear Flows 

22 Simmons, G. F., Differential Equations with Applications and Historical Notes, 

McGraw Hill, 1972. 

75 

Modeling of Turbulent Shear Flows 

22 Simmons, G. F., Differential Equations with Applications and Historical Notes, 

McGraw Hill, 1972. 

75 
1.2 

c 

1.0 --......, 

----~ ------ A* --1{-,_ " " 

., 

----. I ...... x - " " " ~~ 

~ 

, 

::::)'/b 0.8 " " * ~ , " " '\ 

0 , '--' x \ " ... ----. , \ ~IH X '\ 

"ej"ej 

0.6 , \ '--' '\ '\ 

------ A , ----. '\ '\. ., I .;., ~~ , \ ~ '\ \ ::::)'/b c , 

'\\ 0.4 ~u , 

\ '--' 

~IH "ej"ej C 

0.2 c c c 

c 

0.0 

0.0 0.5 1.0 1.5 2.0 

Figure 1. Variation of relative growth rate with convective Mach number, r = 0.1. - -, 

Present: without compressibility corrections; --, Present: with compressibility 

corrections; - --, KC; 0 , Papamoschou and Roshko; A , Goebel et al; ~ , 

Samimy and Elliott; * , Sullins et ali + ,Messersmith et ali X ,Chinzei et ali * , 

Ikawa and Kubota et al. 

76 

1.2 

c 

1.0 --......, 

----~ ------ A* --1{-,_ " " 

., 

----. I ...... x - " " " ~~ 

~ 

, 

::::)'/b 0.8 " " * ~ , " " '\ 

0 , '--' x \ " ... ----. , \ ~IH X '\ 

"ej"ej 

0.6 , \ '--' '\ '\ 

------ A , ----. '\ '\. ., I .;., ~~ , \ ~ '\ \ ::::)'/b c , 

'\\ 0.4 ~u , 

\ '--' 

~IH "ej"ej C 

0.2 c c c 

c 

0.0 

0.0 0.5 1.0 1.5 2.0 

Figure 1. Variation of relative growth rate with convective Mach number, r = 0.1. - -, 

Present: without compressibility corrections; --, Present: with compressibility 

corrections; - --, KC; 0 , Papamoschou and Roshko; A , Goebel et al; ~ , 

Samimy and Elliott; * , Sullins et ali + ,Messersmith et ali X ,Chinzei et ali * , 

Ikawa and Kubota et al. 

76 
00 

Q) 

en 

00 

Q) 

I-< 

~ 

en 

I-< 

~ 

Q) 

..q 

00 

00 

'"0 

....... 

0 

s:::= » 

Q) 

~ 

~ 

~ 

Q) 

~ 

0.014 

0.012 

0.010 

A... 

0.008 

0.006 

0.004 

0.002 

0.000 

0.0 

~ 

~ 

• 

0.5 

" " " " " '* " .. '\. ....... " .. ....... 

• 'A '* 

" 

1.0 1.5 2.0 

Figure 2. Variation of the peak Reynolds shear stress with convective Mach number. 0 

Present: r = 0.1, without compressibility corrections; 0 ,Present: r = 0.: 

with compressibility corrections; - * -, Present: r = 0.2, without compressibili1 

corrections;- + -, Present: r = 0.2, with compressibility corrections;- --, K( 

r = 0.1; • , Elliott and Samimy. 

77 

00 

Q) 

en 

00 

Q) 

I-< 

~ 

en 

I-< 

~ 

Q) 

..q 

00 

00 

'"0 

....... 

0 

s:::= » 

Q) 

~ 

~ 

~ 

Q) 

~ 

0.014 

0.012 

0.010 

A... 

0.008 

0.006 

0.004 

0.002 

0.000 

0.0 

~ 

~ 

• 

0.5 

" " " " " '* " .. '\. ....... " .. ....... 

• 'A '* 

" 

1.0 1.5 2.0 

Figure 2. Variation of the peak Reynolds shear stress with convective Mach number. 0 

Present: r = 0.1, without compressibility corrections; 0 ,Present: r = 0.: 

with compressibility corrections; - * -, Present: r = 0.2, without compressibili1 

corrections;- + -, Present: r = 0.2, with compressibility corrections;- --, K( 

r = 0.1; • , Elliott and Samimy. 

77 
* 

~ 

>"; 

~ 

...... 

u 

0 

....... 

~ 

:> 

~ 

ro 

~ 

S 

"d 

~ 

N ...... ....... 

ro 

S 

I-< 

0 

z 

1.2 

1.0 

0.8 

0.6 

0.4 

0.2 

0.0 • 

-0.2 

-1.0 -0.5 0.0 

Y - YO.5 

bw 

0.5 

• • 

1.0 

Figure 3. Comparison of mean velocity profiles for the shear layer in Case 1 of Elliott and 

Samimy (1991). --, Present; • , experiment. 

78 

----- ----

* 

~ 

>"; 

~ 

...... 

u 

0 

....... 

~ 

:> 

~ 

ro 

~ 

S 

"d 

~ 

N ...... ....... 

ro 

S 

I-< 

0 

z 

1.2 

1.0 

0.8 

0.6 

0.4 

0.2 

0.0 • 

-0.2 

-1.0 -0.5 0.0 

Y - YO.5 

bw 

0.5 

• • 

1.0 

Figure 3. Comparison of mean velocity profiles for the shear layer in Case 1 of Elliott and 

Samimy (1991). --, Present; • , experiment. 

78 

----- ----
0.010 

0.008 

en 

CIJ 0.006 Cl) 

~ 

+" CIJ 

~ 

cO 

Cl) 

...c: CIJ 

CIJ 0.004 "0 

-0 

~ 

>. 

Cl) 

~ 

"0 

Cl) 

0.002 N 

;.:::! 

cO 

S 

~ 

0 

Z 

0.000 • 

-0.002 

-1.0 

•• 

-0.5 

• • 

0.0 

y - YO.5 

Dw 

• 

• • 

0.5 1.0 

Figure 4. Comparison of Reynolds shear stress for the shear layer in Case 1 of Elliott and 

Samimy (1991). --, Present; • , experiment. 

79 

0.010 

0.008 

en 

CIJ 0.006 Cl) 

~ 

+" CIJ 

~ 

cO 

Cl) 

...c: CIJ 

CIJ 0.004 "0 

-0 

~ 

>. 

Cl) 

~ 

"0 

Cl) 

0.002 N 

;.:::! 

cO 

S 

~ 

0 

Z 

0.000 • 

-0.002 

-1.0 

•• 

-0.5 

• • 

0.0 

y - YO.5 

Dw 

• 

• • 

0.5 1.0 

Figure 4. Comparison of Reynolds shear stress for the shear layer in Case 1 of Elliott and 

Samimy (1991). --, Present; • , experiment. 

79 
1 .0 

0.8 

0.6 

0.4 

0.2 

0.0 • • 

-0.20 -0.10 0.00 0.1 0 0.20 

T) 

Figure 5. Mean velocity profiles. - - -, x = 3.82; - - , 4.43; - , 5.99; • , Patel9 . 

80 

1 .0 

0.8 

0.6 

0.4 

0.2 

0.0 • • 

-0.20 -0.10 0.00 0.1 0 0.20 

T) 

Figure 5. Mean velocity profiles. - - -, x = 3.82; - - , 4.43; - , 5.99; • , Patel9 . 

80 
0.15 

0.10 

0.05 

0.00 

0.0 100.0 200.0 300.0 400.0 

x/Bo 

Figure 6. Variation of the wave amplitude with streamwise distance . 

81 

0.15 

0.10 

0.05 

0.00 

0.0 100.0 200.0 300.0 400.0 

x/Bo 

Figure 6. Variation of the wave amplitude with streamwise distance . 

81 
Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Modeling of Near Wall Turbulence 

and Modeling of Bypass Transition 

z. Yang 

1. Motivation and Objective 

1) Modeling of the near wall turbulence. We aim to develop a second order closure for near wall turbulence. As a first step of this project, we try to develop a 

k - E model for near wall turbulence. We require the resulting model to be able to 

handle both near wall turbulence and turbulent flows away from the wall, computationally robust, and applicable for complex flow situations, flow with separation, 

for example. 

2) Modeling of the bypass transition. We aim to develop a bypass transition 

model which contains the effect of intermittency. Thus, the model can be used 

for both the transitional boundary layers and the turbulent boundary layers. We 

require the resulting model to give a good prediction of momentum and heat transfer 

within the transitional boundary and a good prediction of the effect of freestream 

turbulence on transitional boundary layers. 

2. Work Accomplished 

In the past year, progress has been made in both topics mentioned above (i.e., 

modeling of the near wall turbulence and modeling of the bypass transition). In the 

paragraphs below, these two topics will be reported separately. 

2.1 Modeling of Near Wall Turbulence 

Because of the wide range of scales involved in a turbulent flow, DNS (direct 

numerical Simulation) is limited to flows of moderate Reynolds number and simple 

geometry. Turbulence modeling is the only viable approach for the calculation of 

turbulent flows of engineering interest. In turbulence modeling, the k-E model is the 

most widely used model in engineering calculations. The Standard k - E Model1.2 

was devised for high Reynolds number turbulent flows and is traditionally used 

in conjunction with a wall function when it is applied to wall bounded turbulent 

flows. However universal wall functions do not exist in complex flows and it is thus 

necessary to develop a form of k - E model equations which can be integrated down 

to the wall. 

Jones and Launder3 were the first to propose a low Reynolds number k - E model 

for near wall turbulence, which was then followed by a number of similar k - E 

models. A critical evaluation ofthe pre-1985 models was made by Patel et a1. 4 . More 

recently proposed models can be found in Shih5 and Lang and Shih6 . Three major 

deficiencies can be pointed out about existing k-E models. (Some of the models may 

have only one or two of the three deficiencies.) First, a near wall pseudo-dissipation 

rate was introduced to remove the singularity in the dissipation equation at the 

83 Preceding Page Blank 

Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Modeling of Near Wall Turbulence 

and Modeling of Bypass Transition 

z. Yang 

1. Motivation and Objective 

1) Modeling of the near wall turbulence. We aim to develop a second order closure for near wall turbulence. As a first step of this project, we try to develop a 

k - E model for near wall turbulence. We require the resulting model to be able to 

handle both near wall turbulence and turbulent flows away from the wall, computationally robust, and applicable for complex flow situations, flow with separation, 

for example. 

2) Modeling of the bypass transition. We aim to develop a bypass transition 

model which contains the effect of intermittency. Thus, the model can be used 

for both the transitional boundary layers and the turbulent boundary layers. We 

require the resulting model to give a good prediction of momentum and heat transfer 

within the transitional boundary and a good prediction of the effect of freestream 

turbulence on transitional boundary layers. 

2. Work Accomplished 

In the past year, progress has been made in both topics mentioned above (i.e., 

modeling of the near wall turbulence and modeling of the bypass transition). In the 

paragraphs below, these two topics will be reported separately. 

2.1 Modeling of Near Wall Turbulence 

Because of the wide range of scales involved in a turbulent flow, DNS (direct 

numerical Simulation) is limited to flows of moderate Reynolds number and simple 

geometry. Turbulence modeling is the only viable approach for the calculation of 

turbulent flows of engineering interest. In turbulence modeling, the k-E model is the 

most widely used model in engineering calculations. The Standard k - E Model1.2 

was devised for high Reynolds number turbulent flows and is traditionally used 

in conjunction with a wall function when it is applied to wall bounded turbulent 

flows. However universal wall functions do not exist in complex flows and it is thus 

necessary to develop a form of k - E model equations which can be integrated down 

to the wall. 

Jones and Launder3 were the first to propose a low Reynolds number k - E model 

for near wall turbulence, which was then followed by a number of similar k - E 

models. A critical evaluation ofthe pre-1985 models was made by Patel et a1. 4 . More 

recently proposed models can be found in Shih5 and Lang and Shih6 . Three major 

deficiencies can be pointed out about existing k-E models. (Some of the models may 

have only one or two of the three deficiencies.) First, a near wall pseudo-dissipation 

rate was introduced to remove the singularity in the dissipation equation at the 

83 Preceding Page Blank 
Z. Yang 

wall. The definition of the near wall pseudo-dissipation rate was quite arbitrary. 

Second, the model constants were different from those of the Standard k - E Model, 

making the near wall models less capable of handling flows containing both high 

Reynolds number turbulence and near wall turbulence, which often occurs for real 

flow situations. Patel et al. 4 put as the first criterion the ability of the near wall 

models to predict turbulent free shear flows. Third, the variable y+ is used in the 

damping function f Ii- of the eddy viscosity formulae. Since the definition of y+ 

involves U T , the friction velocity, any model containing y+ cannot be used in flows 

with separation. 

Effort is made to propose a new k-E model for near turbulence which is free of the 

three deficiencies mentioned above. In this model, k 1/ 2 is chosen as the turbulent 

velocity scale. The time scale is bounded from below by the Kolmogorov time scale. 

The dissipation equation is reformulated using this time scale and no singularity 

exists at the wall. Thus, it is no longer necessary to introduce the near wall pseudodissipation rate. The model constants used are the same as in the Standard k-E 

Model. Thus, the proposed model will be also suitable for flows away from the wall. 

An earlier version of the model, which contains y+ in the damping function, was 

proposed and reported in Yang and Shih 7 . The model is now improved by using 

Ry = k 1 / 2Y/1/ instead of y+ in the damping function. Hence, the present model can 

be used for flows with separation. 

In the present model, the eddy viscosity is given by 

(1) 

where the time scale T is written as 

(2) 

The first part is the time scale conventionally used for high Reynolds number turbulent flows and the second part is the Kolmogorov time scale. A way from the wall, 

the first part is much large than the second part while near the wall the second part 

dominates, giving the Kolmogorov time scale as the turbulent time scale at the wall. 

The time scale given is bounded from below by the Kolmogorov time scale and is 

always positive. 

The damping function f Ii- is given by 

(3) 

where al = 1.5 x 10-4 , a3 = 5.0 x 10-7 , a5 = 1.0 x 10-10 . The damping function 

is chosen such that the shear stress has the correct near wall asymptotic behavior. 

Away from the wall, f Ii- approaches one as required. 

The modeled transport equations for k and E are 

. 1/T k + U· k . = [(1/ + -)k .J . - < U'u' > U· . - E ] ,] ,] ,] t] t,] , 

(/1.; 

(4) 

84 

Z. Yang 

wall. The definition of the near wall pseudo-dissipation rate was quite arbitrary. 

Second, the model constants were different from those of the Standard k - E Model, 

making the near wall models less capable of handling flows containing both high 

Reynolds number turbulence and near wall turbulence, which often occurs for real 

flow situations. Patel et al. 4 put as the first criterion the ability of the near wall 

models to predict turbulent free shear flows. Third, the variable y+ is used in the 

damping function f Ii- of the eddy viscosity formulae. Since the definition of y+ 

involves U T , the friction velocity, any model containing y+ cannot be used in flows 

with separation. 

Effort is made to propose a new k-E model for near turbulence which is free of the 

three deficiencies mentioned above. In this model, k 1/ 2 is chosen as the turbulent 

velocity scale. The time scale is bounded from below by the Kolmogorov time scale. 

The dissipation equation is reformulated using this time scale and no singularity 

exists at the wall. Thus, it is no longer necessary to introduce the near wall pseudodissipation rate. The model constants used are the same as in the Standard k-E 

Model. Thus, the proposed model will be also suitable for flows away from the wall. 

An earlier version of the model, which contains y+ in the damping function, was 

proposed and reported in Yang and Shih 7 . The model is now improved by using 

Ry = k 1 / 2Y/1/ instead of y+ in the damping function. Hence, the present model can 

be used for flows with separation. 

In the present model, the eddy viscosity is given by 

(1) 

where the time scale T is written as 

(2) 

The first part is the time scale conventionally used for high Reynolds number turbulent flows and the second part is the Kolmogorov time scale. A way from the wall, 

the first part is much large than the second part while near the wall the second part 

dominates, giving the Kolmogorov time scale as the turbulent time scale at the wall. 

The time scale given is bounded from below by the Kolmogorov time scale and is 

always positive. 

The damping function f Ii- is given by 

(3) 

where al = 1.5 x 10-4 , a3 = 5.0 x 10-7 , a5 = 1.0 x 10-10 . The damping function 

is chosen such that the shear stress has the correct near wall asymptotic behavior. 

Away from the wall, f Ii- approaches one as required. 

The modeled transport equations for k and E are 

. 1/T k + U· k . = [(1/ + -)k .J . - < U'u' > U· . - E ] ,] ,] ,] t] t,] , 

(/1.; 

(4) 

84 
Modeling of Near Wall Turbulence and Modeling of Bypass Transition 

£. + Ujf,j = [(1/ + :: )f,jl,j + (-CIt: < U ·iUj > Ui,j - C2t: f)/T + 1/1/TUj,j/,;Ui,j/';' (5) 

Because T is always positive, the dissipation rate equation does not have a singularity at the wall. 

Turbulent channel flows and flat plate boundary layer flows at different Reynolds 

numbers were calculated using the proposed model. The mean velocity, turbulent kinetic energy, turbulent shear stress and turbulent dissipation rate for turbulent channel flow at ReT = 395 and turbulent flat plate boundary layer flow at 

Reo = 1410 are shown in Fig 1 and Fig 2, respectively. DNS data for these cases 

are shown for comparison. Also shown are the predictions using the Jones-Launder 

model and the k - f model proposed by Chien8 . These two models are chosen 

because the Jones-Launder model is the first k - E model for near wall turbulence 

while Chien's model is known to perform quite well for turbulent boundary layer 

flows. Overall, the proposed model is found to give a better prediction. Calculations were also made for turbulent flat plate boundary layers at larger Reynolds 

numbers, turbulent boundary layers with pressure gradient (favorable pressure gradient, adverse pressure gradient, and increasingly adverse pressure.) The results of 

these computation and the comparisons with the available experimental data can 

be 'found in Yang and Shih9 . 

2.2 Modeling of Bypass Transition 

In a quiescent environment, transition is preceeded by the amplification of TollmienSchlichting waves. These waves eventually break down, giving rise to tu'rbulent 

spots, which can be viewed as the onset of transition. In an environment with high 

freestream turbulence, say the flow passing over a turbine blade, turbulent spots 

are formed due to the transport of turbulence from the freestream to the boundary 

layer rather than the T-S wave amplification. This type of transition is called bypass 

transition. Accurate prediction of bypass transitional boundary layers is very important for internal fluid mechanics because a significant proportion of the turbine 

blade is in the a transitional boundary layer region. Furthermore, the performance 

and the life of a turbine are directly related to the peak vales of the momentum and 

heat transfer both of which occur in the transitional boundary layer. 

Priddin10 was the first to notice that the low Reynolds number two equation 

models have the potential to predict transitional flows under the influence of the 

freestream turbulence. This is probably due to the fact that the generation of 

turbulent spots in a boundary layer is a random process and the flow is almost 

fully developed turbulent within a turbulent spot. A detailed calculation procedure 

was given by Rodi and Scheuererll , in which the Lam & Bremhorst low Reynolds 

number k - E model was used. More recently, a comparative study of the performance of existing low Reynolds number k - E models in predicting laminar-turbulent 

transition was made by Fujisawa12 . 

While the low Reynolds number k - E models could mimic transition, the quantitative predictions do not compare very well with the experimental data. This is 

due to the fact that all these low Reynolds number k - E models were originally 

proposed for fully developed turbulent flows and did not take into consideration the 

85 

Modeling of Near Wall Turbulence and Modeling of Bypass Transition 

£. + Ujf,j = [(1/ + :: )f,jl,j + (-CIt: < U ·iUj > Ui,j - C2t: f)/T + 1/1/TUj,j/,;Ui,j/';' (5) 

Because T is always positive, the dissipation rate equation does not have a singularity at the wall. 

Turbulent channel flows and flat plate boundary layer flows at different Reynolds 

numbers were calculated using the proposed model. The mean velocity, turbulent kinetic energy, turbulent shear stress and turbulent dissipation rate for turbulent channel flow at ReT = 395 and turbulent flat plate boundary layer flow at 

Reo = 1410 are shown in Fig 1 and Fig 2, respectively. DNS data for these cases 

are shown for comparison. Also shown are the predictions using the Jones-Launder 

model and the k - f model proposed by Chien8 . These two models are chosen 

because the Jones-Launder model is the first k - E model for near wall turbulence 

while Chien's model is known to perform quite well for turbulent boundary layer 

flows. Overall, the proposed model is found to give a better prediction. Calculations were also made for turbulent flat plate boundary layers at larger Reynolds 

numbers, turbulent boundary layers with pressure gradient (favorable pressure gradient, adverse pressure gradient, and increasingly adverse pressure.) The results of 

these computation and the comparisons with the available experimental data can 

be 'found in Yang and Shih9 . 

2.2 Modeling of Bypass Transition 

In a quiescent environment, transition is preceeded by the amplification of TollmienSchlichting waves. These waves eventually break down, giving rise to tu'rbulent 

spots, which can be viewed as the onset of transition. In an environment with high 

freestream turbulence, say the flow passing over a turbine blade, turbulent spots 

are formed due to the transport of turbulence from the freestream to the boundary 

layer rather than the T-S wave amplification. This type of transition is called bypass 

transition. Accurate prediction of bypass transitional boundary layers is very important for internal fluid mechanics because a significant proportion of the turbine 

blade is in the a transitional boundary layer region. Furthermore, the performance 

and the life of a turbine are directly related to the peak vales of the momentum and 

heat transfer both of which occur in the transitional boundary layer. 

Priddin10 was the first to notice that the low Reynolds number two equation 

models have the potential to predict transitional flows under the influence of the 

freestream turbulence. This is probably due to the fact that the generation of 

turbulent spots in a boundary layer is a random process and the flow is almost 

fully developed turbulent within a turbulent spot. A detailed calculation procedure 

was given by Rodi and Scheuererll , in which the Lam & Bremhorst low Reynolds 

number k - E model was used. More recently, a comparative study of the performance of existing low Reynolds number k - E models in predicting laminar-turbulent 

transition was made by Fujisawa12 . 

While the low Reynolds number k - E models could mimic transition, the quantitative predictions do not compare very well with the experimental data. This is 

due to the fact that all these low Reynolds number k - E models were originally 

proposed for fully developed turbulent flows and did not take into consideration the 

85 
z. Yang 

distinct feature of a transitional boundary layer - intermittency. The intermittency 

of a transitional boundary layer is measured by the intermittency factor which can 

be viewed as the percentage time a transitional boundary layer is in the turbulent 

state due to the passing of a turbulent spot. 

We propose a model for the calculation of transitional boundary layers, which 

takes the effect of intermittency into consideration. The model is based on the k - E 

model for near wall turbulence we have stated above. The effect of intermittency 

is introduced through the following argument: since the percentage of time that a 

transitional boundary layer is turbulent is measured by the intermittency factor, 

a model for transitional boundary layers could be constructed from a model for 

turbulent boundary layers by multiplying all the terms due to turbulence mechanism 

by a weighting factor I, which is linearly related to the intermittency factor. Thus, 

the governing equations for the flat plate transitional boundary layers are, after 

using the boundary layer approximation and the eddy viscosity assumption, 

oU oV _ 0 

ox + oy - , (6) 

U- + V- = - (v +,VT)- , oU oU 0 [ OU] 

ox oy oy oy (7) 

u

ok + v

ok =!..- [(V+IVT)Ok ] +IVT(oU)2 -,E, oX oy oy (Jk oy oy (8) 

U OE +V OE =.i. [(V+IVT)OE ] +, [C1€VT(OU)2 _ C2€C] ~+IVVT(OU)2, (9) 

ox oy oy (J € oy oy T oy 

where x, yare the coordinates along and normal to the plate and U, V are the mean 

velocities in the x , y directions, respectively. 

In order to close the above equations, an expression for the weighting factor I 

is needed. The ~eighting factor is aSsumed to be related to both the freest ream 

turbulent level and the intermittency factor of the boundary layer. The intermittency factor is assumed to be determined by the local state of the boundary layer. 

We use H, the shape factor, to characterize the local state of the boundary layer 

since both the intermittency factor and the shape factor change monotonically from 

the laminar boundary layer to the turbulent boundary layer. Experimental results 

by Abu-Ghannam and Shaw13 are used as a guide to construct this function. The 

weighting factor is also assumed to change in the y direction in such a way that 

outside the boundary layer, the weighting factor is one since the freestream is governed by decaying turbulence. The final expression for the weighting factor is given 

in Yang and Shih14. 

The above system of parabolic equations need to be supplymented by boundary 

conditions at the wall and at the freest ream and by initial conditions at the starting 

point of the calculation. At the wall 

U = V = k = 0, 

86 

z. Yang 

distinct feature of a transitional boundary layer - intermittency. The intermittency 

of a transitional boundary layer is measured by the intermittency factor which can 

be viewed as the percentage time a transitional boundary layer is in the turbulent 

state due to the passing of a turbulent spot. 

We propose a model for the calculation of transitional boundary layers, which 

takes the effect of intermittency into consideration. The model is based on the k - E 

model for near wall turbulence we have stated above. The effect of intermittency 

is introduced through the following argument: since the percentage of time that a 

transitional boundary layer is turbulent is measured by the intermittency factor, 

a model for transitional boundary layers could be constructed from a model for 

turbulent boundary layers by multiplying all the terms due to turbulence mechanism 

by a weighting factor I, which is linearly related to the intermittency factor. Thus, 

the governing equations for the flat plate transitional boundary layers are, after 

using the boundary layer approximation and the eddy viscosity assumption, 

oU oV _ 0 

ox + oy - , (6) 

U- + V- = - (v +,VT)- , oU oU 0 [ OU] 

ox oy oy oy (7) 

u

ok + v

ok =!..- [(V+IVT)Ok ] +IVT(oU)2 -,E, oX oy oy (Jk oy oy (8) 

U OE +V OE =.i. [(V+IVT)OE ] +, [C1€VT(OU)2 _ C2€C] ~+IVVT(OU)2, (9) 

ox oy oy (J € oy oy T oy 

where x, yare the coordinates along and normal to the plate and U, V are the mean 

velocities in the x , y directions, respectively. 

In order to close the above equations, an expression for the weighting factor I 

is needed. The ~eighting factor is aSsumed to be related to both the freest ream 

turbulent level and the intermittency factor of the boundary layer. The intermittency factor is assumed to be determined by the local state of the boundary layer. 

We use H, the shape factor, to characterize the local state of the boundary layer 

since both the intermittency factor and the shape factor change monotonically from 

the laminar boundary layer to the turbulent boundary layer. Experimental results 

by Abu-Ghannam and Shaw13 are used as a guide to construct this function. The 

weighting factor is also assumed to change in the y direction in such a way that 

outside the boundary layer, the weighting factor is one since the freestream is governed by decaying turbulence. The final expression for the weighting factor is given 

in Yang and Shih14. 

The above system of parabolic equations need to be supplymented by boundary 

conditions at the wall and at the freest ream and by initial conditions at the starting 

point of the calculation. At the wall 

U = V = k = 0, 

86 
Modeling of Near Wall Turbulence and Modeling of Bypass Transition 

(10) 

The bounda~y condition for E was obtained by applying the equation for turbulent 

kinetic energy down to the wall. 

At the edge of the boundary layer, the flow variables are given by the freestream 

values, i.e. 

(11) 

The ke, Ee in the above are found from the transport equations for k and E, with 

the condition that the gradient of the flow variables in the y direction vanishes as 

the free stream is approached. Thus, 

(12) 

(13) 

keO and EeO (the values of ke and Ee at the leading edge, for example) are needed. 

keD is obtained from the experiment, and EeO is determined in such a way that the 

resulting ke ( x) profile agrees with the experiment. 

One of the issues in the calculation of transitional boundary layers through the 

low Reynolds number k - E models is the prescription of the initial profiles for 

the turbulence kinetic energy and its dissipation rate, the later of which could not 

be found from the experiment directly. An expression for the initial profiles were 

given in Rodi and Scheuererll . However, computations by Yang and Shih15 which 

tested the effect of the initial conditions on the transition prediction found, in 

agreement with the findings of Patankar and Schmit16 , that the predicted onset of 

the transition is sensitive to the initial profiles. This sensitivity of the results to the 

initial conditions suggests that the only place where the initial conditions could be 

specified unambiguously is at the leading edge. At the leading edge, tIle turbulent 

kinetic energy and its dissipation rate take constant profiles, the values of which are 

determined by the law for the decaying turbulence. 

With the initial co'nditions given at the leading edge and the boundary conditions 

given above, the solutions are marched downstream. Flat plate boundary layers with 

free stream turbulence levels of 3% (Case T3A) and 6% (Case T3B) respectively 

were calculated using the present model. These are the benchmark cases in an 

ongoing project coordinated by Savill17 , testing the capability of turbulence models 

in predicting transitional flows. Fig. 3 shows the variation of skin friction coefficient 

cf against Rex· Results from the experiment is shown for comparison. In addition, 

the prediction of the Launder-Sharma model is also shown in the figure because it 

was reported that among the lower Reynolds number k - E models, the LaunderSharma model performs best for transitional boundary layers. It is clear that the 

present model gives a better prediction. Other features in the transitional boundary 

layers and the calculations of the transitional boundary layers with other levels of 

freest ream turbulence can be found in Ref. 14. 

87 

Modeling of Near Wall Turbulence and Modeling of Bypass Transition 

(10) 

The bounda~y condition for E was obtained by applying the equation for turbulent 

kinetic energy down to the wall. 

At the edge of the boundary layer, the flow variables are given by the freestream 

values, i.e. 

(11) 

The ke, Ee in the above are found from the transport equations for k and E, with 

the condition that the gradient of the flow variables in the y direction vanishes as 

the free stream is approached. Thus, 

(12) 

(13) 

keO and EeO (the values of ke and Ee at the leading edge, for example) are needed. 

keD is obtained from the experiment, and EeO is determined in such a way that the 

resulting ke ( x) profile agrees with the experiment. 

One of the issues in the calculation of transitional boundary layers through the 

low Reynolds number k - E models is the prescription of the initial profiles for 

the turbulence kinetic energy and its dissipation rate, the later of which could not 

be found from the experiment directly. An expression for the initial profiles were 

given in Rodi and Scheuererll . However, computations by Yang and Shih15 which 

tested the effect of the initial conditions on the transition prediction found, in 

agreement with the findings of Patankar and Schmit16 , that the predicted onset of 

the transition is sensitive to the initial profiles. This sensitivity of the results to the 

initial conditions suggests that the only place where the initial conditions could be 

specified unambiguously is at the leading edge. At the leading edge, tIle turbulent 

kinetic energy and its dissipation rate take constant profiles, the values of which are 

determined by the law for the decaying turbulence. 

With the initial co'nditions given at the leading edge and the boundary conditions 

given above, the solutions are marched downstream. Flat plate boundary layers with 

free stream turbulence levels of 3% (Case T3A) and 6% (Case T3B) respectively 

were calculated using the present model. These are the benchmark cases in an 

ongoing project coordinated by Savill17 , testing the capability of turbulence models 

in predicting transitional flows. Fig. 3 shows the variation of skin friction coefficient 

cf against Rex· Results from the experiment is shown for comparison. In addition, 

the prediction of the Launder-Sharma model is also shown in the figure because it 

was reported that among the lower Reynolds number k - E models, the LaunderSharma model performs best for transitional boundary layers. It is clear that the 

present model gives a better prediction. Other features in the transitional boundary 

layers and the calculations of the transitional boundary layers with other levels of 

freest ream turbulence can be found in Ref. 14. 

87 
Z. Yang 

3. Future Plans 

1) Modeling of the near wall turbulence. 

The proposed k - E model is only tested for simple parabolic flows so far. Because 

of the form of the model equations, the proposed model can be used in complex 

flow situations, flow with separation for example. The performance of the proposed 

model in those situations will be tested. 

We will work on the second order closure for near wall turbulence. In particular, 

we will be looking at the effect of the mean flow inhomogeneity on the pressure 

strain correlation. We are hoping to represent this effect rationally, so that the 

ad hoc damping functions currently being used in all the near wall second order 

closures can be avoided. 

2) Modeling of bypass transition. 

We will apply the proposed model to transitional boundary layers with pressure 

gradient and curvature. We will also extend the model to thermal boundary layers. 

4. References 

1 Launder, B.E. & Spalding, D.B., "The Numerical Computation of Turbulent 

Flow," . Computer Methods in Applied Mechanics and Engineering 3, 269-289 

(1974). 

2 Rodi, W., Turbulence Models and Their Application in Hydraulics, Book Pub. of 

International Association for Hydraulic Research, Delft, the Netherlands, 1980. 

3 Jones, W.P., and Launder, B.E., "The Calculation of Low-Reynolds Number 

Phenomena with a Two-Equation Model of Turbulence," International Journal 

of Heat and Mass Transfer 16, 1119-1130 (1973). 

4 Patel, V.C., Rodi, W., and Scheuerer, G., "Turbulence Models for Near-Wall and 

Low-Reynolds Number Flows: a Review," AIAA Journal 23, 1308-1319 (1985). 

5 Shih, T.H., "An Improved k - E Model for Near-Wall Turbulence and Comparison 

with Direct Numerical Simulation," NASA TM 103211 (1990). 

6 Lang, N.J. and Shih, T.H., "A Critical Comparison of Two Equation Turbulence 

Models," NASA TM 105237 (1991). 

7 Yang, Z. and Shih, T.R., "A k - E modeling of near wall turbulence," Proc. 4th 

International Symposium on CFD, 1305-1310 (1991). 

8 Chien, K.Y., "Predictions of Channel and Boundary Layer Flow with a LowReynolds-Number Turbulence Model," AIAA Journal 20, 33-38 (1982). 

9 Yang, Z. & Shih, T.H., "A new time scale based k - E model for near wall 

turbulence," submitted for publication (1992). 

10 Priddin, C.H., The behaviour of the turbulent boundary layer on a curved porous 

wall, PhD Thesis, Imperial College, London, 1975. 

11 Rodi, W. and Scheuerer, S., "Calculation of laminar-turbulence boundary layer 

transition on turbine blades," Proc. 65th AGARD-REP Symposium on Heat 

Transfer and Cooling in Gas Turbine, Bergen, Norway, 18-1 to 18-12 (1985). 

88 

Z. Yang 

3. Future Plans 

1) Modeling of the near wall turbulence. 

The proposed k - E model is only tested for simple parabolic flows so far. Because 

of the form of the model equations, the proposed model can be used in complex 

flow situations, flow with separation for example. The performance of the proposed 

model in those situations will be tested. 

We will work on the second order closure for near wall turbulence. In particular, 

we will be looking at the effect of the mean flow inhomogeneity on the pressure 

strain correlation. We are hoping to represent this effect rationally, so that the 

ad hoc damping functions currently being used in all the near wall second order 

closures can be avoided. 

2) Modeling of bypass transition. 

We will apply the proposed model to transitional boundary layers with pressure 

gradient and curvature. We will also extend the model to thermal boundary layers. 

4. References 

1 Launder, B.E. & Spalding, D.B., "The Numerical Computation of Turbulent 

Flow," . Computer Methods in Applied Mechanics and Engineering 3, 269-289 

(1974). 

2 Rodi, W., Turbulence Models and Their Application in Hydraulics, Book Pub. of 

International Association for Hydraulic Research, Delft, the Netherlands, 1980. 

3 Jones, W.P., and Launder, B.E., "The Calculation of Low-Reynolds Number 

Phenomena with a Two-Equation Model of Turbulence," International Journal 

of Heat and Mass Transfer 16, 1119-1130 (1973). 

4 Patel, V.C., Rodi, W., and Scheuerer, G., "Turbulence Models for Near-Wall and 

Low-Reynolds Number Flows: a Review," AIAA Journal 23, 1308-1319 (1985). 

5 Shih, T.H., "An Improved k - E Model for Near-Wall Turbulence and Comparison 

with Direct Numerical Simulation," NASA TM 103211 (1990). 

6 Lang, N.J. and Shih, T.H., "A Critical Comparison of Two Equation Turbulence 

Models," NASA TM 105237 (1991). 

7 Yang, Z. and Shih, T.R., "A k - E modeling of near wall turbulence," Proc. 4th 

International Symposium on CFD, 1305-1310 (1991). 

8 Chien, K.Y., "Predictions of Channel and Boundary Layer Flow with a LowReynolds-Number Turbulence Model," AIAA Journal 20, 33-38 (1982). 

9 Yang, Z. & Shih, T.H., "A new time scale based k - E model for near wall 

turbulence," submitted for publication (1992). 

10 Priddin, C.H., The behaviour of the turbulent boundary layer on a curved porous 

wall, PhD Thesis, Imperial College, London, 1975. 

11 Rodi, W. and Scheuerer, S., "Calculation of laminar-turbulence boundary layer 

transition on turbine blades," Proc. 65th AGARD-REP Symposium on Heat 

Transfer and Cooling in Gas Turbine, Bergen, Norway, 18-1 to 18-12 (1985). 

88 
Modeling of Near Wall Turbulence and Modeling of Bypass Transition 

12 Fujisawa, N., "Calculation of transitional boundary layers with a refined low 

Reynolds number version of a k - E model of turbulence," Proc. Engineering 

Thrbulence Modeling and Experiment, 23-32 (1990). 

13 Abu-Ghannam, B.J. & Shaw, R., "Natural transition of boundary layers - the 

effects of turbulence, pressure gradient, and flow history," Journal of Mechanical 

Engineering Science 22, 213-228 (1980). 

14 Yang, Z. & Shih, T.R., "A modeling of transitional boundary layers," In preparation (1992). 

15 Yang, Z. and Shih, T.R., "A k - € calculation of transitional boundary layers," 

Transition and Turbulence, Springer-Verlag, 1992. 

16 Schmidt, R.C. and Patankar, S.V., "Two-equation low-Reynolds number turbulence modeling of transitional boundary layer flows characteristic of gas turbine 

blades," NASA CR-4145 (1988). 

17 Savill, A.M., "Thrbulence model predictions for transition under free-stream turbulence," RAeS Transition and Boundary Layer Conference, Cambridge, England 

(1991). 

89 

Modeling of Near Wall Turbulence and Modeling of Bypass Transition 

12 Fujisawa, N., "Calculation of transitional boundary layers with a refined low 

Reynolds number version of a k - E model of turbulence," Proc. Engineering 

Thrbulence Modeling and Experiment, 23-32 (1990). 

13 Abu-Ghannam, B.J. & Shaw, R., "Natural transition of boundary layers - the 

effects of turbulence, pressure gradient, and flow history," Journal of Mechanical 

Engineering Science 22, 213-228 (1980). 

14 Yang, Z. & Shih, T.R., "A modeling of transitional boundary layers," In preparation (1992). 

15 Yang, Z. and Shih, T.R., "A k - € calculation of transitional boundary layers," 

Transition and Turbulence, Springer-Verlag, 1992. 

16 Schmidt, R.C. and Patankar, S.V., "Two-equation low-Reynolds number turbulence modeling of transitional boundary layer flows characteristic of gas turbine 

blades," NASA CR-4145 (1988). 

17 Savill, A.M., "Thrbulence model predictions for transition under free-stream turbulence," RAeS Transition and Boundary Layer Conference, Cambridge, England 

(1991). 

89 
30.0 

u+ 

20.0 

10.0 

0.0 

10°.0 

Z. Yang 

o DNS 

Jones-Launder 

Chien 

Present 

10,·0 

y+ 

Fig la: Mean velocity profile for channel flow at Rer = 395. 

5.0 

'\. 

4.0 

k+ I 

3.0 I - /' I " 

r I, 

2.0 

1.0 

\ 

" " '\. 

"-

"'-

"'-

"'-

o DNS 

Jones-Launder 

Chien 

Present 

0.0 ~--~----~----~----~--~~--~----~--~ 

0.0 100.0 200.0 + 300.0 

y 

400.0 

Fig 1 b: Turbulent energy profile for channel flow at ReT = 395. 

90 

30.0 

u+ 

20.0 

10.0 

0.0 

10°.0 

Z. Yang 

o DNS 

Jones-Launder 

Chien 

Present 

10,·0 

y+ 

Fig la: Mean velocity profile for channel flow at Rer = 395. 

5.0 

'\. 

4.0 

k+ I 

3.0 I - /' I " 

r I, 

2.0 

1.0 

\ 

" " '\. 

"-

"'-

"'-

"'-

o DNS 

Jones-Launder 

Chien 

Present 

0.0 ~--~----~----~----~--~~--~----~--~ 

0.0 100.0 200.0 + 300.0 

y 

400.0 

Fig 1 b: Turbulent energy profile for channel flow at ReT = 395. 

90 
I 

I 

I 

 

 

 

 

Mode ling of Near Wall Turbulence and Modeling of Bypa.$.$ Tran3ition 

1.0 .---~-----.----~----.---~-----.----~----. 

0.8 

-uv+ 

0.6 

0.4 

0.2 

o DNS 

Jones-Launder 

Chien 

Present 

0.0 ~--~~--~----~----~--~~--~----~--~ 

0.0 100.0 200.0 y+ 300.0 

Fig Ic: Sh~ar stress profile for channel flow at ReT = 395. 

0.3 

0.2 

0.1 

0- \ 

I ~, 

I / '$ ~ 

'I ' 

II 

I / 

) 

0.0 

0.0 25.0 50.0 

o DNS 

Jones-Launder 

Chien 

Present 

---

75.0 

400.0 

100.0 

Fig Id: Dissipation rate profile for channel flow at ReT = 395. 

91 

I 

I 

I 

 

 

 

 

Mode ling of Near Wall Turbulence and Modeling of Bypa.$.$ Tran3ition 

1.0 .---~-----.----~----.---~-----.----~----. 

0.8 

-uv+ 

0.6 

0.4 

0.2 

o DNS 

Jones-Launder 

Chien 

Present 

0.0 ~--~~--~----~----~--~~--~----~--~ 

0.0 100.0 200.0 y+ 300.0 

Fig Ic: Sh~ar stress profile for channel flow at ReT = 395. 

0.3 

0.2 

0.1 

0- \ 

I ~, 

I / '$ ~ 

'I ' 

II 

I / 

) 

0.0 

0.0 25.0 50.0 

o DNS 

Jones-Launder 

Chien 

Present 

---

75.0 

400.0 

100.0 

Fig Id: Dissipation rate profile for channel flow at ReT = 395. 

91 
._

u+ 

20.0 

10.0 

0.0 

10°.0 

z. Yang 

o DNS 

Jones-Launder 

Chien 

Present 

101.

0 

y+ 

Fig 2a: Mean velocity profile for channel flow at Reo = 1410. 

5.0 

\ 

\ o DNS 

4.0 0\ Jones-Launder 

k+ 0'\ Chien 0'\ ~ Present 3.0 - ~~ - "-

-~ 

'" 2.0 ~ 

::::-..0 ~ 

~ 

~ 

1.0 ~~, ~, 

~ 

0 0.0 

0.0 250.0 500.0 

y+ 

10+.0 

\ 

750.0 

Fig 2b: Turbulent energy profile for channel flow at Reo = 1410. 

92 

._

u+ 

20.0 

10.0 

0.0 

10°.0 

z. Yang 

o DNS 

Jones-Launder 

Chien 

Present 

101.

0 

y+ 

Fig 2a: Mean velocity profile for channel flow at Reo = 1410. 

5.0 

\ 

\ o DNS 

4.0 0\ Jones-Launder 

k+ 0'\ Chien 0'\ ~ Present 3.0 - ~~ - "-

-~ 

'" 2.0 ~ 

::::-..0 ~ 

~ 

~ 

1.0 ~~, ~, 

~ 

0 0.0 

0.0 250.0 500.0 

y+ 

10+.0 

\ 

750.0 

Fig 2b: Turbulent energy profile for channel flow at Reo = 1410. 

92 
1- ---

M odeling of Near Wall Turbulence and Modeling of Bypa33 TranJition 

1.0 

0.8 

-uv+ 

0.6 

0.4 

0.2 

, 

, 

o DNS 

Jones-Launder 

Chien 

Present 

0.0 .-------------~------------~--~~------~ 

0.0 250.0 500.0 

Fig 2c: Shear stress profile for channel :Bow at Re8 = 1410. 

0.3 

0.2 

0.1 

0.0 

0.0 25.0 50.0 

o DNS 

Jones-Launder 

Chien 

Present 

---

y+ 

75.0 

750.0 

100.0 

Fig 2d: Dissipation rate profile for channel :Bow at Re8 = 1410. 

93 

1- ---

M odeling of Near Wall Turbulence and Modeling of Bypa33 TranJition 

1.0 

0.8 

-uv+ 

0.6 

0.4 

0.2 

, 

, 

o DNS 

Jones-Launder 

Chien 

Present 

0.0 .-------------~------------~--~~------~ 

0.0 250.0 500.0 

Fig 2c: Shear stress profile for channel :Bow at Re8 = 1410. 

0.3 

0.2 

0.1 

0.0 

0.0 25.0 50.0 

o DNS 

Jones-Launder 

Chien 

Present 

---

y+ 

75.0 

750.0 

100.0 

Fig 2d: Dissipation rate profile for channel :Bow at Re8 = 1410. 

93 
10-2.

0 

10-2.

5 

10-2.

0 

10-2.

5 

Z. Yang 

<> EXP 

Launder-Sharma 

-- Present 

I 

I 

I 

I 

~ -6 <> 

Rex 

Fig 3a: Skin friction variation for T3A. 

- ~ __ -_:v 

<> EXP 

Launder-Sharma 

-- Present 

Rex 

Fig 3b: Skin friction variation for T3B. 

94 

10-2.

0 

10-2.

5 

10-2.

0 

10-2.

5 

Z. Yang 

<> EXP 

Launder-Sharma 

-- Present 

I 

I 

I 

I 

~ -6 <> 

Rex 

Fig 3a: Skin friction variation for T3A. 

- ~ __ -_:v 

<> EXP 

Launder-Sharma 

-- Present 

Rex 

Fig 3b: Skin friction variation for T3B. 

94 
Center for Modelling of Turbulence and Transition 

Research Briefs - 1991 

Two Equation Modelling and the 

Pseudo Compressibility Technique 

C. J. Steffen, Jr. 

1. Motivation and Objective 

The primary objective of the Center for Modelling of Turbulence and Transition 

(CMOTT) is to further the understanding of turbulence theory for engineering 

applications. One important foundation for this research is the establishment of 

a data base encompassing the multitude of existing models as well as newly proposed 

ideas. The research effort described in the next few pages is a precursor to an extended 

survey of two equation turbulence models in the presence of a separated shear layer. 

Recently, several authors have examined the performance of two equation models 

in the context of the backward facing step flow. Conflicting results, however, demand 

that further attention is necessary to properly understand the behavior and limitations of this popular technique, especially the low Reynolds number formulations. 

The objective of this research is to validate an incompressible Navier Stokes code 

for use as a numerical test-bed. In turn, this code will be used for analyzing the 

performance of several two equation models. 

2. Work Accomplished 

To date, the validation of the incompressible code DTNS is complete. The details 

of this validation study are documented in reference[l]. The code is based upon 

the pseudo-compressibility technique and incorporates the approximate factorization 

scheme for time integration. Two laminar benchmark flows are used to measure the 

performance and implementation of the numerical methods. The classic solution of 

the Blasius boundary layer is used for validating the flat plate flow, while experimental 

data is incorporated in the validation of backward facing step flow. 

An initial result for the standard high Reynolds number k-€ equations has also been 

calculated to demonstrate an initial performance level of the solution technique for 

the turbulence equations. 

2.1 Numerical Method 

The absence of a variable density in the continuity equation governing incompressible flow complicates the numerical integration procedure. One solution technique 

that has been well received is that of pseudo-compressibility. This idea was first 

put forward by Chorin[2] and enables the equations to be solved using the primitive variables. Recently, Chang and K wak[3], Rizzi and Eriksson[4], K wak and 

Chakravarthy[5]' Michelassi and Shih[6], and Turkel[7] have found this method suitable for resolving incompressible flow . This particular implementation has been va.lidated by Gorski[S- lO] for several different benchmark flows. 

95 

Center for Modelling of Turbulence and Transition 

Research Briefs - 1991 

Two Equation Modelling and the 

Pseudo Compressibility Technique 

C. J. Steffen, Jr. 

1. Motivation and Objective 

The primary objective of the Center for Modelling of Turbulence and Transition 

(CMOTT) is to further the understanding of turbulence theory for engineering 

applications. One important foundation for this research is the establishment of 

a data base encompassing the multitude of existing models as well as newly proposed 

ideas. The research effort described in the next few pages is a precursor to an extended 

survey of two equation turbulence models in the presence of a separated shear layer. 

Recently, several authors have examined the performance of two equation models 

in the context of the backward facing step flow. Conflicting results, however, demand 

that further attention is necessary to properly understand the behavior and limitations of this popular technique, especially the low Reynolds number formulations. 

The objective of this research is to validate an incompressible Navier Stokes code 

for use as a numerical test-bed. In turn, this code will be used for analyzing the 

performance of several two equation models. 

2. Work Accomplished 

To date, the validation of the incompressible code DTNS is complete. The details 

of this validation study are documented in reference[l]. The code is based upon 

the pseudo-compressibility technique and incorporates the approximate factorization 

scheme for time integration. Two laminar benchmark flows are used to measure the 

performance and implementation of the numerical methods. The classic solution of 

the Blasius boundary layer is used for validating the flat plate flow, while experimental 

data is incorporated in the validation of backward facing step flow. 

An initial result for the standard high Reynolds number k-€ equations has also been 

calculated to demonstrate an initial performance level of the solution technique for 

the turbulence equations. 

2.1 Numerical Method 

The absence of a variable density in the continuity equation governing incompressible flow complicates the numerical integration procedure. One solution technique 

that has been well received is that of pseudo-compressibility. This idea was first 

put forward by Chorin[2] and enables the equations to be solved using the primitive variables. Recently, Chang and K wak[3], Rizzi and Eriksson[4], K wak and 

Chakravarthy[5]' Michelassi and Shih[6], and Turkel[7] have found this method suitable for resolving incompressible flow . This particular implementation has been va.lidated by Gorski[S- lO] for several different benchmark flows. 

95 
Christopher J. Steffen) Jr. 

Examine the system of equations involved in the pseudo compressibility method 

and notice that they differ from the incompressible Navier Stokes equations by the 

addition of a time dependant pressure term in the continuity equation. 

~ ap + au + av = 0 

j3 at ax ay 

au + ~ (u 2 + p __ 1 au) + ~ (uv _ ~ au) = 0 at ax Re ox ay Re ay (1) 

av + ~(uv __ 1 av) + ~(v2 + p _ ~ av) = 0 at ax Reax ay Reay 

Here, x and yare the independent variables and Re refers to the Reynolds number. 

The constant j3 is known as the pseudo compressibility parameter. This system 

is hyperbolic in nature while the incompressible flow equations are elliptic. The 

pseudo sound speed, c = Ju2 + j3, is governed by the value of the pa rameter j3 , 

whereas the physical sound speed is infinite. Chang and K wak [3J have shown t hat 

for j3 > 0 the finite speed pseudo waves vanish as time progresses and yi eld the proper 

incompressible solution at the steady state limit. It is through this parameter j3 that 

t he convective and acoustic waves are decoupled, and thus convergence is governed. In 

choosing an optimum value for this parameter, the goal is to avoid giving the viscous 

effects time to react to the passing of t he nonphysical transient pressure waves. Thus 

a lower bound on the acoustic speeds translates into a lower bound on j3. However, 

an upper bound on j3 is strictly scheme dependent. 

The approximate factorization is rather straight forward. Equations (2) can be 

rewritten as 

(2) 

-1[~1] -l[Ju] gl = Re ~ , g2 = Re ~ 

ax ay 

or in generalized coordina.tes (~( , y), T/( x, y)) as seen here in equation (3). 

o a~ 0(- at (q) + a~ \.£I +gl) + BT/ f2+g2) = 0 

- q 

q = -

J 

f - fl + ~yf2 f _ T/xfl + T/yf2 

1- J , 2- J (3) 

_ gl + ~yg2 _ T/ x g1 + 77yg2 

gl = J ' g2 = J 

J = o(C T/ ) 

o(:r, 1) ) 

96 

I 

J 

Christopher J. Steffen) Jr. 

Examine the system of equations involved in the pseudo compressibility method 

and notice that they differ from the incompressible Navier Stokes equations by the 

addition of a time dependant pressure term in the continuity equation. 

~ ap + au + av = 0 

j3 at ax ay 

au + ~ (u 2 + p __ 1 au) + ~ (uv _ ~ au) = 0 at ax Re ox ay Re ay (1) 

av + ~(uv __ 1 av) + ~(v2 + p _ ~ av) = 0 at ax Reax ay Reay 

Here, x and yare the independent variables and Re refers to the Reynolds number. 

The constant j3 is known as the pseudo compressibility parameter. This system 

is hyperbolic in nature while the incompressible flow equations are elliptic. The 

pseudo sound speed, c = Ju2 + j3, is governed by the value of the pa rameter j3 , 

whereas the physical sound speed is infinite. Chang and K wak [3J have shown t hat 

for j3 > 0 the finite speed pseudo waves vanish as time progresses and yi eld the proper 

incompressible solution at the steady state limit. It is through this parameter j3 that 

t he convective and acoustic waves are decoupled, and thus convergence is governed. In 

choosing an optimum value for this parameter, the goal is to avoid giving the viscous 

effects time to react to the passing of t he nonphysical transient pressure waves. Thus 

a lower bound on the acoustic speeds translates into a lower bound on j3. However, 

an upper bound on j3 is strictly scheme dependent. 

The approximate factorization is rather straight forward. Equations (2) can be 

rewritten as 

(2) 

-1[~1] -l[Ju] gl = Re ~ , g2 = Re ~ 

ax ay 

or in generalized coordina.tes (~( , y), T/( x, y)) as seen here in equation (3). 

o a~ 0(- at (q) + a~ \.£I +gl) + BT/ f2+g2) = 0 

- q 

q = -

J 

f - fl + ~yf2 f _ T/xfl + T/yf2 

1- J , 2- J (3) 

_ gl + ~yg2 _ T/ x g1 + 77yg2 

gl = J ' g2 = J 

J = o(C T/ ) 

o(:r, 1) ) 

96 

I 

J 
Two Equation Models and Turbulent Flows 

The third order accurate Chakravarthy and Osher TVD scheme is used to discretize 

the convective terms while central differencing is used for the viscous fluxes. Thus 

the approximate factorization of this method is expressed in terms of flux Jacobians 

A and B as 

~ a- A + a+ A - a- A }D.-* _ acr; + gI") acr; + gi) 

A t + ae + ae + ae v q - ae + aTl 

{

V a- a+ a-} V - + -B+ + -B- + -Bv Sq = - D.q* A t aTl aTl aTl D.t 

(4) 

where V is the cell volume, and D.t = lC:Ji is the timestep. The matrices Av and Bv 

are the viscous flux Jacobians. The interested reader is encouraged to see reference[9] 

for more details. 

The k-t: equations are solved in a similar manner as that seen in equation 2. 

Consider the standard high Reynolds number form of the transport equations of the 

turbulent kinetic energy and dissipation of t urbulent genetic energy as in equation 

(5): 

with 

Vt 

Vk = v + -

Uk 

Vt 

v( = v +-

u( 

and the production of tu'rbulent genetic energy defined as shown below. 

P = Vt [2(u~ + v~) + (uy + vx)2] 

(5) 

(6) 

(7) 

Again, we can translate these equations in a manner similar to that of equation (3) 

and create the following system for generalized coordinates: 

~(qt) + ~(ft+ t) + ~(ft+gt) = st at De 1 1 aTl 2 2 (8) 

This can be solved via. approximate fa.ctoriza.tion as shown here in equation 9. 

(9) 

97 

Two Equation Models and Turbulent Flows 

The third order accurate Chakravarthy and Osher TVD scheme is used to discretize 

the convective terms while central differencing is used for the viscous fluxes. Thus 

the approximate factorization of this method is expressed in terms of flux Jacobians 

A and B as 

~ a- A + a+ A - a- A }D.-* _ acr; + gI") acr; + gi) 

A t + ae + ae + ae v q - ae + aTl 

{

V a- a+ a-} V - + -B+ + -B- + -Bv Sq = - D.q* A t aTl aTl aTl D.t 

(4) 

where V is the cell volume, and D.t = lC:Ji is the timestep. The matrices Av and Bv 

are the viscous flux Jacobians. The interested reader is encouraged to see reference[9] 

for more details. 

The k-t: equations are solved in a similar manner as that seen in equation 2. 

Consider the standard high Reynolds number form of the transport equations of the 

turbulent kinetic energy and dissipation of t urbulent genetic energy as in equation 

(5): 

with 

Vt 

Vk = v + -

Uk 

Vt 

v( = v +-

u( 

and the production of tu'rbulent genetic energy defined as shown below. 

P = Vt [2(u~ + v~) + (uy + vx)2] 

(5) 

(6) 

(7) 

Again, we can translate these equations in a manner similar to that of equation (3) 

and create the following system for generalized coordinates: 

~(qt) + ~(ft+ t) + ~(ft+gt) = st at De 1 1 aTl 2 2 (8) 

This can be solved via. approximate fa.ctoriza.tion as shown here in equation 9. 

(9) 

97 
Christopher J . Steffen, Jr. 

Again, the scheme is written in terms of the turbulence flux Jacobians and the 

approximate source Jacobian, Ht. The coefficients a~ and art are provided to enhance 

the diagonal dominance of the implicit method. While the convective flux terms are 

linear, the linearization of the source terms is not straight forward . Furthermore, th 

low Reynolds number formulation of equation 5 includes other nonlinear correction 

terms in the source vector st. The interested reader is encouraged to see the recent 

paper by Michelassi and Shih[6J. 

2.2 Discussion 

Overall the performance for laminar flow documented in reference [1 J is very encouraging. The flat plate results shown agree with the theoretical Blasius solution 

for several different grid configurations. The laminar back facing step flow was also 

well resolved both in terms of the primary recirculation zone reattachment length, 

and corresponding velocity profiles as seen in figures 1 and 2. However, performa.nce 

of the two dimensional flow code drops off drama.tically when the Reynolds number 

corresponding to the onset of 3D flow is exceeded. Armaly et al [l1J observed th is 

phenomenon occur at a Reynolds number of 400, based upon the hydraulic diameter 

of the inlet channel. Again, the numerical results bear this finding out. 

Figure 1 Reattachment length predictions-comparison with experiment [11]. XI, X2, and X3 are the 

primary reattachment, second ary detachment" and secondary reattachment lengths, respectively. 

20 

18 

16 

14 

12 

[f) 

......... 10 >< 

8 

6 0 

4 a:> 

~ 

2 

0 

0 200 400 600 800 

Reo 

98 

Experiment: 

• XI • 

20 Calcul ation: 

o XI 

o X2 

A X3 

Christopher J . Steffen, Jr. 

Again, the scheme is written in terms of the turbulence flux Jacobians and the 

approximate source Jacobian, Ht. The coefficients a~ and art are provided to enhance 

the diagonal dominance of the implicit method. While the convective flux terms are 

linear, the linearization of the source terms is not straight forward . Furthermore, th 

low Reynolds number formulation of equation 5 includes other nonlinear correction 

terms in the source vector st. The interested reader is encouraged to see the recent 

paper by Michelassi and Shih[6J. 

2.2 Discussion 

Overall the performance for laminar flow documented in reference [1 J is very encouraging. The flat plate results shown agree with the theoretical Blasius solution 

for several different grid configurations. The laminar back facing step flow was also 

well resolved both in terms of the primary recirculation zone reattachment length, 

and corresponding velocity profiles as seen in figures 1 and 2. However, performa.nce 

of the two dimensional flow code drops off drama.tically when the Reynolds number 

corresponding to the onset of 3D flow is exceeded. Armaly et al [l1J observed th is 

phenomenon occur at a Reynolds number of 400, based upon the hydraulic diameter 

of the inlet channel. Again, the numerical results bear this finding out. 

Figure 1 Reattachment length predictions-comparison with experiment [11]. XI, X2, and X3 are the 

primary reattachment, second ary detachment" and secondary reattachment lengths, respectively. 

20 

18 

16 

14 

12 

[f) 

......... 10 >< 

8 

6 0 

4 a:> 

~ 

2 

0 

0 200 400 600 800 

Reo 

98 

Experiment: 

• XI • 

20 Calcul ation: 

o XI 

o X2 

A X3 
Two Equation Models and Turbulent Flows 

Figure 2 Reo=389 velocity profiles-comparison with experiment [11]. 

x/s=O.O 2.55 3.06 4.18 13.57 

4 4 4 

2 2 2 2 2 ....... 

E 

S 0 0 0 0 0 '-' 

>. 

-2 -2 -2 -2 -2 

-4 -4 - 4 -4 -4 

0 80 0 80 0 80 0 80 0 80 

U (em/sec) 

The current thrust of effort is in establishing a comparison between the standard 

high Reynolds number form k-t: model and a version which incorporates the low 

Reynolds number corrections, such as that of Jones and Launder[12]. This model 

is particularly convenient in that the damping functions are independent of the 

parameter, y+. Thus, multiple walls bounding the computational domain do not need 

special attention. An initial calculation using the high Reynolds number approach 

agrees with the published results of Speziale and Thangam[13]. However, the Jones 

and Launder model is proving to be quite challenging. 

3. Future Plans 

Eventually, several two equation models will be examined for their behavior in the 

back facing step flow. The following is a partial list of the two equation models of 

primary interest: 

1. Jones and Launder 

2. Chien 

3. Yang and Shih 

4. Shih and Lumley 

The current research effort will result in an increased unde?standing of the need 

for proper modelling of the nea.r wall region. However, the increase in accuracy 

maybe offset by an unwarranted increase in effort. Further numerical experiments 

are necessary before any conclusions can be drawn. 

4. References 

Steffen, Jr., C. J., "An Investigation of DTNS2D for Use as an Incompressible 

Turbulence Modelling Test-bed," NASA Technical Memorandum 105593 (1992). 

99 

Two Equation Models and Turbulent Flows 

Figure 2 Reo=389 velocity profiles-comparison with experiment [11]. 

x/s=O.O 2.55 3.06 4.18 13.57 

4 4 4 

2 2 2 2 2 ....... 

E 

S 0 0 0 0 0 '-' 

>. 

-2 -2 -2 -2 -2 

-4 -4 - 4 -4 -4 

0 80 0 80 0 80 0 80 0 80 

U (em/sec) 

The current thrust of effort is in establishing a comparison between the standard 

high Reynolds number form k-t: model and a version which incorporates the low 

Reynolds number corrections, such as that of Jones and Launder[12]. This model 

is particularly convenient in that the damping functions are independent of the 

parameter, y+. Thus, multiple walls bounding the computational domain do not need 

special attention. An initial calculation using the high Reynolds number approach 

agrees with the published results of Speziale and Thangam[13]. However, the Jones 

and Launder model is proving to be quite challenging. 

3. Future Plans 

Eventually, several two equation models will be examined for their behavior in the 

back facing step flow. The following is a partial list of the two equation models of 

primary interest: 

1. Jones and Launder 

2. Chien 

3. Yang and Shih 

4. Shih and Lumley 

The current research effort will result in an increased unde?standing of the need 

for proper modelling of the nea.r wall region. However, the increase in accuracy 

maybe offset by an unwarranted increase in effort. Further numerical experiments 

are necessary before any conclusions can be drawn. 

4. References 

Steffen, Jr., C. J., "An Investigation of DTNS2D for Use as an Incompressible 

Turbulence Modelling Test-bed," NASA Technical Memorandum 105593 (1992). 

99 
~ 

Christ.ophe1' J. St. effen, Jr. 

2 Chorin, A. J., "A Numerical Method for Solving In compressible Viscous Flow 

Problems," J. Comput. Phys., 2 , 12- 26 (1967). 

3 Chang, J. 1. C. and K wak, D., "On the Method of Pseudo Compressibili ty for 

Numerically Solving Incompressible Flows," ALA A paper 84-0252 (1984). 

4 Rizzi, A. and Eriksson, 1. E., "Computation of Inviscid Incompressible Flow with 

Rotation," J. Fluid Mech., 153, 275- 312 (1985). 

5 Kwak, D., Chang, J. L. C., Shanks, S. P., and Chakravarthy, S. R., "A Three 

Dimensional Incompressible Navier-Stokes Flow Solver Using Primitive Variables," 

AIAA J., 24, 390- 396 (1986). 

6 Michelassi, V. and T.-H. Shih, , "Elliptic Flow Computation by Low Reynolds 

Number Two Equation Turbulence Models," NASA Technical Memorandum 

105376 (1991). 

7 Turkel, E., "Preconditioned Methods for Solving the Incompressible and Low Speed 

Compressible Equations," J. Comput. Phys., 72 , 277-298 (1987). 

8 Gorski, J. J., "Incompressible Cascade Calculations Using an Upwind Differenced 

TVD Scheme," Advances and Applications in COm1J1dational Fluid Dynamics, 66 

of ASME-FED, edited by O. Baysa.l , 61- 69 (1988). 

9 Gorski, J. J., "TVD Solutions of the Incompressible Navier-Stokes Equations 'With 

an Implicit Multigrid Scheme," Proceedings of th e A IAAj ASMEjSIA Mj A PS l si 

National Fluid Dynamics Congress, 1, 394- 401 (1988). 

10 Gorski , J. J., "Solutions of the Incompressible Navier-Stokes Equations Using 

an Upwind Differenced TVD Scheme," P1'Oceedings of t.h e 11th International 

Conf erence on Numerical M ethods in Fluid Dynamics, 278- 282 (1988). 

11 Armaly, B. F ., Durst, F. , Pereira, J. C. F., and Schonung, B., "Experimental and 

Theoretical Investigation of Backward-Facing Step Flow," J. Flui d Mech., 127, 

473-496 (1983). 

12 Jones, W. P. and Launder, B. E., "The Pred iction of Laminarization with a. TwoEquation Model of Turbulence," Int. J. Heat Mass Transf. , 15 ,301- 314 (1972). 

13 Thangam, S. arid Spezia.le, C. G., "Turbulent Separated Flow Past a Backward 

Facing Step: a Critical Evaluation -of Two-Equation Turbulence Models," NASA 

Contractor Report ]87532 ICASE Report No. 91- 23 (1991). 

100 

~ 

Christ.ophe1' J. St. effen, Jr. 

2 Chorin, A. J., "A Numerical Method for Solving In compressible Viscous Flow 

Problems," J. Comput. Phys., 2 , 12- 26 (1967). 

3 Chang, J. 1. C. and K wak, D., "On the Method of Pseudo Compressibili ty for 

Numerically Solving Incompressible Flows," ALA A paper 84-0252 (1984). 

4 Rizzi, A. and Eriksson, 1. E., "Computation of Inviscid Incompressible Flow with 

Rotation," J. Fluid Mech., 153, 275- 312 (1985). 

5 Kwak, D., Chang, J. L. C., Shanks, S. P., and Chakravarthy, S. R., "A Three 

Dimensional Incompressible Navier-Stokes Flow Solver Using Primitive Variables," 

AIAA J., 24, 390- 396 (1986). 

6 Michelassi, V. and T.-H. Shih, , "Elliptic Flow Computation by Low Reynolds 

Number Two Equation Turbulence Models," NASA Technical Memorandum 

105376 (1991). 

7 Turkel, E., "Preconditioned Methods for Solving the Incompressible and Low Speed 

Compressible Equations," J. Comput. Phys., 72 , 277-298 (1987). 

8 Gorski, J. J., "Incompressible Cascade Calculations Using an Upwind Differenced 

TVD Scheme," Advances and Applications in COm1J1dational Fluid Dynamics, 66 

of ASME-FED, edited by O. Baysa.l , 61- 69 (1988). 

9 Gorski, J. J., "TVD Solutions of the Incompressible Navier-Stokes Equations 'With 

an Implicit Multigrid Scheme," Proceedings of th e A IAAj ASMEjSIA Mj A PS l si 

National Fluid Dynamics Congress, 1, 394- 401 (1988). 

10 Gorski , J. J., "Solutions of the Incompressible Navier-Stokes Equations Using 

an Upwind Differenced TVD Scheme," P1'Oceedings of t.h e 11th International 

Conf erence on Numerical M ethods in Fluid Dynamics, 278- 282 (1988). 

11 Armaly, B. F ., Durst, F. , Pereira, J. C. F., and Schonung, B., "Experimental and 

Theoretical Investigation of Backward-Facing Step Flow," J. Flui d Mech., 127, 

473-496 (1983). 

12 Jones, W. P. and Launder, B. E., "The Pred iction of Laminarization with a. TwoEquation Model of Turbulence," Int. J. Heat Mass Transf. , 15 ,301- 314 (1972). 

13 Thangam, S. arid Spezia.le, C. G., "Turbulent Separated Flow Past a Backward 

Facing Step: a Critical Evaluation -of Two-Equation Turbulence Models," NASA 

Contractor Report ]87532 ICASE Report No. 91- 23 (1991). 

100 
Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Renormalization Group Methods for the 

Reynolds Stress Transport Equations 

R. Rubinstein 

1. Motivation and Objective 

The Yakhot-Orszag renormalization group is used to analyze the pressure gradientvelocity correlation and return to isotropy terms in the Reynolds stress transport 

equations. The perturbation series for the relevant correlations, evaluated to lowest 

order in the €-expansion of the Yakhot-Orszag theory, are infinite series in tensor 

product powers of the mean velocity gradient and its transpose. Formal lowest 

order Pade approximations to the sums of these series produce a rapid pressure 

strain model of the form proposed by Launder, Reece, and Rodi, and a return to 

isotropy model of the form proposed by Rotta. In both cases, the model constants 

are computed theoretically. The predicted Reynolds stress ratios in simple shear 

flows are evaluated and compared with experimental data. The possibility is discussed of deriving higher order nonlinear models by approximating the sums more 

accurately. 

The Yakhot-Orszag renormalization group provides a systematic procedure for deriving turbulence models. Typical applications have included theoretical derivation 

of the universal constants of isotropic turbulence theory, such as the Kolmogorov 

constant, and derivation of two equation models, again with theoretically computed constants and low Reynolds number forms of the equations. Recent work 

has applied this formalism to Reynolds stress modeling, previously in the form of a 

nonlinear eddy viscosity representation of the Reynolds stresses, which can be used 

to model the simplest normal stress effects. The present work attempts to apply 

the Yakhot-Orszag formalism to Reynolds stress transport modeling. 

2. Work Accomplished 

The modelling of the pressure gradient-velocity correlation and return to isotropy 

term in the Reynolds stress transport equation remains an area of active research. 1 ,2,3 

Models will be derived here using the Yakhot-Orszag renormalization group4 partially along the lines of our previous work5 . The result is a model for the rapid 

pressure-strain term of the form proposed by Launder, Reece and Rodi6 (LRR) and 

a model for return to isotropy of the form proposed by Rotta7 with theoretically 

computed constants in good agreement with accepted values. As is usual in investigations of this sort, the priority of Yoshizawa in deriving a pressure strain model 

analytically8 must be noted. 

The analysis requires some new ideas in renormalization group theory recently 

introduced by Yakhot et a19 . As Yakhot et a19 emphasize, the application of the 

renormalization group mode elimination formalism to shear flow creates a double 

perturbation series in powers of €, the parameter of the isotropic theory, and in 

101 

Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Renormalization Group Methods for the 

Reynolds Stress Transport Equations 

R. Rubinstein 

1. Motivation and Objective 

The Yakhot-Orszag renormalization group is used to analyze the pressure gradientvelocity correlation and return to isotropy terms in the Reynolds stress transport 

equations. The perturbation series for the relevant correlations, evaluated to lowest 

order in the €-expansion of the Yakhot-Orszag theory, are infinite series in tensor 

product powers of the mean velocity gradient and its transpose. Formal lowest 

order Pade approximations to the sums of these series produce a rapid pressure 

strain model of the form proposed by Launder, Reece, and Rodi, and a return to 

isotropy model of the form proposed by Rotta. In both cases, the model constants 

are computed theoretically. The predicted Reynolds stress ratios in simple shear 

flows are evaluated and compared with experimental data. The possibility is discussed of deriving higher order nonlinear models by approximating the sums more 

accurately. 

The Yakhot-Orszag renormalization group provides a systematic procedure for deriving turbulence models. Typical applications have included theoretical derivation 

of the universal constants of isotropic turbulence theory, such as the Kolmogorov 

constant, and derivation of two equation models, again with theoretically computed constants and low Reynolds number forms of the equations. Recent work 

has applied this formalism to Reynolds stress modeling, previously in the form of a 

nonlinear eddy viscosity representation of the Reynolds stresses, which can be used 

to model the simplest normal stress effects. The present work attempts to apply 

the Yakhot-Orszag formalism to Reynolds stress transport modeling. 

2. Work Accomplished 

The modelling of the pressure gradient-velocity correlation and return to isotropy 

term in the Reynolds stress transport equation remains an area of active research. 1 ,2,3 

Models will be derived here using the Yakhot-Orszag renormalization group4 partially along the lines of our previous work5 . The result is a model for the rapid 

pressure-strain term of the form proposed by Launder, Reece and Rodi6 (LRR) and 

a model for return to isotropy of the form proposed by Rotta7 with theoretically 

computed constants in good agreement with accepted values. As is usual in investigations of this sort, the priority of Yoshizawa in deriving a pressure strain model 

analytically8 must be noted. 

The analysis requires some new ideas in renormalization group theory recently 

introduced by Yakhot et a19 . As Yakhot et a19 emphasize, the application of the 

renormalization group mode elimination formalism to shear flow creates a double 

perturbation series in powers of €, the parameter of the isotropic theory, and in 

101 
R. Rubinstein 

powers of a dimensionless strain rate, TJ = S K / E, where K denotes the turbulence 

kinetic energy, E denotes the dissipation rate, and S is a measure of the mean 

strain: in Ref. 9, S2 = (aaUm + a

oUn ) °aUm . The present analysis also leads to double Xn Xm Xn 

expansions of this type, with the powers sn replaced by tensors sen) homogeneous of 

degree n in the mean velocity gradient \lU and its transpose. It will be convenient to 

retain the terminology of Ref. 9 and call this expansion the TJ-expansion; when the 

distinction is pertinent, the expansion of Ref. 9 will be called a scalar TJ-expansion. 

The heuristic program of evaluating all scalar amplitudes to lowest order in E 

has proven successful in the past: apparently, the E-expansion is an asymptotic 

series with a sum given very nearly by its first term 10. Unfortunately, there is no 

analogous basis for truncating the TJ-expansion. There are fundamental reasons for 

this distinction between these expansions. The present TJ-expansion is tensorial: 

successively higher order terms do not introduce merely numerical corrections, but 

increasingly complex asymmetries into the theory. Truncation therefore imposes 

a possibly inappropriate symmetry or other constraint on the model. Thus, in 

Ref. 5 the TJ-expansion for the Reynolds stress T was truncated at second order as 

suggested by previous work of Yoshizawall and Speziale12 . Although this type of 

modelling permits unequal normal stresses in a simple shear flow, it is not maximally 

asymmetric: for example, in a flow with mean velocity components Ui (Xl, X2 ), a 

cubic model including a term T rv \lU2 \lUT + \lU\lUT2 would permit nonzero T23 

in the presence of vanishing 8U2/8x3 and 8U3/8x2, an effect which cannot be ruled 

out in advance. 

Although generalizations13 of the Cayley-Hamilton Theorem limit the number 

of independent tensors s(n) , anisotropy and asymmetry cannot exist at all without 

some terms of higher order in TJ; indeed, truncation at lowest order in TJ just produces 

a theory of isotropic turbulence. But the series truncated at any higher order can 

be unsatisfactory in flow regions in which some components (~~~) K / E are large. In , such regions, the truncated series is dominated by its highest order terms. For the 

quadratic stress models of Refs. 5, 11, 12, this domination can produce negative 

normal stresses in the buffer layers of wall bounded flows. Increasing the order of 

truncation obviously exacerbates this problem. 

It follows that finite truncation of the TJ-expansion is theoretically unsatisfactory. 

Yakhot et al9 therefore propose that this expansion must be summed, even if only 

approximately, and have suggested a prototype summation in a different context. 

It should be noted that the same issues arise naturally in Yoshizawa's formalism, 

which also generates infinite series in the mean velocity gradients (and in other 

quantities as well) for correlations of interest in turbulence modeling. Yoshizawa 

has concluded independently that summation of this series is essential and has also 

derived a Reynolds stress transport model by introducing such summations14 . 

In this paper, the perturbation series which the Yakhot-Orszag renormalization 

group generates for the correlation 

(1) 

102 

J 

R. Rubinstein 

powers of a dimensionless strain rate, TJ = S K / E, where K denotes the turbulence 

kinetic energy, E denotes the dissipation rate, and S is a measure of the mean 

strain: in Ref. 9, S2 = (aaUm + a

oUn ) °aUm . The present analysis also leads to double Xn Xm Xn 

expansions of this type, with the powers sn replaced by tensors sen) homogeneous of 

degree n in the mean velocity gradient \lU and its transpose. It will be convenient to 

retain the terminology of Ref. 9 and call this expansion the TJ-expansion; when the 

distinction is pertinent, the expansion of Ref. 9 will be called a scalar TJ-expansion. 

The heuristic program of evaluating all scalar amplitudes to lowest order in E 

has proven successful in the past: apparently, the E-expansion is an asymptotic 

series with a sum given very nearly by its first term 10. Unfortunately, there is no 

analogous basis for truncating the TJ-expansion. There are fundamental reasons for 

this distinction between these expansions. The present TJ-expansion is tensorial: 

successively higher order terms do not introduce merely numerical corrections, but 

increasingly complex asymmetries into the theory. Truncation therefore imposes 

a possibly inappropriate symmetry or other constraint on the model. Thus, in 

Ref. 5 the TJ-expansion for the Reynolds stress T was truncated at second order as 

suggested by previous work of Yoshizawall and Speziale12 . Although this type of 

modelling permits unequal normal stresses in a simple shear flow, it is not maximally 

asymmetric: for example, in a flow with mean velocity components Ui (Xl, X2 ), a 

cubic model including a term T rv \lU2 \lUT + \lU\lUT2 would permit nonzero T23 

in the presence of vanishing 8U2/8x3 and 8U3/8x2, an effect which cannot be ruled 

out in advance. 

Although generalizations13 of the Cayley-Hamilton Theorem limit the number 

of independent tensors s(n) , anisotropy and asymmetry cannot exist at all without 

some terms of higher order in TJ; indeed, truncation at lowest order in TJ just produces 

a theory of isotropic turbulence. But the series truncated at any higher order can 

be unsatisfactory in flow regions in which some components (~~~) K / E are large. In , such regions, the truncated series is dominated by its highest order terms. For the 

quadratic stress models of Refs. 5, 11, 12, this domination can produce negative 

normal stresses in the buffer layers of wall bounded flows. Increasing the order of 

truncation obviously exacerbates this problem. 

It follows that finite truncation of the TJ-expansion is theoretically unsatisfactory. 

Yakhot et al9 therefore propose that this expansion must be summed, even if only 

approximately, and have suggested a prototype summation in a different context. 

It should be noted that the same issues arise naturally in Yoshizawa's formalism, 

which also generates infinite series in the mean velocity gradients (and in other 

quantities as well) for correlations of interest in turbulence modeling. Yoshizawa 

has concluded independently that summation of this series is essential and has also 

derived a Reynolds stress transport model by introducing such summations14 . 

In this paper, the perturbation series which the Yakhot-Orszag renormalization 

group generates for the correlation 

(1) 

102 

J 
R. Rubinstein 

where (0) denotes deviatoric part and (ij) denotes index interchange in the preceding term. 

The next order will produce a term T3 containing cubic products of velocities u<. 

In view of the form of the LRR model, it is reasonable to ask whether a term with 

only one gradient, proportional in the high Reynolds number limit to T'VU might 

occur at this order. Such terms do occur, but they cancel. Evaluation of T3 proves 

to require expansions of the projection operators to second order, leading instead 

to terms S(3) homogeneous of degree three in the mean velocity gradient and its 

transpose. In general, the term Tn of order n has the form s(n)(K/E)n. As noted 

in the Introduction, it will be imperative to include effects of all orders in SK / E 

in the model, but because the terms Tn involve ever higher order derivatives of 

the transverse projection operators, they do not have an obvious law of formation. 

Therefore, an exact summation does not appear feasible. 

A simple approximate summation is obtained by introducing the perturbation 

series5 for UiUj (0) in the form 

(

aU- aU-) v _t + _J = -UiU / O) + ~ s(n) (K/E)n 

ax - ax - L..J 

J t n>2 

and dropping the quadratic terms. The resulting model, 

with 

( ) [ 1 

(0) 2 aUi aUj __ 0 aUj __ 0 aUi IIi" = -K - + - +CT1 UiU ()- +U "U ()-

J 5 aXj aXi _ p axp J P axp 

[ 1 

(0) 

o aUp 0 aUp 

+ C"2 U"U ( )-+U"U ( )- r t p a J p a Xj Xi 

16 CTl = - = .7619 

21 

2 

CT2 = - = .0952 

21 

(5) 

(6) 

agrees with the perturbation series (3) and (4) to terms of order S(3). However, 

unlike the explicit quadratic model which results from simply dropping the 0(S(3») 

terms, this model includes effects of all order in SK/c. The consequences of this 

fact will be discussed later. This type of summation has also been applied by 

Yoshizawal4 • Eqs. (5) and (6) can be compared with Launder, Reece and Rodi's 

"model 1", Eq. (5) with the empirically adjusted constants 

CTl = .7636 CT2 = .1091 (7) 

In this model, the constants CTl and CT2 were not chosen independently; instead, 

to insure some consistency conditions introduced by Rotta7 , Launder, Reece, and 

Rodi set6 

. - ---------

C2 +8 CTl = ---

11 

C- _ 8C2 - 2 - r2 - 11 

103 

R. Rubinstein 

where (0) denotes deviatoric part and (ij) denotes index interchange in the preceding term. 

The next order will produce a term T3 containing cubic products of velocities u<. 

In view of the form of the LRR model, it is reasonable to ask whether a term with 

only one gradient, proportional in the high Reynolds number limit to T'VU might 

occur at this order. Such terms do occur, but they cancel. Evaluation of T3 proves 

to require expansions of the projection operators to second order, leading instead 

to terms S(3) homogeneous of degree three in the mean velocity gradient and its 

transpose. In general, the term Tn of order n has the form s(n)(K/E)n. As noted 

in the Introduction, it will be imperative to include effects of all orders in SK / E 

in the model, but because the terms Tn involve ever higher order derivatives of 

the transverse projection operators, they do not have an obvious law of formation. 

Therefore, an exact summation does not appear feasible. 

A simple approximate summation is obtained by introducing the perturbation 

series5 for UiUj (0) in the form 

(

aU- aU-) v _t + _J = -UiU / O) + ~ s(n) (K/E)n 

ax - ax - L..J 

J t n>2 

and dropping the quadratic terms. The resulting model, 

with 

( ) [ 1 

(0) 2 aUi aUj __ 0 aUj __ 0 aUi IIi" = -K - + - +CT1 UiU ()- +U "U ()-

J 5 aXj aXi _ p axp J P axp 

[ 1 

(0) 

o aUp 0 aUp 

+ C"2 U"U ( )-+U"U ( )- r t p a J p a Xj Xi 

16 CTl = - = .7619 

21 

2 

CT2 = - = .0952 

21 

(5) 

(6) 

agrees with the perturbation series (3) and (4) to terms of order S(3). However, 

unlike the explicit quadratic model which results from simply dropping the 0(S(3») 

terms, this model includes effects of all order in SK/c. The consequences of this 

fact will be discussed later. This type of summation has also been applied by 

Yoshizawal4 • Eqs. (5) and (6) can be compared with Launder, Reece and Rodi's 

"model 1", Eq. (5) with the empirically adjusted constants 

CTl = .7636 CT2 = .1091 (7) 

In this model, the constants CTl and CT2 were not chosen independently; instead, 

to insure some consistency conditions introduced by Rotta7 , Launder, Reece, and 

Rodi set6 

. - ---------

C2 +8 CTl = ---

11 

C- _ 8C2 - 2 - r2 - 11 

103 
Renormalization Group Methods 

where only the constant C2 is arbitrary. By eliminating C2 between these equations, 

there results 

(8) 

which is also satisfied' by the choice of constants in Eq. (6). The LRR model 

corresponds to the choice C2 = .4; Eqs. (5) and (6) correspond instead to the 

choice C2 = 8/21 rv .36. 

The approximate summation used to derive Eq. (5) can be systematically generalized to generate an infinite number of models for IIij. For example, suppose that 

the perturbation series for T is introduced into the cubic terms in the perturbation 

series instead of in the quadratic terms as above. This substitution will produce a 

model which can be written symbolically in the form 

II rv S(I) + S(2) + T(S(I), + S(2),) 

where TS(i)' denotes a sum of matrix products in all possible orders of T and terms 

S(i ). The requirement that the original series agree to order S(4) with the approximation when T is replaced by its perturbation series determines this approximation 

uniquely. 

2.2 The Return to Isotropy Model 

The analytical description of return to isotropy is no less controversial than the 

modeling of the fast pressure strain term 3 . In the usual approach to turbulence 

modeling, in which correlations generated by Reynolds averaging are closed phenomenologically, this process is considered to result partly from the pressure correlation through a "slow" term independent of the mean flow, and partly from the 

deviatoric part of the dissipative correlation (lIO g~: ~~! ). From this viewpoint, the 

analysis in Sect. I is incomplete because it discloses only a t erm proportional to the 

mean velocity gradient, but no slow term. The return to isotropy will be derived 

here by renormalization group methods following a suggestion of Yakhot 17 . 

From the renormalization group viewpoint, it is natural to investigate the return 

to isotropy, even independently of the stress transport equation, by writing the 

perturbation series for 

(9) 

This perturbation series differs from the perturbation series for the Reynolds stresses 

previously reported5 only in the occurence of an additional factor -iw in all frequency integrals. 

The analysis is straightforward. Only the deviatoric terms require attention because the part of the correlation proporational to Oi j contributes to the transport 

equation for K which has been analyzed by Yakhot and Smithl5 . The lowest order 

deviator appears at first order in f/; to lowest order in E 

104 

Renormalization Group Methods 

where only the constant C2 is arbitrary. By eliminating C2 between these equations, 

there results 

(8) 

which is also satisfied' by the choice of constants in Eq. (6). The LRR model 

corresponds to the choice C2 = .4; Eqs. (5) and (6) correspond instead to the 

choice C2 = 8/21 rv .36. 

The approximate summation used to derive Eq. (5) can be systematically generalized to generate an infinite number of models for IIij. For example, suppose that 

the perturbation series for T is introduced into the cubic terms in the perturbation 

series instead of in the quadratic terms as above. This substitution will produce a 

model which can be written symbolically in the form 

II rv S(I) + S(2) + T(S(I), + S(2),) 

where TS(i)' denotes a sum of matrix products in all possible orders of T and terms 

S(i ). The requirement that the original series agree to order S(4) with the approximation when T is replaced by its perturbation series determines this approximation 

uniquely. 

2.2 The Return to Isotropy Model 

The analytical description of return to isotropy is no less controversial than the 

modeling of the fast pressure strain term 3 . In the usual approach to turbulence 

modeling, in which correlations generated by Reynolds averaging are closed phenomenologically, this process is considered to result partly from the pressure correlation through a "slow" term independent of the mean flow, and partly from the 

deviatoric part of the dissipative correlation (lIO g~: ~~! ). From this viewpoint, the 

analysis in Sect. I is incomplete because it discloses only a t erm proportional to the 

mean velocity gradient, but no slow term. The return to isotropy will be derived 

here by renormalization group methods following a suggestion of Yakhot 17 . 

From the renormalization group viewpoint, it is natural to investigate the return 

to isotropy, even independently of the stress transport equation, by writing the 

perturbation series for 

(9) 

This perturbation series differs from the perturbation series for the Reynolds stresses 

previously reported5 only in the occurence of an additional factor -iw in all frequency integrals. 

The analysis is straightforward. Only the deviatoric terms require attention because the part of the correlation proporational to Oi j contributes to the transport 

equation for K which has been analyzed by Yakhot and Smithl5 . The lowest order 

deviator appears at first order in f/; to lowest order in E 

104 
R. Rubinstein 

(10) 

where 

dr 15 VA2 

(11) 

In view of the form of the Rotta model, it is reasonable to seek terms at the next 

order proportional to UiUj. As in Sect. I, such terms do appear, but cancel exactly. 

This apparently ubiquitous cancellation was also obtained by Smith and Reynolds18 

in an analysis of the c transport equation. Accordingly, the second order analysis 

in TJ produces quadratic terms in the velocity gradients. Finite truncation of this 

series violates the requirement that return to isotropy be independent of the mean 

flow. Therefore, we must seek a reasonable approximate summation. The form of 

the lowest order term given in Eqs. (10) and (11) suggests 

Ir- = T1 v (8Ui + 8Uj ) rv _ T 1UiU -(0) 

tJ v 8xj 8Xi V J 

Despite its triviality, this replacement does produce an approximate sum which 

agrees exactly with perturbation theory to lowest order. It therefore can be considered a type of Pade approximation. 

At the infinite Reynolds number asymptotic limit 

, c --(0) II- - = -CR-u-u- lJ K t J (12) 

where, in the Yakhot-Orszag theory, CR = Vic rv 1.6. Equation (12) is therefore 

simply the standard Rotta model with Rotta constant rv 1.6 in agreement with an 

earlier proposal of Yakhot 17. 

A preliminary discussion of higher order summation may be appropriate. By 

analyzing the spectral dynamics of the return to isotropy, Weinstock3 concluded 

that the shear and normal stresses relax at different rates. Although this behavior 

is obviously not accommodated by the Rotta model, it is consistent with the present 

theory: the perturbation series for II' is obtained from the series for T by multiplying 

the term of order n by the factor Cncl K for some constant Cn. The Cn are all 

unequal; therefore, the Rotta model is not exact. Now comparison with the series 

for T shows5 that relaxation of the shear stress is governed by the linear term S(1), 

whereas relaxation of the normal stresses is governed by the quadratic term S(2). 

Since C2 i- C1 , these stresses relax at different rates. The difference is suppressed 

in the Rotta model, which arose in the present formalism by replacing all of the Cn 

by C1 . 

2.3 Algebraic Reynolds Stress Models 

The approximation, due to Rodi2o , of the Reynolds stress transport equation 

by an algebrajc model under the conditions of semi-homogeneous flow (negligible 

diffusion of T and T / K approximately constant) takes the form 

105 

R. Rubinstein 

(10) 

where 

dr 15 VA2 

(11) 

In view of the form of the Rotta model, it is reasonable to seek terms at the next 

order proportional to UiUj. As in Sect. I, such terms do appear, but cancel exactly. 

This apparently ubiquitous cancellation was also obtained by Smith and Reynolds18 

in an analysis of the c transport equation. Accordingly, the second order analysis 

in TJ produces quadratic terms in the velocity gradients. Finite truncation of this 

series violates the requirement that return to isotropy be independent of the mean 

flow. Therefore, we must seek a reasonable approximate summation. The form of 

the lowest order term given in Eqs. (10) and (11) suggests 

Ir- = T1 v (8Ui + 8Uj ) rv _ T 1UiU -(0) 

tJ v 8xj 8Xi V J 

Despite its triviality, this replacement does produce an approximate sum which 

agrees exactly with perturbation theory to lowest order. It therefore can be considered a type of Pade approximation. 

At the infinite Reynolds number asymptotic limit 

, c --(0) II- - = -CR-u-u- lJ K t J (12) 

where, in the Yakhot-Orszag theory, CR = Vic rv 1.6. Equation (12) is therefore 

simply the standard Rotta model with Rotta constant rv 1.6 in agreement with an 

earlier proposal of Yakhot 17. 

A preliminary discussion of higher order summation may be appropriate. By 

analyzing the spectral dynamics of the return to isotropy, Weinstock3 concluded 

that the shear and normal stresses relax at different rates. Although this behavior 

is obviously not accommodated by the Rotta model, it is consistent with the present 

theory: the perturbation series for II' is obtained from the series for T by multiplying 

the term of order n by the factor Cncl K for some constant Cn. The Cn are all 

unequal; therefore, the Rotta model is not exact. Now comparison with the series 

for T shows5 that relaxation of the shear stress is governed by the linear term S(1), 

whereas relaxation of the normal stresses is governed by the quadratic term S(2). 

Since C2 i- C1 , these stresses relax at different rates. The difference is suppressed 

in the Rotta model, which arose in the present formalism by replacing all of the Cn 

by C1 . 

2.3 Algebraic Reynolds Stress Models 

The approximation, due to Rodi2o , of the Reynolds stress transport equation 

by an algebrajc model under the conditions of semi-homogeneous flow (negligible 

diffusion of T and T / K approximately constant) takes the form 

105 
Renormalization Group Methods 

p - c _. -.(0) _ _ (_. _ aUj _. _ aUi)(O) , .. K utUJ - utup 8x

p 

+ u]up 8x

p 

+ IIij + lIt] (13) 

where II and II' depend on T and VU. Explicit solutions for T can be obtained, 

at least in principle, for any such approximation25 • Briefly, one introduces a basis 

for polynomials in VU, and VUT . The basis contains 11 terms of homogeneity 

order n ::; 5. Writing T as a sum of these terms with unknown coefficients and 

substituting in Eq. (13) leads to the explicit expression 

(14) 

where Him) is a scalar function of VU and VUT such that 

when IVUI --t 00. The assumptions made on the approximate summations require 

m + n = 0; thus, T / K is bounded when SK / c --t 00. For example, the familiar eddy 

viscosity formula is replaced in Eq. (29) by a term 

Pope observed25 that the coefficients H(-n) in Eq. (14) would certainly be intractably complex; although they could be explicitly exhibited by symbolic computation, the result would only pertain to the particular implicit equation for the 

Reynolds stresses assumed initially in Eq. (13). Therefore, it is equally reasonable just to postulate simple forms for the functions H( -n). This type of modeling 

could be particularly interesting when applied to the coefficients of the quadratically 

nonlinear models of Refs. 5, 11, and 12. 

2.4 Discussion 

The present analysis of the Reynolds stress transport equation, based on the 

Yakhot-Orszag renormalization group and (tensorial) 7]-expansion summation as 

suggested by Yakhot et a1. 9 , has led to a model transport equation incorporating the 

well-known LRR and Rotta models. The analysis gives theoretical support both to 

these models and to the constants sometimes used with them. More significantly, it 

exhibits the LRR and Rotta models as lowest order approximations, and therefore 

also supports their replacement with higher order nonlinear models which would 

be deduced by more accurate approximate summations. The consistency of the 

analysis with higher order effects like the unequal relaxation rates of shear and 

normal stresses has been discussed. 

3. Future Plans 

The nonlinear eddy viscosity representation of the Reynolds stresses 

106 

Renormalization Group Methods 

p - c _. -.(0) _ _ (_. _ aUj _. _ aUi)(O) , .. K utUJ - utup 8x

p 

+ u]up 8x

p 

+ IIij + lIt] (13) 

where II and II' depend on T and VU. Explicit solutions for T can be obtained, 

at least in principle, for any such approximation25 • Briefly, one introduces a basis 

for polynomials in VU, and VUT . The basis contains 11 terms of homogeneity 

order n ::; 5. Writing T as a sum of these terms with unknown coefficients and 

substituting in Eq. (13) leads to the explicit expression 

(14) 

where Him) is a scalar function of VU and VUT such that 

when IVUI --t 00. The assumptions made on the approximate summations require 

m + n = 0; thus, T / K is bounded when SK / c --t 00. For example, the familiar eddy 

viscosity formula is replaced in Eq. (29) by a term 

Pope observed25 that the coefficients H(-n) in Eq. (14) would certainly be intractably complex; although they could be explicitly exhibited by symbolic computation, the result would only pertain to the particular implicit equation for the 

Reynolds stresses assumed initially in Eq. (13). Therefore, it is equally reasonable just to postulate simple forms for the functions H( -n). This type of modeling 

could be particularly interesting when applied to the coefficients of the quadratically 

nonlinear models of Refs. 5, 11, and 12. 

2.4 Discussion 

The present analysis of the Reynolds stress transport equation, based on the 

Yakhot-Orszag renormalization group and (tensorial) 7]-expansion summation as 

suggested by Yakhot et a1. 9 , has led to a model transport equation incorporating the 

well-known LRR and Rotta models. The analysis gives theoretical support both to 

these models and to the constants sometimes used with them. More significantly, it 

exhibits the LRR and Rotta models as lowest order approximations, and therefore 

also supports their replacement with higher order nonlinear models which would 

be deduced by more accurate approximate summations. The consistency of the 

analysis with higher order effects like the unequal relaxation rates of shear and 

normal stresses has been discussed. 

3. Future Plans 

The nonlinear eddy viscosity representation of the Reynolds stresses 

106 
R. Rubinstein 

T' = U ·U · = ~Kb .. _ C K2 (8Ui + 8Uj) tJ t J 3 tJ v 8 8 [ Xj Xi 

K3 [ (8Ui 8Uj ) (0) (8Ui 8Up 8Uj 8Up) (0) (8Up 8Up) (0)] +- Cr! --- +Cr2 ---+---- +Cr3 ----

[2 8xp 8xp 8xp 8Xj 8xp 8Xi 8Xi 8xj (15) 

in which K is the turbulence kinetic energy, [is the dissipation rate, Cv , Cr !, Cr2 , Cr3 

are constants, and (0) denotes deviatoric part, was proposed by Yoshizawall in order to model normal stress effects in shear flows by means of an explicit formula 

for the stresses. The significance of this formula is not limited to this property: 

Yoshizawa's derivation using a special perturbation expansion, the two-scale direct 

interaction approximation, showed that the expansion could be continued to any 

order in the mean velocity gradient and thereby exhibited the Reynolds stress tensor as the result of an infinite number of increasingly complex interactions between 

the mean velocity field and turbulence. Related expansions are given in Refs. 5, 12. 

The infinite expansion which contains Eq. (15) can be written symbolically as 

Eq. (16) can be considered a decomposition of the Reynolds stress 

(17) 

where 

(184) 

where Sf denotes a symmetric tensor homogeneous of degree n in the mean velocity 

gradient \lU and its transpose, N n denotes the number of linearly independent 

terms of order n (so in Eq. (15), N2 = 3), and the Ain are constants. 

In an analysis of the Reynolds stress transport .equation by renormalization group 

teclmiques35 , we found analogous expansions for the term which governs the return 

to isotropy, 

(19) 

and for the rapid pressure-strain term 

107 

R. Rubinstein 

T' = U ·U · = ~Kb .. _ C K2 (8Ui + 8Uj) tJ t J 3 tJ v 8 8 [ Xj Xi 

K3 [ (8Ui 8Uj ) (0) (8Ui 8Up 8Uj 8Up) (0) (8Up 8Up) (0)] +- Cr! --- +Cr2 ---+---- +Cr3 ----

[2 8xp 8xp 8xp 8Xj 8xp 8Xi 8Xi 8xj (15) 

in which K is the turbulence kinetic energy, [is the dissipation rate, Cv , Cr !, Cr2 , Cr3 

are constants, and (0) denotes deviatoric part, was proposed by Yoshizawall in order to model normal stress effects in shear flows by means of an explicit formula 

for the stresses. The significance of this formula is not limited to this property: 

Yoshizawa's derivation using a special perturbation expansion, the two-scale direct 

interaction approximation, showed that the expansion could be continued to any 

order in the mean velocity gradient and thereby exhibited the Reynolds stress tensor as the result of an infinite number of increasingly complex interactions between 

the mean velocity field and turbulence. Related expansions are given in Refs. 5, 12. 

The infinite expansion which contains Eq. (15) can be written symbolically as 

Eq. (16) can be considered a decomposition of the Reynolds stress 

(17) 

where 

(184) 

where Sf denotes a symmetric tensor homogeneous of degree n in the mean velocity 

gradient \lU and its transpose, N n denotes the number of linearly independent 

terms of order n (so in Eq. (15), N2 = 3), and the Ain are constants. 

In an analysis of the Reynolds stress transport .equation by renormalization group 

teclmiques35 , we found analogous expansions for the term which governs the return 

to isotropy, 

(19) 

and for the rapid pressure-strain term 

107 
Renormalization Group Methods 

+ ~3 .L [Ci2 (S;\1UT + \1USl) (0) + D i2 (S;\1U + \1 UT Sl) ] + ... (20) 

t"S.N2 

Suppose that these sums are introduced into the Reynolds stress transport equation 

i = II' + II - P + D (21) 

where the dot denotes convective derivative, P = T\1UT + \1UT is production and D 

denotes the diffusion term. In order to obtain a Reynolds stress transport equation, 

it is necessary to express the sums (19) and (20) in terms of T and \1U. Although 

the coefficients A, B, C, D can be explicitly exhibited to any order in perturbation 

theory, they do not have an obvious law of formation. Therefore, the sums (19) 

and (20) can only be approximated by polynomials in T and \1U if some hypotheses 

relating the coefficients is introduced. There is no unique hypothesis of this sort, 

but the simplest35 seems to be 

(22) 

which leads to the Rotta return to isotropy model and to an LRR model for the 

rapid term. The approximation expressed by Eq. (22) can be compared to the 

summation introduced in an analogous context by Yakhot et al. 9 , and to Pade 

approximation: it agrees with the perturbation theory of Eqs. (19), (20) to lowest 

order, but includes effects of all order in \1U. 

By evaluating more terms of the perturbation series explicitly and introducing 

an equation like (8) for coefficients of higher order, a hierarchy of models could be 

generated. They would initially be nonlinear in \1U, as advocated by Speziale33 , 

but one might introduce the perturbation series for T' T to obtain a model nonlinear 

in T 1 ,2. However, the close analogy between Eqs. (19) and (20) and Yoshizawa's 

expansion (16) suggests a different approach: namely, use Eqs. (17) and (18) to 

replace SF" in Eqs. (19) and (20) by Tt / Ain. Substitute these modified expressions 

and Eq. (17) into the transport equation Eq. (21), treat Tt as having order I \1U In, and separate the terms oflike order in I \1 U I in ·the standard perturbation theoretic 

fashion. The result is that the terms Tt in the decomposition (17), (18) themselves 

satisfy coupled transport equations. 

For simplicity, let us write an approximate system for the Tn instead of for the 

Tt and assume the most elementary scalar diffusion model. Then the the single 

transport equation for T would be replaced by a system 

K2 

+Cn (Tn- 1\1U + \1UT TlI - 1 ) (0) + Cn _ \12Tn n > 1 2 Dc' - (23) 

Since T = L: Tn, the system (6) should be constrained to satisfy Crow's condition 

and to contain the exact production term following summation over n . Making the 

108 

Renormalization Group Methods 

+ ~3 .L [Ci2 (S;\1UT + \1USl) (0) + D i2 (S;\1U + \1 UT Sl) ] + ... (20) 

t"S.N2 

Suppose that these sums are introduced into the Reynolds stress transport equation 

i = II' + II - P + D (21) 

where the dot denotes convective derivative, P = T\1UT + \1UT is production and D 

denotes the diffusion term. In order to obtain a Reynolds stress transport equation, 

it is necessary to express the sums (19) and (20) in terms of T and \1U. Although 

the coefficients A, B, C, D can be explicitly exhibited to any order in perturbation 

theory, they do not have an obvious law of formation. Therefore, the sums (19) 

and (20) can only be approximated by polynomials in T and \1U if some hypotheses 

relating the coefficients is introduced. There is no unique hypothesis of this sort, 

but the simplest35 seems to be 

(22) 

which leads to the Rotta return to isotropy model and to an LRR model for the 

rapid term. The approximation expressed by Eq. (22) can be compared to the 

summation introduced in an analogous context by Yakhot et al. 9 , and to Pade 

approximation: it agrees with the perturbation theory of Eqs. (19), (20) to lowest 

order, but includes effects of all order in \1U. 

By evaluating more terms of the perturbation series explicitly and introducing 

an equation like (8) for coefficients of higher order, a hierarchy of models could be 

generated. They would initially be nonlinear in \1U, as advocated by Speziale33 , 

but one might introduce the perturbation series for T' T to obtain a model nonlinear 

in T 1 ,2. However, the close analogy between Eqs. (19) and (20) and Yoshizawa's 

expansion (16) suggests a different approach: namely, use Eqs. (17) and (18) to 

replace SF" in Eqs. (19) and (20) by Tt / Ain. Substitute these modified expressions 

and Eq. (17) into the transport equation Eq. (21), treat Tt as having order I \1U In, and separate the terms oflike order in I \1 U I in ·the standard perturbation theoretic 

fashion. The result is that the terms Tt in the decomposition (17), (18) themselves 

satisfy coupled transport equations. 

For simplicity, let us write an approximate system for the Tn instead of for the 

Tt and assume the most elementary scalar diffusion model. Then the the single 

transport equation for T would be replaced by a system 

K2 

+Cn (Tn- 1\1U + \1UT TlI - 1 ) (0) + Cn _ \12Tn n > 1 2 Dc' - (23) 

Since T = L: Tn, the system (6) should be constrained to satisfy Crow's condition 

and to contain the exact production term following summation over n . Making the 

108 
R. Rubinstein 

coefficients ell, er, elf, cD; independent of n reduces the system to a model of the 

LRR form for r = 2: rn. 

This conclusion also follows from Leslie's analysis34 of the direct interaction (DIA) 

equations for shear flow. Leslie suggested a perturbative solution for the (tensor) 

correlation function and Green's function 

u = UO 

G = GO 

+ .. . + .. . (24) 

where un and Gn are of the order I VU I't, and observed that this expansion is 

simultaneously an expansion in powers of the mean strain, and a decomposition into 

symmetry types of increasing complexity. This is also a feature of the expansion 

(16). Substitution of Eq. (20) into the equations of the direct interaction approximation gives a coupled system for the un and Gn in standard fashion. Then in 

principle, by integrating each equation of this system over all wavenumbers and 

introducing the definitions 

rn = J Und k 

we could attempt to obtain coupled transport equations for the Tn. Unfortunately, 

the derivation of equations for single-point quantities from DIA is not entirely 

straightforward, and more heuristic methods like two-scale DIA 8,14 and renormalization group are required. 

The simplest model of this type is a two component model, 

with 

i 1 = - CkT1 - l~K (VU + VUT ) + (ct -1) (T2VUT + VUT2) (0) 

+ ci (r2VU + VUT T2)(0) + cb V

K2 

VT1 

C 

72 = - CjzT2 + (e; -1) (r 1VUT + VUr1)(0) 

+ e? (T1VU + VUT r1) (0) + ebvK2 Vr2 

c 

(25) 

The appearance of T2 in the equation for T1 is required by the production term 

constraint which is violated by direct truncation of the system (20) at the second 

order. The properties of this model for simple shear flow, in which 8Ui /8xj 

S Di1Dj2 follow by setting 

J 

(0) 

T33 

(26) 

109 

R. Rubinstein 

coefficients ell, er, elf, cD; independent of n reduces the system to a model of the 

LRR form for r = 2: rn. 

This conclusion also follows from Leslie's analysis34 of the direct interaction (DIA) 

equations for shear flow. Leslie suggested a perturbative solution for the (tensor) 

correlation function and Green's function 

u = UO 

G = GO 

+ .. . + .. . (24) 

where un and Gn are of the order I VU I't, and observed that this expansion is 

simultaneously an expansion in powers of the mean strain, and a decomposition into 

symmetry types of increasing complexity. This is also a feature of the expansion 

(16). Substitution of Eq. (20) into the equations of the direct interaction approximation gives a coupled system for the un and Gn in standard fashion. Then in 

principle, by integrating each equation of this system over all wavenumbers and 

introducing the definitions 

rn = J Und k 

we could attempt to obtain coupled transport equations for the Tn. Unfortunately, 

the derivation of equations for single-point quantities from DIA is not entirely 

straightforward, and more heuristic methods like two-scale DIA 8,14 and renormalization group are required. 

The simplest model of this type is a two component model, 

with 

i 1 = - CkT1 - l~K (VU + VUT ) + (ct -1) (T2VUT + VUT2) (0) 

+ ci (r2VU + VUT T2)(0) + cb V

K2 

VT1 

C 

72 = - CjzT2 + (e; -1) (r 1VUT + VUr1)(0) 

+ e? (T1VU + VUT r1) (0) + ebvK2 Vr2 

c 

(25) 

The appearance of T2 in the equation for T1 is required by the production term 

constraint which is violated by direct truncation of the system (20) at the second 

order. The properties of this model for simple shear flow, in which 8Ui /8xj 

S Di1Dj2 follow by setting 

J 

(0) 

T33 

(26) 

109 
Renormalization Group Methods 

Then C1 describes the return to isotropy of the shear stress, and C'k the return 

to isotropy of the normal stresses. These relaxation rates can be unequal in this 

theory, an effect predicted by Weinstock's analysis3 of the spectral dynamics of the 

return to isotropy. Weinstock suggested further that the individual normal stresses 

relax. at different rates: this effect is not accommodated by the present model, but 

could occur in a model in which T2 is divided into the three tensor components 

T[, Ti, T; of Eq. (18). 

The inequality of the Rotta constants in the present model can be used to overcome a defect of models of the LRR type, that in semi- homogeneous flows in which 

aij = TijO) j K is approximately constant, the ratio all is too small whereas al2 is 

too biglO . Following Speziale19 , write Eq. (21) as a system of equations for the 

ratios aij and set aij = o. There results, 

[-~ (Ct - 1) + ~Ci ]7Ja12 

all = -"-----::--......,----=---

(-CA + 1) + 7Ja12 (27) 

[~ (Ct - 1) - ~Ci ]7Ja12 

a22 = -=----=-------=--- (-CA + 1) + 7Ja12 

(28) 

[

2 ( 2 ) 4 2] 2 a22 = 3 C1 - 1 - 3C2 JCR 

Eq. (28) shows that setting C'k < Ch both increases all and decreases al2 and 

thus improves the agreement between theory and experiment. The trend required 

here is consistent with Weinstock's findings3 that the Rotta constant for the normal 

stresses should be smaller than the shear constant and should take values close to 

1.0. 

The inequality of the coefficients describing the rapid terms affects the behavior 

of the model under rapid distortion. The rapid distortion analysis of passively 

strained turbulence predicts that a one-component limit state is reached in which 

Tij = 2K bilbjl, 7J = S K j c ~ 00 and P j c is finite 21 . Whether or not this limit 

can in fact occur as an asymptotic state, a stress model should accommodate it 

because it can exist approximately in steady flows as a "spatial transient" in strongly 

110 

Renormalization Group Methods 

Then C1 describes the return to isotropy of the shear stress, and C'k the return 

to isotropy of the normal stresses. These relaxation rates can be unequal in this 

theory, an effect predicted by Weinstock's analysis3 of the spectral dynamics of the 

return to isotropy. Weinstock suggested further that the individual normal stresses 

relax. at different rates: this effect is not accommodated by the present model, but 

could occur in a model in which T2 is divided into the three tensor components 

T[, Ti, T; of Eq. (18). 

The inequality of the Rotta constants in the present model can be used to overcome a defect of models of the LRR type, that in semi- homogeneous flows in which 

aij = TijO) j K is approximately constant, the ratio all is too small whereas al2 is 

too biglO . Following Speziale19 , write Eq. (21) as a system of equations for the 

ratios aij and set aij = o. There results, 

[-~ (Ct - 1) + ~Ci ]7Ja12 

all = -"-----::--......,----=---

(-CA + 1) + 7Ja12 (27) 

[~ (Ct - 1) - ~Ci ]7Ja12 

a22 = -=----=-------=--- (-CA + 1) + 7Ja12 

(28) 

[

2 ( 2 ) 4 2] 2 a22 = 3 C1 - 1 - 3C2 JCR 

Eq. (28) shows that setting C'k < Ch both increases all and decreases al2 and 

thus improves the agreement between theory and experiment. The trend required 

here is consistent with Weinstock's findings3 that the Rotta constant for the normal 

stresses should be smaller than the shear constant and should take values close to 

1.0. 

The inequality of the coefficients describing the rapid terms affects the behavior 

of the model under rapid distortion. The rapid distortion analysis of passively 

strained turbulence predicts that a one-component limit state is reached in which 

Tij = 2K bilbjl, 7J = S K j c ~ 00 and P j c is finite 21 . Whether or not this limit 

can in fact occur as an asymptotic state, a stress model should accommodate it 

because it can exist approximately in steady flows as a "spatial transient" in strongly 

110 
R. Rubinstein 

inhomogeneous regions of very high convective or diffusive transport17 . Moreover, 

models of the LRR form do not capture the transient evolution of rapidly distorted 

flows well. Let us assume that all model coefficients are functions of the " state" 

of turbulence, following Shih and Lumley2; the precise parametrization of the state 

will be left to future investigations. From Eq. (13), it is evident that P / E = -TJa12 

will be of order TJ in the one-component state in which all = 4/3, a22 = -2/3 unless 

2( 1 ) 41 4 -"3 C1 - 1 +"3 C2 = 15 

in this state. But Eq. (13) also shows that all = 4/3, a22 = -2/3 requires 

cf rv 0, 

in this state. These conditions are inconsistent in a model in which Cf = Cf and 

Ci = Ci, but are clearly consistent with the present proposal. 

The advantages of this model are consistency with a systematic perturbation 

theory, the possibility of unequal relaxation rates for normal and shear stresses 

in relaxing strained turbulence, improved agreement with experimental data for 

universal ratios in simple shear flows, and the possibility of accommodating the 

one-component limit of rapid distortion theory. 

4. References 

1 C. G. Speziale, S. Sarkar, and T. Gatski, " Modeling the pressure-strain correlation 

of turbulence - an invariant dynamical systems approach," J. Fluid Mech. to 

appear. 

2 T.R. Shih and J.L. Lumley, " Modeling of pressure correlation terms in Reynolds 

stress and scalar flux equations," Rept. FDA-85-3, Sibley School of Mech. and 

Aero. Eng. Cornell U. (1985). 

3 J. Weinstock, "Theory of the pressure-strain rate: Part II, diagonal elements" J. 

Fluid Mech. 116, 1 (1982). 

4 V. Yakhot and S. A. Orszag, "Renormalization group analysis of turbulence I. 

Basic theory," J. Sci. Compo 1, 3 (1986). 

5 R. Rubinstein and J.M. Barton, " Nonlinear Reynolds stress models and the renormalization group," Phys. Fluids A 2, 1472 (1990). 

6 B. E. Launder, G. Reece, and W. Rodi, "Progress in the development of a 

Reynolds-Stress turbulence closure," J. Fluid Mech. 68 537 (1975). 

7 J. Rotta, " Statistical theory of non-homogeneous turbulence," Z. Phys. 129,547 

(1951). 

8 A. Yoshizawa, "Statistical evaluation of the triple velocity correlation and the 

pressure-velocity correlation in shear turbulence," J. Phys. Soc. Jpn. 51, 2326 

(1982). 

111 

R. Rubinstein 

inhomogeneous regions of very high convective or diffusive transport17 . Moreover, 

models of the LRR form do not capture the transient evolution of rapidly distorted 

flows well. Let us assume that all model coefficients are functions of the " state" 

of turbulence, following Shih and Lumley2; the precise parametrization of the state 

will be left to future investigations. From Eq. (13), it is evident that P / E = -TJa12 

will be of order TJ in the one-component state in which all = 4/3, a22 = -2/3 unless 

2( 1 ) 41 4 -"3 C1 - 1 +"3 C2 = 15 

in this state. But Eq. (13) also shows that all = 4/3, a22 = -2/3 requires 

cf rv 0, 

in this state. These conditions are inconsistent in a model in which Cf = Cf and 

Ci = Ci, but are clearly consistent with the present proposal. 

The advantages of this model are consistency with a systematic perturbation 

theory, the possibility of unequal relaxation rates for normal and shear stresses 

in relaxing strained turbulence, improved agreement with experimental data for 

universal ratios in simple shear flows, and the possibility of accommodating the 

one-component limit of rapid distortion theory. 

4. References 

1 C. G. Speziale, S. Sarkar, and T. Gatski, " Modeling the pressure-strain correlation 

of turbulence - an invariant dynamical systems approach," J. Fluid Mech. to 

appear. 

2 T.R. Shih and J.L. Lumley, " Modeling of pressure correlation terms in Reynolds 

stress and scalar flux equations," Rept. FDA-85-3, Sibley School of Mech. and 

Aero. Eng. Cornell U. (1985). 

3 J. Weinstock, "Theory of the pressure-strain rate: Part II, diagonal elements" J. 

Fluid Mech. 116, 1 (1982). 

4 V. Yakhot and S. A. Orszag, "Renormalization group analysis of turbulence I. 

Basic theory," J. Sci. Compo 1, 3 (1986). 

5 R. Rubinstein and J.M. Barton, " Nonlinear Reynolds stress models and the renormalization group," Phys. Fluids A 2, 1472 (1990). 

6 B. E. Launder, G. Reece, and W. Rodi, "Progress in the development of a 

Reynolds-Stress turbulence closure," J. Fluid Mech. 68 537 (1975). 

7 J. Rotta, " Statistical theory of non-homogeneous turbulence," Z. Phys. 129,547 

(1951). 

8 A. Yoshizawa, "Statistical evaluation of the triple velocity correlation and the 

pressure-velocity correlation in shear turbulence," J. Phys. Soc. Jpn. 51, 2326 

(1982). 

111 
Renormalization Group Methods 

9 V. Yakhot, S. Thangam, T. Gatski, S. A. Orszag, and C. G. Speziale, "Development of turbulence models for shear flows by a double expansion technique," 

submitted to Phys. Fluids A. 

10 V. Yakhot and S. A. Orszag, "Analysis of the E-exparysion in turbulence theory: 

approximate renormalization group for diffusion of a passive scalar in a random 

velocity field," Princeton Univ. preprint (1989). 

11 A. Yoshizawa, "Statistical analysis of the deviation of the Reynolds stress from 

its eddy-viscosity representation," Phys. Fluids 27, 1377 (1984). 

12 C. G. Speziale, "On nonlinear K-I and K-E models of turbulence," J. Fluid Mech. 

178, 459 (1987). 

13 A. J. M. Spencer and R. S. Rivlin, "Further results in the theory of matrix 

polynomials," Arch. Rational Mech. Anal. 4, 214 (1960). 

14 A. Yoshizawa, "Deviation of a model Reynolds stress transport equation using a 

renormalization of the eddy-viscosity-type representation," preprint. 

15 V. Yakhot and L. M. Smith, "The E-expansion for derivation of turbulence models," submitted to Phys. Fluids A. 

16 S. Crow, "Viscoelastic properties of fine grained incompressible turbulence," J. 

Fluid Mech. 41, 81 (1968). 

17 V. Yakhot, private communication. 

18 L. M. Smith and W. C. Reynolds, "A critical assessment of the Yakhot-Orszag 

renormalization group method for deriving turbulence models and statistics," 

submitted to Physics of Fluids A (1991). 

19 C. G. Speziale and N. MacGiolla Mhuiris, "On the prediction of equilibrium st ates 

in homogeneous turbulence," J. Fluid Mech. 209, 591 (1989). 

20 W. Rodi, "A new algebraic relation for computing the Reynolds stresses," Z. Ang. 

Math. Mech. 56, 219 (1976). 

21 W. C. Reynolds, "Fundamentals of turbulence for turbulence modeling and simulation," AGARD Lecture Series No. 86, 1 (1987). 

22 B. E. Launder, "Second-moment closure: present ... and future?," Int. J. Heat 

and Fluid Flow 10, 282 (1989). 

23 R. Rubinstein and J .M. Barton, "Renormalization group analysis of anisotropic 

diffusion in turbulent shear fl.ow," Phys. Fluids A 3, 415 (1991). 

24 A. Yoshizawa, "Statistical modelling of passive scalar diffusion in turbulent shear 

fl.ows," J. Fluid Mech. 195, 541 (1988). 

25 S. G. Pope, "A more general effective-viscosity hypothesis," J. Fluid Mech. 72, 

331 (1975). 

112 

Renormalization Group Methods 

9 V. Yakhot, S. Thangam, T. Gatski, S. A. Orszag, and C. G. Speziale, "Development of turbulence models for shear flows by a double expansion technique," 

submitted to Phys. Fluids A. 

10 V. Yakhot and S. A. Orszag, "Analysis of the E-exparysion in turbulence theory: 

approximate renormalization group for diffusion of a passive scalar in a random 

velocity field," Princeton Univ. preprint (1989). 

11 A. Yoshizawa, "Statistical analysis of the deviation of the Reynolds stress from 

its eddy-viscosity representation," Phys. Fluids 27, 1377 (1984). 

12 C. G. Speziale, "On nonlinear K-I and K-E models of turbulence," J. Fluid Mech. 

178, 459 (1987). 

13 A. J. M. Spencer and R. S. Rivlin, "Further results in the theory of matrix 

polynomials," Arch. Rational Mech. Anal. 4, 214 (1960). 

14 A. Yoshizawa, "Deviation of a model Reynolds stress transport equation using a 

renormalization of the eddy-viscosity-type representation," preprint. 

15 V. Yakhot and L. M. Smith, "The E-expansion for derivation of turbulence models," submitted to Phys. Fluids A. 

16 S. Crow, "Viscoelastic properties of fine grained incompressible turbulence," J. 

Fluid Mech. 41, 81 (1968). 

17 V. Yakhot, private communication. 

18 L. M. Smith and W. C. Reynolds, "A critical assessment of the Yakhot-Orszag 

renormalization group method for deriving turbulence models and statistics," 

submitted to Physics of Fluids A (1991). 

19 C. G. Speziale and N. MacGiolla Mhuiris, "On the prediction of equilibrium st ates 

in homogeneous turbulence," J. Fluid Mech. 209, 591 (1989). 

20 W. Rodi, "A new algebraic relation for computing the Reynolds stresses," Z. Ang. 

Math. Mech. 56, 219 (1976). 

21 W. C. Reynolds, "Fundamentals of turbulence for turbulence modeling and simulation," AGARD Lecture Series No. 86, 1 (1987). 

22 B. E. Launder, "Second-moment closure: present ... and future?," Int. J. Heat 

and Fluid Flow 10, 282 (1989). 

23 R. Rubinstein and J .M. Barton, "Renormalization group analysis of anisotropic 

diffusion in turbulent shear fl.ow," Phys. Fluids A 3, 415 (1991). 

24 A. Yoshizawa, "Statistical modelling of passive scalar diffusion in turbulent shear 

fl.ows," J. Fluid Mech. 195, 541 (1988). 

25 S. G. Pope, "A more general effective-viscosity hypothesis," J. Fluid Mech. 72, 

331 (1975). 

112 
R. Rubinstein 

26 K. Horiuti, "Higher order terms in the anisotropic representation of Reynolds 

stresses," Phys. Fluids A 2, 1708 (1990). 

27 S. Tavoularis and S. C. Corrsin, "Experiments in nearly homogeneous turbulent 

shear flow with a uniform mean temperature gradient Part 1," J. Fluid Mech. 

104, 311 (1981). 

28 M. M. Rogers, P. Moin and W. Reynolds, "The structure and modeling of the 

hydrodynamic and passive scalar fields in homogeneous shear flows," Dept. Mech. 

Eng. Rep. TF-25, Stanford U. (1986). 

29 S. Tavoularis and U. Karnik, "Further experiments on the evolution of turbulent 

stresses and scales in uniformly sheared turbulence," J. Fluid Mech. 204, 457 

(1989). 

30 P. Spalart, "Direct simulation of a turbulent boundary layer up to R() = 1410," 

NASA TM 89407 (1986). 

31 J. Laufer, "The structure of turbulence in fully developed pipe flow," NACA 

Tech. Rep. 1174 (1954). 

32 V. G. garris, J. A. Graham and S. C. Corrsin, "Further experiments in nearly 

homogeneous turbulent shear flow," J. Fluid Mech. 81, 657 (1977). 

33 C. G. Speziale, "Comments on the present state of second-order closure models 

for incompressible flows," Workshop on Engineering Turbulence Modeling, NASA 

CP-10088, 229 (1991). 

34 D. C: Leslie, Developments in the Theory of Turbulence, Clarendon Press, Oxford 

(1973). 

35 R. Rubinstein and J. M. Barton, "Renormalization group analysis of the Reynolds 

stress transport equation," to appear in Physics of Fluids A, (1992). 

113 

R. Rubinstein 

26 K. Horiuti, "Higher order terms in the anisotropic representation of Reynolds 

stresses," Phys. Fluids A 2, 1708 (1990). 

27 S. Tavoularis and S. C. Corrsin, "Experiments in nearly homogeneous turbulent 

shear flow with a uniform mean temperature gradient Part 1," J. Fluid Mech. 

104, 311 (1981). 

28 M. M. Rogers, P. Moin and W. Reynolds, "The structure and modeling of the 

hydrodynamic and passive scalar fields in homogeneous shear flows," Dept. Mech. 

Eng. Rep. TF-25, Stanford U. (1986). 

29 S. Tavoularis and U. Karnik, "Further experiments on the evolution of turbulent 

stresses and scales in uniformly sheared turbulence," J. Fluid Mech. 204, 457 

(1989). 

30 P. Spalart, "Direct simulation of a turbulent boundary layer up to R() = 1410," 

NASA TM 89407 (1986). 

31 J. Laufer, "The structure of turbulence in fully developed pipe flow," NACA 

Tech. Rep. 1174 (1954). 

32 V. G. garris, J. A. Graham and S. C. Corrsin, "Further experiments in nearly 

homogeneous turbulent shear flow," J. Fluid Mech. 81, 657 (1977). 

33 C. G. Speziale, "Comments on the present state of second-order closure models 

for incompressible flows," Workshop on Engineering Turbulence Modeling, NASA 

CP-10088, 229 (1991). 

34 D. C: Leslie, Developments in the Theory of Turbulence, Clarendon Press, Oxford 

(1973). 

35 R. Rubinstein and J. M. Barton, "Renormalization group analysis of the Reynolds 

stress transport equation," to appear in Physics of Fluids A, (1992). 

113 
Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Bypass Transition in 

Compressible Boundary Layers 

J. J. Van der Vegt 

1. Motivation and Objectives 

Transition to turbulence in aerospace applications usually occurs in a strongly disturbed environment. For instance, the effects of free-stream turbulence, roughness 

and obstacles in the boundary layer strongly influence transition. Proper understanding of the mechanisms leading to transition is crucial in the design of aircraft 

wings and gas turbine blades, because lift, drag and heat transfer strongly depend 

on the state of the boundary layer, laminar or turbulent. Unfortunately, most of 

the transition research, both theoretical and experimental, has focused on natural 

transition. Many practical flows, however, defy any theoretical analysis and are extremely difficult to measure. Morkovin5 introduced in his review paper the concept 

of bypass transition as those forms of transition which bypass the known mechanisms of linear and non-linear transition theories and are currently not understood 

by experiments. 

In an effort to better understand the mechanisms leading to transition in an 

disturbed environment, experiments are conducted studying simpler cases, viz. the 

effects of free-stream turbulence on transition on a flat plate, Sohn and Reshotko14 

and Wang et al. 19 . It turns out that these experiments are very difficult to conduct, 

because the generation of free-stream turbulence with sufficiently high fluctuation 

levels and reasonable homogeneity is non trivial. For a discussion see Morkovin5 . 

Serious problems also appear due to the fact that at high Reynolds numbers the 

boundary layers are very thin, especially in the nose region of the plate where the 

transition occurs, which makes the use of very small probes necessary. 

The effects of free-stream turbulence on transition are the subject of this research and are especially important in a gas turbine environment, where turbu.;' 

lence intensities are measured between 5 and 20%, Wang et al.19 . Due to the fact 

that the Reynolds number for turbine blades is considerably lower than for aircraft 

wings, generally a larger portion of the blade will be in a laminar-transitional state. 

Turner15 shows that the effect of free-stream turbulence on transition significantly 

increases when the free-stream turbulence levels become larger than 5% and is accompanied with a significant increase in heat transfer. Recently Rai and Moinll 

presented a direct numerical simulation of transition to turbulence on a flat plate 

in a free-stream with Mach number .1 and turbulence levels at the leading edge 

of about 2.75%. Direct numerical simulations offer a unique opportunity to study 

specific phenomena, while excluding disturbances from other sources. The computations from Rai and Moin show some impressive results, especially regarding 

intermittency and turbulent spots. Their numerical simulation, however, has the 

same problem as with most of the experiments, namely a very low Mach number, 

115 Preceding Page BI·ank 

--- - - - - -

Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Bypass Transition in 

Compressible Boundary Layers 

J. J. Van der Vegt 

1. Motivation and Objectives 

Transition to turbulence in aerospace applications usually occurs in a strongly disturbed environment. For instance, the effects of free-stream turbulence, roughness 

and obstacles in the boundary layer strongly influence transition. Proper understanding of the mechanisms leading to transition is crucial in the design of aircraft 

wings and gas turbine blades, because lift, drag and heat transfer strongly depend 

on the state of the boundary layer, laminar or turbulent. Unfortunately, most of 

the transition research, both theoretical and experimental, has focused on natural 

transition. Many practical flows, however, defy any theoretical analysis and are extremely difficult to measure. Morkovin5 introduced in his review paper the concept 

of bypass transition as those forms of transition which bypass the known mechanisms of linear and non-linear transition theories and are currently not understood 

by experiments. 

In an effort to better understand the mechanisms leading to transition in an 

disturbed environment, experiments are conducted studying simpler cases, viz. the 

effects of free-stream turbulence on transition on a flat plate, Sohn and Reshotko14 

and Wang et al. 19 . It turns out that these experiments are very difficult to conduct, 

because the generation of free-stream turbulence with sufficiently high fluctuation 

levels and reasonable homogeneity is non trivial. For a discussion see Morkovin5 . 

Serious problems also appear due to the fact that at high Reynolds numbers the 

boundary layers are very thin, especially in the nose region of the plate where the 

transition occurs, which makes the use of very small probes necessary. 

The effects of free-stream turbulence on transition are the subject of this research and are especially important in a gas turbine environment, where turbu.;' 

lence intensities are measured between 5 and 20%, Wang et al.19 . Due to the fact 

that the Reynolds number for turbine blades is considerably lower than for aircraft 

wings, generally a larger portion of the blade will be in a laminar-transitional state. 

Turner15 shows that the effect of free-stream turbulence on transition significantly 

increases when the free-stream turbulence levels become larger than 5% and is accompanied with a significant increase in heat transfer. Recently Rai and Moinll 

presented a direct numerical simulation of transition to turbulence on a flat plate 

in a free-stream with Mach number .1 and turbulence levels at the leading edge 

of about 2.75%. Direct numerical simulations offer a unique opportunity to study 

specific phenomena, while excluding disturbances from other sources. The computations from Rai and Moin show some impressive results, especially regarding 

intermittency and turbulent spots. Their numerical simulation, however, has the 

same problem as with most of the experiments, namely a very low Mach number, 

115 Preceding Page BI·ank 

--- - - - - -
Jacobus J. Van der Vegt 

while many applications operate in the transonic regime. Due the nature of their 

numerical scheme, a non-conservation formulation of the Navier-Stokes equations, 

it is a non-trivial extension to compute flow fields in the transonic regime. 

This project aims at better understanding the effects of large free-stream turbulence in compressible boundary layers at Mach numbers both in the subsonic 

and transonic regime using direct numerical simulations. The present project aims 

at computing the flow over a flat plate and curved surface. This research will 

provide data which can be used to clarify mechanisms leading to transition in an 

environment with high free stream turbulence. This information is useful for the 

development of turbulence models, which are of great importance for CFD applications, and are currently unreliable for more complex flows, such as transitional 

flows. 

2. Accomplishments 

Direct simulations of transition in compressible flows with both shocks and boundary layers requires an extremely accurate and efficient scheme. Several conflicting 

requirements present a serious challenge which cannot be met by existing numerical 

schemes: 

• The small grid spacing in the boundary layer makes an implicit scheme necessary, 

because an explicit scheme would have a severe time step limitation. Implicit 

schemes usually are not time accurate and rather "dissipative. 

• Higher order accurate schemes are necessary but higher order accurate schemes 

generally do not give non-oscillatory solutions around discontinuities, such as 

shocks. Many of the popular non-oscillatory shock capturing schemes, such as 

TVD (Total Variation Diminishing) methods, are only first order accurate in 

multi-dimensional flows and even in one-dimension they reduce to first order at 

non-sonic local extrema. 

In order to satisfy these conflicting requirements a significant effort has been made 

to improve and combine several successful numerical schemes. A fully implicit and 

time accurate code for the solution of the three-dimensional compressible NavierStokes equations in general geometries has been written and tested. Higher order 

accuracy and shock capturing are implemented using an Essentially Non-Oscillatory 

(ENO) scheme. Time accuracy is obtained using a Newton method. 

In the next section a brief description of the numerical scheme will be given 

followed by the discussion of a series of tests aimed at validating the code. 

2.1 Numerical Scheme 

The compressible Navier-Stokes equations are solved using a finite volume method .• 

A detailed discussion of finite volume and difference methods can be found in 

Vinokur18 . The integral formulation of the Navier-Stokes equations, assuming all 

variables are continuous in time, is given by: 

~ r U dV + j n . FdS = 0 

ut JV(t) JS(t) 

(2.1.1) 

116 

Jacobus J. Van der Vegt 

while many applications operate in the transonic regime. Due the nature of their 

numerical scheme, a non-conservation formulation of the Navier-Stokes equations, 

it is a non-trivial extension to compute flow fields in the transonic regime. 

This project aims at better understanding the effects of large free-stream turbulence in compressible boundary layers at Mach numbers both in the subsonic 

and transonic regime using direct numerical simulations. The present project aims 

at computing the flow over a flat plate and curved surface. This research will 

provide data which can be used to clarify mechanisms leading to transition in an 

environment with high free stream turbulence. This information is useful for the 

development of turbulence models, which are of great importance for CFD applications, and are currently unreliable for more complex flows, such as transitional 

flows. 

2. Accomplishments 

Direct simulations of transition in compressible flows with both shocks and boundary layers requires an extremely accurate and efficient scheme. Several conflicting 

requirements present a serious challenge which cannot be met by existing numerical 

schemes: 

• The small grid spacing in the boundary layer makes an implicit scheme necessary, 

because an explicit scheme would have a severe time step limitation. Implicit 

schemes usually are not time accurate and rather "dissipative. 

• Higher order accurate schemes are necessary but higher order accurate schemes 

generally do not give non-oscillatory solutions around discontinuities, such as 

shocks. Many of the popular non-oscillatory shock capturing schemes, such as 

TVD (Total Variation Diminishing) methods, are only first order accurate in 

multi-dimensional flows and even in one-dimension they reduce to first order at 

non-sonic local extrema. 

In order to satisfy these conflicting requirements a significant effort has been made 

to improve and combine several successful numerical schemes. A fully implicit and 

time accurate code for the solution of the three-dimensional compressible NavierStokes equations in general geometries has been written and tested. Higher order 

accuracy and shock capturing are implemented using an Essentially Non-Oscillatory 

(ENO) scheme. Time accuracy is obtained using a Newton method. 

In the next section a brief description of the numerical scheme will be given 

followed by the discussion of a series of tests aimed at validating the code. 

2.1 Numerical Scheme 

The compressible Navier-Stokes equations are solved using a finite volume method .• 

A detailed discussion of finite volume and difference methods can be found in 

Vinokur18 . The integral formulation of the Navier-Stokes equations, assuming all 

variables are continuous in time, is given by: 

~ r U dV + j n . FdS = 0 

ut JV(t) JS(t) 

(2.1.1) 

116 
Bypass Transition in Compressible Boundary Layers 

Here V(t) and surface S(t) are the volume and outer surface of the domain n 

and n an outward unit normal vector at S. The vector U represents the conserved 

variables: (p, pu, pv, pw, e)T, with p the density, u, v and w the velocity components, 

and e the total energy. The tensor F is defined as F = £ + V, with £ the inviscid 

contribution defined as: 

pU pv pw 

pu2 +p puv puw 

E1 = puv E2 = pv2 +p E3 = pvw 

puw pvw pw2 +p 

(e + p)u (e + p)v (e + p)w 

and V the viscous contribution: 

V 2 = ( 72 0) ; V 3 = (0) 73 

. U· 72 + q2 U • 73 + q3 

The shear stress tensor T, with components (71,72,73) is defined as: 

1 

T = Re (JL(Vu + VuT ) + '\(V . u)1) 

and the heat flux q as: 

K,VT 

q = ...,.---....,------,--

(/-l)ReM~Pr 

(2.1.2) 

(2.1.3) 

(2.1.4) 

(2.1.5) 

The variables p, T, JL, A and K, represent the pressure, temperature, first and 

second viscosity coefficient and thermal conductivity, respectively. The coefficients 

Re, M(X), and Pr are the Reynolds, Mach, and Prandtl numbers. All variables are 

non-dimensionalized using free-stream variables and a characteristic length L. 

The Navier-Stokes equations are solved using a finite volume method because we 

seek a weak solution in order to capture shocks in high Reynolds number flows. 

The finite volume method is also the most natural way to satisfy the conservation 

properties of the differential equations. After subdividing the volume V into a set of 

disjunct cells we obtain the finite volume discretization for a cell with index i, j, k: 

a (- ) ~l l ~2 ~2 ~3 3 -

at VU i j k + F .+ 1 . k - F. 1 . k + F. .+ 1 k - F. . 1 k + F . . k+ 1 - F. . k 1 = 0 ' , t 2,J, t- 2 ,J, t,J 2' t,J- 2 , t,J, 2 t,J, -2 

(2.1.6) 

where a barred quantity with index i,j, k is an average of the unbarred quantity 

over the cell with index i, j, k and indices i ± ~, j ± ~ and k ± ~ refer to values at 

the cell faces. The fluxes :pi at the cell faces are defined as: 

(2.1.7) 

with Sl the cell face in the direction i. The computation of the cell face SI and 

volume V has to be done with great care in order satisfy the geometric conservation 

law, for details see Vinokur19 : 

117 

Bypass Transition in Compressible Boundary Layers 

Here V(t) and surface S(t) are the volume and outer surface of the domain n 

and n an outward unit normal vector at S. The vector U represents the conserved 

variables: (p, pu, pv, pw, e)T, with p the density, u, v and w the velocity components, 

and e the total energy. The tensor F is defined as F = £ + V, with £ the inviscid 

contribution defined as: 

pU pv pw 

pu2 +p puv puw 

E1 = puv E2 = pv2 +p E3 = pvw 

puw pvw pw2 +p 

(e + p)u (e + p)v (e + p)w 

and V the viscous contribution: 

V 2 = ( 72 0) ; V 3 = (0) 73 

. U· 72 + q2 U • 73 + q3 

The shear stress tensor T, with components (71,72,73) is defined as: 

1 

T = Re (JL(Vu + VuT ) + '\(V . u)1) 

and the heat flux q as: 

K,VT 

q = ...,.---....,------,--

(/-l)ReM~Pr 

(2.1.2) 

(2.1.3) 

(2.1.4) 

(2.1.5) 

The variables p, T, JL, A and K, represent the pressure, temperature, first and 

second viscosity coefficient and thermal conductivity, respectively. The coefficients 

Re, M(X), and Pr are the Reynolds, Mach, and Prandtl numbers. All variables are 

non-dimensionalized using free-stream variables and a characteristic length L. 

The Navier-Stokes equations are solved using a finite volume method because we 

seek a weak solution in order to capture shocks in high Reynolds number flows. 

The finite volume method is also the most natural way to satisfy the conservation 

properties of the differential equations. After subdividing the volume V into a set of 

disjunct cells we obtain the finite volume discretization for a cell with index i, j, k: 

a (- ) ~l l ~2 ~2 ~3 3 -

at VU i j k + F .+ 1 . k - F. 1 . k + F. .+ 1 k - F. . 1 k + F . . k+ 1 - F. . k 1 = 0 ' , t 2,J, t- 2 ,J, t,J 2' t,J- 2 , t,J, 2 t,J, -2 

(2.1.6) 

where a barred quantity with index i,j, k is an average of the unbarred quantity 

over the cell with index i, j, k and indices i ± ~, j ± ~ and k ± ~ refer to values at 

the cell faces. The fluxes :pi at the cell faces are defined as: 

(2.1.7) 

with Sl the cell face in the direction i. The computation of the cell face SI and 

volume V has to be done with great care in order satisfy the geometric conservation 

law, for details see Vinokur19 : 

117 
Jacobus J . Van der Vegt 

Flux Approximation 

The crucial part in the development of a finite volume method is the approximation of the fluxes at the cell faces. The flux F;+l is computed using the Osher 2 

approximate Riemann solver. The first order accurate conservative flux is given by: 

(2.1.8) 

with equivalent expressions for the other two directions. The integral is computed 

along a path in phase space, connecting the points with index (i, j, k) and (i + 1, j, k). 

Along each subpath a Riemann problem is solved, which is used to determine the 

intermediate states. In this way exact expressions can be derived for the path integrals. More details about the implementation of the Osher scheme can be found in 

Osher and Solomon 6 , Osher and Chakravarthy 7, Chakravarthy and Osherl and Rai 

and Chakravarthylo. The Osher approximate Riemann solver is the most accurate 

approximate Riemann solver and satisfies the entropy condition, contrary to the 

Roe approximate Riemann solver which needs an entropy fix to eliminate steady 

expansion shocks. The Osher scheme captures steady shocks in at most two points. 

The most important reason for the choice of the Osher scheme, however, has 

been its low numerical dissipation in boundary layers, Koren4 . Most schemes for 

the Euler equations are very dissipative in the boundary layer and not well suited 

for direct numerical simulations. In earlier work, Van der Vegtl7 , modifications to 

flux vector splitting schemes were discussed to alleviate this problem, but although 

significant improvement was achieved on steady laminar boundary layers, it was 

not possible to reach accuracy levels necessary for direct simulations. 

Higher order spatial accuracy 

Direct simulations require a high accuracy which cannot be achieved with standard second order schemes. It is fairly straightforward to derive higher order accurate finite difference schemes, but shock capturing then will not be possible. The 

development of higher order accurate, multi-dimensional finite volume schemes, 

capable of shock capturing still is an area of active research, and has been an important subject in this project. A significant effort has been made to combine the 

Osher approximate Riemann solver, discussed in the previous section, with an ENO 

scheme. Results of this work are described in Van der Vegtl \ where the different 

ENO methods are discussed and results of various tests are discussed. 

Higher order accuracy of a finite volume method can be defined is various ways. 

One approach is to define higher order accuracy with respect to the cell averaged 

values. This resembles most closely the finite volume description, which gives equations for the cell averaged values. Another definition of higher order accuracy uses 

the point values at the cell center, which is used in conservative higher order finite 

difference methods. Both approaches are being used. For subsonic flows currently 

the fifth order scheme, developed by Rai9 , is used, which is based on a Taylor series 

expansion of the flux vector along the lines presented by Osher and Chakravarthy8. 

This method is a conservative finite difference scheme. It has the benefit that it is 

118 

_J 

Jacobus J . Van der Vegt 

Flux Approximation 

The crucial part in the development of a finite volume method is the approximation of the fluxes at the cell faces. The flux F;+l is computed using the Osher 2 

approximate Riemann solver. The first order accurate conservative flux is given by: 

(2.1.8) 

with equivalent expressions for the other two directions. The integral is computed 

along a path in phase space, connecting the points with index (i, j, k) and (i + 1, j, k). 

Along each subpath a Riemann problem is solved, which is used to determine the 

intermediate states. In this way exact expressions can be derived for the path integrals. More details about the implementation of the Osher scheme can be found in 

Osher and Solomon 6 , Osher and Chakravarthy 7, Chakravarthy and Osherl and Rai 

and Chakravarthylo. The Osher approximate Riemann solver is the most accurate 

approximate Riemann solver and satisfies the entropy condition, contrary to the 

Roe approximate Riemann solver which needs an entropy fix to eliminate steady 

expansion shocks. The Osher scheme captures steady shocks in at most two points. 

The most important reason for the choice of the Osher scheme, however, has 

been its low numerical dissipation in boundary layers, Koren4 . Most schemes for 

the Euler equations are very dissipative in the boundary layer and not well suited 

for direct numerical simulations. In earlier work, Van der Vegtl7 , modifications to 

flux vector splitting schemes were discussed to alleviate this problem, but although 

significant improvement was achieved on steady laminar boundary layers, it was 

not possible to reach accuracy levels necessary for direct simulations. 

Higher order spatial accuracy 

Direct simulations require a high accuracy which cannot be achieved with standard second order schemes. It is fairly straightforward to derive higher order accurate finite difference schemes, but shock capturing then will not be possible. The 

development of higher order accurate, multi-dimensional finite volume schemes, 

capable of shock capturing still is an area of active research, and has been an important subject in this project. A significant effort has been made to combine the 

Osher approximate Riemann solver, discussed in the previous section, with an ENO 

scheme. Results of this work are described in Van der Vegtl \ where the different 

ENO methods are discussed and results of various tests are discussed. 

Higher order accuracy of a finite volume method can be defined is various ways. 

One approach is to define higher order accuracy with respect to the cell averaged 

values. This resembles most closely the finite volume description, which gives equations for the cell averaged values. Another definition of higher order accuracy uses 

the point values at the cell center, which is used in conservative higher order finite 

difference methods. Both approaches are being used. For subsonic flows currently 

the fifth order scheme, developed by Rai9 , is used, which is based on a Taylor series 

expansion of the flux vector along the lines presented by Osher and Chakravarthy8. 

This method is a conservative finite difference scheme. It has the benefit that it is 

118 

_J 
---- ---- -----

Bypass Transition in Compressible Boundary Layers 

simple to implement in more dimensions, but does not allow shock capturing. Osher and Chakravarthy8 demonstrated how to make these schemes TVD and allow 

shock capturing, but they are not very useful and only first order accurate globally. 

The scheme therefore is used in its unlimited form, limiting its application to flows 

without discontinuities such as shocks. The scheme also is rather expensive and 

work is in progress to improve its efficiency. 

For flows with shocks research has been carried out to develop higher order accurate ENO schemes. ENO schemes use an adaptive stencil where a searching 

algorithm tries to find that part of the flow field surrounding a cell which is the 

smoothest. Then a conservative, higher order accurate interpolation method is used 

to "reconstruct" the point values from the cell averaged values. Due to the fact that 

the interpolation process only uses data from the smooth part of the flow field numerical oscillations will be minimized. In this way uniform higher order accuracy 

can be obtained. The first ENO methods were developed by Harten et a1. 3 , and 

later modifications were proposed by Shu and Osher12,13. In Van der Vegt17 the 

different methods were compared and it was found that the ENO scheme, using 

primitive function reconstruction from cell averaged variables with the CauchyKowalewski procedure for time integration combined with the Osher approximate 

Riemann solver, was the most accurate and robust. In one dimension it has been 

successfully used up to fifth order accuracy, but due to the fact that in multidimensional flows currently dimension splitting is used, its accuracy is limited to 

second order in more than one dimension. Work to extend this scheme to higher 

order accuracy in multi-dimensional flows is in progress. 

Time integration 

Due to the very small gridspacing necessary at the wall and in critical layers 

explicit time integration would result in a serious time step limitation. To alleviate 

this problem implicit time integration has to be used, but most implicit time integration schemes make assumptions in the implicit part which reduce or eliminate 

time accuracy. The development of implicit and time accurate numerical schemes 

therefore has been a significant part of this research. Time accuracy is obtained 

using the Newton method discussed in Rai9 , which solves the non-linear system of 

equations in the implicit time integration scheme using a Newton method. Rai uses 

this method also to reduce the error caused by approximate factorization. We do 

not use approximate factorization but solve the whole matrix system iteratively, see 

Van der Vegt 16 , and need the Newton scheme only to reduce the error due to the 

time linearization. This iterative scheme also has the benefit that it is not necessary to use an exact linearization of the flux vectors, which can be very difficult 

and time consuming to obtain. First order Steger-Warming flux vector splitting is 

used in the implicit scheme, while a higher order accurate spatial discretization is 

used for the explicit part. At each time step the Newton iteration is performed 

such that the accuracy of the higher order explicit part is maintained. The use of 

an approximate linearization, however, limits the maximum time step and work is 

in progress to evaluate if a more accurate linearization would improve the performance and robustness of the scheme. Especially at high Mach numbers there still 

119 

---- ---- -----

Bypass Transition in Compressible Boundary Layers 

simple to implement in more dimensions, but does not allow shock capturing. Osher and Chakravarthy8 demonstrated how to make these schemes TVD and allow 

shock capturing, but they are not very useful and only first order accurate globally. 

The scheme therefore is used in its unlimited form, limiting its application to flows 

without discontinuities such as shocks. The scheme also is rather expensive and 

work is in progress to improve its efficiency. 

For flows with shocks research has been carried out to develop higher order accurate ENO schemes. ENO schemes use an adaptive stencil where a searching 

algorithm tries to find that part of the flow field surrounding a cell which is the 

smoothest. Then a conservative, higher order accurate interpolation method is used 

to "reconstruct" the point values from the cell averaged values. Due to the fact that 

the interpolation process only uses data from the smooth part of the flow field numerical oscillations will be minimized. In this way uniform higher order accuracy 

can be obtained. The first ENO methods were developed by Harten et a1. 3 , and 

later modifications were proposed by Shu and Osher12,13. In Van der Vegt17 the 

different methods were compared and it was found that the ENO scheme, using 

primitive function reconstruction from cell averaged variables with the CauchyKowalewski procedure for time integration combined with the Osher approximate 

Riemann solver, was the most accurate and robust. In one dimension it has been 

successfully used up to fifth order accuracy, but due to the fact that in multidimensional flows currently dimension splitting is used, its accuracy is limited to 

second order in more than one dimension. Work to extend this scheme to higher 

order accuracy in multi-dimensional flows is in progress. 

Time integration 

Due to the very small gridspacing necessary at the wall and in critical layers 

explicit time integration would result in a serious time step limitation. To alleviate 

this problem implicit time integration has to be used, but most implicit time integration schemes make assumptions in the implicit part which reduce or eliminate 

time accuracy. The development of implicit and time accurate numerical schemes 

therefore has been a significant part of this research. Time accuracy is obtained 

using the Newton method discussed in Rai9 , which solves the non-linear system of 

equations in the implicit time integration scheme using a Newton method. Rai uses 

this method also to reduce the error caused by approximate factorization. We do 

not use approximate factorization but solve the whole matrix system iteratively, see 

Van der Vegt 16 , and need the Newton scheme only to reduce the error due to the 

time linearization. This iterative scheme also has the benefit that it is not necessary to use an exact linearization of the flux vectors, which can be very difficult 

and time consuming to obtain. First order Steger-Warming flux vector splitting is 

used in the implicit scheme, while a higher order accurate spatial discretization is 

used for the explicit part. At each time step the Newton iteration is performed 

such that the accuracy of the higher order explicit part is maintained. The use of 

an approximate linearization, however, limits the maximum time step and work is 

in progress to evaluate if a more accurate linearization would improve the performance and robustness of the scheme. Especially at high Mach numbers there still 

119 
Jacobus J . Van der Vegt 

are convergence problems for large Courant numbers when the scheme is used to 

obtain steady state solutions. 

2.2 Results and Discussion 

Several computations have been carried out to test the code and the various 

numerical schemes. In order to test the Osher approximate Riemann solver and 

the ENO schemes a large number of shock tube calculations have been carried out, 

a detailed description of this work can be found in Van der Vegt 17 . The different 

ENO schemes tested were the ENO method from Harten et a1. 3 , using primitive 

function reconstruction and either Runge-Kutta time integration or the CauchyKowalewski procedure, and the Shu and Osher flux-ENO scheme. In all cases the 

Osher approximate Riemann solver was used and the effect of the ordering of the 

eigenvalues, viz. natural and reversed ordering, has been investigated. Four cases 

with different difficulties were tested, see Table 1. The performance of the different 

schemes was reasonable in most cases, but it turned out that the ENO scheme 

with primitive function reconstruction and the Cauchy-Kowalewski procedure for 

time integration (ENO-CK) was the most accurate and robust. Some of the results 

obtained with this method, are shown in Figures 1 and 2. Figure 1 shows a left 

moving shock followed by a contact discontinuity and a right moving expansion 

wave. A difficult problem for the ENO schemes in case A is the fact that in the 

initial stages there are not enough grid points available in the region between the 

shock and contact or shock-shock. The ENO scheme searches for the stencil which 

gives the smoothest part of the flow field around a grid point or cell. In these 

cases there exist in both directions a discontinuity and there are not enough points 

available to build a non-oscillatory higher order reconstruction. This problem exist 

for all ENO schemes but the ENO Cauchy Kowalewski scheme is the least sensitive 

for it. The other methods have mild to strong oscillations in these areas. One way 

to solve this problem is to reduce the accuracy locally till enough grid points are 

available to create a higher order reconstruction, but this is a problem which still 

needs further attention. Case B, Figure 3, shows a left moving expansion wave and 

a right moving contact discontinuity and shock. One of the problems in this case is 

the appearance of a sonic point, which gives a small jump of O( 6x) at first order. 

The shock tube tests showed that it is possible to use a higher order scheme for 

flows with discontinuities, but the convergence of these higher order schemes can 

Case PL PR UL UR TL TR 

A 15000 98400 0 0 1378 4390 

B 988000 9930 0 a 2438 2452 

C 10000 100000 a 0 2627 272 

E 573 22300 2200 0 199 546 

Table 1. Initial Conditions Shock Tube Tests. 

120 I 

I 

I 

J 

Jacobus J . Van der Vegt 

are convergence problems for large Courant numbers when the scheme is used to 

obtain steady state solutions. 

2.2 Results and Discussion 

Several computations have been carried out to test the code and the various 

numerical schemes. In order to test the Osher approximate Riemann solver and 

the ENO schemes a large number of shock tube calculations have been carried out, 

a detailed description of this work can be found in Van der Vegt 17 . The different 

ENO schemes tested were the ENO method from Harten et a1. 3 , using primitive 

function reconstruction and either Runge-Kutta time integration or the CauchyKowalewski procedure, and the Shu and Osher flux-ENO scheme. In all cases the 

Osher approximate Riemann solver was used and the effect of the ordering of the 

eigenvalues, viz. natural and reversed ordering, has been investigated. Four cases 

with different difficulties were tested, see Table 1. The performance of the different 

schemes was reasonable in most cases, but it turned out that the ENO scheme 

with primitive function reconstruction and the Cauchy-Kowalewski procedure for 

time integration (ENO-CK) was the most accurate and robust. Some of the results 

obtained with this method, are shown in Figures 1 and 2. Figure 1 shows a left 

moving shock followed by a contact discontinuity and a right moving expansion 

wave. A difficult problem for the ENO schemes in case A is the fact that in the 

initial stages there are not enough grid points available in the region between the 

shock and contact or shock-shock. The ENO scheme searches for the stencil which 

gives the smoothest part of the flow field around a grid point or cell. In these 

cases there exist in both directions a discontinuity and there are not enough points 

available to build a non-oscillatory higher order reconstruction. This problem exist 

for all ENO schemes but the ENO Cauchy Kowalewski scheme is the least sensitive 

for it. The other methods have mild to strong oscillations in these areas. One way 

to solve this problem is to reduce the accuracy locally till enough grid points are 

available to create a higher order reconstruction, but this is a problem which still 

needs further attention. Case B, Figure 3, shows a left moving expansion wave and 

a right moving contact discontinuity and shock. One of the problems in this case is 

the appearance of a sonic point, which gives a small jump of O( 6x) at first order. 

The shock tube tests showed that it is possible to use a higher order scheme for 

flows with discontinuities, but the convergence of these higher order schemes can 

Case PL PR UL UR TL TR 

A 15000 98400 0 0 1378 4390 

B 988000 9930 0 a 2438 2452 

C 10000 100000 a 0 2627 272 

E 573 22300 2200 0 199 546 

Table 1. Initial Conditions Shock Tube Tests. 

120 I 

I 

I 

J 
Bypass Transition in Compressible Boundary Layers 

1 .00 1.00 

0 .80 0 .60 

0 .60 0 .60 

0 .40 '::---------::"::---------;:"::---' 

O~ O~ ~7 0 .40 '::-------::'":::----------::-':~ 

0.3 0 .5 07 

Fig. 1. Case A, density at t = .2, 2nd and 5th order ENO-CK. 

1.0 ,---..,,-~--~--~--____, 1 .0 ,----rr-~-----~--___, 

0 .6 0 .6 

0 .6 0 .6 

0 .4 0.4 

0 .2 0 .2 

Fig. 2. Case B, density at t = .2, 2nd and 5th order ENO-CK. 

be at most first order around these discontinuities. Tests of all the ENO schemes 

on smooth solutions showed that they all reached the proper level of accuracy. Test 

are currently underway to check if the ENO schemes give higher order accuracy in 

regions outside discontinuities as they are supposed to. This is an important test 

to see if these methods are capable of shock-turbulence interaction simulations. 

In order to test the shock capturing properties of the code the flow field around 

a circular cylinder at Mach 8 has been computed. Although the flow field was twodimensional the three-dimensional code was used to check if the flow field remained 

exactly two-dimensional and the geometric conservation law was satisfied. Figure 

3. presents contours of the pressure at a spanwise station along the cylinder. The 

solution is the same at all stations. The flow field consists of a strong bow shock 

where the Mach number changes from 8 to about 2.8 behind the shock at the 

symmetry line. Apart from the strong shock another aspect of this case is the 

121 

Bypass Transition in Compressible Boundary Layers 

1 .00 1.00 

0 .80 0 .60 

0 .60 0 .60 

0 .40 '::---------::"::---------;:"::---' 

O~ O~ ~7 0 .40 '::-------::'":::----------::-':~ 

0.3 0 .5 07 

Fig. 1. Case A, density at t = .2, 2nd and 5th order ENO-CK. 

1.0 ,---..,,-~--~--~--____, 1 .0 ,----rr-~-----~--___, 

0 .6 0 .6 

0 .6 0 .6 

0 .4 0.4 

0 .2 0 .2 

Fig. 2. Case B, density at t = .2, 2nd and 5th order ENO-CK. 

be at most first order around these discontinuities. Tests of all the ENO schemes 

on smooth solutions showed that they all reached the proper level of accuracy. Test 

are currently underway to check if the ENO schemes give higher order accuracy in 

regions outside discontinuities as they are supposed to. This is an important test 

to see if these methods are capable of shock-turbulence interaction simulations. 

In order to test the shock capturing properties of the code the flow field around 

a circular cylinder at Mach 8 has been computed. Although the flow field was twodimensional the three-dimensional code was used to check if the flow field remained 

exactly two-dimensional and the geometric conservation law was satisfied. Figure 

3. presents contours of the pressure at a spanwise station along the cylinder. The 

solution is the same at all stations. The flow field consists of a strong bow shock 

where the Mach number changes from 8 to about 2.8 behind the shock at the 

symmetry line. Apart from the strong shock another aspect of this case is the 

121 
'. 

Jacobus 1- Van der Vegt 

~ r. /' 

,jf!'/ / , S?' / I ~ " / .' ~ / " / 

":-: / I /' 

; .' I / ' ( I / / / 

/ //I/ l //,/ 1; t, 

//~/ // 

/r,{." f/ /f l /: ! /~ ,: / ,,I " / :-,' __ 

,1/////;': ! 

J I J) I • , . I 

, , /; I I I / I' ' I// l , /I//: .,' . f i ' I J ' .' / / .'" l/;l//// .' 

j,! t/j l/f//! / 

I f;:'I'!:;l; , ;; /1 ; ; 1:" " ;' I f ; i : / .' ! ,! / , ' 

, j , I

J

' f' J J / : /! ;, ) ' J ' , I ' . ' , ! I ' I 'f ' " ' 

/:

' : ,:,' 1' '/ J,l l( ,." , J • r • I I ' ./ .,/ 

, I (' .i i if I /' I i j :' / :' :' ,' / 

j ;, !, i ) I,"! : " ' i I : : If! ; , : :' .' / 

\ : : r , ( .' . I I I' ',. / 

i -'\ ' ' . f " /' It ; .. .' ,.' -"" ~ ~;:li ./ I/f/,'"" ,. t l :. \ I : { .= t .' / .' / .: .: ,. . . ' ".' i , \ \ \ ' I, I .i ! i I I ! , i -' .' ",' " ,\\ \ \\\\:[ l!i //// /,,'/</'~<",/, \ \ '\ \ \. \ " '\ \ iii" jl" ;' " .r / /' /" / .,r-' \ ' .' , 1" ( ' I', ./ . , 

\ \. \ \ \ \ " ; \ : i i (j ! j //' > ',' ,/," , ..-

\ 

\ \ \ ' \ \ "" i : , ' j! i (, i '/" .' '\' ,\ \ \ ': \ \', : JI,: /ll/i " ' \ \\ \\\\; \\ :1' /.I I' 

~\ \

\ \ \, \ \ '\ \ \, :' j J i ' \\ ' \' \,.' ,' \ ,\\ \1\\, : : ' 

\ ~\\ \\ \'l; , \ \ ' >' \ ,\ 

\ \\. 

\' 

Figure 3. Pressure field of flow around a cylinder at Mach 8 

fact that the flow field in the stagnation region ahead of the cylinder is subsonic. 

The sonic line is at about 45 degrees with the flow angle and a smooth transition 

is observed from the subsonic to the supersonic region. This case has also been 

computed by Osher and Chakravarthy7 and the results compare well. To test the 

ability of the code to simulate transitional flows which is a crucial test before bypass 

transition can be simulated currently computations are done on natural transition 

in a flat plate boundary layer. A comparion is made with the results of Fasel et 

122 

--------

'. 

Jacobus 1- Van der Vegt 

~ r. /' 

,jf!'/ / , S?' / I ~ " / .' ~ / " / 

":-: / I /' 

; .' I / ' ( I / / / 

/ //I/ l //,/ 1; t, 

//~/ // 

/r,{." f/ /f l /: ! /~ ,: / ,,I " / :-,' __ 

,1/////;': ! 

J I J) I • , . I 

, , /; I I I / I' ' I// l , /I//: .,' . f i ' I J ' .' / / .'" l/;l//// .' 

j,! t/j l/f//! / 

I f;:'I'!:;l; , ;; /1 ; ; 1:" " ;' I f ; i : / .' ! ,! / , ' 

, j , I

J

' f' J J / : /! ;, ) ' J ' , I ' . ' , ! I ' I 'f ' " ' 

/:

' : ,:,' 1' '/ J,l l( ,." , J • r • I I ' ./ .,/ 

, I (' .i i if I /' I i j :' / :' :' ,' / 

j ;, !, i ) I,"! : " ' i I : : If! ; , : :' .' / 

\ : : r , ( .' . I I I' ',. / 

i -'\ ' ' . f " /' It ; .. .' ,.' -"" ~ ~;:li ./ I/f/,'"" ,. t l :. \ I : { .= t .' / .' / .: .: ,. . . ' ".' i , \ \ \ ' I, I .i ! i I I ! , i -' .' ",' " ,\\ \ \\\\:[ l!i //// /,,'/</'~<",/, \ \ '\ \ \. \ " '\ \ iii" jl" ;' " .r / /' /" / .,r-' \ ' .' , 1" ( ' I', ./ . , 

\ \. \ \ \ \ " ; \ : i i (j ! j //' > ',' ,/," , ..-

\ 

\ \ \ ' \ \ "" i : , ' j! i (, i '/" .' '\' ,\ \ \ ': \ \', : JI,: /ll/i " ' \ \\ \\\\; \\ :1' /.I I' 

~\ \

\ \ \, \ \ '\ \ \, :' j J i ' \\ ' \' \,.' ,' \ ,\\ \1\\, : : ' 

\ ~\\ \\ \'l; , \ \ ' >' \ ,\ 

\ \\. 

\' 

Figure 3. Pressure field of flow around a cylinder at Mach 8 

fact that the flow field in the stagnation region ahead of the cylinder is subsonic. 

The sonic line is at about 45 degrees with the flow angle and a smooth transition 

is observed from the subsonic to the supersonic region. This case has also been 

computed by Osher and Chakravarthy7 and the results compare well. To test the 

ability of the code to simulate transitional flows which is a crucial test before bypass 

transition can be simulated currently computations are done on natural transition 

in a flat plate boundary layer. A comparion is made with the results of Fasel et 

122 

--------
Bypass Transition in Compressible Boundary Layers 

Figure 4. Vertical velocity field in a flat plate boundary layer at Mach 0.08 with 

periodic suction and blowing (vertical direction enlarged 20 times). 

al. 2 , which computed transition in an incompressible boundary layer. In order to 

make the comparison as accurate as possible a very low Mach number, '.9 ) = .08 

was chosen. This Mach number is approximately the lower limit for the numerical 

scheme and despite the fact that the computations are fully implicit a severe time 

step limitation is imposed by the sound waves. To start the simulation first a steady 

laminar boundary layer is computed, which is a non-trivial task because a very high 

accuracy is needed in this computation. The disturbances at the beginning of the 

plate have an amplitude of 10-4 and transition simulations require a very low numerical dissipation. The disturbances are generated by periodic suction and blowing 

in a strip somewhat downstream of the inflow boundary. This is done because there 

123 

Bypass Transition in Compressible Boundary Layers 

Figure 4. Vertical velocity field in a flat plate boundary layer at Mach 0.08 with 

periodic suction and blowing (vertical direction enlarged 20 times). 

al. 2 , which computed transition in an incompressible boundary layer. In order to 

make the comparison as accurate as possible a very low Mach number, '.9 ) = .08 

was chosen. This Mach number is approximately the lower limit for the numerical 

scheme and despite the fact that the computations are fully implicit a severe time 

step limitation is imposed by the sound waves. To start the simulation first a steady 

laminar boundary layer is computed, which is a non-trivial task because a very high 

accuracy is needed in this computation. The disturbances at the beginning of the 

plate have an amplitude of 10-4 and transition simulations require a very low numerical dissipation. The disturbances are generated by periodic suction and blowing 

in a strip somewhat downstream of the inflow boundary. This is done because there 

123 
Jacobus J. Van der Vegt 

are always numerical disturbances due to the fact that the inflow boundary conditions are not perfect. The disturbances are generated in a region of the boundary 

layer which is linearly stable. When they move downstream first the transients are 

damped and after they move in the unstable region the Tolmien Schlichting waves, 

which are the most unstable two-dimensional waves, are amplified. In order to accommodate for the fact that the subsonic outflow boundary conditions, which are 

essentially inviscid, are not perfect in a boundary layer a buffer layer is added to the 

plate to damp as much as possible the reflections coming from this boundary layer. 

This is the same procedure as used by Rai and Moinll . The number of grid points 

in this computation is 340 x 82. It turned out that the most efficient way to obtain 

the steady boundary layer was to first start with the second order scheme running 

at a Courant number of 120 using the implicit Euler time integration and to switch 

to the fifth order scheme after most of the transients are damped out. The fifth 

order scheme has a very low dissipation and would otherwise take a long time to 

converge. The maximum CFL number for which the fifth order scheme is stable is 

approximately 160. Ater the steady state was obtained periodic suction and blowing 

were added and the fifth order scheme was used with the Newton time integration 

scheme at a CFL of 60. Figure 4. shows a preliminary result of this comput ation 

and it clearly shows the gradual build up of the boundary layer instability. The 

results are currently analysed to make a comparison with the incompressible results 

of Fasel et al. 2 . 

3. Future Plans 

Further testing of the code will have to be done for more complicated cases. 

Currently a comparison with the results of stability of a flat plate boundary layer 

at low Mach number with the results from Fasel et al. 2 is being completed. If 

this comparison and equivalent tests for high Mach number boundary layers are 

satisfactory a simulation of bypass transition on a subsonic flat plate boundary 

layer will be made, followed by simulations of a boundary layer on a curved plate. 

Also work has to be continued to develop higher order accurate ENO schemes for 

multi-dimensional flows. This is crucial for direct simulations of transonic flows 

with both shocks and turbulence. 

4. Referen ces 

1 Chakravarthy, S.R. & Osher, S., "Numerical Experiments with Osher Upwind 

Scheme for the Euler Equations", AIAA Journal, 21, 1241-1248, (1983). 

2 Fasel, H. F., Rist, U. and Konzelmann, U., "Numerical Investigation of the ThreeDimensional Development in Boundary-Layer Transition", AIAA Journal, 28, 

29-37, (1990). 

3 Harten, A., Engquist, B., Osher, S. & Chakravarthy, S.R., "Uniformly High Order 

Accurate Essentially Non-Oscillatory Schemes III" , Journal of Computational 

Physics, 71 , 231-303, (1987). 

4 Koren, H., "Upwind Discretization of the Steady Navier-Stokes Equations" , International Journal for Numerical Methods in Fluids, II, 99-117, (1990). 

124 

Jacobus J. Van der Vegt 

are always numerical disturbances due to the fact that the inflow boundary conditions are not perfect. The disturbances are generated in a region of the boundary 

layer which is linearly stable. When they move downstream first the transients are 

damped and after they move in the unstable region the Tolmien Schlichting waves, 

which are the most unstable two-dimensional waves, are amplified. In order to accommodate for the fact that the subsonic outflow boundary conditions, which are 

essentially inviscid, are not perfect in a boundary layer a buffer layer is added to the 

plate to damp as much as possible the reflections coming from this boundary layer. 

This is the same procedure as used by Rai and Moinll . The number of grid points 

in this computation is 340 x 82. It turned out that the most efficient way to obtain 

the steady boundary layer was to first start with the second order scheme running 

at a Courant number of 120 using the implicit Euler time integration and to switch 

to the fifth order scheme after most of the transients are damped out. The fifth 

order scheme has a very low dissipation and would otherwise take a long time to 

converge. The maximum CFL number for which the fifth order scheme is stable is 

approximately 160. Ater the steady state was obtained periodic suction and blowing 

were added and the fifth order scheme was used with the Newton time integration 

scheme at a CFL of 60. Figure 4. shows a preliminary result of this comput ation 

and it clearly shows the gradual build up of the boundary layer instability. The 

results are currently analysed to make a comparison with the incompressible results 

of Fasel et al. 2 . 

3. Future Plans 

Further testing of the code will have to be done for more complicated cases. 

Currently a comparison with the results of stability of a flat plate boundary layer 

at low Mach number with the results from Fasel et al. 2 is being completed. If 

this comparison and equivalent tests for high Mach number boundary layers are 

satisfactory a simulation of bypass transition on a subsonic flat plate boundary 

layer will be made, followed by simulations of a boundary layer on a curved plate. 

Also work has to be continued to develop higher order accurate ENO schemes for 

multi-dimensional flows. This is crucial for direct simulations of transonic flows 

with both shocks and turbulence. 

4. Referen ces 

1 Chakravarthy, S.R. & Osher, S., "Numerical Experiments with Osher Upwind 

Scheme for the Euler Equations", AIAA Journal, 21, 1241-1248, (1983). 

2 Fasel, H. F., Rist, U. and Konzelmann, U., "Numerical Investigation of the ThreeDimensional Development in Boundary-Layer Transition", AIAA Journal, 28, 

29-37, (1990). 

3 Harten, A., Engquist, B., Osher, S. & Chakravarthy, S.R., "Uniformly High Order 

Accurate Essentially Non-Oscillatory Schemes III" , Journal of Computational 

Physics, 71 , 231-303, (1987). 

4 Koren, H., "Upwind Discretization of the Steady Navier-Stokes Equations" , International Journal for Numerical Methods in Fluids, II, 99-117, (1990). 

124 
Bypass Transition in Compressible Boundary Layers 

5 Morkovin, M. V., "Bypass Transition to Thrbulence and Research Desiderata in 

"Transition in Turbines", NASA Conf. Publ. 2386, 161-204, (1984). 

6 Osher, S. & Solomon, F., "Upwind Difference Schemes for Hyperbolic Systems of 

Conservation Laws", Mathematics of Computation, 38, 339-374, (1982). 

7 Osher, S. & Chakravarthy, S.R., "Upwind Schemes and Boundary Conditions 

with Applications to Euler Equations in General Geometries", Journal of Computational Physics, 50,447-481, (1983). 

8 Osher, S. & Chakravarthy, S.R., "Very High Order TVD Schemes", Mathematics 

and its Applications, 2, 229-274, (1986). 

9 Rai, M. M., "Navier-Stokes Simulations of Blade-Vortex Interaction using HighOrder Accurate Upwind Schemes", AIAA paper 87-0543, (1987). 

10 Rai, M.M. & Chakravarthy, S.R., "An Implicit Form for the Osher Upwind 

Scheme", AIAA Journal, 24, 735-743, (1986). 

11 Rai, M. M. Moin, P., " Direct Numerical Simulation of Transition and Turbulence 

in a Spatially Evolving Boundary Layer", AIAA paper 91-1607-CP, (1991). 

12 Shu, C.-W. & Osher, S., "Efficient Implementation of Essentially Non-Oscillatory 

Shock-Capturing Schemes", Journal of Computational Physics, 77, 439-471, (1988) .• 

13 Shu, C.-W. & Osher, S., "Efficient Implementation of Essentially Non-Oscillatory 

Shock Capturing Schemes, II", Journal of Computational Physics, 83, 32-78, 

(1989). 

14 Sohn, K. H. and Reshotko, E., "Experimental Study of Boundary Layer Transition with Elevated Freestream Turbulence on a Heated Plate, NASA Contractor 

Report 187068, (1991). 

15 Thrner, A. B., "Local Heat Transfer Measurements on a Gas Thrbine Blade", J. 

of Mech. Eng. Sciences, 13, 1-12, (1971). 

16 Van der Vegt, J. J. W. , "Assessment of Flux Vector Splitting for Viscous Compressible Flows", AIAA paper 91-0243, (1991). 

17 Van der Vegt, J.J.W., "ENO-Osher Scheme for Euler Equationsll, submitted for 

publication, (1992). 

18 Vinokur, M., "An Analysis of Finite-Difference and Finite-Volume Formulations 

of Conservation Laws", Journal of Computational Physics, 81, 1-52, (1989). 

19 Wang, T., Simon, T. W. and Buddhavarapu, J., "Heat Transfer and Fluid Mechanics Measurements in Transitional Boundary Layer Flows", in "Transition in 

Turbines", NASA Conf. Pub!. 2386, 69-79, (1984). 

125 

--------- ---

Bypass Transition in Compressible Boundary Layers 

5 Morkovin, M. V., "Bypass Transition to Thrbulence and Research Desiderata in 

"Transition in Turbines", NASA Conf. Publ. 2386, 161-204, (1984). 

6 Osher, S. & Solomon, F., "Upwind Difference Schemes for Hyperbolic Systems of 

Conservation Laws", Mathematics of Computation, 38, 339-374, (1982). 

7 Osher, S. & Chakravarthy, S.R., "Upwind Schemes and Boundary Conditions 

with Applications to Euler Equations in General Geometries", Journal of Computational Physics, 50,447-481, (1983). 

8 Osher, S. & Chakravarthy, S.R., "Very High Order TVD Schemes", Mathematics 

and its Applications, 2, 229-274, (1986). 

9 Rai, M. M., "Navier-Stokes Simulations of Blade-Vortex Interaction using HighOrder Accurate Upwind Schemes", AIAA paper 87-0543, (1987). 

10 Rai, M.M. & Chakravarthy, S.R., "An Implicit Form for the Osher Upwind 

Scheme", AIAA Journal, 24, 735-743, (1986). 

11 Rai, M. M. Moin, P., " Direct Numerical Simulation of Transition and Turbulence 

in a Spatially Evolving Boundary Layer", AIAA paper 91-1607-CP, (1991). 

12 Shu, C.-W. & Osher, S., "Efficient Implementation of Essentially Non-Oscillatory 

Shock-Capturing Schemes", Journal of Computational Physics, 77, 439-471, (1988) .• 

13 Shu, C.-W. & Osher, S., "Efficient Implementation of Essentially Non-Oscillatory 

Shock Capturing Schemes, II", Journal of Computational Physics, 83, 32-78, 

(1989). 

14 Sohn, K. H. and Reshotko, E., "Experimental Study of Boundary Layer Transition with Elevated Freestream Turbulence on a Heated Plate, NASA Contractor 

Report 187068, (1991). 

15 Thrner, A. B., "Local Heat Transfer Measurements on a Gas Thrbine Blade", J. 

of Mech. Eng. Sciences, 13, 1-12, (1971). 

16 Van der Vegt, J. J. W. , "Assessment of Flux Vector Splitting for Viscous Compressible Flows", AIAA paper 91-0243, (1991). 

17 Van der Vegt, J.J.W., "ENO-Osher Scheme for Euler Equationsll, submitted for 

publication, (1992). 

18 Vinokur, M., "An Analysis of Finite-Difference and Finite-Volume Formulations 

of Conservation Laws", Journal of Computational Physics, 81, 1-52, (1989). 

19 Wang, T., Simon, T. W. and Buddhavarapu, J., "Heat Transfer and Fluid Mechanics Measurements in Transitional Boundary Layer Flows", in "Transition in 

Turbines", NASA Conf. Pub!. 2386, 69-79, (1984). 

125 

--------- ---
Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Runge-Kutta Methods Combined 

with Compact Difference Schemes 

for the Unsteady Euler Equations 

Sheng-Tao Yu 

1. Motivation and Objective 

Recent development using compact difference schemes to solve the Navier-Stokes 

equations show spectral-like accuracyl,2. In this paper, we report further study 

of the numerical characteristics of various combinations of the Runge-Kutta (RK) 

methods and compact difference schemes to calculate the unsteady Euler equations. 

Conventionally, the accuracy of finite difference schemes is assessed based on the 

evaluations of dissipative error. The objectives are reducing the numerical damping and, at the same time, preserving numerical stability. While this approach has 

tremendous success solving steady flows, numerical characteristics of unsteady calculations remain largely unclear. For unsteady flows, in addition to the dissipative 

errors, phase velocity and harmonic content of the numerical results are of concern. 

As a result of the discretization procedure, the simulated unsteady flow motions 

actually propagate in a dispersive numerical medium. Consequently, the dispersion 

characteristics of the numerical schemes which relate the phase velocity and wave 

number may greatly impact the numerical accuracy. The objective of the present 

paper is to assess the numerical accuracy of the simulated results. To this end, 

the Fourier analysis is performed to provide the dispersive correlations of various 

numerical schemes. 

First, a detailed investigation of the existing RK methods is carried out. A 

generalized form of an N-step RK method is derived. With this generalized form, 

the criteria are derived for the three and four-step RK methods to be third and 

fourth-order time accurate for the non-linear equations, e.g., flow equations. These 

criteria are then applied to commonly used RK methods such as Jameson's 3-

step and 4-step3,4 schemes and Wray's algorithm5 to identify the accuracy of the 

methods. For the spatial discretization, compact difference schemes are 'presented. 

The schemes are formulated in the operator-type6 to render themselves suitable for 

the Fourier analyses. The results of the analyses provide CFL limits, the numerical 

dispersion relations, and the artificial damping required for stable and time-accurate 

solutions. 

Finally, the performance of the numerical methods is demonstrated by numerical examples. The first case is a quasi-one-dimensional calculation of the acoustic 

admittance in a converging nozzle. The CFD results are compared with Tsien's 

analytical solution 7

; the harmonic content of this flow field is limited to one frequency mode. All numerical schemes of concern provide accurate solutions. The 

second case is a one-dimensional simulation of a shocked sound wave. The harmonic 

127 Preceding Page Blank 

Center for Modeling of Turbulence and Transition 

Research Briefs - 1991 

Runge-Kutta Methods Combined 

with Compact Difference Schemes 

for the Unsteady Euler Equations 

Sheng-Tao Yu 

1. Motivation and Objective 

Recent development using compact difference schemes to solve the Navier-Stokes 

equations show spectral-like accuracyl,2. In this paper, we report further study 

of the numerical characteristics of various combinations of the Runge-Kutta (RK) 

methods and compact difference schemes to calculate the unsteady Euler equations. 

Conventionally, the accuracy of finite difference schemes is assessed based on the 

evaluations of dissipative error. The objectives are reducing the numerical damping and, at the same time, preserving numerical stability. While this approach has 

tremendous success solving steady flows, numerical characteristics of unsteady calculations remain largely unclear. For unsteady flows, in addition to the dissipative 

errors, phase velocity and harmonic content of the numerical results are of concern. 

As a result of the discretization procedure, the simulated unsteady flow motions 

actually propagate in a dispersive numerical medium. Consequently, the dispersion 

characteristics of the numerical schemes which relate the phase velocity and wave 

number may greatly impact the numerical accuracy. The objective of the present 

paper is to assess the numerical accuracy of the simulated results. To this end, 

the Fourier analysis is performed to provide the dispersive correlations of various 

numerical schemes. 

First, a detailed investigation of the existing RK methods is carried out. A 

generalized form of an N-step RK method is derived. With this generalized form, 

the criteria are derived for the three and four-step RK methods to be third and 

fourth-order time accurate for the non-linear equations, e.g., flow equations. These 

criteria are then applied to commonly used RK methods such as Jameson's 3-

step and 4-step3,4 schemes and Wray's algorithm5 to identify the accuracy of the 

methods. For the spatial discretization, compact difference schemes are 'presented. 

The schemes are formulated in the operator-type6 to render themselves suitable for 

the Fourier analyses. The results of the analyses provide CFL limits, the numerical 

dispersion relations, and the artificial damping required for stable and time-accurate 

solutions. 

Finally, the performance of the numerical methods is demonstrated by numerical examples. The first case is a quasi-one-dimensional calculation of the acoustic 

admittance in a converging nozzle. The CFD results are compared with Tsien's 

analytical solution 7

; the harmonic content of this flow field is limited to one frequency mode. All numerical schemes of concern provide accurate solutions. The 

second case is a one-dimensional simulation of a shocked sound wave. The harmonic 

127 Preceding Page Blank 
S.-T. Yu 

content is complex and distinct differences between various schemes are observed. 

The results are also compared with the analytical solution provided by Morse and 

IngardB. In the one-dimensional cases, details of the numerical methods in setting 

up the initial conditions and the perturbation on the computational boundary are 

described. 

The third case is a two-dimensional simulation of a Lamb vortex9 in an uniform 

flow. This calculation provides a realistic assessment of various finite difference 

schemes in terms of the conservation of the vortex strength and the harmonic content 

after travelling a substantial distance. The numerical implementation of Giles' nonreflective equations17 coupled with the characteristic equations as the boundary 

condition is discussed in detail. Finally, the single vortex calculation is extended 

to simulate vortex pairinglO . For the distance between two vortices less than a 

threshold value, numerical results show crisp resolution of the vortex merging. 

2. Work Accomplished 

2.1 Numerical Method 

The Euler equations in Cartesian coordinates can be cast into a vector form: 

(1) 

where Q is the unknown vector and Ei is the inviscid flux in the Xi direction. 

The Runge-Kutta algorithm is applied as the temporal discretization and the second, fourth, and sixth-order compact difference schemes are applied to the spatial 

discretization. 

2.2.1 The Runge-Kutta Method 

The use of the Runge-Kutta methods for flow equations stems from the application of the methods to solve ordinary differential equations (ODEs). An ODE has 

one independent variable and its solution is obtained by integrating the equation 

from its initial condition. When one applies the Runge-Kutta method to the flow 

equations, time is treated as the independent variable as is in an ODE, and the 

convective terms are taken as the inhomogeneous part of the equations, such as 

~~ = R(Q). (2) 

Notice that the boldface symbols which represent vectors have been temporarily 

dropped for typographic convenience. In addition, all results in the following discussion are valid for both scalar and vector equations. 

The Runge-Kutta methods have algorithms of the form 

(3) 

128 

S.-T. Yu 

content is complex and distinct differences between various schemes are observed. 

The results are also compared with the analytical solution provided by Morse and 

IngardB. In the one-dimensional cases, details of the numerical methods in setting 

up the initial conditions and the perturbation on the computational boundary are 

described. 

The third case is a two-dimensional simulation of a Lamb vortex9 in an uniform 

flow. This calculation provides a realistic assessment of various finite difference 

schemes in terms of the conservation of the vortex strength and the harmonic content 

after travelling a substantial distance. The numerical implementation of Giles' nonreflective equations17 coupled with the characteristic equations as the boundary 

condition is discussed in detail. Finally, the single vortex calculation is extended 

to simulate vortex pairinglO . For the distance between two vortices less than a 

threshold value, numerical results show crisp resolution of the vortex merging. 

2. Work Accomplished 

2.1 Numerical Method 

The Euler equations in Cartesian coordinates can be cast into a vector form: 

(1) 

where Q is the unknown vector and Ei is the inviscid flux in the Xi direction. 

The Runge-Kutta algorithm is applied as the temporal discretization and the second, fourth, and sixth-order compact difference schemes are applied to the spatial 

discretization. 

2.2.1 The Runge-Kutta Method 

The use of the Runge-Kutta methods for flow equations stems from the application of the methods to solve ordinary differential equations (ODEs). An ODE has 

one independent variable and its solution is obtained by integrating the equation 

from its initial condition. When one applies the Runge-Kutta method to the flow 

equations, time is treated as the independent variable as is in an ODE, and the 

convective terms are taken as the inhomogeneous part of the equations, such as 

~~ = R(Q). (2) 

Notice that the boldface symbols which represent vectors have been temporarily 

dropped for typographic convenience. In addition, all results in the following discussion are valid for both scalar and vector equations. 

The Runge-Kutta methods have algorithms of the form 

(3) 

128 
Compact Differencing for the Euler Equations 

where the increment function R(Qn, tlt) is a suitable chosen approximation to the 

inhomogeneous part of the equation, that is, R( Q). In general, the calculation of 

the increment function R is subdivided into N steps on the interval tn ~ t ~ tn+!. 

And the final increment function it is a weighted average of the inhomogeneous 

terms evaluated at the different steps on the interval tn :s; t :s; tn+1, that is 

Ql = Qn + ~t(auRn), 

Q2 = Qn + tlt(a21Rn + a22 R1 ), 

Q3 = Qn + tlt(a31 Rn + a 32 R 1 + a33R2), (4) 

where the superscript n, 1, 2, ... , and n+ 1 denote the time steps on the time interval 

t n :s; tl :s; t2 :s; ... :s; tN :s; tn+!, and aij is the weighting factor for the step i and 

term j. There are 2:~1 i weighting coefficients to be determined and an infinite 

number of coefficient sets can be chosen. However, certain criteria must be met for 

the algorithm to retain high-order accuracy. 

In what follows, the criteria of the coefficient set of a 3-step Runge-Kutta method 

to be third-order accurate is given. To proceed, we follow the conventional approach 

and expand all inhomogeneous terms Ri in Eqn. (4) to a Taylor's series about Rn 

and drop all terms in which the exponent of tlt is greater than 3. The result 

is compared with its analytical counterpart by equating terms in like powers of 

tlt. The result is tabulated in Table 1. For the convenience of the discussion, the 

following simplification of symbols is activated: R denotes R( Qn), Q denotes Qn, 

R' denotes (8Rj8Q)n, and R" denotes (82Rj8Q2)n. In addition to the equality of 

the coefficients of all the powers of tlt, we also want the equality of the coefficients 

of the functions of R, R', and R". As a result, we find the criteria of the coefficients 

for the 3-step RK methods to be third-order accurate as, 

a31 + a32 + a33 = 1, 

1 

aU a32 + a33(a21 + (22) = 2' 

2 2 1 

au a32 + (a21 + (22) a33 = 3' 

1 

au a22 a33 = "6. 

(5a) 

(5b) 

(5c) 

(5d) 

Equations (5a) and (5b) are the criteria of first and second order accuracy, respectively. The remaining equations are of third order term. Since four equations 

contain six unknowns, the system is underdetermined, and two of the coefficients 

may be chosen arbitrarily. The obvious choice is to let the two coefficients be null 

to reduce intermediate storage and numerical operations. According to Eqn. (5d), 

129 

Compact Differencing for the Euler Equations 

where the increment function R(Qn, tlt) is a suitable chosen approximation to the 

inhomogeneous part of the equation, that is, R( Q). In general, the calculation of 

the increment function R is subdivided into N steps on the interval tn ~ t ~ tn+!. 

And the final increment function it is a weighted average of the inhomogeneous 

terms evaluated at the different steps on the interval tn :s; t :s; tn+1, that is 

Ql = Qn + ~t(auRn), 

Q2 = Qn + tlt(a21Rn + a22 R1 ), 

Q3 = Qn + tlt(a31 Rn + a 32 R 1 + a33R2), (4) 

where the superscript n, 1, 2, ... , and n+ 1 denote the time steps on the time interval 

t n :s; tl :s; t2 :s; ... :s; tN :s; tn+!, and aij is the weighting factor for the step i and 

term j. There are 2:~1 i weighting coefficients to be determined and an infinite 

number of coefficient sets can be chosen. However, certain criteria must be met for 

the algorithm to retain high-order accuracy. 

In what follows, the criteria of the coefficient set of a 3-step Runge-Kutta method 

to be third-order accurate is given. To proceed, we follow the conventional approach 

and expand all inhomogeneous terms Ri in Eqn. (4) to a Taylor's series about Rn 

and drop all terms in which the exponent of tlt is greater than 3. The result 

is compared with its analytical counterpart by equating terms in like powers of 

tlt. The result is tabulated in Table 1. For the convenience of the discussion, the 

following simplification of symbols is activated: R denotes R( Qn), Q denotes Qn, 

R' denotes (8Rj8Q)n, and R" denotes (82Rj8Q2)n. In addition to the equality of 

the coefficients of all the powers of tlt, we also want the equality of the coefficients 

of the functions of R, R', and R". As a result, we find the criteria of the coefficients 

for the 3-step RK methods to be third-order accurate as, 

a31 + a32 + a33 = 1, 

1 

aU a32 + a33(a21 + (22) = 2' 

2 2 1 

au a32 + (a21 + (22) a33 = 3' 

1 

au a22 a33 = "6. 

(5a) 

(5b) 

(5c) 

(5d) 

Equations (5a) and (5b) are the criteria of first and second order accuracy, respectively. The remaining equations are of third order term. Since four equations 

contain six unknowns, the system is underdetermined, and two of the coefficients 

may be chosen arbitrarily. The obvious choice is to let the two coefficients be null 

to reduce intermediate storage and numerical operations. According to Eqn. (5d), 

129 
S.-T. Yu 

none of aii where i = 1,2,3, could be zero, and one can set the two of the three remaining coefficients to be zero. Therefore, at least one of the intermediate steps has 

two non-zero coefficients. Consequently, one needs to store two steps of intermediate 

solutions for the 3-step, third-order RK methods. 

is 

A 3-step Runge-Kutta method proposed by Jameson et a1. 3 to solve flow equations 

Ql = Qn + tltRn, 

Q2 = Qn + ~t (Rn + Rl ), 

Qn+l = Qn + tlt (Rn + R2). 

2 

(6) 

It can be shown that the weighting coefficients of the 3-step method satisfy only 

Eqns. (5a) and (5b). And the method is second-order accurate in time. 

Wray5 proposed another 3-steP--ID@.thod, 

Ql = Qn + tlt( 18

5 

Rn ), 

Q2 = Ql + tlt( ~ Rn _ 17 Rl) 12 60' (7) 

Qn+l = Q2 + tlt(~Rn _ ~R2) . 4 12 

This formula may be manipulated to fit the generalized form as proposed in Eqn. 

(4), and we obtain, 

Ql = Qn + tlt( 18

5 

Rn ), 

Q2 = Qn + tlt(lRn + 152R1), 

Qn+l = Qn + tlt( ~Rn + ~R2). 4 4 

(8) 

Wray's coefficients match all the equations in Eqn. (5) and therefore the scheme is 

third-order accurate. In this formula, a32 is set to zero and two sets of solutions 

are needed in the second and the third steps. The calculation can be carried out by 

either the vectorized algorithm proposed by Wray, or straightforward calculation 

according to Eqns. (7) and (8). 

A similar procedure can be applied to the 4-step Runge-Kutta methods, and the 

criteria for the scheme to be fourth-order accurate are: 

a41 + a42 + a43 + a44 = 1, (9a) 

1 

CXU a 42 + (a21 + CX22)CX43 + (a31 + a32 + a33)a44 = 2' (9b) 

130 

S.-T. Yu 

none of aii where i = 1,2,3, could be zero, and one can set the two of the three remaining coefficients to be zero. Therefore, at least one of the intermediate steps has 

two non-zero coefficients. Consequently, one needs to store two steps of intermediate 

solutions for the 3-step, third-order RK methods. 

is 

A 3-step Runge-Kutta method proposed by Jameson et a1. 3 to solve flow equations 

Ql = Qn + tltRn, 

Q2 = Qn + ~t (Rn + Rl ), 

Qn+l = Qn + tlt (Rn + R2). 

2 

(6) 

It can be shown that the weighting coefficients of the 3-step method satisfy only 

Eqns. (5a) and (5b). And the method is second-order accurate in time. 

Wray5 proposed another 3-steP--ID@.thod, 

Ql = Qn + tlt( 18

5 

Rn ), 

Q2 = Ql + tlt( ~ Rn _ 17 Rl) 12 60' (7) 

Qn+l = Q2 + tlt(~Rn _ ~R2) . 4 12 

This formula may be manipulated to fit the generalized form as proposed in Eqn. 

(4), and we obtain, 

Ql = Qn + tlt( 18

5 

Rn ), 

Q2 = Qn + tlt(lRn + 152R1), 

Qn+l = Qn + tlt( ~Rn + ~R2). 4 4 

(8) 

Wray's coefficients match all the equations in Eqn. (5) and therefore the scheme is 

third-order accurate. In this formula, a32 is set to zero and two sets of solutions 

are needed in the second and the third steps. The calculation can be carried out by 

either the vectorized algorithm proposed by Wray, or straightforward calculation 

according to Eqns. (7) and (8). 

A similar procedure can be applied to the 4-step Runge-Kutta methods, and the 

criteria for the scheme to be fourth-order accurate are: 

a41 + a42 + a43 + a44 = 1, (9a) 

1 

CXU a 42 + (a21 + CX22)CX43 + (a31 + a32 + a33)a44 = 2' (9b) 

130 
Compact Differencing for the Euler Equations 

2 (2 )2 1 

au a42 + a21 + a22) a43 + (a31 + a32 + a33 a44 = 3' 

1 

au a22 a43 + [all a32 + (a21 + a22 )a33] a44 = 6' 

3 ( 3 ( )3 1 all a42 + a21 + a22) a43 + a31 + a32 + a33 a44 = 4' 

1 2 2 all a22 a43 + (a21 + a22 )0:11 a22 a43 

+~ [(a2l + a22)2 a33 + a~la32] a44 

1 

+(a31 + a32 + a33) [aU a 32 + (a21 + a22)a33] a44 = 6' 

1 

all a22 a 33 a 44 = 24' 

(9c) 

(9d) 

(ge) 

(9J) 

(9g) 

Equations (9a) and (9b) are for first and second-order accuracy, respectively. Equations (9c) and (9d) are for third-order accuracy. The remaining equations are for 

the fourth-order terms. Here, seven equations contain ten unknowns, and three of 

the coefficients may be chosen arbitrarily. 

A 4-step RK method attributed to Kuttall for solving ODEs was adopted by 

Jameson et al. 3 to solve the flow equations. The algorithm can be expressed as, 

QI = Qn D.t R n 

+ 2 ' 

Q2 = Qn D.t RI 

+ 2 ' (10) 

Q3 = Qn + D.tR2, 

Qn+l = Qn + D.t (Rn + 2Rl + 2R2 + R3) 6 . 

The coefficients satisfy Eqn. (9) and the algorithm is fourth-order accurate. However, this method requires all four intermediate solutions in the final step. As a 

result, the use of this scheme for large-scale calculations is undesirable. 

Later on, Jameson and Baker4 proposed another 4-step algorithm, 

(11) 

This scheme was proposed to calculate the steady state solutions and the transient 

solutions were not of concern. For that purpose, the algorithm is convenient to 

131 

-- - - - ------ - ---

Compact Differencing for the Euler Equations 

2 (2 )2 1 

au a42 + a21 + a22) a43 + (a31 + a32 + a33 a44 = 3' 

1 

au a22 a43 + [all a32 + (a21 + a22 )a33] a44 = 6' 

3 ( 3 ( )3 1 all a42 + a21 + a22) a43 + a31 + a32 + a33 a44 = 4' 

1 2 2 all a22 a43 + (a21 + a22 )0:11 a22 a43 

+~ [(a2l + a22)2 a33 + a~la32] a44 

1 

+(a31 + a32 + a33) [aU a 32 + (a21 + a22)a33] a44 = 6' 

1 

all a22 a 33 a 44 = 24' 

(9c) 

(9d) 

(ge) 

(9J) 

(9g) 

Equations (9a) and (9b) are for first and second-order accuracy, respectively. Equations (9c) and (9d) are for third-order accuracy. The remaining equations are for 

the fourth-order terms. Here, seven equations contain ten unknowns, and three of 

the coefficients may be chosen arbitrarily. 

A 4-step RK method attributed to Kuttall for solving ODEs was adopted by 

Jameson et al. 3 to solve the flow equations. The algorithm can be expressed as, 

QI = Qn D.t R n 

+ 2 ' 

Q2 = Qn D.t RI 

+ 2 ' (10) 

Q3 = Qn + D.tR2, 

Qn+l = Qn + D.t (Rn + 2Rl + 2R2 + R3) 6 . 

The coefficients satisfy Eqn. (9) and the algorithm is fourth-order accurate. However, this method requires all four intermediate solutions in the final step. As a 

result, the use of this scheme for large-scale calculations is undesirable. 

Later on, Jameson and Baker4 proposed another 4-step algorithm, 

(11) 

This scheme was proposed to calculate the steady state solutions and the transient 

solutions were not of concern. For that purpose, the algorithm is convenient to 

131 

-- - - - ------ - ---
S.-T. Yu 

program and no intermediate solution needs to be stored. For the present investigation, however, the weighting coefficients of this method satisfy only part of Eqn. 

(9) and the algorithm is second-order accurate. Nevertheless, this formulation is 

favorable compared to a 2-step, second-order RK method because part of the third 

and fourth-order terms are satisfied, namely, Eqns. (9d) and (9g). Consequently, a 

larger marching step, i.e., a larger CFL number, could be used. 

2.2.2 Compact Difference Schemes 

The remaining task of discretizing the flow equations is the spatial differencing 

of the inviscid fluxes. An effective manner for generating a high-order, central 

difference scheme is the compact difference method. The scheme is obtained by 

using only three and five points to achieve fourth-order and sixth-order accuracy 

in space, respectively. The gain in the accuracy is not based on the involvement of 

more points as in the conventional approach, but on implicitly solving the derivatives 

simultaneously at all locations. According to Hermite's generalization of a Taylor's 

series,12 one can get 

where the superscript I represents the spatial derivatives. Equation (12) is the 

fourth-order method and Eqn. (13) is the sixth-order one. When the fourthorder method is used in the interior nodes, a third-order biased implicit scheme13 

is adopted for grid nodes at the computational boundary, such as 

I I 1 3 

2u1 + 4U2 = ~x (-SUI + 4U2 + U3) + O(~x ), 

I I 1 3 

2um ax + 4umax -1 = ~x (5umax - 4Umax -1 - Umax -2) + O(~x ). 

(14) 

When the sixth-order method is used in the interior nodes, the fourth-order scheme, 

Eqn. (12), is used at locations one grid node away from the boundary and the 

third-order biased scheme is used at the boundary. The application of the implicit 

compact difference schemes with the appropriate boundary conditions involves the 

inversion of a scalar tridiagonal matrix. The inversion of the matrix incurs little 

penalty in terms of CPU time. 

2.3 Fourier Analysis 

By definition, any function, u(x, t), which is continuous, periodic, and square 

summable can be expressed in a Fourier series expansion at a constant time, 

00 

u(x, t) = L u(k, t)ei27rkx/L (15) 

k=- oc 

132 

S.-T. Yu 

program and no intermediate solution needs to be stored. For the present investigation, however, the weighting coefficients of this method satisfy only part of Eqn. 

(9) and the algorithm is second-order accurate. Nevertheless, this formulation is 

favorable compared to a 2-step, second-order RK method because part of the third 

and fourth-order terms are satisfied, namely, Eqns. (9d) and (9g). Consequently, a 

larger marching step, i.e., a larger CFL number, could be used. 

2.2.2 Compact Difference Schemes 

The remaining task of discretizing the flow equations is the spatial differencing 

of the inviscid fluxes. An effective manner for generating a high-order, central 

difference scheme is the compact difference method. The scheme is obtained by 

using only three and five points to achieve fourth-order and sixth-order accuracy 

in space, respectively. The gain in the accuracy is not based on the involvement of 

more points as in the conventional approach, but on implicitly solving the derivatives 

simultaneously at all locations. According to Hermite's generalization of a Taylor's 

series,12 one can get 

where the superscript I represents the spatial derivatives. Equation (12) is the 

fourth-order method and Eqn. (13) is the sixth-order one. When the fourthorder method is used in the interior nodes, a third-order biased implicit scheme13 

is adopted for grid nodes at the computational boundary, such as 

I I 1 3 

2u1 + 4U2 = ~x (-SUI + 4U2 + U3) + O(~x ), 

I I 1 3 

2um ax + 4umax -1 = ~x (5umax - 4Umax -1 - Umax -2) + O(~x ). 

(14) 

When the sixth-order method is used in the interior nodes, the fourth-order scheme, 

Eqn. (12), is used at locations one grid node away from the boundary and the 

third-order biased scheme is used at the boundary. The application of the implicit 

compact difference schemes with the appropriate boundary conditions involves the 

inversion of a scalar tridiagonal matrix. The inversion of the matrix incurs little 

penalty in terms of CPU time. 

2.3 Fourier Analysis 

By definition, any function, u(x, t), which is continuous, periodic, and square 

summable can be expressed in a Fourier series expansion at a constant time, 

00 

u(x, t) = L u(k, t)ei27rkx/L (15) 

k=- oc 

132 
Compact Differencing for the Euler Equations 

where L is the period of the function u(x, t), k is the wave number, and i is .j=I. 

The Fourier coefficient is defined as, 

1 1L/2 . u(k, t) = - u(x, t)e-t27rkx/Ldx . 

L -L/2 

(16) 

In the Fourier analysis of a finite difference scheme, functions are defined at discrete 

points. The discrete Fourier series and its coefficients are defined analogous to their 

continuous counterparts, 

K-l 

uj = 2:= un(k)ei27rkj/K, j = 0, ±1, ±2,'" ,±oo, 

k=O 

K 

un(k) = ~ L uje-i27rkj/K, k = 0,1"", K - l. 

j=l 

(17) 

Here, the length of the computational domain L is decomposed into K grid nodes 

(L = K!:lx). The superscript n denotes the time step and the subscript j is the 

spatial location index. Similar to the continuous function, the algebraic system in 

terms of the function exp( i27r kj / K) is periodic over the computational domain L 

(or K) and orthonormal, such as 

ei27rkj / K = ei27rk(j+K)/ K 

K-l 

~ '" ei27rkd/Ke-i27rk2j/K = 0 K ~ k1k2 

j=O 

(18) 

where Ok1k2 is the Kronecker delta. Therefore, the establishment of the discrete 

Fourier series and coefficients is self-sufficient, and is not an approximation of its 

continuous counterpart. 

As shown in Eqns. (17) and (18), the harmonic content of the discretized equation 

is limited to the number of grid nodes used in the computational domain. A discrete 

solution uj at a location (j) and time (n) is a linear combination of K wave modes. 

The Fourier analysis is performed by substituting each wave mode of the discrete 

Fourier expansion, Eqn. (17), into the discretized flow equations to calculate the 

amplification factor, g(k), which is defined as 

(19) 

The procedure is repeated for all wave modes (k = 0,1"", K - 1) and the full 

spectrum of the amplification factor is obtained. In this process, we map the function u in terms of spatial variable x on the interval [-L/2, L/2] to the wave number 

space on [-7r,7r] assuming that the analysis is local for an infinite and periodic domain. Therefore, the solution of the amplification factor on the interval [-Jr,O] is 

133 

Compact Differencing for the Euler Equations 

where L is the period of the function u(x, t), k is the wave number, and i is .j=I. 

The Fourier coefficient is defined as, 

1 1L/2 . u(k, t) = - u(x, t)e-t27rkx/Ldx . 

L -L/2 

(16) 

In the Fourier analysis of a finite difference scheme, functions are defined at discrete 

points. The discrete Fourier series and its coefficients are defined analogous to their 

continuous counterparts, 

K-l 

uj = 2:= un(k)ei27rkj/K, j = 0, ±1, ±2,'" ,±oo, 

k=O 

K 

un(k) = ~ L uje-i27rkj/K, k = 0,1"", K - l. 

j=l 

(17) 

Here, the length of the computational domain L is decomposed into K grid nodes 

(L = K!:lx). The superscript n denotes the time step and the subscript j is the 

spatial location index. Similar to the continuous function, the algebraic system in 

terms of the function exp( i27r kj / K) is periodic over the computational domain L 

(or K) and orthonormal, such as 

ei27rkj / K = ei27rk(j+K)/ K 

K-l 

~ '" ei27rkd/Ke-i27rk2j/K = 0 K ~ k1k2 

j=O 

(18) 

where Ok1k2 is the Kronecker delta. Therefore, the establishment of the discrete 

Fourier series and coefficients is self-sufficient, and is not an approximation of its 

continuous counterpart. 

As shown in Eqns. (17) and (18), the harmonic content of the discretized equation 

is limited to the number of grid nodes used in the computational domain. A discrete 

solution uj at a location (j) and time (n) is a linear combination of K wave modes. 

The Fourier analysis is performed by substituting each wave mode of the discrete 

Fourier expansion, Eqn. (17), into the discretized flow equations to calculate the 

amplification factor, g(k), which is defined as 

(19) 

The procedure is repeated for all wave modes (k = 0,1"", K - 1) and the full 

spectrum of the amplification factor is obtained. In this process, we map the function u in terms of spatial variable x on the interval [-L/2, L/2] to the wave number 

space on [-7r,7r] assuming that the analysis is local for an infinite and periodic domain. Therefore, the solution of the amplification factor on the interval [-Jr,O] is 

133 
S.-T. Yu 

the complex conjugate of that on [0,7r] . For this reason, the results of our Fourier 

analyses are presented on the interval [0,7r]. 

In the present investigation, one-dimensional equations are considered for the 

Fourier analysis. In addition, we can perform a similarity transformation to transform the one-dimensional Euler equations to their characteristic form, i. e., three 

decoupled scalar equations. Consequently, a scalar, advective equation on a periodic domain is adopted as the model equation in our analysis, 

AU A au = 0 

at + ox ' (20) 

where the phase velocity (A) is equivalent to the eigenvalues of the Euler equations, 

namely u - c, u + c, and u where u is velocity and c is the speed of sound. The 

phase speed (A) is treated as a parameter in the Fourier analysis to avoid the 

Fourier convolution, therefore the analysis is linear. For the unsteady calculations, 

the requirement of the time resolution of the flow field restricts the time marching 

step. In other words, the variations of flow properties between time steps are small. 

Thus, linear analysis is a viable tool. 

In what follows, the procedure to obtain the amplification factor of the model 

equation discretized by Runge-K utta methods and compact differences is illustrated. 

First, the generalized forms of the amplification factor for the 3 and 4-step RungeKutta methods are derived. These representations of the amplification factors are 

independent of the spatial discretization schemes. From the equations of Wray's 

3-step scheme, Eqn.(7), we have 

where 

8 

gl = 1 + 15 Z, 

1 5 

g2 = 1 + - Z + _ Zgl 4 12 ' 

1 3 2 

9 = 1 + -

44' Z + - Zg 

glQn = Q1, 

g2Qn = Q2, 

gQn = Qn+1. 

(21) 

(22) 

The variable Z represents the spatial discretization applied to the convective term ( 

-AOU/OX ). By substituting the amplification factors of the intermediate steps, gl 

and g2, into the last step of Wray's algorithm, we obtain the amplification factor, 

g, for the 3-step scheme, 

1 1 

9 = 1 + Z + "2 Z2 + (3 Z3 . (23) 

134 

S.-T. Yu 

the complex conjugate of that on [0,7r] . For this reason, the results of our Fourier 

analyses are presented on the interval [0,7r]. 

In the present investigation, one-dimensional equations are considered for the 

Fourier analysis. In addition, we can perform a similarity transformation to transform the one-dimensional Euler equations to their characteristic form, i. e., three 

decoupled scalar equations. Consequently, a scalar, advective equation on a periodic domain is adopted as the model equation in our analysis, 

AU A au = 0 

at + ox ' (20) 

where the phase velocity (A) is equivalent to the eigenvalues of the Euler equations, 

namely u - c, u + c, and u where u is velocity and c is the speed of sound. The 

phase speed (A) is treated as a parameter in the Fourier analysis to avoid the 

Fourier convolution, therefore the analysis is linear. For the unsteady calculations, 

the requirement of the time resolution of the flow field restricts the time marching 

step. In other words, the variations of flow properties between time steps are small. 

Thus, linear analysis is a viable tool. 

In what follows, the procedure to obtain the amplification factor of the model 

equation discretized by Runge-K utta methods and compact differences is illustrated. 

First, the generalized forms of the amplification factor for the 3 and 4-step RungeKutta methods are derived. These representations of the amplification factors are 

independent of the spatial discretization schemes. From the equations of Wray's 

3-step scheme, Eqn.(7), we have 

where 

8 

gl = 1 + 15 Z, 

1 5 

g2 = 1 + - Z + _ Zgl 4 12 ' 

1 3 2 

9 = 1 + -

44' Z + - Zg 

glQn = Q1, 

g2Qn = Q2, 

gQn = Qn+1. 

(21) 

(22) 

The variable Z represents the spatial discretization applied to the convective term ( 

-AOU/OX ). By substituting the amplification factors of the intermediate steps, gl 

and g2, into the last step of Wray's algorithm, we obtain the amplification factor, 

g, for the 3-step scheme, 

1 1 

9 = 1 + Z + "2 Z2 + (3 Z3 . (23) 

134 
Compact Differencing for the Euler Equations 

It is interesting to note that Eqn. (23) can be directly derived from the Taylor's 

series expansion by adopting the invariance property of the time and spatial derivatives of the model equation, i.e., Z = -Aau/ax = au/at. This is valid because the 

coefficients satisfy Eqn. (5), which is deduced from the Taylor's series expansion up 

to the third-order term. On the other hand, the amplification factor of the 3-step 

scheme proposed by Jameson et al., Eqn. (6), can be derived as 

1 1 

9 = 1 + Z + "2 Z2 + "4 Z3 . (24) 

It is obvious that the scheme is not third-order accurate. 

A similar analysis can be applied to the 4-step methods. Identical forms of the 

amplification factors of both 4-step methods of concern (Eqns. (10) and (11)) are 

obtained, such as 

1 1 1 

9 = 1 + Z + - Z2 + - Z3 + - Z4 

2 6 24 ' (25) 

Unlike the case of the 3-step schemes, the effect of the order of accuracy of these 

two 4-step schemes does not appear in the expression of the amplification factor. 

This is because the amplification factor is derived based on the linearized equation. 

Only by using Eqns. (5) and (9), which take into account nonlinear terms, can one 

justify the order of the accuracy of the RK schemes. 

The remaining task is to derive the explicit form of the spatial discretization 

operation, Z, of compact difference schemes. The fourth-order compact difference 

method, Eqn. (12), can be cast into the operator-type by defining 

(26) 

where Ui could be any flow property of interest at grid point i. As a result, the 

fourth-order method can be rewritten as, 

( 8

2

) (au) 1 4 1 + "6 ax i = 26.x (Ui+l - ui-d + O(6.x ). (27) 

This equation allows us to express (au/ax)i in an explicit form, 

(28) 

To proceed, we substitute this explicit, discretized form, Eqn. (32), into the model 

equation, and we obtain 

Z(4) = _ 6Fsin(k)i 

4 + 2cos(k), (29) 

135 

Compact Differencing for the Euler Equations 

It is interesting to note that Eqn. (23) can be directly derived from the Taylor's 

series expansion by adopting the invariance property of the time and spatial derivatives of the model equation, i.e., Z = -Aau/ax = au/at. This is valid because the 

coefficients satisfy Eqn. (5), which is deduced from the Taylor's series expansion up 

to the third-order term. On the other hand, the amplification factor of the 3-step 

scheme proposed by Jameson et al., Eqn. (6), can be derived as 

1 1 

9 = 1 + Z + "2 Z2 + "4 Z3 . (24) 

It is obvious that the scheme is not third-order accurate. 

A similar analysis can be applied to the 4-step methods. Identical forms of the 

amplification factors of both 4-step methods of concern (Eqns. (10) and (11)) are 

obtained, such as 

1 1 1 

9 = 1 + Z + - Z2 + - Z3 + - Z4 

2 6 24 ' (25) 

Unlike the case of the 3-step schemes, the effect of the order of accuracy of these 

two 4-step schemes does not appear in the expression of the amplification factor. 

This is because the amplification factor is derived based on the linearized equation. 

Only by using Eqns. (5) and (9), which take into account nonlinear terms, can one 

justify the order of the accuracy of the RK schemes. 

The remaining task is to derive the explicit form of the spatial discretization 

operation, Z, of compact difference schemes. The fourth-order compact difference 

method, Eqn. (12), can be cast into the operator-type by defining 

(26) 

where Ui could be any flow property of interest at grid point i. As a result, the 

fourth-order method can be rewritten as, 

( 8

2

) (au) 1 4 1 + "6 ax i = 26.x (Ui+l - ui-d + O(6.x ). (27) 

This equation allows us to express (au/ax)i in an explicit form, 

(28) 

To proceed, we substitute this explicit, discretized form, Eqn. (32), into the model 

equation, and we obtain 

Z(4) = _ 6Fsin(k)i 

4 + 2cos(k), (29) 

135 
S.-T. Yu 

where F is the CFL number which is defined as F = )"'6.t / 6. x, and k is the normalized wave number (k = 21fk/ K) . 

It is interesting to note that if the solution reaches a steady state solu tion, i.e., the 

time derivative term is zero, the operator (1 + 82 /6)-1 in the discretized equation 

becomes futile and the spatial discretization is represented by (Ui+1 - ui-d/26.x, 

which is only second-order accurate. However, the steady state solution of the onedimensional wave equation is a constant and the spatial accuracy is meaningless. 

On the other hand, the accuracy of multi-dimensional calculations is more complex. 

For example, consider a two-dimensional version of the model equation discretized 

by the fourth-order compact difference method, and we have 

1 

( )

-1 2 - s:2 au 8 U· · - u · _ . u u· · - U: .. _ +)... (1 + ) (t+1,) t 1,)) + )... 1 + -...J{ ( t. J+1 ~'J-1 ) = o. 

at x 6 26.x y 6 26.y 

(30) 

Again, the operators (1 + 5;/6) and (1 + 5U6) are scalar tridiagonal matrices with 

the dimensions IL x IL and JL x JL, respectively. IL and JL are the numbers 

of the grid nodes in the x and y directions of the computational domain. When 

IL = JL, two operators are identical. We then multiply Eqn. (30) by the operator 

(1 + 8;/6) and obtain the steady state equation as 

\ Ui+1 ,j - Ui-1,j \ Ui ,j+1 - Ui,j-1 

Ax 26.x + Ay 26.y = O. (31 ) 

Therefore, the steady state solution is only second-order accurate. 

Similarly, the spatial discretization of the sixth-order compact difference scheme 

can be represented in the operator-type such as, 

au ( 82

) -1 1 6 

ax = 1 + 5 606.x (Ui+2 + 28ui+1 - 28ui-1 - Ui-2) + O(6.x ). (32) 

And we obtain, 

Z(6) = _ F[4 sin(k) cos(k ~ + 56 sin(k )]i . 

12[2cos(k) + 3] (33) 

Again, when solving the steady state solution with same number of grid nodes in 

each direction of the computational domain, the spatial discretization is represented 

by (Ui+2+28ui+1 -28ui-1 -Ui-2)/606.x. Unlike the case of the fourth-order scheme, 

this representation does not match any conventional central difference scheme and 

it is at most second-order accurate. 

According to the above discussion, we can obtain the amplification factor g(k) for 

various combinations of the Runge-Kutta methods and compact difference schemes. 

The amplification factor g(k) is a complex number and can be expressed as g(k) = 

136 

S.-T. Yu 

where F is the CFL number which is defined as F = )"'6.t / 6. x, and k is the normalized wave number (k = 21fk/ K) . 

It is interesting to note that if the solution reaches a steady state solu tion, i.e., the 

time derivative term is zero, the operator (1 + 82 /6)-1 in the discretized equation 

becomes futile and the spatial discretization is represented by (Ui+1 - ui-d/26.x, 

which is only second-order accurate. However, the steady state solution of the onedimensional wave equation is a constant and the spatial accuracy is meaningless. 

On the other hand, the accuracy of multi-dimensional calculations is more complex. 

For example, consider a two-dimensional version of the model equation discretized 

by the fourth-order compact difference method, and we have 

1 

( )

-1 2 - s:2 au 8 U· · - u · _ . u u· · - U: .. _ +)... (1 + ) (t+1,) t 1,)) + )... 1 + -...J{ ( t. J+1 ~'J-1 ) = o. 

at x 6 26.x y 6 26.y 

(30) 

Again, the operators (1 + 5;/6) and (1 + 5U6) are scalar tridiagonal matrices with 

the dimensions IL x IL and JL x JL, respectively. IL and JL are the numbers 

of the grid nodes in the x and y directions of the computational domain. When 

IL = JL, two operators are identical. We then multiply Eqn. (30) by the operator 

(1 + 8;/6) and obtain the steady state equation as 

\ Ui+1 ,j - Ui-1,j \ Ui ,j+1 - Ui,j-1 

Ax 26.x + Ay 26.y = O. (31 ) 

Therefore, the steady state solution is only second-order accurate. 

Similarly, the spatial discretization of the sixth-order compact difference scheme 

can be represented in the operator-type such as, 

au ( 82

) -1 1 6 

ax = 1 + 5 606.x (Ui+2 + 28ui+1 - 28ui-1 - Ui-2) + O(6.x ). (32) 

And we obtain, 

Z(6) = _ F[4 sin(k) cos(k ~ + 56 sin(k )]i . 

12[2cos(k) + 3] (33) 

Again, when solving the steady state solution with same number of grid nodes in 

each direction of the computational domain, the spatial discretization is represented 

by (Ui+2+28ui+1 -28ui-1 -Ui-2)/606.x. Unlike the case of the fourth-order scheme, 

this representation does not match any conventional central difference scheme and 

it is at most second-order accurate. 

According to the above discussion, we can obtain the amplification factor g(k) for 

various combinations of the Runge-Kutta methods and compact difference schemes. 

The amplification factor g(k) is a complex number and can be expressed as g(k) = 

136 
1-- - 

_ .. _-, 

Compact Differencing for the Euler Equations 

exp[iw(k)], where w is the normalized frequency and is defined as w = 27rW6.t/T, 

W is the frequency, T is the the time period of function uj, and the phase speed 

(..\) is equal to LIT. Here, the disSipative and dispersive artifacts of the numerical 

schemes can be assessed: 

1. Dissipation. The normalized frequency w is a complex number (w = 0: + i(3) and 

its imaginary part represents the magnitude of the amplification factor, Le., 

g(k) = eiw = e-{3eia 

IgCk)1 = e-{3 (34) 

The magnitude of the amplification factor is the artificial dissipation. When 

/g/ 2 I, the scheme is unstable. For the calculations of unsteady flows, we want 

Igl to be less than unity but very close to it to ensure numerical stability with 

minimum artificial dissipation. In the following section, we plot Igi against k to 

illustrate the artificial dissipation. 

2. Dispersion. According to Eqn. (34), the relation of o:(k) and k represents the 

artificial dispersion. We plot (} = 0:1 F against k to show phase velocities. Notice 

that the model equation is dispersionless and the phase velocity is a constant, Le., 

A. After being normalized by the CFL number, the exact solution is a straight 

line with 45° angle on the plot of 0: against k. 

Figure 1 shows the results of the Fourier analysis of the third-order Runge-Kutta 

CRK3) method combined with fourth (CD4), and sixth (CD6) order compact difference schemes and the conventional second-order central difference scheme (CD2). 

The figures show the dissipative as well as dispersive effects at CFL numbers from 

0.4 to 1.4 with an increment of 0.2 between neighboring curves. Figure 1a shows 

the dissipation of the RK3-CD6 scheme. The method is unstable for CFL numbers 

greater than 0.8. As the order of spatial differencing decreases (compare Figs. la, 

1c, and Ie), the limit of the CFL number increases for stable calculation. 

Figure 1 b shows the dispersive effect of the RK3-CD6 scheme. For a CFL number 

of 0.4, the phase velocity is correct for wave numbers up to 1.8. The phase velocities 

are slower than they should be at large wave numbers. Increasing the CFL number 

makes phase velocities deviate from the 45° straight line at smaller wave numbers. 

Decreasing the CFL number merges the curves together to reach an asymptotic 

curve. However, the dispersive effect at high wave numbers does not improve. 

Comparing Figs. 1 b, Id, and If shows that the increase of the order of the spatial 

differencing reduces the numerical dispersion at high wave numbers. Specifically, a 

significant improvement is achieved by changing the spatial differencing from CD2 

to CD4, whereas only a limited gain is obtained by switching from CD4 to CD6. 

Figure 2 shows the results of the Fourier analyses ofthe fourth-order Runge-Kutta 

(RK4) method combined with various central difference schemes. The CFL numbers 

are the same as that in Fig. 1. Similar to the case of RK3, for the same CFL number 

and wave number, e.g., CFL = 1.4, w = 1.5, higher-order spatial discretization 

137 

1-- - 

_ .. _-, 

Compact Differencing for the Euler Equations 

exp[iw(k)], where w is the normalized frequency and is defined as w = 27rW6.t/T, 

W is the frequency, T is the the time period of function uj, and the phase speed 

(..\) is equal to LIT. Here, the disSipative and dispersive artifacts of the numerical 

schemes can be assessed: 

1. Dissipation. The normalized frequency w is a complex number (w = 0: + i(3) and 

its imaginary part represents the magnitude of the amplification factor, Le., 

g(k) = eiw = e-{3eia 

IgCk)1 = e-{3 (34) 

The magnitude of the amplification factor is the artificial dissipation. When 

/g/ 2 I, the scheme is unstable. For the calculations of unsteady flows, we want 

Igl to be less than unity but very close to it to ensure numerical stability with 

minimum artificial dissipation. In the following section, we plot Igi against k to 

illustrate the artificial dissipation. 

2. Dispersion. According to Eqn. (34), the relation of o:(k) and k represents the 

artificial dispersion. We plot (} = 0:1 F against k to show phase velocities. Notice 

that the model equation is dispersionless and the phase velocity is a constant, Le., 

A. After being normalized by the CFL number, the exact solution is a straight 

line with 45° angle on the plot of 0: against k. 

Figure 1 shows the results of the Fourier analysis of the third-order Runge-Kutta 

CRK3) method combined with fourth (CD4), and sixth (CD6) order compact difference schemes and the conventional second-order central difference scheme (CD2). 

The figures show the dissipative as well as dispersive effects at CFL numbers from 

0.4 to 1.4 with an increment of 0.2 between neighboring curves. Figure 1a shows 

the dissipation of the RK3-CD6 scheme. The method is unstable for CFL numbers 

greater than 0.8. As the order of spatial differencing decreases (compare Figs. la, 

1c, and Ie), the limit of the CFL number increases for stable calculation. 

Figure 1 b shows the dispersive effect of the RK3-CD6 scheme. For a CFL number 

of 0.4, the phase velocity is correct for wave numbers up to 1.8. The phase velocities 

are slower than they should be at large wave numbers. Increasing the CFL number 

makes phase velocities deviate from the 45° straight line at smaller wave numbers. 

Decreasing the CFL number merges the curves together to reach an asymptotic 

curve. However, the dispersive effect at high wave numbers does not improve. 

Comparing Figs. 1 b, Id, and If shows that the increase of the order of the spatial 

differencing reduces the numerical dispersion at high wave numbers. Specifically, a 

significant improvement is achieved by changing the spatial differencing from CD2 

to CD4, whereas only a limited gain is obtained by switching from CD4 to CD6. 

Figure 2 shows the results of the Fourier analyses ofthe fourth-order Runge-Kutta 

(RK4) method combined with various central difference schemes. The CFL numbers 

are the same as that in Fig. 1. Similar to the case of RK3, for the same CFL number 

and wave number, e.g., CFL = 1.4, w = 1.5, higher-order spatial discretization 

137 
S.-T. Yu 

introduces more artificial damping (see Figs. 2a, 2c, and 2e) and therefore reduces 

the CFL number limit for stable calculation. Again, the dispersive error at high 

wave numbers decreases as the order of the spatial differencing increases (see Figs. 

2b, 2d, and 2f). 

Figures. lc and 2c can be compared to show the difference of the dissipation 

effects between the RK3 and RK4 methods. For the same CFL and wave numbers, 

the RK4 method introduces more artificial damping, and a larger CFL number 

could be used. On the other hand, Figs. Id and 2d show that an increase of the 

order of the time marching scheme does not improve the dispersive effect at high 

wave numbers. 

From the above discussion, it is clear that reliable solutions of the finite difference 

schemes are at low wave numbers. For example, for the RK4-CD6 method at CFL 

= 0.8 (see Figs. 2a and 2b) the solution with wave numbers less than 1/37f (6 grid 

nodes per wave) is fairly accurate. Numerical solutions with higher wave numbers 

(wave length less than 6 grid nodes) suffer significant dispersive and dissipative 

errors. On the other hand, for the conventional RK4-CD2 method at the same 

CFL, 12 to 16 grid nodes per wave are needed for an accurate solution. 

It is interesting to note that compact difference schemes have no dissipative effect 

at the highest wave number resolved by a given numerical grid, i.e., two grid nodes 

per wave (see Figs. la, 1c, Ie, 2a, 2c, and 2e). Nevertheless, a significant dispersive 

error is introduced to these highest-wave-number waves and cause the even-odd decoupling of the numerical solutions. Furthermore, applying the compact difference 

scheme twice to calculate the viscous terms of the N avier Stokes equations does not 

eliminate the erroneous oscillation, owing to the linearity of the operation. These 

high-wave-number waves continue oscillating with erroneous phase speeds throughout the course of computation and eventually destroy the solution. It is therefore 

appropriate to impose a small amount of high-order artificial damping to filter out 

t hese waves while at the same time keeping the resolution at low wave modes intact. 

Figure 3 shows the dissipation and dispersion effects of the RK4-CD6 method at 

CFL=O.8 with various amounts of sixth-order artificial damping, defined as 

'TJ A.D. = S[Ui +3 + U'i-3 - 6(Ui+2 + Ui-2) + 15(ui+3 + Ui-3) - 20Ui]. (35) 

The range of'TJ is from 0.01 to 0.05 with an increment of 0.01 between the neighboring curves. Comparison between Fig. 3 and Figs. 2a and 2b shows that no 

additional damping at low wave numbers is introduced into the system by the imposed artificial dissipation for 'TJ :::; 0.03, whereas the undesirable high-wave-number 

waves are dissipated. 

2.4 Numerical Examples 

2.4.1 Acoustic Admittance of A Nozzle Flow 

The first case is a forced oscillatory quasi-one-dimensional flow in a converging 

nozzle. The governing equations are 

138 

S.-T. Yu 

introduces more artificial damping (see Figs. 2a, 2c, and 2e) and therefore reduces 

the CFL number limit for stable calculation. Again, the dispersive error at high 

wave numbers decreases as the order of the spatial differencing increases (see Figs. 

2b, 2d, and 2f). 

Figures. lc and 2c can be compared to show the difference of the dissipation 

effects between the RK3 and RK4 methods. For the same CFL and wave numbers, 

the RK4 method introduces more artificial damping, and a larger CFL number 

could be used. On the other hand, Figs. Id and 2d show that an increase of the 

order of the time marching scheme does not improve the dispersive effect at high 

wave numbers. 

From the above discussion, it is clear that reliable solutions of the finite difference 

schemes are at low wave numbers. For example, for the RK4-CD6 method at CFL 

= 0.8 (see Figs. 2a and 2b) the solution with wave numbers less than 1/37f (6 grid 

nodes per wave) is fairly accurate. Numerical solutions with higher wave numbers 

(wave length less than 6 grid nodes) suffer significant dispersive and dissipative 

errors. On the other hand, for the conventional RK4-CD2 method at the same 

CFL, 12 to 16 grid nodes per wave are needed for an accurate solution. 

It is interesting to note that compact difference schemes have no dissipative effect 

at the highest wave number resolved by a given numerical grid, i.e., two grid nodes 

per wave (see Figs. la, 1c, Ie, 2a, 2c, and 2e). Nevertheless, a significant dispersive 

error is introduced to these highest-wave-number waves and cause the even-odd decoupling of the numerical solutions. Furthermore, applying the compact difference 

scheme twice to calculate the viscous terms of the N avier Stokes equations does not 

eliminate the erroneous oscillation, owing to the linearity of the operation. These 

high-wave-number waves continue oscillating with erroneous phase speeds throughout the course of computation and eventually destroy the solution. It is therefore 

appropriate to impose a small amount of high-order artificial damping to filter out 

t hese waves while at the same time keeping the resolution at low wave modes intact. 

Figure 3 shows the dissipation and dispersion effects of the RK4-CD6 method at 

CFL=O.8 with various amounts of sixth-order artificial damping, defined as 

'TJ A.D. = S[Ui +3 + U'i-3 - 6(Ui+2 + Ui-2) + 15(ui+3 + Ui-3) - 20Ui]. (35) 

The range of'TJ is from 0.01 to 0.05 with an increment of 0.01 between the neighboring curves. Comparison between Fig. 3 and Figs. 2a and 2b shows that no 

additional damping at low wave numbers is introduced into the system by the imposed artificial dissipation for 'TJ :::; 0.03, whereas the undesirable high-wave-number 

waves are dissipated. 

2.4 Numerical Examples 

2.4.1 Acoustic Admittance of A Nozzle Flow 

The first case is a forced oscillatory quasi-one-dimensional flow in a converging 

nozzle. The governing equations are 

138 
I 

Compact Differencing for the Euler Equations 

BQ BE_H 

at + ax - (36) 

where 

E = (pu~: p) a, H = (po~~) , (e + p)u 

(37) 

p is density, p is pressure, and e is the total energy defined as e = p( Cv T + ~ U 2 ). Cv 

is the constant volume specific heat. The variable a is the cross sectional area and 

is prescribed as a function of x. The theoretical solution of the acoustic admittance 

of a choked nozzle was provided by Tsien 7 under the assumption that the velocity 

of the base flow is a linear function of axial location as shown in Fig. 4. The nozzle 

shape can be inversely derived according to Tsien's assumption, and we have 

~ 1 (2 , -1 2) 2(,,-1) 

a=- --+--M 

M ,+1 ,+1 (38) 

where, is the specific heat ratio and M is the Mach number which can be expressed 

as 

(39) 

The superscript * denotes the property at the nozzle throat. According to Tsien's 

derivation, the linearized quasi-one-dimensional equations can be manipulated to 

the following form under the isentropic condition, 

d2 P ( i(3) dP . (2 + i(3) z(1-z)--2 1+-- z--z(3 P=O 

dz 2 1 +, dz 2(, + 1) ( 40) 

(, + 1 )(1 - z): - (, - 1 + i(3)P + (2 + i(3)U = 0 ( 41) 

where 

(42) 

u and p are the velocity and pressure of the base flow, (3 is the normalized frequency 

which is defined as (3 = w(l - z)/(c* - u), and T is the non-dimensionalized time 

which is defined as T = c*t / x*. The independent variable z can be expressed 

in different forms due to the linearity between the base flow velocity u and axial 

location x, and we have 

139 

I 

Compact Differencing for the Euler Equations 

BQ BE_H 

at + ax - (36) 

where 

E = (pu~: p) a, H = (po~~) , (e + p)u 

(37) 

p is density, p is pressure, and e is the total energy defined as e = p( Cv T + ~ U 2 ). Cv 

is the constant volume specific heat. The variable a is the cross sectional area and 

is prescribed as a function of x. The theoretical solution of the acoustic admittance 

of a choked nozzle was provided by Tsien 7 under the assumption that the velocity 

of the base flow is a linear function of axial location as shown in Fig. 4. The nozzle 

shape can be inversely derived according to Tsien's assumption, and we have 

~ 1 (2 , -1 2) 2(,,-1) 

a=- --+--M 

M ,+1 ,+1 (38) 

where, is the specific heat ratio and M is the Mach number which can be expressed 

as 

(39) 

The superscript * denotes the property at the nozzle throat. According to Tsien's 

derivation, the linearized quasi-one-dimensional equations can be manipulated to 

the following form under the isentropic condition, 

d2 P ( i(3) dP . (2 + i(3) z(1-z)--2 1+-- z--z(3 P=O 

dz 2 1 +, dz 2(, + 1) ( 40) 

(, + 1 )(1 - z): - (, - 1 + i(3)P + (2 + i(3)U = 0 ( 41) 

where 

(42) 

u and p are the velocity and pressure of the base flow, (3 is the normalized frequency 

which is defined as (3 = w(l - z)/(c* - u), and T is the non-dimensionalized time 

which is defined as T = c*t / x*. The independent variable z can be expressed 

in different forms due to the linearity between the base flow velocity u and axial 

location x, and we have 

139 
S.-T. Yu 

x 

z=-

x* 

U 

c* 

(J + 1)M2 

2 + (, - 1)M2 

( 43) 

With Eqn. (43), it is clear that P and U are functions of the Mach number (M) 

with prescribed frequency ({3). Equation (40) is a hypergeometric equation14 that 

can be solved by a power series expansion. The converged solution does not exist 

in the supersonic region because the Mach number is greater than unity. U(z) can 

be easily solved with P(z) known as shown in Eqn. (41). Finally, the acoustic 

admittance function defined as A(z) = U(z)/ P(z) can be obtained as a function of 

the Mach number. 

In what follows, the procedure of the CFD calculation to compare with Tsien's 

solution is illustrated. First, the base flow field is obtained by solving the quasione-dimensional equations, Eqns. (36) and (37), using the RK4-CD2 method with 

the nozzle area ratio prescribed by Eqns. (38) and (39). The results are checked 

by the area Mach number relation15 and the solution is accurate up to five decimal 

digits. The perturbation at the inlet is obtained by specifying sinusoidal pressure 

fluctuations in terms of magnitude and frequency. With the prescribed pressure 

and isentropic correlation, the temperature fluctuation is also determined. Numerically, these boundary conditions are enforced by defining a vector k = k( Q) at the 

upstream boundary, such as 

(44) 

where 6 and 6 are the specified values of p and T. To proceed, Equation (44) is 

linearized to become a function of ~Q , such as 

(45) 

where k n+1 is equal to the specified pressure and temperature at the time step 

n + 1 and ok/oQ is a 3 x 3 matrix. In order to close the system, the null entry 

in the vector k may be filled by the out-running characteristic relation deduced 

from the flow equations. Numerically, the similarity transformation is applied to 

the discretized flow equations (see Eqn. (4)), and we get 

LM- 1 (Qi _ Qn) = LM-1~t L Q ik R k- 1, 

k=l (46) 

i = 1,···,N 

140 

S.-T. Yu 

x 

z=-

x* 

U 

c* 

(J + 1)M2 

2 + (, - 1)M2 

( 43) 

With Eqn. (43), it is clear that P and U are functions of the Mach number (M) 

with prescribed frequency ({3). Equation (40) is a hypergeometric equation14 that 

can be solved by a power series expansion. The converged solution does not exist 

in the supersonic region because the Mach number is greater than unity. U(z) can 

be easily solved with P(z) known as shown in Eqn. (41). Finally, the acoustic 

admittance function defined as A(z) = U(z)/ P(z) can be obtained as a function of 

the Mach number. 

In what follows, the procedure of the CFD calculation to compare with Tsien's 

solution is illustrated. First, the base flow field is obtained by solving the quasione-dimensional equations, Eqns. (36) and (37), using the RK4-CD2 method with 

the nozzle area ratio prescribed by Eqns. (38) and (39). The results are checked 

by the area Mach number relation15 and the solution is accurate up to five decimal 

digits. The perturbation at the inlet is obtained by specifying sinusoidal pressure 

fluctuations in terms of magnitude and frequency. With the prescribed pressure 

and isentropic correlation, the temperature fluctuation is also determined. Numerically, these boundary conditions are enforced by defining a vector k = k( Q) at the 

upstream boundary, such as 

(44) 

where 6 and 6 are the specified values of p and T. To proceed, Equation (44) is 

linearized to become a function of ~Q , such as 

(45) 

where k n+1 is equal to the specified pressure and temperature at the time step 

n + 1 and ok/oQ is a 3 x 3 matrix. In order to close the system, the null entry 

in the vector k may be filled by the out-running characteristic relation deduced 

from the flow equations. Numerically, the similarity transformation is applied to 

the discretized flow equations (see Eqn. (4)), and we get 

LM- 1 (Qi _ Qn) = LM-1~t L Q ik R k- 1, 

k=l (46) 

i = 1,···,N 

140 
Compact Differencing for the Euler Equations 

where i = 1, · .. ,N represents the N-step RK method. Here, M-1 is the eigenvector 

matrix of J aco bian matrix A = 8E / 8Q, and L is a -selection matrix with zeros and 

ones on the diagonal in such a fashion that the proper outrunning characteristics 

are selected. By combining the imposed conditions, Eqn. (45), with the outrunning 

characteristic relations, Eqn.( 46), we form the complete equation at the boundary 

point as, 

(LM-1 + :~) (Qi _ Qn) = LM-1L'>t ~ "ikRk- 1 + k n+1 _ kn, (47) 

i = I,··· ,N 

For the supersonic out-flow condition, Eqn. (46) is used with the selection matrix L 

equal to an identity matrix. In both cases, the out-running characteristic equations 

are solved with one-sided difference as shown in Eqn. (14). In other words, the 

characteristic boundary conditions are always discretized by an upwinding scheme 

which is physically sound and the numerical stability is enhanced. These boundary 

conditions are applied at each of the Runge-Kutta stages. 

The acoustic admittance is a complex number and can be written as A = IAlei9 . 

In the present paper, a small pressure perturbation of 1.1% (p' = O.Ol1p) is imposed 

at the nozzle inlet. The length of the converging part of the nozzle is 0.9 L * (see 

Fig. 4) and the inlet Mach number is about 0.09. The frequency ofthe perturbation 

is set at f3 = 6, which corresponds to about 2000 Hz. 

Figure 5 shows the comparisons between the CFD results of the RK4-CD6 method 

and the theoretical solution of the acoustic admittance in terms of the magnitude 

IAI and the phase angle e in the subsonic region of the nozzle. Both the magnitude 

and the phase angle of the acoustic admittance decrease as the flow speeds up. As 

shown in the figure, perfect agreement is obtained for the comparison of IAI, while 

the predicted phase angle is slightly off due to the resolution of the numerical grid 

for the phase angle. In this case, the harmonic content of the solution is limited to 

one frequency with a wave length comparable to the computational domain which 

is resolved by 61 grid nodes. Therefore, all numerical schemes of concern provide 

accurate solutions. The numerical errors of IAI and e are tabulated in Table 2. 

There is slight advantage in using the higher-order schemes for the prediction of 

IAlj however, no obvious advantage of using the higher-order scheme for the phase 

angle calculation is observed. 

2.4.2 Shocked Sound Waves 

The second case is the propagation of shocked sound waves in a tube with a 

periodic boundary condition. The governing equations are the same as in the first 

case, namely, Eqns. (36) and (37), with cross section area (a) equal to a constant. 

This case is interesting for its complex harmonic content compared to the first 

case. In addition, the capability of the high-order compact difference schemes for 

141 

Compact Differencing for the Euler Equations 

where i = 1, · .. ,N represents the N-step RK method. Here, M-1 is the eigenvector 

matrix of J aco bian matrix A = 8E / 8Q, and L is a -selection matrix with zeros and 

ones on the diagonal in such a fashion that the proper outrunning characteristics 

are selected. By combining the imposed conditions, Eqn. (45), with the outrunning 

characteristic relations, Eqn.( 46), we form the complete equation at the boundary 

point as, 

(LM-1 + :~) (Qi _ Qn) = LM-1L'>t ~ "ikRk- 1 + k n+1 _ kn, (47) 

i = I,··· ,N 

For the supersonic out-flow condition, Eqn. (46) is used with the selection matrix L 

equal to an identity matrix. In both cases, the out-running characteristic equations 

are solved with one-sided difference as shown in Eqn. (14). In other words, the 

characteristic boundary conditions are always discretized by an upwinding scheme 

which is physically sound and the numerical stability is enhanced. These boundary 

conditions are applied at each of the Runge-Kutta stages. 

The acoustic admittance is a complex number and can be written as A = IAlei9 . 

In the present paper, a small pressure perturbation of 1.1% (p' = O.Ol1p) is imposed 

at the nozzle inlet. The length of the converging part of the nozzle is 0.9 L * (see 

Fig. 4) and the inlet Mach number is about 0.09. The frequency ofthe perturbation 

is set at f3 = 6, which corresponds to about 2000 Hz. 

Figure 5 shows the comparisons between the CFD results of the RK4-CD6 method 

and the theoretical solution of the acoustic admittance in terms of the magnitude 

IAI and the phase angle e in the subsonic region of the nozzle. Both the magnitude 

and the phase angle of the acoustic admittance decrease as the flow speeds up. As 

shown in the figure, perfect agreement is obtained for the comparison of IAI, while 

the predicted phase angle is slightly off due to the resolution of the numerical grid 

for the phase angle. In this case, the harmonic content of the solution is limited to 

one frequency with a wave length comparable to the computational domain which 

is resolved by 61 grid nodes. Therefore, all numerical schemes of concern provide 

accurate solutions. The numerical errors of IAI and e are tabulated in Table 2. 

There is slight advantage in using the higher-order schemes for the prediction of 

IAlj however, no obvious advantage of using the higher-order scheme for the phase 

angle calculation is observed. 

2.4.2 Shocked Sound Waves 

The second case is the propagation of shocked sound waves in a tube with a 

periodic boundary condition. The governing equations are the same as in the first 

case, namely, Eqns. (36) and (37), with cross section area (a) equal to a constant. 

This case is interesting for its complex harmonic content compared to the first 

case. In addition, the capability of the high-order compact difference schemes for 

141 
S.-T. Yu 

shock capturing can also be studied. At time equal to zero, a sinusoidal pressure 

distribution is given. Because-of the periodic boundary condition, only one cycle 

resolved by 61 grid nodes is imposed in the computational domain. According to 

the isentropic condition, the distributions of temperature, density, and speed of 

sound are also determined. The velocity profile is determined by the simple wave 

correlation 16, such as 

( p(x) dp 

U(X) = Jp p(X)C(X) ' 

2, p-=r ,.-1 2=.! fJ = - ---= (p(X)2=Y - P 2,. ) , 

,-I ,p 

(48) 

where the average flow properties are denoted by a bar. With the simple wave 

correlation, the wave forms of all flow properties are in phase and the initial condition of the present CFD computation matches the theoretical analysis provided 

by Morse and IngardB. It is interesting to note that the simple wave correlation is 

an extension of a linear, plane, acoustic wave. For a variation of pressure less than 

5%, the plane wave relations could be adopted, such as 

T( x) = t ( 1 + ,~1 p'1x)) , 

p( x) = p ( 1 + p~;)) , 

p'(x) u(x) = c(x)-_ , ,p 

(49) 

where p' (x) is the prescribed pressure fluctuation. As shown in Eqns. (48) and (49), 

the wave speeds u + c, u - c and u vary as a result of the flow property distribution. 

The distortion of the wave form is a cumulative effect resulting from the wave speed 

distribution. For simple waves, i.e., all flow properties are in phase, the wave crest 

will quickly overtake the trough and form a shock. 

Figure 6 shows the time history of the pressure fluctuation at one end of the 

computational domain for various finite difference schemes. According to Morse andIngard, the first shock appears after about two cycles for the case of a 10% pressure 

perturbation (p'lp = O.I)B. All schemes of concern predict the wave steepening rate 

correctly. After the wave shocked, the flow evolution is no longer isentropic and the 

kinetic energy is gradually converted to thermal energy due to the existence of the 

shock wave. As a result, the strength of the shock wave diminishes as time passes. 

The shock front is a combination of many wave modes travelling at the same 

speed. The dispersion error introduced by the finite difference schemes will cause 

the high-wave-number waves to travel with erroneous speeds. As shown in Fig. 6, 

the methods of RK4-CD6 and RK4-CD4 with a small amount of the sixth-order 

artificial damping ('T} = 0.02) crisply resolve the shock except for the over-shoots. 

142 

S.-T. Yu 

shock capturing can also be studied. At time equal to zero, a sinusoidal pressure 

distribution is given. Because-of the periodic boundary condition, only one cycle 

resolved by 61 grid nodes is imposed in the computational domain. According to 

the isentropic condition, the distributions of temperature, density, and speed of 

sound are also determined. The velocity profile is determined by the simple wave 

correlation 16, such as 

( p(x) dp 

U(X) = Jp p(X)C(X) ' 

2, p-=r ,.-1 2=.! fJ = - ---= (p(X)2=Y - P 2,. ) , 

,-I ,p 

(48) 

where the average flow properties are denoted by a bar. With the simple wave 

correlation, the wave forms of all flow properties are in phase and the initial condition of the present CFD computation matches the theoretical analysis provided 

by Morse and IngardB. It is interesting to note that the simple wave correlation is 

an extension of a linear, plane, acoustic wave. For a variation of pressure less than 

5%, the plane wave relations could be adopted, such as 

T( x) = t ( 1 + ,~1 p'1x)) , 

p( x) = p ( 1 + p~;)) , 

p'(x) u(x) = c(x)-_ , ,p 

(49) 

where p' (x) is the prescribed pressure fluctuation. As shown in Eqns. (48) and (49), 

the wave speeds u + c, u - c and u vary as a result of the flow property distribution. 

The distortion of the wave form is a cumulative effect resulting from the wave speed 

distribution. For simple waves, i.e., all flow properties are in phase, the wave crest 

will quickly overtake the trough and form a shock. 

Figure 6 shows the time history of the pressure fluctuation at one end of the 

computational domain for various finite difference schemes. According to Morse andIngard, the first shock appears after about two cycles for the case of a 10% pressure 

perturbation (p'lp = O.I)B. All schemes of concern predict the wave steepening rate 

correctly. After the wave shocked, the flow evolution is no longer isentropic and the 

kinetic energy is gradually converted to thermal energy due to the existence of the 

shock wave. As a result, the strength of the shock wave diminishes as time passes. 

The shock front is a combination of many wave modes travelling at the same 

speed. The dispersion error introduced by the finite difference schemes will cause 

the high-wave-number waves to travel with erroneous speeds. As shown in Fig. 6, 

the methods of RK4-CD6 and RK4-CD4 with a small amount of the sixth-order 

artificial damping ('T} = 0.02) crisply resolve the shock except for the over-shoots. 

142 
Compact Differencing for the Euler Equations 

These over-shoots are caused by the Gibbs phenomenon and can be fixed only by 

TVD type shock-capturing schemes. Almost no difference can be observed between 

the results of the CD4 and CD6 methods. On the other hand, the method of RK4-

CD6 without background filtering shows that significant high-wave-number waves 

lag behind the shock front because the compact difference scheme introduces no 

dissipative but high dispersive effects on the highest wave number waves. As shown 

in the figure, these oscillations eventually contaminate the whole solution. For the 

conventional RK4-CD2 method, results show significant oscillations of moderate 

wave numbers behind the shock front because of dispersion errors. 

Figure 7 shows the normalized power spectrums of the pressure profiles after 

about 17 cycles calculated by different methods. The analytical solution is plotted 

as the solid line. The power of each wave mode is roughly inversely proportional 

to the square of the wave number (ex: 1/n2 ). Since 61 grid nodes are used, only 

30 Fourier modes are resolved for the power spectrum (the other 30 modes are the 

complex conjugates). Clearly, the method of RK4-CD2 has significant errors in low 

wave modes. On the other hand, the methods of RK4-CD4 and RK4-CD6 compare 

well with the analytical solution. 

2.4.3 Vortex Propagation in an Uniform Flow 

A Lamb vortex propagated in an uniform flow is chosen as a two-dimensional 

numerical example. The vortex can be characterized by the circulation r and the 

core radius a. The azimuthal velocity Uo at a distance r from the vortex center is 

given as, 

r r 

u - ---:--- () - 27f' r2 + a 2 ' (50) 

The flow near the vortex center is a rigid-body rotation (uo ex: r). The flow far 

outside the core is irrotational (U() ex: 1/r) with Uo decreasing as r increases. Eqn. 

(50) is a continuous function to connect the two extremes. With the prescribed 

velocity field, the pressure and density distributions of the vortex can be determined 

by the momentum and the energy equations, 

op u~ --par - r' 

2 _,_!!" + U o = ho 

,-lp 2 ' 

(51) 

(52) 

where ho is the total enthalpy and is set to be a constant such as ho = ,pl(t - l)p 

with the free stream condition denoted by a bar. To proceed, substitute Eqns. (50) 

and (52) into Eqn. (51) and integrate the equation over r. As a result, the pressure 

distribution is obtained. Consequently, the density distribution and the whole flow 

field is determined. The solutions of this stationary vortex can be superimposed 

to any uniform flow with arbitrary speed. Physically, this process may be interpreted as a stationary vortex being observed from a moving coordinate system with 

constant velocity. Thus, the vortex in an uniform flow can be constructed as, 

143 

Compact Differencing for the Euler Equations 

These over-shoots are caused by the Gibbs phenomenon and can be fixed only by 

TVD type shock-capturing schemes. Almost no difference can be observed between 

the results of the CD4 and CD6 methods. On the other hand, the method of RK4-

CD6 without background filtering shows that significant high-wave-number waves 

lag behind the shock front because the compact difference scheme introduces no 

dissipative but high dispersive effects on the highest wave number waves. As shown 

in the figure, these oscillations eventually contaminate the whole solution. For the 

conventional RK4-CD2 method, results show significant oscillations of moderate 

wave numbers behind the shock front because of dispersion errors. 

Figure 7 shows the normalized power spectrums of the pressure profiles after 

about 17 cycles calculated by different methods. The analytical solution is plotted 

as the solid line. The power of each wave mode is roughly inversely proportional 

to the square of the wave number (ex: 1/n2 ). Since 61 grid nodes are used, only 

30 Fourier modes are resolved for the power spectrum (the other 30 modes are the 

complex conjugates). Clearly, the method of RK4-CD2 has significant errors in low 

wave modes. On the other hand, the methods of RK4-CD4 and RK4-CD6 compare 

well with the analytical solution. 

2.4.3 Vortex Propagation in an Uniform Flow 

A Lamb vortex propagated in an uniform flow is chosen as a two-dimensional 

numerical example. The vortex can be characterized by the circulation r and the 

core radius a. The azimuthal velocity Uo at a distance r from the vortex center is 

given as, 

r r 

u - ---:--- () - 27f' r2 + a 2 ' (50) 

The flow near the vortex center is a rigid-body rotation (uo ex: r). The flow far 

outside the core is irrotational (U() ex: 1/r) with Uo decreasing as r increases. Eqn. 

(50) is a continuous function to connect the two extremes. With the prescribed 

velocity field, the pressure and density distributions of the vortex can be determined 

by the momentum and the energy equations, 

op u~ --par - r' 

2 _,_!!" + U o = ho 

,-lp 2 ' 

(51) 

(52) 

where ho is the total enthalpy and is set to be a constant such as ho = ,pl(t - l)p 

with the free stream condition denoted by a bar. To proceed, substitute Eqns. (50) 

and (52) into Eqn. (51) and integrate the equation over r. As a result, the pressure 

distribution is obtained. Consequently, the density distribution and the whole flow 

field is determined. The solutions of this stationary vortex can be superimposed 

to any uniform flow with arbitrary speed. Physically, this process may be interpreted as a stationary vortex being observed from a moving coordinate system with 

constant velocity. Thus, the vortex in an uniform flow can be constructed as, 

143 
S.-T. Yu 

u = it + u', 

V = V + VI, 

(53) 

where the velocities of the background flow are denoted by a bar and the superscript 

I denotes the vortex velocities specified by Eqn. (50). The pressure and density 

distribution of the moving vortex is the same as that of the stationary vortex and 

may be obtained from the solutions of Eqns. (51) and (52). 

The boundary condition of the present case is an extension of the characteristic 

type treatment discussed in Case 1. Essentially, only one-dimensional characteristics (derived from two-dimensional flow equations) normal to the computational 

boundary are considered. For the purposes of this discussion, the subsonic out-flow 

condition is considered. The coupled equations of three out-running characteristics and one specified boundary condition, similar to that in Eqn. (47), should be 

solved. For steady state calculations, a back pressure (Pb) is specified to regulate 

the flow rate, such as 

k= (D (54) 

Similarly, the dimension of vectors Q and R is four and the matrix M-l is a 4 x 4 

eigenvector matrix for the flux vector normal to the computational boundary. 

For a non-reflective boundary condition, Giles' formulation17 instead of the back 

pressure is used to fill the entry for the specified boundary condition, such as 

(55) 

where y is in the direction parallel to the computational boundary. The variables 

Ci, i = 1, ... ,4 are the characteristic variables and can be obtained by the similarity 

transformation from the non-conservative form equations as illustrated by Giles. In 

our case, the characteristic variables are derived from the conservative-form equations using the same eigenvector matrix M-l as in the aforementioned discussion. 

Giles' non-reflective formulation, Eqn. (55), is relatively simple to used with an 

existing one-dimensional characteristic boundary condition. Nonetheless, according 

to Giles' analysis, some two-dimensional effect is considered in the equation. Numerically, Eqn. (55) may be discretized :::J.ccording to the finite difference scheme 

of the interior nodes and combined with the discretized out-running characteristic 

equations (the two-dimensional version of Eqn. (46)) to form the complete subsonic 

out-flow boundary condition. It is interestillg to note, however, according to Huff's 

study18 and our experience, that stretching the numerical grid nodes downstream 

to dampen the outgoing unsteady waves is just as effective. 

144 

S.-T. Yu 

u = it + u', 

V = V + VI, 

(53) 

where the velocities of the background flow are denoted by a bar and the superscript 

I denotes the vortex velocities specified by Eqn. (50). The pressure and density 

distribution of the moving vortex is the same as that of the stationary vortex and 

may be obtained from the solutions of Eqns. (51) and (52). 

The boundary condition of the present case is an extension of the characteristic 

type treatment discussed in Case 1. Essentially, only one-dimensional characteristics (derived from two-dimensional flow equations) normal to the computational 

boundary are considered. For the purposes of this discussion, the subsonic out-flow 

condition is considered. The coupled equations of three out-running characteristics and one specified boundary condition, similar to that in Eqn. (47), should be 

solved. For steady state calculations, a back pressure (Pb) is specified to regulate 

the flow rate, such as 

k= (D (54) 

Similarly, the dimension of vectors Q and R is four and the matrix M-l is a 4 x 4 

eigenvector matrix for the flux vector normal to the computational boundary. 

For a non-reflective boundary condition, Giles' formulation17 instead of the back 

pressure is used to fill the entry for the specified boundary condition, such as 

(55) 

where y is in the direction parallel to the computational boundary. The variables 

Ci, i = 1, ... ,4 are the characteristic variables and can be obtained by the similarity 

transformation from the non-conservative form equations as illustrated by Giles. In 

our case, the characteristic variables are derived from the conservative-form equations using the same eigenvector matrix M-l as in the aforementioned discussion. 

Giles' non-reflective formulation, Eqn. (55), is relatively simple to used with an 

existing one-dimensional characteristic boundary condition. Nonetheless, according 

to Giles' analysis, some two-dimensional effect is considered in the equation. Numerically, Eqn. (55) may be discretized :::J.ccording to the finite difference scheme 

of the interior nodes and combined with the discretized out-running characteristic 

equations (the two-dimensional version of Eqn. (46)) to form the complete subsonic 

out-flow boundary condition. It is interestillg to note, however, according to Huff's 

study18 and our experience, that stretching the numerical grid nodes downstream 

to dampen the outgoing unsteady waves is just as effective. 

144 
Compact Differencing for the Euler Equations 

For the in-flow conditions, the characteristic-type treatment combined with Giles' 

equation (different from Eqn. (55)) may be adopted. For the present calculations, 

however, the upstream condition is relatively insensitive to various forms of nonreflective treatment as long as the proper out-running characteristic equation is 

selected and solved with the prescribed incoming conditions similar to that in Eqn. 

(47). In the present case, constant total pressure and total temperature are prescribed as the forcing boundary conditions upstream. 

As in the one-dimensional cases, dissipative and dispersive effects of various 

schemes are assessed. The prescribed 'vortex flow field contains a broad band of 

frequencies due to the distribution of the azimuthal velocity. Theoretically, all wave 

modes travels at the same speed to ensure the integrity of the vortex structure. 

For numerical methods with dispersive error, the shape of the vortex could deform, 

even break up in the later stage of the time marching procedure. In addition, the 

dissipation effect of finite difference schemes can be evaluated by the conservation 

of the sharp pressure dip at the center of the vortex propagated in the numerical 

grid. 

In the present calculations, the Mach number of the background flow is 0.4. The 

grid size is 301 x 91 in the streamwise and transverse directions, respectively. Uniform grids are used in the axial direction and the transverse grids are stretched near 

the outside boundary. The CFL number calculated based on the background flow 

is about 0.7 for all calculations. As discussed before, a small amount of background 

filtering (7] = 0.02) is applied for all calculations. The core radius (a) is about 1 cm 

and is resolved by about 4 grid nodes. 

Figure 8 shows the vorticity and Mach number contours of the initial condition 

prescribed by Eqns. (50) - (53). Figure 9 shows the contours after the eddy propagates about 60 core radii downstream as simulated by various numerical schemes. 

The comparison between Fig. 8 and Fig. 9 shows that the structure of the eddy is 

retained by the compact difference schemes (CD4 and CD6). In contrast, the eddy 

predicted by the second-order central difference is shattered due to the excessive 

dispersive error. 

Figure 10 shows pressure distributions of the eddy at various instances. In this 

figure, the x axis represents the streamwise locations non-dimensionalized by the 

core radius of the vortex and the y axis is the pressure. For both CD4 and CD6 

methods, the pressure at the vortex center increases about 1 % through the process. 

In comparison, the results of the second-order scheme show pressure fluctuations 

with an overall increase about 3 %. The pressure fluctuation predicted by the 

second-order scheme is due to the deviation of the vortex path. 

2.4.4 Vortex Pairing 

Finally, the calculation of the single vortex is extended to the simulation of the 

vortex pairing. The vortex pairing is the controlling mechanism for the growth of 

the mixing layerlO . In theory, vortex pairing occurs when the distance between 

two vortices is less than a threshold value. Unfortunately, no theoretical analysis 

145 

Compact Differencing for the Euler Equations 

For the in-flow conditions, the characteristic-type treatment combined with Giles' 

equation (different from Eqn. (55)) may be adopted. For the present calculations, 

however, the upstream condition is relatively insensitive to various forms of nonreflective treatment as long as the proper out-running characteristic equation is 

selected and solved with the prescribed incoming conditions similar to that in Eqn. 

(47). In the present case, constant total pressure and total temperature are prescribed as the forcing boundary conditions upstream. 

As in the one-dimensional cases, dissipative and dispersive effects of various 

schemes are assessed. The prescribed 'vortex flow field contains a broad band of 

frequencies due to the distribution of the azimuthal velocity. Theoretically, all wave 

modes travels at the same speed to ensure the integrity of the vortex structure. 

For numerical methods with dispersive error, the shape of the vortex could deform, 

even break up in the later stage of the time marching procedure. In addition, the 

dissipation effect of finite difference schemes can be evaluated by the conservation 

of the sharp pressure dip at the center of the vortex propagated in the numerical 

grid. 

In the present calculations, the Mach number of the background flow is 0.4. The 

grid size is 301 x 91 in the streamwise and transverse directions, respectively. Uniform grids are used in the axial direction and the transverse grids are stretched near 

the outside boundary. The CFL number calculated based on the background flow 

is about 0.7 for all calculations. As discussed before, a small amount of background 

filtering (7] = 0.02) is applied for all calculations. The core radius (a) is about 1 cm 

and is resolved by about 4 grid nodes. 

Figure 8 shows the vorticity and Mach number contours of the initial condition 

prescribed by Eqns. (50) - (53). Figure 9 shows the contours after the eddy propagates about 60 core radii downstream as simulated by various numerical schemes. 

The comparison between Fig. 8 and Fig. 9 shows that the structure of the eddy is 

retained by the compact difference schemes (CD4 and CD6). In contrast, the eddy 

predicted by the second-order central difference is shattered due to the excessive 

dispersive error. 

Figure 10 shows pressure distributions of the eddy at various instances. In this 

figure, the x axis represents the streamwise locations non-dimensionalized by the 

core radius of the vortex and the y axis is the pressure. For both CD4 and CD6 

methods, the pressure at the vortex center increases about 1 % through the process. 

In comparison, the results of the second-order scheme show pressure fluctuations 

with an overall increase about 3 %. The pressure fluctuation predicted by the 

second-order scheme is due to the deviation of the vortex path. 

2.4.4 Vortex Pairing 

Finally, the calculation of the single vortex is extended to the simulation of the 

vortex pairing. The vortex pairing is the controlling mechanism for the growth of 

the mixing layerlO . In theory, vortex pairing occurs when the distance between 

two vortices is less than a threshold value. Unfortunately, no theoretical analysis 

145 
5.-T. Yu 

is available for compressible flows. In the present paper, the RK4-CD6 method is 

used to simulate the pairing process to demonstrate the resolution of the high-order 

compact difference scheme. 

The initial condition is specified by two identical vortices placed 5 core rarlii apart 

in a quiescent gas. The core radius is 1 cm and circulation is 15 m2 / s. At the center 

of each vortex, there is a pressure deficit about 15% compared to the ambient gas. 

The grid size is 201 x 201. Uniform grids are used at the center of the computational 

domain to resolve the vortices. The grids are slightly stretched in all four directions 

to prevent erroneous wave reflection. In addition, one-dimensional characteristic 

equations combined with Giles' unsteady, subsonic, out-flow equation is solved on 

the boundary as the non-reflective boundary condition. 

Figure 11 shows the contours of the vorticity magnitude at various stages of the 

vortex interaction. The whole sequence is about one and a half revolutions. Finally, 

a single larger vortex emerges as the result of the vortex pairing interaction. Figure 

12 shows the corresponding Mach number contours for the same flow . 

3. Concluding Remarks 

In this work, the quasi-one-dimensional and two-dimensional Euler solvers using various combinations of the Runge Kutta methods and the compact difference 

schemes were developed for numerical simulations of unsteady flows. The accuracy 

of the finite difference schemes is assessed by Fourier analysis and numerical examples in terms of numerical dissipation and dispersion. The dispersive characteristic 

is improved by high-order compact difference schemes compared to the second-order 

central difference. The increase of the order of time stepping scheme, on the other 

hand, enlarges the CFL limit for stable computations. In particular, significant improvement of the dispersive effect is obtained by adopting the fourth-order compact 

scheme (6 to 8 grid nodes to resolve a wave) instead of the conventional second-order 

central difference (12 to 16 grid nodes for one wave). The use of the sixth-order 

compact scheme (5 to 8 grid nodes for one wave), however, gains little improvement 

compared with the fourth-order scheme. It was also found that the compact difference schemes have no dissipative but high dispersive effects to the highest-wavenumber waves resolved by a given numerical grid. Consequently, a small amount of 

background filtering is necessary to dissipate the high-wave-number waves and, at 

the same time, keep the low-wave-number solution intact. Other issues such as the 

order of accuracy of the Runge-Kutta schemes for nonlinear equations are analyzed. 

Specifically, the criteria for the 3 and 4-step methods to be third and fourth-order 

accurate are derived. The accuracy of the compact difference schemes for the steady 

state solution is also addressed. 

In general, simulation of unsteady flow provides an overwhelming amount of information. It is our experience that the initial and boundary conditions must be 

carefully set up to obtain interpretable and physically meaningful solutions. For 

practical purposes, Giles' non-reflective equations combined with one-dimensional 

characteristic equations and their implementation to the present numerical scheme 

146 

5.-T. Yu 

is available for compressible flows. In the present paper, the RK4-CD6 method is 

used to simulate the pairing process to demonstrate the resolution of the high-order 

compact difference scheme. 

The initial condition is specified by two identical vortices placed 5 core rarlii apart 

in a quiescent gas. The core radius is 1 cm and circulation is 15 m2 / s. At the center 

of each vortex, there is a pressure deficit about 15% compared to the ambient gas. 

The grid size is 201 x 201. Uniform grids are used at the center of the computational 

domain to resolve the vortices. The grids are slightly stretched in all four directions 

to prevent erroneous wave reflection. In addition, one-dimensional characteristic 

equations combined with Giles' unsteady, subsonic, out-flow equation is solved on 

the boundary as the non-reflective boundary condition. 

Figure 11 shows the contours of the vorticity magnitude at various stages of the 

vortex interaction. The whole sequence is about one and a half revolutions. Finally, 

a single larger vortex emerges as the result of the vortex pairing interaction. Figure 

12 shows the corresponding Mach number contours for the same flow . 

3. Concluding Remarks 

In this work, the quasi-one-dimensional and two-dimensional Euler solvers using various combinations of the Runge Kutta methods and the compact difference 

schemes were developed for numerical simulations of unsteady flows. The accuracy 

of the finite difference schemes is assessed by Fourier analysis and numerical examples in terms of numerical dissipation and dispersion. The dispersive characteristic 

is improved by high-order compact difference schemes compared to the second-order 

central difference. The increase of the order of time stepping scheme, on the other 

hand, enlarges the CFL limit for stable computations. In particular, significant improvement of the dispersive effect is obtained by adopting the fourth-order compact 

scheme (6 to 8 grid nodes to resolve a wave) instead of the conventional second-order 

central difference (12 to 16 grid nodes for one wave). The use of the sixth-order 

compact scheme (5 to 8 grid nodes for one wave), however, gains little improvement 

compared with the fourth-order scheme. It was also found that the compact difference schemes have no dissipative but high dispersive effects to the highest-wavenumber waves resolved by a given numerical grid. Consequently, a small amount of 

background filtering is necessary to dissipate the high-wave-number waves and, at 

the same time, keep the low-wave-number solution intact. Other issues such as the 

order of accuracy of the Runge-Kutta schemes for nonlinear equations are analyzed. 

Specifically, the criteria for the 3 and 4-step methods to be third and fourth-order 

accurate are derived. The accuracy of the compact difference schemes for the steady 

state solution is also addressed. 

In general, simulation of unsteady flow provides an overwhelming amount of information. It is our experience that the initial and boundary conditions must be 

carefully set up to obtain interpretable and physically meaningful solutions. For 

practical purposes, Giles' non-reflective equations combined with one-dimensional 

characteristic equations and their implementation to the present numerical scheme 

146 
Compact Differencing for the Euler Equations 

were illustrated in detail. In addition, the initial conditions of the simple wave, 

plane acoustic wave, and the Lamb vortex were also provided. Finally, as illustrated in the numerical examples, for flows of simple harmonic content, e.g., one 

frequency in Case 1, the conventional second-order central difference scheme is adequate provided enough grid nodes are used to resolve the very wave mode. On 

the other hand, for flows of complex harmonic content, the use of the Runge-Kutta 

method combined high-order compact difference schemes shows crisp resolution of 

unsteady flows. 

4. References 

1 Lele, S. "Direct Numerical Simulations of Compressible Free Shear Flows," AIAA 

Paper 89-0374 (1989). 

2 Lele, S. "Compact Finite Difference Schemes with Spectral-Like Resolution," 

CTR Manuscript 107, Center for Turbulence Research, Stanford University, C. 

A. (1990). 

3 Jameson, A., Schmidt, W. and Turkel, E. "Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes," 

AIAA Paper 81-1259 (1981). 

4 Jameson, A. and Baker, T. J. "Solution of Euler Equations for Complex Configuration," AIAA Paper 83-1929 (1983). 

5 Wray, A. A., "Very Low Storage Time Advancement Schemes," Internal Report, 

NASA Ames Research Center, Moffett Field, C. A. (1986). 

6 Ciment, M., Leventhal, S. H., and Weinberg, B., J. Comput. Phys. 28, 135 

(1978). 

7 Tsien, H. S., J. Am. Rocket Soc. 22, 139 (1952). 

8 Morse, P. M. and Ingard, K. V., "Theoretical Acoustics," McGraw-Hill, 874 

(1968). 

9 Liu, N.-S., Davoudzadeh, F., Briley, W. R., and Shamroth, S. J., J. Fluids Eng. 

12, 501 (1990). 

10 Winant, C. D. and Browand, F. K., J. Fluid Mech. 63, 237 (1974). 

11 Carnahan, B., Luther, A. A., and Wilkes, J. 0 ., "Applied Numerical Methods," 

John Wiley fj Sons, Inc., 363 (1969). 

12 Colatz, L., "The Numerical Treatment of Differential Equations," Springer Verlag, 538 (1966). 

13 Adam, Y., J. Comput. Phys. 24, 10 (1977). 

14 Williams, F. A., "Combustion Theory" Benjamin & Cummings, 306 (1985). 

15 Anderson, J. D., "Modern Compressible Flow," McGraw-Hill, 128 (1982). 

16 Lighthill, J., "Waves in Fluids," Cambridge University Press, 142 (1978). 

17 Giles, M. B., AIAA J. 28, 12, 2050 (1990). 

18 Huff, D. L., "Pressure Waves Propagation Studies for Oscillating Cascades," 

NASA-TM-105406 (1992). 

147 

Compact Differencing for the Euler Equations 

were illustrated in detail. In addition, the initial conditions of the simple wave, 

plane acoustic wave, and the Lamb vortex were also provided. Finally, as illustrated in the numerical examples, for flows of simple harmonic content, e.g., one 

frequency in Case 1, the conventional second-order central difference scheme is adequate provided enough grid nodes are used to resolve the very wave mode. On 

the other hand, for flows of complex harmonic content, the use of the Runge-Kutta 

method combined high-order compact difference schemes shows crisp resolution of 

unsteady flows. 

4. References 

1 Lele, S. "Direct Numerical Simulations of Compressible Free Shear Flows," AIAA 

Paper 89-0374 (1989). 

2 Lele, S. "Compact Finite Difference Schemes with Spectral-Like Resolution," 

CTR Manuscript 107, Center for Turbulence Research, Stanford University, C. 

A. (1990). 

3 Jameson, A., Schmidt, W. and Turkel, E. "Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes," 

AIAA Paper 81-1259 (1981). 

4 Jameson, A. and Baker, T. J. "Solution of Euler Equations for Complex Configuration," AIAA Paper 83-1929 (1983). 

5 Wray, A. A., "Very Low Storage Time Advancement Schemes," Internal Report, 

NASA Ames Research Center, Moffett Field, C. A. (1986). 

6 Ciment, M., Leventhal, S. H., and Weinberg, B., J. Comput. Phys. 28, 135 

(1978). 

7 Tsien, H. S., J. Am. Rocket Soc. 22, 139 (1952). 

8 Morse, P. M. and Ingard, K. V., "Theoretical Acoustics," McGraw-Hill, 874 

(1968). 

9 Liu, N.-S., Davoudzadeh, F., Briley, W. R., and Shamroth, S. J., J. Fluids Eng. 

12, 501 (1990). 

10 Winant, C. D. and Browand, F. K., J. Fluid Mech. 63, 237 (1974). 

11 Carnahan, B., Luther, A. A., and Wilkes, J. 0 ., "Applied Numerical Methods," 

John Wiley fj Sons, Inc., 363 (1969). 

12 Colatz, L., "The Numerical Treatment of Differential Equations," Springer Verlag, 538 (1966). 

13 Adam, Y., J. Comput. Phys. 24, 10 (1977). 

14 Williams, F. A., "Combustion Theory" Benjamin & Cummings, 306 (1985). 

15 Anderson, J. D., "Modern Compressible Flow," McGraw-Hill, 128 (1982). 

16 Lighthill, J., "Waves in Fluids," Cambridge University Press, 142 (1978). 

17 Giles, M. B., AIAA J. 28, 12, 2050 (1990). 

18 Huff, D. L., "Pressure Waves Propagation Studies for Oscillating Cascades," 

NASA-TM-105406 (1992). 

147 
5.-T. Yu 

Table 1. The Accuracy of the 3-Step Runge-Kutta Methods. 

!:It Expansion 

o R 

1 R 

2 lRR' 

3 I(R2 R" + RR,2) 

3-step R-K methods 

R 

(Q31 + Q32 + Q33)R 

[allQ32 + Q33(a21 + an)]RR' 

~[QrlQ32 + (Q21 +Q22) 2Q33]R2R" 

+Qll Qn Q33 RR,2 

Table 2. The Relative Error of the Acoustic Admittance Calculation (%). 

Numerical Schemes 

RK4-CD6 

RK4-CD4 

RK4-CD2 

Error of IAI 

0.45 

0.52 

1.65 

148 

Error of e 

3.6 

3.3 

4.1 

5.-T. Yu 

Table 1. The Accuracy of the 3-Step Runge-Kutta Methods. 

!:It Expansion 

o R 

1 R 

2 lRR' 

3 I(R2 R" + RR,2) 

3-step R-K methods 

R 

(Q31 + Q32 + Q33)R 

[allQ32 + Q33(a21 + an)]RR' 

~[QrlQ32 + (Q21 +Q22) 2Q33]R2R" 

+Qll Qn Q33 RR,2 

Table 2. The Relative Error of the Acoustic Admittance Calculation (%). 

Numerical Schemes 

RK4-CD6 

RK4-CD4 

RK4-CD2 

Error of IAI 

0.45 

0.52 

1.65 

148 

Error of e 

3.6 

3.3 

4.1 
1.4 

1.3 

1.2 

0' 

1.1 

1.0 

0.9 

0.0 1.0 2.0 3.0 k 

a. Dissipation of RK3-CD6 

1.4 

1.3 

1.2 

1.1 

0.9 '---'--_'---'-_'---'-_L.. 

0.0 1.0 2.0 3.0 

c. Dissipation of RK3-CD4 

l.4 

1.3 

1.2 

l.1 

0.4 

0.9 L.._...l----l_....l-.---1._..!.-..-'-

0.0 1.0 2.0 .3.0 

e. Dissipation of RK.3-CD2 

(~ 

3.0 

2.5 

2.0 

1.5 

1.0 

0.5 

0.0 

0.0 1.0 2.0 3.0 

.3.0 

2.5 

2.0 

1.5 

1.0 

0.5 

b. Dispersion of RK3-CD6 

1.4 

0.0 1L-_---l_....L..----1._...I.-~ 

0.0 1.0 2.0 3.0 

d . D:spersion of RK3-CD4 

.3.0 

2.5 

2.0 

, .5 

1.0 

0.5 

1.4 

0.0 ~ __ L.._--'-_1__--'-_.u. 

0.0 l.0 2.0 3 .0 

f . D:sDersion of RK3-CD2 

k 

Fig. 1 Dissipation and dispersion characteristics of the RK3 time-stepping combined with 

various spatial discretization schemes. 

149 

1.4 

1.3 

1.2 

0' 

1.1 

1.0 

0.9 

0.0 1.0 2.0 3.0 k 

a. Dissipation of RK3-CD6 

1.4 

1.3 

1.2 

1.1 

0.9 '---'--_'---'-_'---'-_L.. 

0.0 1.0 2.0 3.0 

c. Dissipation of RK3-CD4 

l.4 

1.3 

1.2 

l.1 

0.4 

0.9 L.._...l----l_....l-.---1._..!.-..-'-

0.0 1.0 2.0 .3.0 

e. Dissipation of RK.3-CD2 

(~ 

3.0 

2.5 

2.0 

1.5 

1.0 

0.5 

0.0 

0.0 1.0 2.0 3.0 

.3.0 

2.5 

2.0 

1.5 

1.0 

0.5 

b. Dispersion of RK3-CD6 

1.4 

0.0 1L-_---l_....L..----1._...I.-~ 

0.0 1.0 2.0 3.0 

d . D:spersion of RK3-CD4 

.3.0 

2.5 

2.0 

, .5 

1.0 

0.5 

1.4 

0.0 ~ __ L.._--'-_1__--'-_.u. 

0.0 l.0 2.0 3 .0 

f . D:sDersion of RK3-CD2 

k 

Fig. 1 Dissipation and dispersion characteristics of the RK3 time-stepping combined with 

various spatial discretization schemes. 

149 
CFL 0.4 

1.0 3.0 

0.9 2.5 

0.8 2.0 

01 <<::\ 1.5 

0.7 

1.0 

0.6 0.5 

0.5 k 0.0 k 

0.0 1.0 2.0 3.0 0.0 1.0 2.0 3 .0 

o. Dissipation of RK4-CD6 b. Dispersion of RK4-CD6 

0.4 

1.0 3.0 

0.9 2.5 

0.8 2.0 

1.5 

0.7 

1.0 

0.6 0.5 

0.5 0.0 

0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 

c. Dissipation of RK4-C04 d . Dispersion of RK4-CD4 

0.4 

1.0 3.0 

0.9 1.4 2.5 

0.8 2.0 

1.5 

0.7 

1.0 0.4 .4 

0.6 0.5 

0.5 0.0 

0.0 1.0 2.0 3.0 0.0 1.0 2. 0 3 .0 

e. Dissipation of RK4-CD2 f. Dispersion of R:<4-CD2 

Fig. 2 Dissipation and dispersion characteristics of the RK4 time-stepping combined with 

various spatial discretization schemes. 

150 

- - ,--

CFL 0.4 

1.0 3.0 

0.9 2.5 

0.8 2.0 

01 <<::\ 1.5 

0.7 

1.0 

0.6 0.5 

0.5 k 0.0 k 

0.0 1.0 2.0 3.0 0.0 1.0 2.0 3 .0 

o. Dissipation of RK4-CD6 b. Dispersion of RK4-CD6 

0.4 

1.0 3.0 

0.9 2.5 

0.8 2.0 

1.5 

0.7 

1.0 

0.6 0.5 

0.5 0.0 

0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 

c. Dissipation of RK4-C04 d . Dispersion of RK4-CD4 

0.4 

1.0 3.0 

0.9 1.4 2.5 

0.8 2.0 

1.5 

0.7 

1.0 0.4 .4 

0.6 0.5 

0.5 0.0 

0.0 1.0 2.0 3.0 0.0 1.0 2. 0 3 .0 

e. Dissipation of RK4-CD2 f. Dispersion of R:<4-CD2 

Fig. 2 Dissipation and dispersion characteristics of the RK4 time-stepping combined with 

various spatial discretization schemes. 

150 

- - ,--
1.00 

0 .95 1] = 0.01 

0 .90 

~ 0.85 

0 .80 

0 .75 

0 .70 0.05 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

k 

2.0 0.05 

1.5 1] = 0.01 

~ 1.0 

0 .5 

0.0 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

k 

Fig. 3 Dissipation and dispersion characteristics of the RK4-CD6 method with vanous 

amount of t he sixth-order numerical damping. 

151 

1.00 

0 .95 1] = 0.01 

0 .90 

~ 0.85 

0 .80 

0 .75 

0 .70 0.05 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

k 

2.0 0.05 

1.5 1] = 0.01 

~ 1.0 

0 .5 

0.0 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

k 

Fig. 3 Dissipation and dispersion characteristics of the RK4-CD6 method with vanous 

amount of t he sixth-order numerical damping. 

151 
----------/ 

Fig. 4 Configuration of Tsien's converging-diverging nozzle [7J. 

152 

x 

----------/ 

Fig. 4 Configuration of Tsien's converging-diverging nozzle [7J. 

152 

x 
6 

5 

(l) 

-g 4 

0... 

E 3 

o 

2 

0.4 0.8 

x/x* 

6. 

45 

-

Q) 

CJl 40 \ \ 

C 

0 35 

(l) 

(f) 

0 

...c 30 

0.-

25 

20 

0.4 0.8 

x/x* 

Fig. 5 Acoustic admittance calculation using the RK4-CD6 method for the inlet perturbation 

as f3 = 6, p' 1ft = 0.011. The solid line is Tsien's theorem and the triangles are the 

CFD results. (a) Magnitude. (b )Phase angle. 

153 

6 

5 

(l) 

-g 4 

0... 

E 3 

o 

2 

0.4 0.8 

x/x* 

6. 

45 

-

Q) 

CJl 40 \ \ 

C 

0 35 

(l) 

(f) 

0 

...c 30 

0.-

25 

20 

0.4 0.8 

x/x* 

Fig. 5 Acoustic admittance calculation using the RK4-CD6 method for the inlet perturbation 

as f3 = 6, p' 1ft = 0.011. The solid line is Tsien's theorem and the triangles are the 

CFD results. (a) Magnitude. (b )Phase angle. 

153 
0.20 

0.10 

Q.. 

::--- 0.00 Q.. 

-0.10 

-0.20 L...-__ ---L. ___ ...l-__ ----' ___ --L.. __ ___l 

0.20 

0.10 

Q.. 

::--- 0.00 Q.. 

-0.10 

o 20000 40000 

(0) RK4-CD6 with A.D. 

-0.20 L...-__ ---L ___ --'-___ ...l-___ ..!.-__ --..J 

0.20 

0 . 10 

Q.. 

::--- 0.00 Q.. 

-0.10 

o 20000 40000 

(b) RK4-CD6 without A.D. 

-0.20 l..-__ --L ___ -L-__ ----' ___ ~ __ ___l 

0.20 

Q.. 

::--- 0.00 Q.. 

-0. 10 

o 20000 40000 

(C) RK4-CD4 with A.D. 

-0.20 l..-__ --L ___ -L-__ ----' ___ ---!... __ ----l 

o 20000 40000 

(d) RK4-CD2 with A.D. 

Fig. 6 Time histories of the pressure fluctuations of the N-wave calculation at one end of the 

periodic domain by various numerical schemes. 

154 

0.20 

0.10 

Q.. 

::--- 0.00 Q.. 

-0.10 

-0.20 L...-__ ---L. ___ ...l-__ ----' ___ --L.. __ ___l 

0.20 

0.10 

Q.. 

::--- 0.00 Q.. 

-0.10 

o 20000 40000 

(0) RK4-CD6 with A.D. 

-0.20 L...-__ ---L ___ --'-___ ...l-___ ..!.-__ --..J 

0.20 

0 . 10 

Q.. 

::--- 0.00 Q.. 

-0.10 

o 20000 40000 

(b) RK4-CD6 without A.D. 

-0.20 l..-__ --L ___ -L-__ ----' ___ ~ __ ___l 

0.20 

Q.. 

::--- 0.00 Q.. 

-0. 10 

o 20000 40000 

(C) RK4-CD4 with A.D. 

-0.20 l..-__ --L ___ -L-__ ----' ___ ---!... __ ----l 

o 20000 40000 

(d) RK4-CD2 with A.D. 

Fig. 6 Time histories of the pressure fluctuations of the N-wave calculation at one end of the 

periodic domain by various numerical schemes. 

154 
Fig. 7 Power spectrum of the pressure distribution after about 17 cycles of the N-wave propagation by various numerical schemes. 

155 

Fig. 7 Power spectrum of the pressure distribution after about 17 cycles of the N-wave propagation by various numerical schemes. 

155 
Fig. 8 Vorticity and Mach number contours of an analytical Lamb vortex. 

156 

Fig. 8 Vorticity and Mach number contours of an analytical Lamb vortex. 

156 
RK4-CD6 

RK4-CD4 

RK4-CD2 

d Mac tours of t he lamb vortex after travellIng - about 60 core 

h number con n-cal schemes_ - -t an _ nume FIg_ - 9 VortIcI Y predIcte _ d b,- varIouS _ 

diameters 157 

RK4-CD6 

RK4-CD4 

RK4-CD2 

d Mac tours of t he lamb vortex after travellIng - about 60 core 

h number con n-cal schemes_ - -t an _ nume FIg_ - 9 VortIcI Y predIcte _ d b,- varIouS _ 

diameters 157 
1.00 

0 .98 

.§ 0.96 0 

...... 0.94 CI.l 

... 

~ 0.92 (Il 

~ 0 .90 a. 

0.88 

0 .86 

0 20 40 60 80 

core radii 

a. RK4-CD6 

1.00 

0.98 

.§ 0 .9 6 0 

...... 0.94 CI.l 

... 

~ 0 .92 (Il 

~ 0.90 a. 

0.88 

0 .86 

0 20 40 60 80 

core radii 

b. RK4-C04 

1.00 

0.98 

.§ 0 .96 0 

0 .94 CI.l 

... 

~ 0.92 (Il 

~ 0.90 a. 

0 .88 

0 .86 

0 20 40 60 80 

core radii 

c. RK4 - CD2 

Fig. 10 Vortex pressure profiles at the center line at various instances predicted by different 

numerical schemes. 

158 

1.00 

0 .98 

.§ 0.96 0 

...... 0.94 CI.l 

... 

~ 0.92 (Il 

~ 0 .90 a. 

0.88 

0 .86 

0 20 40 60 80 

core radii 

a. RK4-CD6 

1.00 

0.98 

.§ 0 .9 6 0 

...... 0.94 CI.l 

... 

~ 0 .92 (Il 

~ 0.90 a. 

0.88 

0 .86 

0 20 40 60 80 

core radii 

b. RK4-C04 

1.00 

0.98 

.§ 0 .96 0 

0 .94 CI.l 

... 

~ 0.92 (Il 

~ 0.90 a. 

0 .88 

0 .86 

0 20 40 60 80 

core radii 

c. RK4 - CD2 

Fig. 10 Vortex pressure profiles at the center line at various instances predicted by different 

numerical schemes. 

158 
II 

h 

r d 

Fig. 1/ The vorticity magnitude contours for the vortex pairing. 

159 

II 

h 

r d 

Fig. 1/ The vorticity magnitude contours for the vortex pairing. 

159 
e 

(' 

Fig. II The vorticity magnitude contours for the vortex pairing. 

160 

e 

(' 

Fig. II The vorticity magnitude contours for the vortex pairing. 

160 
1

... 

/ 

d 

c. 

Fig. /2 The Mach number contours for the vortex pairing. 

161 

1... 

/ 

d 

c. 

Fig. /2 The Mach number contours for the vortex pairing. 

161 
Center for M o d e I" tng of Turbule Research B,"iefs -1991 nce and T,"ansition 

Position Chart 

Appendix A 

Organization - 1991 

163 Preceding Page Blank 

Center for M o d e I" tng of Turbule Research B,"iefs -1991 nce and T,"ansition 

Position Chart 

Appendix A 

Organization - 1991 

163 Preceding Page Blank 
C ente7' f07' Modeling o f TU1'btd ence and T,'u n siti on 

R e s e a1'ch BTie f s -1991 

ICOMP Director 

Coordinator 

CMOTT Technical Leader 

Advisors 

Dr. Louis A. Povinelli 

Deputy Chief 

Internal Fluid Mechanics Division 

NASA Lewis Research Center 

Dr. Meng-Sing Liou 

Senior Scientist 

Internal Fluid Mechanics Division 

NASA Lewis Research Center 

Dr. Tsan-Hsing Shih 

Senior Research Associate 

ICOMP, NASA Lewis Research Center 

Dr. Marvin E. Goldstein 

Chief Scientist 

NASA Lewis Research Center 

Professor John L. Lumley 

Sibley School of Mechanical and 

Aerospace Engineering 

Cornell University 

Professor Eli Reshotko 

Department of Mechanical and 

Aerospace Engineering 

Case Western Reserve University 

164 

_. __ ---.J 

C ente7' f07' Modeling o f TU1'btd ence and T,'u n siti on 

R e s e a1'ch BTie f s -1991 

ICOMP Director 

Coordinator 

CMOTT Technical Leader 

Advisors 

Dr. Louis A. Povinelli 

Deputy Chief 

Internal Fluid Mechanics Division 

NASA Lewis Research Center 

Dr. Meng-Sing Liou 

Senior Scientist 

Internal Fluid Mechanics Division 

NASA Lewis Research Center 

Dr. Tsan-Hsing Shih 

Senior Research Associate 

ICOMP, NASA Lewis Research Center 

Dr. Marvin E. Goldstein 

Chief Scientist 

NASA Lewis Research Center 

Professor John L. Lumley 

Sibley School of Mechanical and 

Aerospace Engineering 

Cornell University 

Professor Eli Reshotko 

Department of Mechanical and 

Aerospace Engineering 

Case Western Reserve University 

164 

_. __ ---.J 
Cente,' fO'B' Mod eli'/'}'g90! TIL·/'bulence and T"ansition R e sea1'c1i ·,.te Js -1 J 

Current Member Listing 

Names/Term Affiliation Research Areas 

Duncan, Beverly Sverdrup Tech., Inc. Multiple-Scale Thrbulence 

7/1991- present Models 

Hsu, Andrew T. Sverdrup Tech., Inc. PDF Thrbulence Modeling, 

5/1990 - present DNS 

Liou, Meng-Sing NASA LeRc CFD Algorithms, 

5/1990 - present High-Speed Flow 

Liou, William W. ICOMP Compressible Flow 

11/1990 - present Modeling, Weakly 

Nonlinear Wave Models 

Rubinstein, Robert Sverdrup Tech., Inc. Analytical Theories of 

7/1991 - present Turbulence 

Shabbir, Aamir ICOMP Buoyancy Effects on 

5/1990 - present Turbulence, Turbulence 

Modeling 

Shih, Tsan-Hsing ICOMP Thrbulence Modeling 

5/1990 - present 

Steffen, Christopher J. Jr. NASA LeRc Upwind Algorithms 

10/1990 - present Incompressible Flow 

Two-Equation 

Turbulence Models 

Van der Vegt, Jacobus J. ICOMP DNS Compressible Flows 

10/1991 - present High Order Shock 

Capturing Schemes 

Yang, Zhigang ICOMP Modeling of Bypass 

7/1990 - present Transition 

Yu, Sheng-Tao Sverdrup Tech., Inc. Modeling of Chemical 

3/1991 - present Reacting Flows, DNS 

Zhu, Jiang ICOMP Application of Thrbulence 

4/1992 - present Models in Complex Flows 

165 

Cente,' fO'B' Mod eli'/'}'g90! TIL·/'bulence and T"ansition R e sea1'c1i ·,.te Js -1 J 

Current Member Listing 

Names/Term Affiliation Research Areas 

Duncan, Beverly Sverdrup Tech., Inc. Multiple-Scale Thrbulence 

7/1991- present Models 

Hsu, Andrew T. Sverdrup Tech., Inc. PDF Thrbulence Modeling, 

5/1990 - present DNS 

Liou, Meng-Sing NASA LeRc CFD Algorithms, 

5/1990 - present High-Speed Flow 

Liou, William W. ICOMP Compressible Flow 

11/1990 - present Modeling, Weakly 

Nonlinear Wave Models 

Rubinstein, Robert Sverdrup Tech., Inc. Analytical Theories of 

7/1991 - present Turbulence 

Shabbir, Aamir ICOMP Buoyancy Effects on 

5/1990 - present Turbulence, Turbulence 

Modeling 

Shih, Tsan-Hsing ICOMP Thrbulence Modeling 

5/1990 - present 

Steffen, Christopher J. Jr. NASA LeRc Upwind Algorithms 

10/1990 - present Incompressible Flow 

Two-Equation 

Turbulence Models 

Van der Vegt, Jacobus J. ICOMP DNS Compressible Flows 

10/1991 - present High Order Shock 

Capturing Schemes 

Yang, Zhigang ICOMP Modeling of Bypass 

7/1990 - present Transition 

Yu, Sheng-Tao Sverdrup Tech., Inc. Modeling of Chemical 

3/1991 - present Reacting Flows, DNS 

Zhu, Jiang ICOMP Application of Thrbulence 

4/1992 - present Models in Complex Flows 

165 
C ent eT f01' M o deling of TU1'bul ence and T"ansition 

R es ea1'ch B1'ie fs -1991 

Appendix B 

eM OTT Biweekly Seminars / Technical Meetings 

The purpose of these seminars is to exchange ideas and opinions on the latest 

developments and current state of turbulence and transition research. The speakers 

are invited from within and without of the NASA LeRC, including foreign speakers. 

The seminars were intended not noly to keep the members informed of the latest 

development of local turbulence and transition modeling research but also to increase 

interactions between group members and other researchers at the NASA LeRc. 

The following is the meeting schedule and the abstract of the semainars during 

the reporting period. 

167 Preceding Page Blank 

C ent eT f01' M o deling of TU1'bul ence and T"ansition 

R es ea1'ch B1'ie fs -1991 

Appendix B 

eM OTT Biweekly Seminars / Technical Meetings 

The purpose of these seminars is to exchange ideas and opinions on the latest 

developments and current state of turbulence and transition research. The speakers 

are invited from within and without of the NASA LeRC, including foreign speakers. 

The seminars were intended not noly to keep the members informed of the latest 

development of local turbulence and transition modeling research but also to increase 

interactions between group members and other researchers at the NASA LeRc. 

The following is the meeting schedule and the abstract of the semainars during 

the reporting period. 

167 Preceding Page Blank 
Date: 

To: 

From: 

Subject: 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

July 3, 1991 

CMOTT Members and SVR and IFMD Staff 

William W. Liou (6682) 

CMOTT Biweekly Meeting 

The following is a tentati ve schedule for the CMOTT biweekly get-together from July 

10, 1991 to August 28, 1991. 

The presentations will be informal and active participation is expected from the attendants. Soda and snack will be served in the meetings. These meetings complement the 

CMOTT Seminar Series, which are mainly formal presentations. 

We would also appreciate some contributions from you. Subjects related to either the 

theoretical, experimental or computational aspects of turbulence and transition modeling 

are welcomed. Those who are willing to share their experience in these areas can contact 

me or Dr. T .-H. Shih at 6680 for further arrangement. 

The meeting will start at 4:00 p.m. in Room 228, Sverdrup Building. 

July 10, 1991 

July 24, 1991 

J. Lepicovsky (61-6753) 

LDV Measurement of Large Structures in a Tone 

Excited Turbulent Jet 

C. R. Wang (5865) 

Computations of Turbulence in a Shock/Turbulent 

Boundary Layer Interaction Flow 

August 7, 1991 A. Hsu (61-664.8) 

PDF Turbulence Model and Its Applications 

August 28, 1991 C. Steffen (8508) 

DTNS: An Accurate and Efficient Testbed for 

Incompressible Flow Turbulence Modeling 

168 

Date: 

To: 

From: 

Subject: 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

July 3, 1991 

CMOTT Members and SVR and IFMD Staff 

William W. Liou (6682) 

CMOTT Biweekly Meeting 

The following is a tentati ve schedule for the CMOTT biweekly get-together from July 

10, 1991 to August 28, 1991. 

The presentations will be informal and active participation is expected from the attendants. Soda and snack will be served in the meetings. These meetings complement the 

CMOTT Seminar Series, which are mainly formal presentations. 

We would also appreciate some contributions from you. Subjects related to either the 

theoretical, experimental or computational aspects of turbulence and transition modeling 

are welcomed. Those who are willing to share their experience in these areas can contact 

me or Dr. T .-H. Shih at 6680 for further arrangement. 

The meeting will start at 4:00 p.m. in Room 228, Sverdrup Building. 

July 10, 1991 

July 24, 1991 

J. Lepicovsky (61-6753) 

LDV Measurement of Large Structures in a Tone 

Excited Turbulent Jet 

C. R. Wang (5865) 

Computations of Turbulence in a Shock/Turbulent 

Boundary Layer Interaction Flow 

August 7, 1991 A. Hsu (61-664.8) 

PDF Turbulence Model and Its Applications 

August 28, 1991 C. Steffen (8508) 

DTNS: An Accurate and Efficient Testbed for 

Incompressible Flow Turbulence Modeling 

168 
Date: 

To: 

From: 

Subject: 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

September 4, 1991 

CMOTT Members and SVR and IFMD Staff 

William W. Liou (6682) 

CMOTT Biweekly Meeting 

The following is a tentative schedule for the CMOTT biweekly get- together from 

September 11, 1991 to October 23, 1991 . 

The presentations will be informal and active participation is expected from the attendants. Soda and snack will be served in the meetings. These meetings complement the 

CMOTT Seminar Series, which are mainly formal presentations. 

We would also appreciate some contributions from you. Subjects related to either the 

theoretical, experimental or computational aspects of turbulence and transition modeling 

are welcomed. Those who are willing to share their experience in these areas can contact 

me or Dr. T.-H. Shih at 6680 for further arrangement. 

The meeting will start at 4:00 p.m. in Room 228, Sverdrup Building. 

Sept. 11, 1991 

Sept. 25, 1991 

T. Bui (5639) 

Implementation of the Chien Low-Re k-E Models into the 

Proteus Code 

K. Ahn (5965) 

A 2-D Oscillating Flow Analysis Using Quasi-steady 

Turbulence Model 

October 9, 1991 J. Schwab (8446) 

Variable-density Turbulence Modeling for Turbomachinery 

October 23, 1991 M. Mawid (5965) 

Multiphase Turbulent Combustion 

169 

Date: 

To: 

From: 

Subject: 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

September 4, 1991 

CMOTT Members and SVR and IFMD Staff 

William W. Liou (6682) 

CMOTT Biweekly Meeting 

The following is a tentative schedule for the CMOTT biweekly get- together from 

September 11, 1991 to October 23, 1991 . 

The presentations will be informal and active participation is expected from the attendants. Soda and snack will be served in the meetings. These meetings complement the 

CMOTT Seminar Series, which are mainly formal presentations. 

We would also appreciate some contributions from you. Subjects related to either the 

theoretical, experimental or computational aspects of turbulence and transition modeling 

are welcomed. Those who are willing to share their experience in these areas can contact 

me or Dr. T.-H. Shih at 6680 for further arrangement. 

The meeting will start at 4:00 p.m. in Room 228, Sverdrup Building. 

Sept. 11, 1991 

Sept. 25, 1991 

T. Bui (5639) 

Implementation of the Chien Low-Re k-E Models into the 

Proteus Code 

K. Ahn (5965) 

A 2-D Oscillating Flow Analysis Using Quasi-steady 

Turbulence Model 

October 9, 1991 J. Schwab (8446) 

Variable-density Turbulence Modeling for Turbomachinery 

October 23, 1991 M. Mawid (5965) 

Multiphase Turbulent Combustion 

169 
Date : 

To: 

From: 

Subject: 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

November 4, 1991 

CMOTT Members and SVR and IFMD Staff 

William W. Liou (3-6682) 

CMOTT Biweekly Meeting 

The following is a tentative schedule for the ClviOTT biweekly get- together from 

Novernber 6, 1991 to D ecember 18, 1991. 

This will be the last session of the CMOTT group-meetings/informal- seminars this 

year but the series will resume in mid-J anuary 1992. Thank you for your patience and 

participation through out the year. The group-meetings/informal-scminars of CMOTT 

are meant not only to serve CMOTT members but also to provide an informal forum for 

those who are involved in transition/turbulence predictions. I thank all the speakers and 

participants who have made these objectives "realizable". Now, we arc planing for 1992. 

If you have any suggestions or like to give a talk or two in the corning year, please call me 

or Dr. T. H. Shih at 3-6680. In the mean time, don't forget to ma.rk your calendar for the 

following talks ! 

HAPPY HOLIDAYS fff 

The meeting will start at 4:00 p.m. in [{oom 228, SverdruJl Building. 

Nov. 6, 1991 W. Liou (3-6682) 

Weakly Nonlinear Models for Thrbule nt Free Shear Flows (2) 

- A Self-Contained Energy Transfer Model 

Nov 20, 1991 D. Ashpis (3-8317) 

DNS of Disturbances in Boundary Laye r Flow 

Dec. 4, 1991 B. Rubinstein (61-6612) 

Analytical Theory of Thrbulence and Thrbule nce Mode ling: 

TSDIA and RNG 

Dec. 18, 1991 B. Duncan (61-2998) 

A N ew Three-Equation Mode l for Turbule nce 

170 

Date : 

To: 

From: 

Subject: 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

November 4, 1991 

CMOTT Members and SVR and IFMD Staff 

William W. Liou (3-6682) 

CMOTT Biweekly Meeting 

The following is a tentative schedule for the ClviOTT biweekly get- together from 

Novernber 6, 1991 to D ecember 18, 1991. 

This will be the last session of the CMOTT group-meetings/informal- seminars this 

year but the series will resume in mid-J anuary 1992. Thank you for your patience and 

participation through out the year. The group-meetings/informal-scminars of CMOTT 

are meant not only to serve CMOTT members but also to provide an informal forum for 

those who are involved in transition/turbulence predictions. I thank all the speakers and 

participants who have made these objectives "realizable". Now, we arc planing for 1992. 

If you have any suggestions or like to give a talk or two in the corning year, please call me 

or Dr. T. H. Shih at 3-6680. In the mean time, don't forget to ma.rk your calendar for the 

following talks ! 

HAPPY HOLIDAYS fff 

The meeting will start at 4:00 p.m. in [{oom 228, SverdruJl Building. 

Nov. 6, 1991 W. Liou (3-6682) 

Weakly Nonlinear Models for Thrbule nt Free Shear Flows (2) 

- A Self-Contained Energy Transfer Model 

Nov 20, 1991 D. Ashpis (3-8317) 

DNS of Disturbances in Boundary Laye r Flow 

Dec. 4, 1991 B. Rubinstein (61-6612) 

Analytical Theory of Thrbulence and Thrbule nce Mode ling: 

TSDIA and RNG 

Dec. 18, 1991 B. Duncan (61-2998) 

A N ew Three-Equation Mode l for Turbule nce 

170 
Date: 

To: 

From: 

Subject: 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

January 30, 1992 

CMOTT Members and SVR and IFMD Staff 

William W. Liou (3-6682) 

CMOTT Biweekly Meeting 

The following is a tentative program for the CMOTT biweekly get-together/seminar from 

February 5, 1992 to March 18, 1992. Ybu are welcomed to join us. 

Thanks to the your suggestions, we have made a few changes from last year's format. First, 

we have scheduled the CMOTT Seminar Series, which are mainly formal presentations, into 

the biweekly time frame of the CMOTT group-meeting/informal-talks. Also, the abstract 

of each presentation, formal or informal, will be distributed about one week prior to its 

scheduled date. Again, if you are interested in giving a presentation, please contact us. 

The meeting will run from 1:30-2:15 PM in Room 145, Sverdrup Building, unless otherwise 

noted. 

(1) Feb. 5, 1992 D. Davis 

Weakly Nonlinear Vortex/Wave Interactions in 

Incompressible Cross-flow Boundary Layers in Transition 

(2) Feb. 19, 1992 Z. Yang 

A Modeling of Bypass Transition 

(3) March 4, 1992 K. Zaman 

Effect of "Delta-Tabs" on the Evolution of Axisymmetric 

Jets 

(4) March 18, 1992 Professor R. M. C. So, Arizona State University 

Near Wall Heat Transfer Modeling 

171 

Date: 

To: 

From: 

Subject: 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

January 30, 1992 

CMOTT Members and SVR and IFMD Staff 

William W. Liou (3-6682) 

CMOTT Biweekly Meeting 

The following is a tentative program for the CMOTT biweekly get-together/seminar from 

February 5, 1992 to March 18, 1992. Ybu are welcomed to join us. 

Thanks to the your suggestions, we have made a few changes from last year's format. First, 

we have scheduled the CMOTT Seminar Series, which are mainly formal presentations, into 

the biweekly time frame of the CMOTT group-meeting/informal-talks. Also, the abstract 

of each presentation, formal or informal, will be distributed about one week prior to its 

scheduled date. Again, if you are interested in giving a presentation, please contact us. 

The meeting will run from 1:30-2:15 PM in Room 145, Sverdrup Building, unless otherwise 

noted. 

(1) Feb. 5, 1992 D. Davis 

Weakly Nonlinear Vortex/Wave Interactions in 

Incompressible Cross-flow Boundary Layers in Transition 

(2) Feb. 19, 1992 Z. Yang 

A Modeling of Bypass Transition 

(3) March 4, 1992 K. Zaman 

Effect of "Delta-Tabs" on the Evolution of Axisymmetric 

Jets 

(4) March 18, 1992 Professor R. M. C. So, Arizona State University 

Near Wall Heat Transfer Modeling 

171 
Date: 

To: 

From: 

Subject: 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

March 26, 1992 

CMOTT Members and SVR and IFMD Staff 

William W. Liou (3-6682) 

CMOTT Biweekly Meeting 

The following is a tentative program for the CMOTT biweekly get-together/seminar from 

April 1, 1992 to May 13, 1992. You are welcomed to join us. Also, if you are interested 

in giving a presentation, please let us know. 

The meeting will run from 1:30-2:15 PM in Room 228, Sverdrup Building, unless otherwise 

noted. 

(5) April 1, 1992 J. Van der Vegt 

The Development of an ENO-Osher Scheme for Direct 

Simulation of Compressible Flows 

(6) April 15, 1992 J. Goodrich 

Unsteady Time Asymptotic State: Incompressible Results, 

New Directions for Algorithms 

(7) April 29, 1992 T.-H. Shih 

Kolmogorov Behavior of Near-Wall Turbulence and 

Its Application in Turbulence Modeling 

(8) May 13, 1992 Z. Yang 

A Modeling of Bypass Transition 

172 

Date: 

To: 

From: 

Subject: 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

March 26, 1992 

CMOTT Members and SVR and IFMD Staff 

William W. Liou (3-6682) 

CMOTT Biweekly Meeting 

The following is a tentative program for the CMOTT biweekly get-together/seminar from 

April 1, 1992 to May 13, 1992. You are welcomed to join us. Also, if you are interested 

in giving a presentation, please let us know. 

The meeting will run from 1:30-2:15 PM in Room 228, Sverdrup Building, unless otherwise 

noted. 

(5) April 1, 1992 J. Van der Vegt 

The Development of an ENO-Osher Scheme for Direct 

Simulation of Compressible Flows 

(6) April 15, 1992 J. Goodrich 

Unsteady Time Asymptotic State: Incompressible Results, 

New Directions for Algorithms 

(7) April 29, 1992 T.-H. Shih 

Kolmogorov Behavior of Near-Wall Turbulence and 

Its Application in Turbulence Modeling 

(8) May 13, 1992 Z. Yang 

A Modeling of Bypass Transition 

172 
1----

I 

Date: 

To: 

From: 

Subject: 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

June 1, 1992 

CMOTT Members and SVR and IFMD Staff 

William W. Liou (3-6682) 

CMOTTBiweekly Meeting 

The following is· a tentative program for the CMOTT biweekly get-together/seminar from 

June 10, 1992 to July 22, 1992. You are welcomed to joill us. 

The talks will be informal and active participation will be expected from the audience. 

Also, if you are interested in giving a presentation abou t t he progress or some results of 

your own work 011 turbulence or transition, please let us kllow. 

The meeting will run from 1:30-2:15 PM in Room 228, Sverdrup Buildillg, unless otherwise 

noted. 

(9) June 10, 1992 J. Zhu 

Finite Volume Computations in Incompressible Flows 

with Complex Geometries 

(10) June 24, 1992 J. Lee 

RPL US Code and Standard k - E Models and Applications 

(11) july 8, 1992 R. Mankbadi 

Unsteady Turbulent Flows 

(12) july 22, 1992 D. Rigby 

The Effect of Spanwise Variations in Momentum on 

Leading Edge Heat Transfer 

173 

1----

I 

Date: 

To: 

From: 

Subject: 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

June 1, 1992 

CMOTT Members and SVR and IFMD Staff 

William W. Liou (3-6682) 

CMOTTBiweekly Meeting 

The following is· a tentative program for the CMOTT biweekly get-together/seminar from 

June 10, 1992 to July 22, 1992. You are welcomed to joill us. 

The talks will be informal and active participation will be expected from the audience. 

Also, if you are interested in giving a presentation abou t t he progress or some results of 

your own work 011 turbulence or transition, please let us kllow. 

The meeting will run from 1:30-2:15 PM in Room 228, Sverdrup Buildillg, unless otherwise 

noted. 

(9) June 10, 1992 J. Zhu 

Finite Volume Computations in Incompressible Flows 

with Complex Geometries 

(10) June 24, 1992 J. Lee 

RPL US Code and Standard k - E Models and Applications 

(11) july 8, 1992 R. Mankbadi 

Unsteady Turbulent Flows 

(12) july 22, 1992 D. Rigby 

The Effect of Spanwise Variations in Momentum on 

Leading Edge Heat Transfer 

173 
CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1) 

Weakly Nonlinear Vortex/Wave Interactions in 

Incompressible Crossflow Boundary layers in 

Transition 

by 

Dominic Davis 

ICOMP 

Wed., 5 February, 1992 

1:30-2:15 PM 

Room 145, SVR Building 

ABSTRACT 

The instability of an incompressible three-dimensional boundary layer 

is considered theoretically and computationally in the context of vortex/wave interactions. Specifically the work centres on two low-amplitude, 

lower-branch Tollmien-Schlichting {TS) waves which mutually interact 

to induce a weak longitudinal vortex flow;the vortex motion,in turn,gives 

rise to significant wave-modulation via wall-shear forcing. The characteristic Reynolds number is taken as a large parameter and,a!; a consequence,the TS waves and the vortex are governed primarily by tripledeck theory. The nonlinear interaction is captured by a viscous partialdifferential system for the vortex coupled with a pair of amplitude equations for the wave pressures. Computations were performed for relatively 

small crossflow values.Three distinct possibilities were found to emerge 

for the nonlinear behaviour of the flow solution downstream - an algebraic finite-distance singularity,far- downstream decay or repeated oscillations - depending on the various parameter values,the input amplitudes 

and the wave angles. 

174 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1) 

Weakly Nonlinear Vortex/Wave Interactions in 

Incompressible Crossflow Boundary layers in 

Transition 

by 

Dominic Davis 

ICOMP 

Wed., 5 February, 1992 

1:30-2:15 PM 

Room 145, SVR Building 

ABSTRACT 

The instability of an incompressible three-dimensional boundary layer 

is considered theoretically and computationally in the context of vortex/wave interactions. Specifically the work centres on two low-amplitude, 

lower-branch Tollmien-Schlichting {TS) waves which mutually interact 

to induce a weak longitudinal vortex flow;the vortex motion,in turn,gives 

rise to significant wave-modulation via wall-shear forcing. The characteristic Reynolds number is taken as a large parameter and,a!; a consequence,the TS waves and the vortex are governed primarily by tripledeck theory. The nonlinear interaction is captured by a viscous partialdifferential system for the vortex coupled with a pair of amplitude equations for the wave pressures. Computations were performed for relatively 

small crossflow values.Three distinct possibilities were found to emerge 

for the nonlinear behaviour of the flow solution downstream - an algebraic finite-distance singularity,far- downstream decay or repeated oscillations - depending on the various parameter values,the input amplitudes 

and the wave angles. 

174 
CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-2) 

A k-E Model for Near Wall Turbulence and 

its Application in Turbulent Boundary layer 

with/without Pressure Gradient 

by 

z. Yang 

ICOMP 

VVed., 19 February, 1992 

1:30-2:15 PM 

Room 145, SVR Building 

ABSTRACT 

A k - € model is proposed for turbulent wall bounded flows. In this 

model, turbulent velocity scale and turbulent time scale are used to 

define the eddy viscosity. The time scale used is bounded from below 

by the Kolmogorov time scale. The dissipation equation is reformulated 

using this time scale, removing the singularity of the high Reynolds 

number k - € equation at the wall and rendering the introduction of 

the pseudo-dissipation unnecessary. The damping function used in the 

eddy viscosity is chosen to be a function of Ry = Icl~2y instead of y+. 

Thus, the model could be used for flows with separation. The model 

constants used are the same as the model constants in the commonly 

used high turbulent Reynolds standard k - € model. Thus, the proposed 

model would reduce to the standard k - € model when it is far away 

from the wall. Boundary layer flows at zero pressure gradient, favorable 

pressure gradient, adverse pressure gradient and increasingly adverse 

pressure gradient are calculated respectively. The comparisons of model 

predictions and the available experimental data are found to be good. 

175 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-2) 

A k-E Model for Near Wall Turbulence and 

its Application in Turbulent Boundary layer 

with/without Pressure Gradient 

by 

z. Yang 

ICOMP 

VVed., 19 February, 1992 

1:30-2:15 PM 

Room 145, SVR Building 

ABSTRACT 

A k - € model is proposed for turbulent wall bounded flows. In this 

model, turbulent velocity scale and turbulent time scale are used to 

define the eddy viscosity. The time scale used is bounded from below 

by the Kolmogorov time scale. The dissipation equation is reformulated 

using this time scale, removing the singularity of the high Reynolds 

number k - € equation at the wall and rendering the introduction of 

the pseudo-dissipation unnecessary. The damping function used in the 

eddy viscosity is chosen to be a function of Ry = Icl~2y instead of y+. 

Thus, the model could be used for flows with separation. The model 

constants used are the same as the model constants in the commonly 

used high turbulent Reynolds standard k - € model. Thus, the proposed 

model would reduce to the standard k - € model when it is far away 

from the wall. Boundary layer flows at zero pressure gradient, favorable 

pressure gradient, adverse pressure gradient and increasingly adverse 

pressure gradient are calculated respectively. The comparisons of model 

predictions and the available experimental data are found to be good. 

175 
CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-3) 

Effect of Tabs on the Evolution of Axisymmetric Jets 

by 

Khairul Zaman 

IFMD 

W ed., 4 March, 1992 

1:30-2:15 PM 

Room 145, SVR Building 

ABSTRACT 

Vortex generators, in the form of small tabs at the nozzle exit, can have 

a profound influence on the evolution of an axisymmetric jet. Using 

tabs of certain shapes, the jet cross section can be distorted almost 

arbitrarily. Such distortion is accompanied by elimination of screech 

noise from supersonic jets and a significant increase in jet mixing. Key 

resurts obtained so far, covering a jet Mach number range of 0.3 and 

1.B, will be summarized in this presentation. Observations will be made 

on the mechanisms of the effect including the likely vorticity dynamics 

in the flow. 

176 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-3) 

Effect of Tabs on the Evolution of Axisymmetric Jets 

by 

Khairul Zaman 

IFMD 

W ed., 4 March, 1992 

1:30-2:15 PM 

Room 145, SVR Building 

ABSTRACT 

Vortex generators, in the form of small tabs at the nozzle exit, can have 

a profound influence on the evolution of an axisymmetric jet. Using 

tabs of certain shapes, the jet cross section can be distorted almost 

arbitrarily. Such distortion is accompanied by elimination of screech 

noise from supersonic jets and a significant increase in jet mixing. Key 

resurts obtained so far, covering a jet Mach number range of 0.3 and 

1.B, will be summarized in this presentation. Observations will be made 

on the mechanisms of the effect including the likely vorticity dynamics 

in the flow. 

176 
CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-4) 

Near-Wall Modeling of Turbulent Heat Transfer 

by 

Professor Ronald M. C. So 

Mechanical and Aerospace Engineering 

Arizona State University 

Wed., 18 March, 1992 

1:30-·2:30 PM 

Room 119, Building 5 

ABSTRACT 

A near-wall two-equation model for turbulent heat fluxes is derived from 

the temperature variance and its dissipation-rate equations and the assumption of gradient transport. The near-wall asymptotics of each term 

in the exact equations are examined and used to derive near-wall correction functions that render the modeled equations consistent with 

these behavior. Thus modeled, the equations are used to calculate 

fully-developed pipe and channel flows with hear transfer. It is found 

that the proposed two-equation model yields asymptotically correct nearwall behavior for the normal heat flux, the temperature variance and its 

near-wall' budget and correct limiting wall values for these properties 

compared to direct simulation data a nd measurements obtained under 

different wall boundary conditions. 

CONTACT: T.-H. Shih, PABX 3-5698 

177 

---- ----------------------------------------

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-4) 

Near-Wall Modeling of Turbulent Heat Transfer 

by 

Professor Ronald M. C. So 

Mechanical and Aerospace Engineering 

Arizona State University 

Wed., 18 March, 1992 

1:30-·2:30 PM 

Room 119, Building 5 

ABSTRACT 

A near-wall two-equation model for turbulent heat fluxes is derived from 

the temperature variance and its dissipation-rate equations and the assumption of gradient transport. The near-wall asymptotics of each term 

in the exact equations are examined and used to derive near-wall correction functions that render the modeled equations consistent with 

these behavior. Thus modeled, the equations are used to calculate 

fully-developed pipe and channel flows with hear transfer. It is found 

that the proposed two-equation model yields asymptotically correct nearwall behavior for the normal heat flux, the temperature variance and its 

near-wall' budget and correct limiting wall values for these properties 

compared to direct simulation data a nd measurements obtained under 

different wall boundary conditions. 

CONTACT: T.-H. Shih, PABX 3-5698 

177 

---- ----------------------------------------
CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-5) 

The Development of an END-Osher Scheme for 

Direct Simulation of Compressible Flows 

by 

J aap Van der Vegt 

ICOMP 

Wed., April 1, 1992 

1:30--2:30 PM 

Room 228, SVR Building 

ABSTRACT 

Direct simulation of turbulence and transition in compressible wall 

bounded flows presents an alternative to investigate important physical 

phenomena which are difficult to measure or study otherwise. It also 

provides data useful for turbulence modeling. A new program is being developed which solves the three-dimensional compressible NavierStokes equations using a higher order, fully implicit and time accurate 

ENO scheme together with Osher flux splitting. In this presentation an 

overview will be given of the numerical scheme and several test cases, 

both for supersonic and subsonic flow, will be presented and further 

improvements will be discussed. 

178 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-5) 

The Development of an END-Osher Scheme for 

Direct Simulation of Compressible Flows 

by 

J aap Van der Vegt 

ICOMP 

Wed., April 1, 1992 

1:30--2:30 PM 

Room 228, SVR Building 

ABSTRACT 

Direct simulation of turbulence and transition in compressible wall 

bounded flows presents an alternative to investigate important physical 

phenomena which are difficult to measure or study otherwise. It also 

provides data useful for turbulence modeling. A new program is being developed which solves the three-dimensional compressible NavierStokes equations using a higher order, fully implicit and time accurate 

ENO scheme together with Osher flux splitting. In this presentation an 

overview will be given of the numerical scheme and several test cases, 

both for supersonic and subsonic flow, will be presented and further 

improvements will be discussed. 

178 
CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-6) 

Unsteady Time Asymptotic States: Incompressible Results 

and New Directions for Algorithms 

by 

John Goodrich 

IFMD 

Wed., April 15, 1992 

1:30-2:15 PM 

Room 228, SVR Building 

ABSTRACT 

Unsteady time asymptotic flow states for high Reynolds number viscous incompressible flow problems are presented. Discrete frequency flows are shown for 

the square driven cavity, with periodic cases for Re = 9000 and Re = 9600, and 

with aperiodic cases for Re = 9700 and Re = 10000. The algorithm for these calculations is based on the fourth order POE for incompressible fluid flow which 

uses the streamfunction as the only dependent variable, it is second order accurate in space and time, and it has a stability constraint of CFL :S 1. The 

algorithm is extremely robust with respect to Reynolds number. 

The direct numerical simulation of transition and turbulence requires numerical methods to be more than second order accurate in order to accurately 

represent the relevant scales of the physical processes. Recently developed 

finite difference algorithms are presented for unsteady convection equations, 

including the advection and inviscid Burgers equation in one space dimension, 

and the wave equation treated as a system, with remarks on diffusion equations and extension to higher space dimensions. The new algorithms that will 

be discussed all use local stencils, they range from third to seventh order in 

accuracy, they all have the same order of accuracy in both space and time, and 

they are all one step explicit methods (except for diffusion equations). Since all 

of the algorithms use a small local stencil, the number of degrees of freedom of 

known data required for higher order accuracy is obtained by higher information 

density than just the solution data. The use of a two point stencil (for some 

of the methods) allows for arbitrary grid spacing, though a convective stability 

constraint must be observed at each grid point. The use of local data for an 

explicit algorithm with high order accuracy avoids the requirement of using a 

global solution method such as compact differencing or spline based algorithms. 

There will be computational results for all of the algorithms that are presented. 

179 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-6) 

Unsteady Time Asymptotic States: Incompressible Results 

and New Directions for Algorithms 

by 

John Goodrich 

IFMD 

Wed., April 15, 1992 

1:30-2:15 PM 

Room 228, SVR Building 

ABSTRACT 

Unsteady time asymptotic flow states for high Reynolds number viscous incompressible flow problems are presented. Discrete frequency flows are shown for 

the square driven cavity, with periodic cases for Re = 9000 and Re = 9600, and 

with aperiodic cases for Re = 9700 and Re = 10000. The algorithm for these calculations is based on the fourth order POE for incompressible fluid flow which 

uses the streamfunction as the only dependent variable, it is second order accurate in space and time, and it has a stability constraint of CFL :S 1. The 

algorithm is extremely robust with respect to Reynolds number. 

The direct numerical simulation of transition and turbulence requires numerical methods to be more than second order accurate in order to accurately 

represent the relevant scales of the physical processes. Recently developed 

finite difference algorithms are presented for unsteady convection equations, 

including the advection and inviscid Burgers equation in one space dimension, 

and the wave equation treated as a system, with remarks on diffusion equations and extension to higher space dimensions. The new algorithms that will 

be discussed all use local stencils, they range from third to seventh order in 

accuracy, they all have the same order of accuracy in both space and time, and 

they are all one step explicit methods (except for diffusion equations). Since all 

of the algorithms use a small local stencil, the number of degrees of freedom of 

known data required for higher order accuracy is obtained by higher information 

density than just the solution data. The use of a two point stencil (for some 

of the methods) allows for arbitrary grid spacing, though a convective stability 

constraint must be observed at each grid point. The use of local data for an 

explicit algorithm with high order accuracy avoids the requirement of using a 

global solution method such as compact differencing or spline based algorithms. 

There will be computational results for all of the algorithms that are presented. 

179 
CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-7) 

Kolmogorov Behavior of Near-Wall Turbulence 

and Its Application in Turbulence Modeling 

by 

Tsan-Hsing Shih 

ICOMP 

Wed., April 29, 1992 

1:30-2:15 PM 

Room 228, SVR Building 

ABSTRACT 

The near-wall behavior of turbulence is re-examined in a way different from 

that proposed by Hanjalic and Launder[l] and followers[2],[3].[4].[5[. It is shown 

that at a certain distance from the wall, all energetic large eddies will reduce 

to Kolmogorov eddies (the smallest eddies in turbulence). All the important 

wall parameters, such as friction velocity, viscous length scale, and mean strain 

rate at the wall, are characterized by Kolmogorov microscales. According to this 

Kolmogorov behavior of near-wall turbulence, the turbulence quantities, such 

as turbulent kinetic energy, dissipation rate, etc. at the location where the 

large eddies become "Kolmogorov" eddies, can be estimated by using both direct numerical simulation CONS) data and asymptotic analysis of near-wall 

turbulence. This information will provide useful boundary conditions for the 

turbulent transport equations. As an example, the concept is incorporated in 

the standard k-f. model which is then applied to channel and boundary layer 

flows. Using appropriate boundary conditions (based on Kollllogorov behavior 

of near-wall turbulence), there is no need for any wall-modification to the kf. equations (including model constants). Results compare very well with the 

ONS and experimental data. 

180 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-7) 

Kolmogorov Behavior of Near-Wall Turbulence 

and Its Application in Turbulence Modeling 

by 

Tsan-Hsing Shih 

ICOMP 

Wed., April 29, 1992 

1:30-2:15 PM 

Room 228, SVR Building 

ABSTRACT 

The near-wall behavior of turbulence is re-examined in a way different from 

that proposed by Hanjalic and Launder[l] and followers[2],[3].[4].[5[. It is shown 

that at a certain distance from the wall, all energetic large eddies will reduce 

to Kolmogorov eddies (the smallest eddies in turbulence). All the important 

wall parameters, such as friction velocity, viscous length scale, and mean strain 

rate at the wall, are characterized by Kolmogorov microscales. According to this 

Kolmogorov behavior of near-wall turbulence, the turbulence quantities, such 

as turbulent kinetic energy, dissipation rate, etc. at the location where the 

large eddies become "Kolmogorov" eddies, can be estimated by using both direct numerical simulation CONS) data and asymptotic analysis of near-wall 

turbulence. This information will provide useful boundary conditions for the 

turbulent transport equations. As an example, the concept is incorporated in 

the standard k-f. model which is then applied to channel and boundary layer 

flows. Using appropriate boundary conditions (based on Kollllogorov behavior 

of near-wall turbulence), there is no need for any wall-modification to the kf. equations (including model constants). Results compare very well with the 

ONS and experimental data. 

180 
CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-8) 

A Modeling of Bypass Transition 

by 

Z. Yang 

ICOMP 

Wed., May 13, 1992 

1:30-2:15 PM 

Room 228, SVR Building 

ABSTRACT 

A model for the calculation of bypass transitional boundary layers due to the 

freestream turbulence is proposed. The model combines a near wall k-f. model 

proposed for the fully developed turbulent flows with the intermittency of the 

transitional boundary layers. The intermittency factor is assumed to be a function of both the free stream turbulence and the shape factor of the boundary 

layer. Transitional boundary layers over a flat plate with different freestream 

turbulence level are calculated using the proposed model. It is found that the 

model calculations agree well with the experimental data and give a better prediction compared with other low Reynolds number k - f. models, which do not 

incorporate the i nterm ittency effect. 

181 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-8) 

A Modeling of Bypass Transition 

by 

Z. Yang 

ICOMP 

Wed., May 13, 1992 

1:30-2:15 PM 

Room 228, SVR Building 

ABSTRACT 

A model for the calculation of bypass transitional boundary layers due to the 

freestream turbulence is proposed. The model combines a near wall k-f. model 

proposed for the fully developed turbulent flows with the intermittency of the 

transitional boundary layers. The intermittency factor is assumed to be a function of both the free stream turbulence and the shape factor of the boundary 

layer. Transitional boundary layers over a flat plate with different freestream 

turbulence level are calculated using the proposed model. It is found that the 

model calculations agree well with the experimental data and give a better prediction compared with other low Reynolds number k - f. models, which do not 

incorporate the i nterm ittency effect. 

181 
CENTER FOR MODELING OF 

TURBULENCE AND TRANSI TION 

Biweekly Meeting Series (1992-9) 

Finite-Volume Computations of Incompressible 

Flows with Complex Geometries 

by 

J. Zhu 

ICOMP 

W ed ., June 10 , 1992 

1:30-2:30 PM 

Room 228, SVR Building 

ABSTRACT 

A brief review is given of finite-volume procedures developed at the 

Institute for Hydromechanics, University of Karlsruhe, for calculating 

incompressible elliptic flows with complex boundaries. The procedures 

include: numerical grid generation, higher-order bounded convection 

schemes, zonal solution, simulation of two-phase flows, and near-wall 

turbulence modelling. Various application examples will be given. 

182 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSI TION 

Biweekly Meeting Series (1992-9) 

Finite-Volume Computations of Incompressible 

Flows with Complex Geometries 

by 

J. Zhu 

ICOMP 

W ed ., June 10 , 1992 

1:30-2:30 PM 

Room 228, SVR Building 

ABSTRACT 

A brief review is given of finite-volume procedures developed at the 

Institute for Hydromechanics, University of Karlsruhe, for calculating 

incompressible elliptic flows with complex boundaries. The procedures 

include: numerical grid generation, higher-order bounded convection 

schemes, zonal solution, simulation of two-phase flows, and near-wall 

turbulence modelling. Various application examples will be given. 

182 
CENTER FOR MODELING OF 

TURBULENCE A1VD TRANSITION 

Biweekly Meeting Series (1992-10) 

Development of the RPlUS code with 

Standard k - € model and Their Applications 

by 

Jinho Lee 

Sverdrup Technology, Inc. 

Wed., June 24, 1992 

1:30-2:30 PM 

Room 228, SVR B uild ing 

ABSTRACT 

T he primary goal of this research effort is to develop a CFD tool which can be used in 

a variety of practical supersonic/hypersonic propulsion device development/analysis environments. One focus of this work has been to develop and validate the Reactive Propulsion 

code based on LU Scheme(RPLUS). T his effort also includes the development of turbulence models which can be used in the predictions of highly complex flow environments 

inside of combustors. 

This presentation will cover only a small part. of a larger development effort and focus 

will primarily on the analysis, implementation, and development of the turbulence model 

capabilities of the RPLUS code. 

Some of the issues which will be covered are; formulation of the turbulence models, 

the numerical technique used to solve the turbulence model equations, and modeling of 

compressibility effects. The primary numerical technique used in the RPLUS code is 

the LU-SSOR(LU scheme based on Successive Symmetric Over Relaxation) technique. 

Therefore, the turbulent kinetic energy transport and dissipation transport equations are 

also solved using the LU-SSOR numerical t.echnique. 

Both two and three dimensional turbulence model development are being developed 

for the RPLUS code. However, the majority of the presentation will focus on the development of the two dimensional k-€ models for the RPLUS code. Issues regarding compressible wall-function boundary conditions and compressibility effects will be addressed. 

Both low and high Reynolds number forms of the k-€ models are being developed. The 

"standard" low Reynolds number model of Launder-Sharma and Chien has been used in 

this study. The problems of primary interests are supersonic turbulent boundary-layers, 

shock-wave/boundary-layer interactions, arld shear-layers in two or three dimensional environments. 

183 

_J 

CENTER FOR MODELING OF 

TURBULENCE A1VD TRANSITION 

Biweekly Meeting Series (1992-10) 

Development of the RPlUS code with 

Standard k - € model and Their Applications 

by 

Jinho Lee 

Sverdrup Technology, Inc. 

Wed., June 24, 1992 

1:30-2:30 PM 

Room 228, SVR B uild ing 

ABSTRACT 

T he primary goal of this research effort is to develop a CFD tool which can be used in 

a variety of practical supersonic/hypersonic propulsion device development/analysis environments. One focus of this work has been to develop and validate the Reactive Propulsion 

code based on LU Scheme(RPLUS). T his effort also includes the development of turbulence models which can be used in the predictions of highly complex flow environments 

inside of combustors. 

This presentation will cover only a small part. of a larger development effort and focus 

will primarily on the analysis, implementation, and development of the turbulence model 

capabilities of the RPLUS code. 

Some of the issues which will be covered are; formulation of the turbulence models, 

the numerical technique used to solve the turbulence model equations, and modeling of 

compressibility effects. The primary numerical technique used in the RPLUS code is 

the LU-SSOR(LU scheme based on Successive Symmetric Over Relaxation) technique. 

Therefore, the turbulent kinetic energy transport and dissipation transport equations are 

also solved using the LU-SSOR numerical t.echnique. 

Both two and three dimensional turbulence model development are being developed 

for the RPLUS code. However, the majority of the presentation will focus on the development of the two dimensional k-€ models for the RPLUS code. Issues regarding compressible wall-function boundary conditions and compressibility effects will be addressed. 

Both low and high Reynolds number forms of the k-€ models are being developed. The 

"standard" low Reynolds number model of Launder-Sharma and Chien has been used in 

this study. The problems of primary interests are supersonic turbulent boundary-layers, 

shock-wave/boundary-layer interactions, arld shear-layers in two or three dimensional environments. 

183 

_J 
CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-11) 

Unsteady Turbulent Flows 

by 

Reda R. Mankbadi 

NASA Lewis Research Center 

Wed., July 8, 1992 

1 :30-2:30 PM 

Room 228, SVR Building 

ABSTRACT 

Current research activities emphasize computation/modelling of turbulent flows when basic flow is time-periodic. Wall-bounded flows and free 

shear flows exhibit different features when the basic flow is unsteady; 

and as such, different approaches are used to model them. 

(A) In wall-bounded, oscillatory flows, two approaches are used to 

model turbulence: (I) Turbulence is assumed to behave in a quasi-steady 

manner, and steady-state models are directly extended to the unsteady 

case. This approach fails at high frequencies of oscillations. (II) Rapid 

distortion theory (ROT) is successfully adapted to aid in turbulence 

modelling of highly unsteady flows (high frequencies). The eddy viscosity 

hypothesis is replaced by the ratio of turbulent stresses/kinetic energy; 

which is given by ROT as a function of the accumulated rate of strain. 

(8) In free shear flows (naturally unsteady, or excited to be unsteady), two approaches are investigated: (I) The large-scale (organized, 

coherent) component is modelled as instability waves interacting with 

each other as well as with the mean flow and the fine-scale (random, 

background) turbulence. Integrated kinetic-energy equations are then 

obtained for each scale of motion. The approach is successful in predicting results in good agreements with experiments in which excitation 

devices are used to control jet mixing and turbulence. (II) The other 

approach adopted is Large-Eddy Simulations (LES) with application to 

predicting the far-field noise of a supersonic jet. 

CONTACT: William W. Lion, PAnx 3-GG82 

184 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-11) 

Unsteady Turbulent Flows 

by 

Reda R. Mankbadi 

NASA Lewis Research Center 

Wed., July 8, 1992 

1 :30-2:30 PM 

Room 228, SVR Building 

ABSTRACT 

Current research activities emphasize computation/modelling of turbulent flows when basic flow is time-periodic. Wall-bounded flows and free 

shear flows exhibit different features when the basic flow is unsteady; 

and as such, different approaches are used to model them. 

(A) In wall-bounded, oscillatory flows, two approaches are used to 

model turbulence: (I) Turbulence is assumed to behave in a quasi-steady 

manner, and steady-state models are directly extended to the unsteady 

case. This approach fails at high frequencies of oscillations. (II) Rapid 

distortion theory (ROT) is successfully adapted to aid in turbulence 

modelling of highly unsteady flows (high frequencies). The eddy viscosity 

hypothesis is replaced by the ratio of turbulent stresses/kinetic energy; 

which is given by ROT as a function of the accumulated rate of strain. 

(8) In free shear flows (naturally unsteady, or excited to be unsteady), two approaches are investigated: (I) The large-scale (organized, 

coherent) component is modelled as instability waves interacting with 

each other as well as with the mean flow and the fine-scale (random, 

background) turbulence. Integrated kinetic-energy equations are then 

obtained for each scale of motion. The approach is successful in predicting results in good agreements with experiments in which excitation 

devices are used to control jet mixing and turbulence. (II) The other 

approach adopted is Large-Eddy Simulations (LES) with application to 

predicting the far-field noise of a supersonic jet. 

CONTACT: William W. Lion, PAnx 3-GG82 

184 
CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-12) 

The Effect of Spanwise Variations in 

Momentum on leading Edge Heat Transfer 

by 

David Rigby 

Sverdrup Tech. INc. 

Wed., July 22, 1992 

1:30-2:30 PM 

Room 228, SVR Building 

ABSTRACT 

A study of the effect of spanwise variation in momentum on leading 

edge heat transfer is discussed. Numerical and experimental results 

are presented for a circular leading edge and for a 3:1 elliptical leading 

edge. Direct comparison of the two-dimensional results, that is with no 

spanwise variations, to the analytical results of Frossling is very good. 

The numerical calculation, using the PARC3D code, solves the threedimensional Navier-Stokes equations, assuming steady laminar flow on 

the leading edge region. Experimentally, increases in spanwise averaged 

heat transfer coefficient as high as 50% above the two-dimensional value 

were observed. Numerically, the heat transfer coefficient was seen to 

increase by as much as 25% percent. In general, the circular leading 

edge, under the same flow conditions, produced a higher heat transfer 

rate than the elliptical leading edge. As a percentage of the respective 

two-dimensional values, the circular and elliptical leading edges showed 

similar sensitivity to spanwise variations in momentum. By equating the 

root mean square of the amplitude of the spanwise variation in momentum to the turbulence intensity, a qualitative comparison between the 

present work and turbulent results was possible. 

CONTACT : William W. Liou, PABX 3-6682 

185 

CENTER FOR MODELING OF 

TURBULENCE AND TRANSITION 

Biweekly Meeting Series (1992-12) 

The Effect of Spanwise Variations in 

Momentum on leading Edge Heat Transfer 

by 

David Rigby 

Sverdrup Tech. INc. 

Wed., July 22, 1992 

1:30-2:30 PM 

Room 228, SVR Building 

ABSTRACT 

A study of the effect of spanwise variation in momentum on leading 

edge heat transfer is discussed. Numerical and experimental results 

are presented for a circular leading edge and for a 3:1 elliptical leading 

edge. Direct comparison of the two-dimensional results, that is with no 

spanwise variations, to the analytical results of Frossling is very good. 

The numerical calculation, using the PARC3D code, solves the threedimensional Navier-Stokes equations, assuming steady laminar flow on 

the leading edge region. Experimentally, increases in spanwise averaged 

heat transfer coefficient as high as 50% above the two-dimensional value 

were observed. Numerically, the heat transfer coefficient was seen to 

increase by as much as 25% percent. In general, the circular leading 

edge, under the same flow conditions, produced a higher heat transfer 

rate than the elliptical leading edge. As a percentage of the respective 

two-dimensional values, the circular and elliptical leading edges showed 

similar sensitivity to spanwise variations in momentum. By equating the 

root mean square of the amplitude of the spanwise variation in momentum to the turbulence intensity, a qualitative comparison between the 

present work and turbulent results was possible. 

CONTACT : William W. Liou, PABX 3-6682 

185 
CenteT' for Mod eling of Tnro'ulence and Transit.ion 

Resea.rch Briefs - 1991 

Appendix C 

List of Member's Publications 

'. -- --

Barton, J. M., Rubinstein, R. and Kirtley, K. R. "Nonlinear Reynolds Stress Model 

for Turbulent Shear Flows, " AIAA Paper No. 91-0609, (Jan. 1991). 

Barton, J. M. and Rubinstein, R., "Nonlinerar Algebraic Reynolds Stress Model for 

Anisotropic Turbulent Flows, " 11th U. S. National Congress of Applied Mechanics, 

TUcson, AZ (May 1990). 

Barton, J. M. and Rubinstein, R., "Renormalization Group Theory and Turbulence 

Modeling, " 2nd International Workshop on Chaos and TUrbulence, Tsukuba, Japan 

(Jan. 1992). 

Barton, J. M. and Rubinstein, R., "Renormalization Group Analysis of TUrbulenceDriven Secondary Fows, " 11th Australian Fluid Mechanics Conference, Hobart, 

Australia (Dec. 1992). 

Brown, S., Leibovich, S. and Z. Yang, "On the linear instability of the HallStewartson vortex," Theoretical and Computational Fluid Dynamics, 2, 27-46, 

(1990). 

Duncan, B. S., Liou, W. W. and Shih, T.-H., "A Multiple-scale turbulence model 

for incompressible flow," NASA TM (to appear) (1992). 

Duncan, B.S., Lumley, J.L., Shih, T.H. and To, W.M., "A new model for the 

turbulent dissipation" International Conference of Fluid Mechanics and Theoretical 

Physics, Bejing, China (1992). 

Hsu, A.T. and Liou, M., "Computational analysis of underexpanded jets in the 

hypersonic regime," Journal of Propulsion and Power, 7, No.2, (1991). 

Hsu, A.T., "Progress in the development of PDF turbulence models for combustion," 10th NASP Symposium, April 23-16, 1991, Monterey, California. 

Hsu, A.T., "A study of hydrogen diffusion flames using PDF turbulence model," 

AIAA 91-1780, AIAA 22nd Fluid Dynamics Conference, June 24-26, 1991, Honolulu, Hawaii. 

Hsu, A.T. and Chen, J.Y, "A continuous mixing Model for PDF simulations and 

its applications to com busting shear flows," 8th International Symposium on TUrbulence Shear Flows, Sept. 9-11, 1991, Munich, Germany. 

Lang, N.J. and Shih, T.-H., "A critical comparison of two-equation turbulence 

models," NASA TM 105237, (1991). 

Liou, M. S. and Steffen, Jr., C. J., "A New Flux Splitting Scheme," NASA TM 

104404 (1991). 

Preceding Page Blank 

187 

-----------------------

CenteT' for Mod eling of Tnro'ulence and Transit.ion 

Resea.rch Briefs - 1991 

Appendix C 

List of Member's Publications 

'. -- --

Barton, J. M., Rubinstein, R. and Kirtley, K. R. "Nonlinear Reynolds Stress Model 

for Turbulent Shear Flows, " AIAA Paper No. 91-0609, (Jan. 1991). 

Barton, J. M. and Rubinstein, R., "Nonlinerar Algebraic Reynolds Stress Model for 

Anisotropic Turbulent Flows, " 11th U. S. National Congress of Applied Mechanics, 

TUcson, AZ (May 1990). 

Barton, J. M. and Rubinstein, R., "Renormalization Group Theory and Turbulence 

Modeling, " 2nd International Workshop on Chaos and TUrbulence, Tsukuba, Japan 

(Jan. 1992). 

Barton, J. M. and Rubinstein, R., "Renormalization Group Analysis of TUrbulenceDriven Secondary Fows, " 11th Australian Fluid Mechanics Conference, Hobart, 

Australia (Dec. 1992). 

Brown, S., Leibovich, S. and Z. Yang, "On the linear instability of the HallStewartson vortex," Theoretical and Computational Fluid Dynamics, 2, 27-46, 

(1990). 

Duncan, B. S., Liou, W. W. and Shih, T.-H., "A Multiple-scale turbulence model 

for incompressible flow," NASA TM (to appear) (1992). 

Duncan, B.S., Lumley, J.L., Shih, T.H. and To, W.M., "A new model for the 

turbulent dissipation" International Conference of Fluid Mechanics and Theoretical 

Physics, Bejing, China (1992). 

Hsu, A.T. and Liou, M., "Computational analysis of underexpanded jets in the 

hypersonic regime," Journal of Propulsion and Power, 7, No.2, (1991). 

Hsu, A.T., "Progress in the development of PDF turbulence models for combustion," 10th NASP Symposium, April 23-16, 1991, Monterey, California. 

Hsu, A.T., "A study of hydrogen diffusion flames using PDF turbulence model," 

AIAA 91-1780, AIAA 22nd Fluid Dynamics Conference, June 24-26, 1991, Honolulu, Hawaii. 

Hsu, A.T. and Chen, J.Y, "A continuous mixing Model for PDF simulations and 

its applications to com busting shear flows," 8th International Symposium on TUrbulence Shear Flows, Sept. 9-11, 1991, Munich, Germany. 

Lang, N.J. and Shih, T.-H., "A critical comparison of two-equation turbulence 

models," NASA TM 105237, (1991). 

Liou, M. S. and Steffen, Jr., C. J., "A New Flux Splitting Scheme," NASA TM 

104404 (1991). 

Preceding Page Blank 

187 

-----------------------
Appendix C 

Liou, W. W. and Shih, T.-H., "A two-scale model for compressible turbulent flows," 

NASA TM (to appear) (1992). 

Liou, W. W. and Shih, T.-H., "On the basic equations for sencond-order modeling 

of compressible turbulence," NASA TM 105277 (1991). 

Liou, W. W. and Morris, P. J., "Weakly nonlinear models for turbulent mixing in 

a plane mixing layer," Phy. Fluids (to appear). 

Liou, W. W. and Morris, P. J., "The eigenvalue specturm of the Rayleigh equation 

for a plane shear layer," Int. J. Num. Fluids, (to appear). 

Liou, W. W., "A new energy transfer model for turbulent free shear flow," ASA 

TM (to appear) (1992). 

Mansour, N N. and Shih, T.-H. and Reynolds, W. C., "The effects of rotation on 

initially anisotropic homogeneous flows," Physics of fluid A, 3,10,2421-2425 (1991). 

Michelassi, V. and Shih, T .-H., "Low Reynolds number two-equation modeling of 

turbulent flows," NASA TM 104368, (1991). 

Michelassi, V. and Shih, T.-H., "Elliptic flow computation by low Reynolds number 

two-equation turbulence models ," NASA TM 105376, (1991). 

Moin, P., Shih, T.-H., Driver, D. and Mansour, ., "Direct numerical simulation of 

a three-dimensional turbulent boundary layer," Physics of Fluids A 2, 10, 1846-1853 

(1990). 

Rubinstein, R. and Barton, J. M., "Nonlinear Reynolds stress models and the renormalization group," Phys. Fluids A 2, 1472 (1990). 

Rubinstein, R. and Barton, J. M., «Renormalization group analysis of anisotropic 

diffusion in turbulent shear flow," Phys. Fluids A 3, 415 (1991). 

Rubinstein, R. and Barton, J. M., "Renormalization group analysis of the Reynolds 

stress transport equation," to appear in Physics of Fluids A, (July 1992). 

Shabbir, Munich. A., "Experimental balances for the second moments for a buoyant plume and their implication on turbulence modeling," Eighth Symposium on 

TUrbulent Shear Flows, 27-1-1 to 27-1-6 (1991). 

Shabbir, A. and Shih, T.-H., «Critical comparison of second order turbulence models 

in homogeneous flows," Submitted to AIAA Annual meeting in Reno, Jan. 1992. 

Shih, T.-H. and Luml~y, J.L., "Kolmogorov behavior of near-wall turbulence and 

its application in turbulence modeling," NASA TM 105663 (1992) 

188 

Appendix C 

Liou, W. W. and Shih, T.-H., "A two-scale model for compressible turbulent flows," 

NASA TM (to appear) (1992). 

Liou, W. W. and Shih, T.-H., "On the basic equations for sencond-order modeling 

of compressible turbulence," NASA TM 105277 (1991). 

Liou, W. W. and Morris, P. J., "Weakly nonlinear models for turbulent mixing in 

a plane mixing layer," Phy. Fluids (to appear). 

Liou, W. W. and Morris, P. J., "The eigenvalue specturm of the Rayleigh equation 

for a plane shear layer," Int. J. Num. Fluids, (to appear). 

Liou, W. W., "A new energy transfer model for turbulent free shear flow," ASA 

TM (to appear) (1992). 

Mansour, N N. and Shih, T.-H. and Reynolds, W. C., "The effects of rotation on 

initially anisotropic homogeneous flows," Physics of fluid A, 3,10,2421-2425 (1991). 

Michelassi, V. and Shih, T .-H., "Low Reynolds number two-equation modeling of 

turbulent flows," NASA TM 104368, (1991). 

Michelassi, V. and Shih, T.-H., "Elliptic flow computation by low Reynolds number 

two-equation turbulence models ," NASA TM 105376, (1991). 

Moin, P., Shih, T.-H., Driver, D. and Mansour, ., "Direct numerical simulation of 

a three-dimensional turbulent boundary layer," Physics of Fluids A 2, 10, 1846-1853 

(1990). 

Rubinstein, R. and Barton, J. M., "Nonlinear Reynolds stress models and the renormalization group," Phys. Fluids A 2, 1472 (1990). 

Rubinstein, R. and Barton, J. M., «Renormalization group analysis of anisotropic 

diffusion in turbulent shear flow," Phys. Fluids A 3, 415 (1991). 

Rubinstein, R. and Barton, J. M., "Renormalization group analysis of the Reynolds 

stress transport equation," to appear in Physics of Fluids A, (July 1992). 

Shabbir, Munich. A., "Experimental balances for the second moments for a buoyant plume and their implication on turbulence modeling," Eighth Symposium on 

TUrbulent Shear Flows, 27-1-1 to 27-1-6 (1991). 

Shabbir, A. and Shih, T.-H., «Critical comparison of second order turbulence models 

in homogeneous flows," Submitted to AIAA Annual meeting in Reno, Jan. 1992. 

Shih, T.-H. and Luml~y, J.L., "Kolmogorov behavior of near-wall turbulence and 

its application in turbulence modeling," NASA TM 105663 (1992) 

188 
List of Member's Publications 

Shih, T.-H., Chen, J.-Y. and Lumley, J. L., "Second order modeling offree turbulent 

shear flows," AIAA Journal, 30, 6, 1553-1560 (1992). 

Shih, T.-H. and Lumley, J.L., "A critical comparison of second order closures with 

direct numerical simulation of homogeneous turbulence," NASA TM 105351 (1991). 

Shih, T.-H. and Hsu, A.-T., "An improved k - € model for near-wall turbulence," 

AIAA paper, 91-0611 (1991). 

Shih, T.-H. Shih, Shabbir, A. and Lumley, J.L., "Advances in Modeling the pressure 

correlation terms in the second moment equations," The Lumley Symposium: Resent Developments in Thrbulence, November, 12-13, 1990, ICASE, NASA Langley 

Research Center, Edited by Gatski, T.B., Sarkar, S., Speziale, C.G. 

Shih, T.-H., "An improved k - E model for near-wall turbulence and comparison 

with direct numerical simulation," NASA TM 103221 (1990). 

Shih, T.-H. and Mansour, N. N., "Modeling of near-wall turbulence," Engineering 

Turbulence Modeling and Experiments, September 24-28, 1990, Dubrovnik, Yugoslavia, Editors: W.Rodi, E.N. Ganic. 

Shih, T.-H. and Reynolds, W. C., "A spectrum model for weakly anisotropic turbulence," Physics of Fluids A, 2, 8, (1990). 

Steffen, Jr., C. J., "An Investigation of DTNS2D for Use as an Incompressible 

Turbulence Modelling Test-Bed," NASA TM 105593 (1992). 

Steffen, Jr., C. J. and Beard, 1. M., "Incompressible Navier Stokes Solutions Using 

the Pseudo Compressibility Technique," Proceeding of the 23rd Annual Pittsburgh 

Conference on Modeling and Simulation, in print (1992). 

Van der Vegt, Jaap, "ENO-Osher schems for Euler equations," submitted for publication (1992). 

Van der Vegt, Jaap, "Overview of the Osher approximate Riemann solver for threedimensional flows," NASA TM, in print (1992). 

Yang, Z. and Leibovich, S "Nonlinear dynamics near the stability margin in rotating 

pipe flow," J. Fluid Mech. 233, 329-347, (1991). 

Yang, Z. and Leibovich, S., "Unstable viscous wall mo9.es in rotating pipe flow," 

AIAA Paper 91-1801, (1991). Also submitted to Physics Fluids A for publication. 

Yang, Z. and Shih, T.R., "A k - E modeling of near wall turbulence," Proc. of 4th 

IntI. Sym. on CFD, Davis, CA, 1305-1310, (1991). Also available as NASA TM 

105238. 

189 

List of Member's Publications 

Shih, T.-H., Chen, J.-Y. and Lumley, J. L., "Second order modeling offree turbulent 

shear flows," AIAA Journal, 30, 6, 1553-1560 (1992). 

Shih, T.-H. and Lumley, J.L., "A critical comparison of second order closures with 

direct numerical simulation of homogeneous turbulence," NASA TM 105351 (1991). 

Shih, T.-H. and Hsu, A.-T., "An improved k - € model for near-wall turbulence," 

AIAA paper, 91-0611 (1991). 

Shih, T.-H. Shih, Shabbir, A. and Lumley, J.L., "Advances in Modeling the pressure 

correlation terms in the second moment equations," The Lumley Symposium: Resent Developments in Thrbulence, November, 12-13, 1990, ICASE, NASA Langley 

Research Center, Edited by Gatski, T.B., Sarkar, S., Speziale, C.G. 

Shih, T.-H., "An improved k - E model for near-wall turbulence and comparison 

with direct numerical simulation," NASA TM 103221 (1990). 

Shih, T.-H. and Mansour, N. N., "Modeling of near-wall turbulence," Engineering 

Turbulence Modeling and Experiments, September 24-28, 1990, Dubrovnik, Yugoslavia, Editors: W.Rodi, E.N. Ganic. 

Shih, T.-H. and Reynolds, W. C., "A spectrum model for weakly anisotropic turbulence," Physics of Fluids A, 2, 8, (1990). 

Steffen, Jr., C. J., "An Investigation of DTNS2D for Use as an Incompressible 

Turbulence Modelling Test-Bed," NASA TM 105593 (1992). 

Steffen, Jr., C. J. and Beard, 1. M., "Incompressible Navier Stokes Solutions Using 

the Pseudo Compressibility Technique," Proceeding of the 23rd Annual Pittsburgh 

Conference on Modeling and Simulation, in print (1992). 

Van der Vegt, Jaap, "ENO-Osher schems for Euler equations," submitted for publication (1992). 

Van der Vegt, Jaap, "Overview of the Osher approximate Riemann solver for threedimensional flows," NASA TM, in print (1992). 

Yang, Z. and Leibovich, S "Nonlinear dynamics near the stability margin in rotating 

pipe flow," J. Fluid Mech. 233, 329-347, (1991). 

Yang, Z. and Leibovich, S., "Unstable viscous wall mo9.es in rotating pipe flow," 

AIAA Paper 91-1801, (1991). Also submitted to Physics Fluids A for publication. 

Yang, Z. and Shih, T.R., "A k - E modeling of near wall turbulence," Proc. of 4th 

IntI. Sym. on CFD, Davis, CA, 1305-1310, (1991). Also available as NASA TM 

105238. 

189 
Appendix C 

Yang, Z. and Shih, T.R., "A k - ( calculation of transitional boundary layers," 

To appear in Transition and Turbulence, Springer-Verlag, 1992. Also available as 

NASA TM 105604. 

Yang, Z. and Shih, T.R. "A new time scale based k - ( model for near wall turbulence," Submitted for publication. It is also to appear as a NASA TM (1992). 

Yang, Z. and Shih, T. H., "A modeling of transitional boundary layers," In preparation. 

Zhu, G., Lai, M.-C and Shih, T.-H., "Second-order closure modeling of turbulent 

buoyant wall plumes," NASA TM (in print) (1992). 

190 

Appendix C 

Yang, Z. and Shih, T.R., "A k - ( calculation of transitional boundary layers," 

To appear in Transition and Turbulence, Springer-Verlag, 1992. Also available as 

NASA TM 105604. 

Yang, Z. and Shih, T.R. "A new time scale based k - ( model for near wall turbulence," Submitted for publication. It is also to appear as a NASA TM (1992). 

Yang, Z. and Shih, T. H., "A modeling of transitional boundary layers," In preparation. 

Zhu, G., Lai, M.-C and Shih, T.-H., "Second-order closure modeling of turbulent 

buoyant wall plumes," NASA TM (in print) (1992). 

190 
1-

REPORT DOCUMENTATION PAGE 

Form Approved 

OMB No. 0704-0188 

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, 

gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this 

collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for information Operations and Reports, 1215 Jefferson 

Davis Highway, Su~e 1204, Arlington, VA 22202-4302, and to the OffICe of Management and Budget, Paperwor1< Reduction Project (0704-0188), Washington, DC 20503. 

1. AGENCY USE ONLY (Leave blank) /2. REPORT DATE /3. 

REPORT TYPE AND DATES COVERED 

September 1992 Technical Memorandum 

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS 

Center for Modeling of Turbulence and Transition (CMOTT) 

Research Briefs-l 992 

6. AUTHOR(S) WU-505-62-21 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION 

REPORT NUMBER 

National Aeronautics and Space Administration 

Lewis Research Center E-7274 

Cleveland, Ohio 44135-3191 

9. SPONSORING/MONITORING AGENCY NAMES(S) AND ADDRESS(ES) 10. SPONSORINGfMONITORING 

AGENCY REPORT NUMBER 

National Aeronautics and Space Administration 

NASA TM-105834 Washington, D.C. 20546-0001 

ICOMP-92-12; CMOTT-92-D8 

11. SUPPLEMENTARY NOTES 

Report compiled by William W. Liou and T.-H. Shih, NASA Lewis Research Center (work funded under NASA 

Cooperative Agreement NCC3-233). ICOMP Program Director: Louis A. Povinelli, (216) 433-5818. 

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRI8UTION CODE 

Unclassified - Unlimited 

Subject Category 34 

13. ABSTRACT (Maximum 200 words) 

This research brief contains the progress reports of the Research Staff of the Center for Modeling of Turbulence and Transition 

(CMOTf) from May 1991 to May 1992. It is intended as an annual report to the Institute for Computational Mechanics in Propulsion 

and NASA Lewis Research Center. A separate report entitled, "Workshop on Engineering Turbulence Modeling," covering some of 

the 1991 CMOTT Summer research activities was released earlier this year. The main objective of the CMOTT is to develop, validate 

and implement the turbulence and transition models for practical engineering flows. The flows of interest are three-dimensional, 

incompressible and compressible flows with chemical reaction. During this period, the research covers two-equation (e.g., k-E) and 

algebraic Reynolds-stress models, second moment closure models, probability density function (pdt) models, Renormalization Group 

Theory (RNG), Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS). Last year was CMOTT's second year in 

operation. CMOTT now has eleven members from ICOMP, NASA LeRC, and Sverdrup Technology Inc., working on various aspects 

of turbulence and transition modeling in collaboration with NASA-Lewis scientists and Case Western Reserve University faculty 

members. The CMOTT members have been continuously and actively involved in the international and national turbulence research 

activities. A biweekly CMOTT seminar series has been conducted with speakers invited from within and without the NASA Lewis 

Research Center, including foreign speakers. The current CMOTf roster and its organization are listed in Appendix A. Listed in 

Appendix B are the abstracts and the scientific and technical issues discussed in biweekly CMOTT seminars. Appendix C gives a list 

of references which are the papers contributed by CMOTT members in the last two years. 

14. SUBJECT TERMS 15. NUMBER OF PAGES 

Turbulence models 192 

16. PRICE CODE 

A09 

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT 

OF REPORT OF THIS PAGE OF ABSTRACT 

Unclassified Unclassified Unclassified 

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89) 

Prescribed by ANSI Std. Z39-18 

298-102 

--- ----- --. - ---

1-

REPORT DOCUMENTATION PAGE 

Form Approved 

OMB No. 0704-0188 

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, 

gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this 

collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for information Operations and Reports, 1215 Jefferson 

Davis Highway, Su~e 1204, Arlington, VA 22202-4302, and to the OffICe of Management and Budget, Paperwor1< Reduction Project (0704-0188), Washington, DC 20503. 

1. AGENCY USE ONLY (Leave blank) /2. REPORT DATE /3. 

REPORT TYPE AND DATES COVERED 

September 1992 Technical Memorandum 

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS 

Center for Modeling of Turbulence and Transition (CMOTT) 

Research Briefs-l 992 

6. AUTHOR(S) WU-505-62-21 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION 

REPORT NUMBER 

National Aeronautics and Space Administration 

Lewis Research Center E-7274 

Cleveland, Ohio 44135-3191 

9. SPONSORING/MONITORING AGENCY NAMES(S) AND ADDRESS(ES) 10. SPONSORINGfMONITORING 

AGENCY REPORT NUMBER 

National Aeronautics and Space Administration 

NASA TM-105834 Washington, D.C. 20546-0001 

ICOMP-92-12; CMOTT-92-D8 

11. SUPPLEMENTARY NOTES 

Report compiled by William W. Liou and T.-H. Shih, NASA Lewis Research Center (work funded under NASA 

Cooperative Agreement NCC3-233). ICOMP Program Director: Louis A. Povinelli, (216) 433-5818. 

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRI8UTION CODE 

Unclassified - Unlimited 

Subject Category 34 

13. ABSTRACT (Maximum 200 words) 

This research brief contains the progress reports of the Research Staff of the Center for Modeling of Turbulence and Transition 

(CMOTf) from May 1991 to May 1992. It is intended as an annual report to the Institute for Computational Mechanics in Propulsion 

and NASA Lewis Research Center. A separate report entitled, "Workshop on Engineering Turbulence Modeling," covering some of 

the 1991 CMOTT Summer research activities was released earlier this year. The main objective of the CMOTT is to develop, validate 

and implement the turbulence and transition models for practical engineering flows. The flows of interest are three-dimensional, 

incompressible and compressible flows with chemical reaction. During this period, the research covers two-equation (e.g., k-E) and 

algebraic Reynolds-stress models, second moment closure models, probability density function (pdt) models, Renormalization Group 

Theory (RNG), Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS). Last year was CMOTT's second year in 

operation. CMOTT now has eleven members from ICOMP, NASA LeRC, and Sverdrup Technology Inc., working on various aspects 

of turbulence and transition modeling in collaboration with NASA-Lewis scientists and Case Western Reserve University faculty 

members. The CMOTT members have been continuously and actively involved in the international and national turbulence research 

activities. A biweekly CMOTT seminar series has been conducted with speakers invited from within and without the NASA Lewis 

Research Center, including foreign speakers. The current CMOTf roster and its organization are listed in Appendix A. Listed in 

Appendix B are the abstracts and the scientific and technical issues discussed in biweekly CMOTT seminars. Appendix C gives a list 

of references which are the papers contributed by CMOTT members in the last two years. 

14. SUBJECT TERMS 15. NUMBER OF PAGES 

Turbulence models 192 

16. PRICE CODE 

A09 

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT 

OF REPORT OF THIS PAGE OF ABSTRACT 

Unclassified Unclassified Unclassified 

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89) 

Prescribed by ANSI Std. Z39-18 

298-102 

--- ----- --. - ---