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NASA Technical Memorandum 106383

ICOMP-93-44; CMOTT-93-16

Center for Modeling of Turbulence and

Transition: Research Briefs 1993

January 1994
Center for Modeling of Turbulence and Transition i

Research Briefs - 1993

Table of Contents

Page

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Research Activities at the Center for Modeling of Turbulence and Transition

Tsan-Hsing Shih . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Eddy Viscosity Models for Turbulent and Transitional Flows

Zhigang Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Testing New Two-Equation Turbulence Models for Complex Flows

Jiang Zhu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Low Ret k — e Models and the Backward-Facing Step Flow

Christopher J. Steffen, Jr. . . . . . . . . . . . . . . . . . . . . . . . . 63

Modeling of Homogeneous Scalar Turbulence

Aamir Shabbir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

A Multiple-Scale Turbulence Model for Incompressible Flow

Beverly Duncan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Modeling of Turbulent Shear Flows

William W. Liou . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

PDF Models for Compressible Reactive Flows and DNS

Andrew T. Hsu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Modeling of Turbulent, Reacting Flows by PDF Methods

Andrew T. Norris . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

A Time Dependent Generalization of the Nonlinear Eddy Viscosity

Representation of Turbulence

Robert Rubinstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

High-Order Accurate Osher Schemes with Application to Compressible

Boundary Layer Stability

Jacobus J. Van der Vegt . . . . . . . . . . . . . . . . . . . . . . . . . 157

Calculation of Waves in Fluids Using A High-Order

Compact Difference Scheme

Sheng-Tao Yu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
ii

Center for Modeling of Turbulence and Transition

Research Briefs -1993

Visitors' Research Activities

J.-Y. Chen ( University of California at Berkeley ) . . . . . . . . . . . . . . 215

P. G. Huang ( Eloret Institute ) . . . . . . . . . . . . . . . . . . . . . . 216

Arne V. Johansson ( Royal Institute of Technology, Sweden) . . . . . . . . . 216

Chaoqun Liu ( University of Colorado ) . . . . . . . . . . . . . . . . . . 216

Appendix A : Organization - 1993 . . . . . . . . . . . . . . . . . . . 219

Appendix B : CMOTT Biweeky Seminars . . . . . . . . . . . . . . . 223

Appendix C : List of Member's Publications . . . . . . . . . . . . . . 251
Center for Modeling of Turbulence and Transition 111

Research Briefs -1993

Preface

This research brief contains the progress reports of the research staff of the Center for

Modeling of Turbulence and Transition (CMOTT) from June 1992 to July 1993. It is also

an annual report to the Institute for Computational Mechanics in Propulsion located at

Ohio Aerospace Institute and NASA Lewis Research Center.

The main objectives of the research activities at CMOTT are to develop, validate and implement turbulence and transition models for flows of interest in propulsion

systems. Currently, our research covers eddy viscosity one- and two-equation models,

Reynolds-stress algebraic equation models, Reynolds-stress transport equation models,

non-equilibrium multiple-scale models, bypass transition models, joint scalar probability density function models and Renormalization Group Theory and Direct Interaction

Approximation methods. Some numerical simulations (LES and DNS) have also been carried out to support the development of turbulence modeling. Last year was CMOTT's

third year in operation. During this period, in addition to the above mentioned research,

CMOTT has also hosted the following programs: an eighteen-hour short course on "Turbulence Fundamentals and Computational Modeling (Part I)" given by CMOTT at

the NASA Lewis Research Center; a productive summer visitor research program that has

generated many encouraging results; collaborative programs with industry customers (e.g.

P. & W. and RocketDyne) to help improve their turbulent flow calculations for propulsion

system designs; a biweekly CMOTT seminar series with speakers from within and without the NASA Lewis Research Center, including foreign speakers. In addition, CMOTT

members have been actively involved in the national and international turbulence research

activities.

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

Appendix B are the abstracts of the biweekly CMOTT seminars. Appendix C lists the

papers contributed by CMOTT members.

Tsan-Hsing Shih
IV
Center for Modeling of Turbulence and Transition 1

Research Briefs - 1993

Research Activities at the Center for

Modeling of Turbulence and Transition

Tsan-Hsing Shih

1. Introduction

2. General developments

2.1 Turbulent constitutive relations

2.2 Mechanical and scalar dissipation equations

2.3 Eddy viscosity transport equation

3. One-point closure schemes

3.1 One-equation eddy viscosity transport equation model

3.2 Galilean and tensorial invariant realizable k-e model

3.3 Reynolds stress algebraic equation model

3.4 Scalar flux algebraic equation model

3.5 Reynolds stress transport equation model

3.6 Non-equilibrium multiple-scale model

3.7 Bypass transition model

3.8 Joint scalar PDF model

4. RNG and DIA

5. Numerical simulation

Abstract

The main research activities at the Center for Modeling of Turbulence and Transition (CMOTT) are described. The research objective of CMOTT is to improve

and/or develop turbulence and transition models for propulsion systems. The flows

of interest in propulsion systems can be both compressible and incompressible,

three dimensional, bounded by complex wall geometries, chemically reacting, and

involve "bypass" transition. The most relevant turbulence and transition models for

the above flows are one- and two-equation eddy viscosity models, Reynolds stress

algebraic- and transport-equation models, pdf models, and multiple-scale models.

All these models are classified as one-point closure schemes since only one-point (in

time and space) turbulent correlations, such as second moments (Reynolds stresses

and turbulent heat fluxes) and third moments (ut uj uk , ui O2 ), are involved. In

computational fluid dynamics, all turbulent quantities are one-point correlations.

Therefore, the study of one-point turbulent closure schemes is the focus of our

turbulence research. However, other research, such as the renormalization group

theory, the direct interaction approximation method and numerical simulations are

also pursued to support the development of turbulence modeling.
2 Tsan-Hsing Shih

1. Introduction

The center for modeling of turbulence and transition was established as a special

focus group within the Institute for Computational Mechanics in Propulsion at

NASA Lewis Research Center in 1990. Its objective is to improve and/or develop

turbulence and transition models for computational fluid dynamics (CFD) applied

in propulsion systems. With the advance of computer technology and algorithms,

accurate turbulence and transition modeling becomes the pacing item for improving

flow calculations used in propulsion system design in all its key elements. The flows

of interest in propulsion systems are, in general, very complex since there are wallbounded three-dimensional complex geometries, chemical reactions, compressibility

and transition, etc. In order to accurately predict these flows one must correctly

model the turbulent stresses and scalar fluxes which are one-point (in time and

space) turbulent correlations. For flows with finite rate chemical reactions, accurate

modeling of the production rate of species is crucial for turbulent flow calculations.

Based on the above considerations, turbulence modeling activities at CMOTT are

focused on one-point closure schemes, that is, using the moment closure schemes for

the turbulent velocity field and the joint scalar pdf method for the reacting scalar

field.

There are various moment closure schemes which have been developed for various engineering applications. However, in practice, one often finds that the existing models need to be improved and/or re-developed in order to reasonably

simulate complex flow structures appearing in propulsion systems. For this purpose, CMOTT devotes itself to improving and/or re-developing these moment closure schemes which include eddy viscosity (one- and two-equation) models, second

moment algebraic- and transport-equation models, non-equilibrium multiple-scale

models, and bypass transition models. In addition, other studies supporting the

development of one-point closure schemes have been also carried out (for example,

studies on renormalization group theory (RNG), direct interaction approximation

(DIA), direct numerical simulation (DNS) and large eddy simulation (LES)).

In this report, we first describe the general development of turbulent constitutive

relations, turbulent mechanical and thermal dissipation and a new eddy viscosity

equation. Second, we describe the detailed developments on each moment closure

scheme and the pdf method. Then the RNG and DIA methods and finally, the

numerical simulation of particular turbulence phenomena, such as rotation and

bypass transition, etc., are considered.

Each research subject is the joint project of several CMOTT researchers and

visitors. In describing research activities, the names of involved researchers will be

mentioned for reference.

I

I
Research Activities at CMOTT 3

2. General Developments

2.1 Turbulent Constitutive Relations

Reynolds stress

Using the invariant theory in continuum mechanics and Generalized CayleyHamilton formulas for tensor products, a turbulent constitutive relation (or a general turbulence model) for any turbulent correlations can be obtained, in principle.

Therefore, this theory provides an avenue to develop better turbulence models than

those existing. For example, a commonly used constitutive relation for Reynolds

stresses ui uj (in terms of the mean deformation rate tensor Ui,j and the turbulent

velocity and length scales characterized by the turbulent kinetic energy k and its

dissipation rate e) is

2

— Uiuj = Cμ E U( i,j + Uj,i) - 2 k6ij

The effective eddy viscosity vT defined as

2

VT 

= 

_ u^u7

 = CC11,E for i 0 (2.1.2)

Ui 7j + U7 t

is isotropic since vT is a scalar quantity. However, the invariant theory enables us

to formulate the following general model (Shih and Lumley l , Johansson 2 ):

2

uiuj = 

2 

kSij + 2a2 K (Ui,j + Uj,i — 2 Ui,isi7)

3

+2a46(U?j+U^i- 3^lsij)

3

• 2a6 K (Ui, k Uj,k — I H26ij)

3

• 2a7 K ( Uk, iUk,j — 1H2bij)

4

• 2a8 

K 

(Ui,kUj k + U^kUj,k 31136ij)

4

• 2alo 

K 

(Uk, iUk

2 

,7 + Uk, 7 U2

2

^3 k,i — 3

H 

3 Si7

•) (2.1.3)

5

• 2a12 

E 

(Zli,k Uj, k — 3 H46ij )

5

• 2a13 

K (

U k,i Uk,j — 1 N46ij)

5

• 2a14 

K (

Ui,k Ul,k U2j + Uj,kUl,kU1i 3rl56i.j)

6

• 2a16 E ( Ui,k Ui k Ui j + Uj,k Ui k Ui i — 3n6bi;)

7

+ 2a18 

s (Ui,kU1 , k UI12 U^,,,^ + Uj,k U1,kU12 n U m. — 3H7bij)
4 Tsan-Hsing Shih

where

Ill = Ui,kUk,i

I'l2 — Ui,k Ui,k , 113 — Ui,k Ui,k + 1`14 = Ui,kUi,k

11.5 = Ui, kUi,kUii , rI6 = Ui,k Ul,k

U

l i 117 = Ui

, kUc ,kUlrn,U2m

(2.1.4)

From Eq.(2.1.3), the effective eddy viscosity

—uiui

( VT) ii — Ui j + U, t

(2.1.5)

is no longer a scalar and, hence, is an anisotropic eddy viscosity. It is noticed that

the first two terms on the right hand side of Eq.(2.1.3) represent the standard k -E

eddy viscosity model (2.1.1) and that the first five terms of Eq.(2.1.3) are of the same

form as the models derived from both the two-scale DIA approach (Yoshizawa 3 ) and

the RNG method (Rubinstein and Barton 4).

Eq.(2.1.3) is a general model for u iuj . It contains 11 undetermined coefficients

which are, in general, scalar functions of various invariants of the tensors in question, such as Si;Si; (strain rate) and QijQij (rotation rate) which are (11 2 +111)/2

and (142 — 11 1 )/2 respectively. The detailed forms of these scalar functions must

be determined by other model constraints, for example, realizability, and by experimental data. Eq.(2.1.3) contains 12 terms; however, its quadratic tensorial form

may be sufficient for practical applications. We will see later in section 3.3 that the

constitutive relation (2.1.3) has a significant impact on the development of Reynolds

stress algebraic equation models.

Turbulent scalar flux 9ui

We assume the following functional form:

Bui = Fi( Ui,j, T,i, k, E, 9 2 , Ee) (2.1.6)

where e2 is the variance of a fluctuating scalar and Ee is its dissipation rate.

Eq.(2.1.6) indicates that the scalar flux depends on not only the mean scalar gradient Ti , but also the mean velocity gradient Ui,; and the scales of both velocity and

scalar fluctuations characterized by k, s, B2 , Ee.

Applying the invariant theory, we may obtain the following general constitutive

i
Research Activities at CMOTT 5

relation for Bui:

Bui = a, k(—.—) k 92 1 /Z T'i + k2 ( k 92 ) 1/2 (a2 U

i,j + a3 Uj,i)T ,j E Ee

((

E E Ee

+ E2 ( k 82 )

1/2 (a4Ui,kUk , j + a5Uj , kUk , i + a6 Ui,k Uj , k + a7 Uk,i Uk,j )z,j

E E Ee

42

+ Ea ( 

8B )

1/2 (a8Ui ,kUjk +a9U?k Uj , k+a10Uk , iUk,j +allUk,i Uk , j )Ta

+ k5 (k 92 ) 112 (a12Ui k U

?

+a13U?. k

U

k,7

2 .

E4 E Eg \

7,k ,z 

+ a14 Ui,kU1,k U12 j + a15 Uj,kU1,k Ul,i )T,j

+ 0 ( k 92) 1/2(a

16 Ui,kU?k U12j + a17Uj , kU1 kU12i)

E5 E Ee

l

7 2

+ (kEg E 9 )1/2a18Ui,kUl,kUi

sE ,nU^,,, T j

(2.1.7)

The coefficients a 1 — a18 are, in general, functions of the time scale ratio E /Zco

and the other invariants formed by the tensors in question, for example, T kTk,

Ti Ui,jTj , etc.. Again, Eq.(2.1.7) implies that the effective eddy diffusivity

bT)i = —Bui

Ti

is not isotropic. It is noticed that the first term on the right hand side of Eq.(2.1.7)

is the standard eddy diffusion model, and the models derived from the two-scale

DIA (Yoshizawa5 ) and the RNG method (Rubinstein and Barton 6 ) are similar to

the first two terms of Eq.(2.1.7). In practice, a form containing the first two terms

on the right hand side of Eq.(2.1.7) may suffice. Further development of this model

for turbulent heat transfer is described in Section 3.4.

The Researchers involved with the subject in this section are T.-H. Shih, J. Zhu,

A. Shabbir, J.L. Lumley t and A. Johansson.t

2.2 Mechanical and Scalar Dissipation Equation

Mechanical dissipation E

In turbulence modeling, we often need turbulent characteristic velocity and length

scales. While the turbulent kinetic energy k is used to characterize the velocity scale,

the mechanical dissipation rate E and the scalar dissipation rate Ee are used to characterize the length scales for mechanical and scalar fields, respectively. Comparing

with the turbulent kinetic energy equation, the exact dissipation rate equation is

t Professor, Cornell University, Ithaca, NY

t Professor, Royal Institute of Technology, Stockholm, Sweden
6 Tsan-Hsing Shih

very complicated. In this equation, all the terms which represent important turbulence physics (for example, turbulent diffusion, generation and destruction) are

unknown and are of complex forms that are all related to small scales of turbulence.

Therefore, in the literature, the exact dissipation equation is not considered as a

useful equation to work with. Instead, one creates a model equation by assuming an

analogy to the turbulent kinetic energy equation, i.e., one assumes that the model

dissipation rate equation also has generation and destruction terms which are assumed to be proportional respectively to the production and dissipation terms in

the turbulent kinetic energy equation over the period of large eddy turn-over time

characterized by kle. The resulting model dissipation rate equation is written as

E ,t + UiE ,i =VE,ii — (iui),i

E E2 (2.2.1)

— Cel ^ ui uj Ui,a — Ce2 -

Recently, Lumley? proposed a dissipation rate equation based on the concept of

spectral energy transfer caused by interactions between eddies of different sizes.

This model equation mimics the physics of statistical energy transfer from large

eddies to small eddies and is of a different form than equation (2.2.1).

In this study, we explore another rational way to obtain the model dissipation

rate equation which contains certain important physics and hope it will work better

than the existing one. The idea is that first, there is a relationship between the

dissipation rate E and the mean-square vorticity fluctuation Wiwi at high Reynolds

numbers or in homogeneous turbulence:

E = VWiwi

and second, all the terms appearing in the Wiwi equation have more clear physical

meanings than that in the E equation so that the Wiwi equation is easier to model.

Once the wiwi equation is modeled, a model dissipation rate equation will be readily

obtained.

The exact equation for Wiwi is

Wi wti Wi wi Wiwi

( 2 ),t + Uj( 2 ) ,j = v( 2 ),jj — 2(U WiWi),j +Wi2Ui,jQj (2.2.2)

— 2UjWiQi j + WtiWj Ui

, j + wiwjui,j — VWi,jwi,j

where ui and Ui are the fluctuating and mean velocities, and wi and Qi are the

fluctuating and mean vorticity which are defined by

Wi = EijkUk,j Qi = EijkUk,j (2.2.3)

Tennekes and Lumley8 clearly described the physical meaning of each term in equation (2.2.2). Order of magnitude analysis shows that the first, third, fourth and fifth

terms on the right hand side of Eq.(2.2.2) become small compared with all other
Research Activities at CMOTT 7

terms in the equation as the turbulent Reynolds number increases. The sixth and

seventh terms are the production due to fluctuating vortex stretching and the dissipation due to the viscosity of the fluid. As the turbulent Reynolds number increases

these last two terms become dominant and the balance between them determines the

evolution of vorticity fluctuations. Neglecting terms wi ui , j S23 ) —ujwi S2 i j , wiwjui,j

and v(' i2 , ) ,jj , the evolution of Wiwi at large Reynolds number will be described by

the following equation,

Wi wi Wiwi 1

( 2 ),t + Uj( 2 ),j = — 2 ( ujwiWi),j + wiwj ui,j — vwi,jwi,j (2.2.4)

To model wiwj ui,j — vwi,jwi,j , let us first estimate wiwjui,j . we define an

anisotropic tensor by.:

Wiwj 1

bij = 2 — — Sij (2.2.5)

Wk 3

then wiwj ui,j can be written as

wiwjui,j — b^jw2Ui,j (2.2.6)

We expect that the vortex stretching tends to align vortex lines with the strain rate

so that the anisotropy by. would be proportional to the strain rate s ij , i. e.,

.

M'.

si

oc —' where s = (2sijsij)112, sij = (

ui,j + uj, i)/2

(2.2.7)

s

This leads to the following model:

wiwj ui,j a wk(2s ij sij ) 1 1 2 a: 2s ij s ij (2.2.8)

where we have assumed that wk and (2s ijsij ) 1/2 are well correlated.

Using the relation, wi = Eijkuk,j, it is not difficult to show that at large turbulent

Reynolds number,

wiwi 2sij sij (2.2.9)

and Eq.(2.2.8) can be also written as

W z 2_wk iWjui , j a wk w.i — _ (2.2.10)

Wi

Equation (2.2.10) indicates that this term is of the order (u3 /13 )R3/2 as it should

be. On the other hand, from eq.(2.2.4) the term wiwjui j — vwi,jwi, j must be of the

order (u3 /13 )Rt which is the order of magnitude of all the other terms in Eq.(2.2.4),

therefore the term —vwi,jwi,j must cancel the term (2.2.10) or (2.2.8) such that the

difference of these two terms is smaller than the term (2.2.10) or (2.2.8) by an order
8 Tsan-Hsing Shih

of Rt1/2. This suggests that the combination Wiw; u-j ; — vwi ;wi ; can be modeled by

the following two terms:

W2 W? w 

Z W2S (2.2.11)

v +rW,

because the ratio of k/v to w? and the ratio of s to S are of order Rt /2 , where

k -z^ u2 is the turbulent kinetic energy and S is the mean strain rate (2Si;Si;)1/2_

Equation ( 2.2.11) does give the right order of magnitude for wiw;ui,; — vWi,;Wi,;.

Therefore, the dynamical equation for fluctuating vorticity (2.2.4) at large Reynolds

number can be modeled as

2 z

Wi wi Wiwi 1

2k.

Wk Wi

( 2 )>t + U^( 2 ),^ _ — 2 (u.7WiWi ),.Y + C.1 S — QA (2.2.12)

v +rW

Using E = vwiwi , we readily obtain the following model dissipation rate equation,

2

E,t + U;s,; = — (u; E ),; + C^11 S e — Q,2 6 (2.2.13)

1£ + vE

where C,,, 1 and C,2 are the model coefficients which are expected to be constant at

large Reynolds number.

It should be noticed that Eq.(2.2.13) is different from the standard E equation

(2.2.1) by both the generation and destruction terms. First, the Reynolds stresses

do not appear in the generation term and the new form of the generation term is

similar to that proposed by Lumley? which is based on the concept of spectral energy

transfer. Second, the destruction term is well behaved so that equation (2.2.13) will

not have a singularity anywhere in the flow field. We expect that equation (2.2.13)

will be numerically much more robust than equation (2.2.1).

Equation (2.2.13) can be applied to any level of turbulence modeling including

second order closure models; however the turbulent transport term ( ui),i needs

to be modeled differently at different levels of turbulence modeling. In an eddy

viscosity model, the term (eui) ,i will be modeled as

(2.2.14)

The coefficients C, 1 , Cli,2 , a, and the eddy viscosity VT must be calibrated using

experimental data ( Shih et a1.9)

Scalar dissipation Eo

A similar analysis leads to the following model scalar dissipation rate equation:

EEO

Eo,t + U 2.2.15 j E o,j _ — (u;Ee)J + COI Eg + CB2 Pr_1 ^ 2

EEO — Cas k + vE ( )

where 4^ = T iT i and T is the mean scalar quantity, such as, the mean temperature. Further development of heat transfer model is described in Section 3.4.

The Researchers involved with the subject in this section are T.-H. Shih, W. Liou,

A. Shabbir and Z. Yang.
Research Activities at CMOTT 9

2.3 Eddy Viscosity Transport Equation

In eddy viscosity models, one accepts the following simple constitutive relation

ui u; = —2vT Si; + 

2

3

kbi; (2.3.1)

and assumes that the eddy viscosity is characterized by some kind of velocity and

length scales u' and f:

11T OC U f (2.3.2)

In two-equation k-e eddy viscosity models, for example, one specifies that

U

1 oc k 2 , e a 

k2

(2.3.3)

e

and, hence, the eddy viscosity is assumed as

k2

VT = CN, 

e

(2.3.4)

The eddy viscosity assumption (2.3.4) is commonly adopted in two-equation models.

Egs.(2.3.1) and (2.3.4) together with appropriate k and e equations have been widely

used in engineering calculations. However, for cases where the mean flow changes

quickly or has a strong mean stream-line curvature or rotation, etc., this kind of

model does not work very well, since the assumption (2.3.4) is too simple to account

for the effect of the above mean flow structure on eddy viscosity.

The main purpose of this study is to drop the assumption (2.3.4) and to derive

an exact equation for VT based on Eq.(2.3.1) and other exact turbulence equations

(i.e. first principles). In this way, we hope that some important turbulent physics

can be brought into the eddy viscosity and that a physically sound turbulence eddy

viscosity can be calculated.

Using Eq.(2.3.1), we may write for incompressible flows

ui uj uiuj = 2vT S2 + 3 k2 , where S2 = 2SijSij (2.3.5)

Differentiating both sides, we obtain

D _ Sij D VT D 2

Dt IT

S2 Dt u

zu

j 2S2 Dt 

S (2.3.6)

The equation for ui uj can be written as

D

u,

Dt

= Dig + Pik + IIij — eij + Coif

(2.3.7)
10 Tsan-Hsing Shih

where

Dij = [UUiuj — 26iUjuk],k

Pi = —uiukUj,k — ujukUi,k

1 Ilij = --P,iuj +p,jui P

Eij = 2vui,kuj,k

Coij = —2Eim.kQmukuj — 2Ejm.k2mukui

Inserting Eq.(2.3.7) into Eq.(2.3.6), we obtain an exact transport equation for eddy

viscosity

D vT = — SZ ( Dij Pij IIij — Eij +Coij) — v a D ( ) Dt S2 2S Dt S

2 2.3.8

In this equation, all the important turbulence physics in the Reynolds stress equation, such as Reynolds stress diffusion term Dij , production term Pij , pressurevelocity gradient correlation term Il ij and dissipation tensor Eij , are involved.

Comparing with the standard eddy viscosity assumption (2.3.4), this exact eddy

viscosity equation (2.3.8) contains very rich turbulence physics. This equation also

implies that a second order closure model will naturally lead to a corresponding

eddy viscosity model.

Now, as an example, we use Launder Reece and Rodi 's 10 model and a gradient

transport model for the triple velocity correlation ( - 26iUiUk = T uiuj k ) to derive

a model equation for VT. The resulting equation is

D UT 3 UT UT kS k UT 2VTSijSij,kk

Dt VT = [( U + Q ) UT, k l ,k + (U + 2 ^) S2 + (U U) S2 (2.3.9)

4 E SikSkjSji UT D 2

+ 15 

k — Ci UT + 2(C2 — 2)vT S2 2S2 Dt S

Note that the Coriolis terms do not explicitly appear in this equation; however

the rotation effect on VT could be carried over through the mean flow field. In

addition, we also note that there are no extra model coefficients introduced in

Eq.(2.3.9). All model coefficients (v, Cl and C2 ) are brought in from the second

order closure model. The values of these model coefficients may need adjustment in

model applications. Note that Eq.(2.3.9) is not a self-consistent equation since the

turbulent kinetic energy k and its dissipation rate E are also involved. Eq.(2.3.9)

together with k-E transport equations will provide a new three-equation model which

may better represent the effect of mean flow structure as well as mean flow history

on the eddy viscosity.

The Researchers involved with the subject in this section are T.-H. Shih, Z Yang,

and W. Liou.

3. One-Point Closure Schemes

In this section, we describe the developments on each of the moment closure

scheme and the pdf method which are of concern at CMOTT. The first two sections
Research Activities at CMOTT 11

3.1 and 3.2 describe the one- and two-equation isotropic eddy viscosity models. Sections 3.3 and 3.4 describe the new developments on Reynolds stress and scalar flux

algebraic equation models. Section 3.5 assesses Reynolds stress transport equation

models. Section 3.6 describes a multiple-scale model for non-equilibrium turbulence.

Section 3.7 is about transition models. Finally, in Section 3.8 the pdf method for

turbulent chemical reaction is described.

3.1 One- equation eddy viscosity model

Recently developed one-equation eddy viscosity models are either based on the

assumption (Baldwin and Barth"):

2

VT = Cp k

b

or created according to computational experience (Spalart and Allmaras 12 ). Both of

them are successful in some flow calculations. This scheme is quite attractive in CFD

because one only needs to solve one scalar VT equation without bothering about

other turbulence quantities. However, comparing with k-e two equation models, the

above mentioned one-equation VT models do not contain any more turbulent physics.

In fact, Baldwin and Barth's model is, basically, a change of dependent variable

based on Eq. (3.1.1) plus some extra approximations. Therefore, in principle, we

should not expect any superior performance over two-equation models. However, if

we do not use the assumption (3.1.1), there is the possibility to improve and extend

the capability of one-equation eddy viscosity models.

The objective of this study at CMOTT is to derive a physically sound eddy

viscosity equation which contains rich turbulent physics and accounts for various

effects from mean flow structures.

Note that in Section 2.3 we have already derived an exact equation for the eddy

viscosity (2.3.8) and also a model equation (2.3.9) which is based on the Reynolds

stress transport equation model of Launder, Reece and Rodi (LRR). All turbulent

physics contained in the Reynolds stress equation can be brought into the eddy

viscosity equation. Therefore, in principle, the transport equation (2.3.9) should be

better than existing one-equation models based on Eq.(3.1.1). However, Eq.(2.3.9)

is not self-consistent because k and e are also involved. To make Eq.(2.3.9) selfconsistent, we must model k and k/e in terms of VT and S. In most shear flows,

the energy-containing eddy turn-over time k/s is of the same order as the mean

flow time scale S-1 , so that Elk oc S is a reasonable model. In addition, a crude

dimensional analysis gives k oc vTS and this is, of course, reasonable only for shear

flows. After the above considerations, the resulting self-consistent one-equation

model is:

Dt VT = 

[ ( v + 

vT )

(VT) , k l ,k —

CIO 

(vT),k(VT),k + Cv1S 11T

+ 2(C,2 — 2 )1/T S2

— VT D 52

S2 2S2 Dt
12 Tsan-Hsing Shih

where the diffusion terms from the Reynolds stress equation (2.3.7) have been manipulated and approximated. Eq.(3.1.2) clearly exhibits the various effects of the

mean flow on the eddy viscosity.

The model coefficients Ci1 , C,.2 and v can be determined by using the experimental data of homogeneous shear flows, free shear flows and boundary layer flows

as well as the relations in the inertial sublayer. Extensive tests of this model in

various flows are carrying out at the CMOTT.

The Researchers involved with the subject in this section are T.-H. Shih, W. Liou,

Z. Yang and J. Zhu.

3.2 Galilean and tensorial invariant realizable k-E model

The two-equation k-E eddy viscosity model is one of the most widely used turbulence models in engineering calculations. The k-E model has versions for high

Reynolds numbers and for low Reynolds numbers. For wall bounded turbulent

flows, the high Reynolds number k-E model (for example, Launder and Spalding 13)

must be applied together with a wall function as its boundary condition, while the

low Reynolds number k-E model (for example, Jones and Launder14 ) can be integrated to the wall. The high Reynolds number k-E model of Launder and Spalding

is considered as a standard k-E model. We notice that even though the model dissipation rate equation is created by assuming an analogy with the turbulent kinetic

energy, there was not much modification until Lumley ? and Shih et al.' For near

wall turbulence, in addition to Jones and Launder's model, there are many other

versions of low Reynolds number k-E models (such as Chien15 , Shih and Lumley"',

Yang and Shih 17) which have made better performance over Jones and Launder's

model.

There are, probably, four or five issues worth mentioning about existing low

Reynolds number k-E models: the model constants are not consistent with those

in the high Reynolds number k-E model; the wall correction terms and damping

functions are related to the wall distance so that models are not tensorial invariant;

a nonrealistic dissipation rate near the wall is introduced; they are not always

realizable since normal stress could become negative; and finally, they do not work

very well for boundary layer flows with various pressure gradients.

The objective of this study at CMOTT is to overcome the above mentioned

problems. First, we propose a vorticity dynamics based dissipation rate equation

as a part of high Reynolds number k-E base model- 9 Second, based on the invariant

theory, inhomogeneous terms for the dissipation rate equation are proposed which

enable the model to better respond to the change of pressure gradients (Yang and

Shih 18 ). Third, the wall distance parameter is removed from the damping function

so that the model is tensorially invariant (Yang and Shih 19 ). The model constants

are consistent with those in the high Reynolds number k-E model. Finally, the

non-negativity of normal Reynolds stresses, the realizability condition, is imposed.

The Researchers involved with the subject in this section are Z. Yang, T.-H. Shih

and C. Steffen.
Research Activities at CMOTT 13

3.3 Reynolds stress algebraic equation model

All eddy viscosity models including one- and two-equation models are isotropic.

For the flows where anisotropy is important, for example, the secondary flows driven

by turbulent normal stresses in a square duct or curved duct, eddy viscosity models do not produce correct flow structures. To overcome this intrinsic deficiency of

isotropic eddy viscosity models, one proposes a Reynolds stress algebraic equation

model which will provide an effective anisotropic eddy viscosity. The first such a

model was proposed by Rodi 20 and it achieved some success in the prediction of

anisotropic related flow structure. However, Rodi's formulation is a set of algebraic

non-linear system equations for Reynolds stresses and it often creates numerical difficulty in obtaining a converged solution. Recently, Taulbee 21 obtained an explicit

algebraic expression for the Reynolds stress using Pope S21 tensor expansion formulation and solved this numerical difficulty. However, in general, Rodi's formulation

assumes that the ratio uiu;/k is constant and, of course, this is not really true for

most turbulent flows of interest. Therefore, sometimes, this Reynolds stress algebraic equation model produces even worse results than the isotropic eddy viscosity

models for cases where eddy viscosity models are appropriate.

Alternative ways for obtaining effective anisotropic eddy viscosity models have

been tried by a few researchers, for example, the DIA method by Yoshizawa3,

the RNG method by Rubinstein and Barton 4 and invariant theory by Shih and

Lumley.'. It is interesting to point out that the RNG and DIA methods result in

the same formulation and that this formulation is the first five terms of a general

constitutive relation Eq.(2.1.3) except that the model coefficients are different.

One of our goals at CMOTT is to search for an effective anisotropic eddy viscosity model for complex turbulent flows where the nonequilibrium of turbulence

is not very severe so that the constitutive relation (2.1.3) is more or less valid. We

have explored the potential capability of Eq.(2.1.3) and found that a truncation of

Eq.(2.1.3) up to the quadratic terms of the mean velocity gradients is sufficient for

various flows of interest. The model coefficients are determined such that realizability for the normal stresses is ensured. The detailed analysis is described by Shih et

al. 23

The quadratic version of Eq.(2.1.3) together with the standard k-e transport

equations, successfully predicts many complex flows as well as simple flows which

include backward-facing step flows; confined coflowing jets; confined swirling coaxial

jets; flows in 180° curved duct; flows in a diffuser and a nozzle; boundary layer

flows with pressure gradient and turbulent free shear flows. See references 23-25 for

detailed results.

The Researchers involved with the subject in this section are J. Zhu and T.-H.

Shih.

3.4 Scalar flux algebraic equation model

In parallel with Reynolds stress algebraic equation model, we have also tried to

develop an effective anisotropic scalar eddy diffusivity model for scalar (heat) fluxes

based on the new constitutive relation (2.1.7) and the new thermal dissipation rate
14 Tsan-Hsing Shih

equation (2.2.16). We have determined that it seems sufficient to truncate Eq.(2.1.7)

up to linear terms of the mean velocity gradient, i.e.,

Bui

 = alk(k 02 ) 1/2 + k2 (k 

e2 ^ 112 ( a2 UiJ +a'3Ud,i)Z,? (3.4.1)

E Ee E E Eg

This equation indicates that the heat flux and the mean temperature gradient are

not necessarily in alignment due to the distortion of the flow field. This means that

the effective scalar eddy diffusivity is anisotropic.

Eq.(3.4.1) together with the B2 and Ee equations will be a closed set of model

equations for turbulent heat fluxes. The model coefficients are calibrated from

homogeneous flows. Detailed analysis and a few model tests are described in this

research briefs by A. Shabbir.

The Researchers involved with the subject in this section are A. Shabbir and

T.-H. Shih.

3.5 Reynolds stress transport equation model

The Reynolds stress transport equation model is considered as a next generation

of advanced turbulence modeling for engineering applications. In principle, the

second moment equations describe various effects of the mean flow and external

agencies on the evolution of turbulence, hence, are the most attractive way (also

the simplest correct way) to study turbulent flows.

Various closure models for second moment equations have been developed. The

success of these closures are marginal and vary with each flow. To identify the

sources of their deficiencies, one often uses simple flows where the specific model

term in the second moment equations can be isolated, hence, the corresponding

model can be checked against experimental data or direct numerical simulation

(DNS). For example, using pre-distorted anisotropic homogeneous relaxation flows,

we may check the return-to-isotropy models with experimental data or DNS. However, for other flows, several model terms, such as, triple velocity correlations, rapid

and slow pressure-strain correlations, etc., simultaneously exist and can not be isolated in the experiments. In these cases (for example, in a homogeneous shear flow

or a channel flow) only DNS can provide all the information for simultaneously

checking various models.

We have examined various existing closure models using experimental data as

well as DNS data (Shin et al. 16 and Shih and Lumley27 ). Conclusive statements are

difficult to draw at this time. However, the following remarks can be made about

various closures for the second moment equations, i.e., the triple velocity correlation

Tijk, the rapid and slow pressure related correlations TIP, Il!^, and the dissipation

rate tensor Eij:

a) Ti; k . All the existing models, such as Daly and Harlow 21 , LRR10 , Lumley29,

etc., are not very satisfactory for highly inhomogeneous flows, such as flow near the

wall. However, for flows where the inhomogeneity is not very high, the above closure

models become close to each other and also closer to the DNS data. In addition,

the triple velocity correlations in these situations are usually small comparing with
Research Activities at CMOTT 15

other terms in the equation, so that modeling of this term is not as critical as other

terms for the results of turbulent flow calculations, except for the flow near the wall.

b) 117. It is very clear from all the available DNS data that nonlinear models, such

as, Shih and Lumley 30 are much better than linear models, such as SSG". It seems

also that the following constitutive relation

r

Ilj = F( ui uj, Ui,9

is quite appropriate, i.e., its dependence on turbulent Reynolds number and other

parameters is quite weak and can be neglected. However, one deficiency of this form

observed by Reynolds 32 is that it can not take the rotation effect into account.

c) IIi^. This term is usually modeled together with the dissipation tensor ei; and

the combination of the two is called the return -to-isotropy term. All existing models are unsatisfactory at the present time. They are far from "universal', i.e., their

performance varies from flow to flow. It is noticed that some strange behavior of

return-to-isotropy (for example, for some pre-distorted flow relaxation, turbulence

evolves toward anisotropy before it returns to isotropy) occurs and cannot be possibly modeled with the following constitutive relation:

II J = F(ui uj , k, e)

In addition, the behavior of return -to-isotropy was found to depend not only on the

Reynolds stresses at the present time but also on their history according to DNS

data (Lee33 ). It may be also necessary to include triple velocity correlations into

the above constitutive relations from the definition of IIi^. The term II7^ seems

highly dependent on the turbulent Reynolds number and slowly approaches to its

asymptote as Reynolds number goes to infinity, so that, in general, one should not

exclude its dependence on turbulent Reynolds number even for moderate Reynolds

numbers. In addition, 11?^ is also noticeably affected by the mean strain rate according to the DNS data34 , so that, in general, the mean strain rate should be also

considered in the constitutive relation. In short, much more research is needed for

developing a better model of 11j.

The Researchers involved with the subject in this section are T.-H. Shih and A.

Shabbir.

3.6 Non-equilibrium multiple-scale model

To consider the effect of the nonequilibrium of energy spectrum on turbulent

quantities, such as k, e and ui uj , etc., Hanjelic et al. 35 are the first to propose a

partition in the turbulent energy spectrum. Because of the nonequilibrium, the rate

that energy enters the low wave number region, EP , does not equal to the energy

transfer rate from low wave numbers to high wave numbers, e t . Therefore it is

reasonable to describe the evolution of the energy contained in low wave number

region, kP , and high wave number region k t , separately. As a result, the time scale

or the length scale defined by different energy transfer rates will be different and

this multiple-scale concept reflects the nonequilibrium effect of turbulence.
16 Tsan-Hsing Shih

We think that this concept would be more appropriate for compressible flows

because the compressibility often creates nonequilibrium interactions between large

and small eddies. We first modify Hanjelic et al.'s model, test it in various free

shear flows and boundary layer flows and then extend it to compressible flows by

consideration of the effects of compressibility on the equations for kP and _-P . The

proposed model is tested in both compressible free shear flows and boundary layer

flows. For detailed analysis and flow calculations see the report by Duncan et al.36

and Liou and Shih37

The Researchers involved with the subject in this section are W. Liou, T.-H. Shih

and B. Duncan.

3.7 Bypass transition model

The onset of turbulence transition in the propulsion system is often highly influenced by the free stream turbulence. This transition process does not go through

the linear instability but is mainly controlled by nonlinear processes. Therefore, it

is sometimes called "bypass" transition. Because of this highly nonlinear process of

transition, turbulence models may be used to predict it. In fact, many two-equation

models, for example, k-e eddy viscosity models of Launder and Sharma38 , Chien15,

etc., do mimic bypass transition on a flat plate when the free stream has a certain

amount of turbulent intensity. However, to obtain an accurate prediction of bypass

transition, the study of the bypass transition process and physics is needed. The

conventional turbulence models must be modified to take into account the intermittent phenomena of transitional flows.

We have proposed transition models based on a two-equation turbulence model

using an intermittency factor to modify either the eddy viscosity or modeled k-e

equations. Successful results for a flat plate boundary layer under various freestream turbulence intensities are obtained. For details see the report by Yang and

Shih39.

The Researchers involved with the subject in this section are Z. Yang and T.-H.

Shih.

3.8 Joint scalar PDF model

One of the critical problems in turbulent combustion is how to treat the interaction between the chemical reaction on the turbulence. The estimation of the

production rate of compositions based on the mean flow temperature would be in a

very large error for flows with finite rate chemical reactions. The reason is that the

production rate of compositions depends not only on the mean values of temperature T and compositions Ci , but also very much depends on the detailed fluctuations

of temperature B and compositions c i . The moment closure scheme of modeling the

production rate of compositions in terms of the mean flow temperature, the mean

compositions and various correlations consisting of the fluctuating temperature and

composition, such as B 2 , Bci ,ci c

j

, ... has not been successful. However, the PDF

method allows us to treat chemical reaction exactly without modeling (Pope").

Therefore, for the study of turbulent combustion problems, we use the joint scalar

PDF transport equation for the scalar field and the moment closure schemes for
Research Activities at CMOTT 17

the velocity field and develop a hybrid solver consisting of a Monte Carlo scheme

and a conventional CFD method. For detailed description of this procedure and its

applications see Hsu " and Hsu et a1.42

The Researchers involved with the subject in this section are A. Hsu, A.T. Norris

and J.Y. Chent

4. RNG and DIA

In developing one point turbulence models, conventional modeling methods can

be supplemented by "non-conventional" methods such as renormalization group

theory (RNG) and the direct interaction approximation (DIA). These are two point

theories formulated in wavevector or fourier space; one point models are derived

by integration over wavevectors. This approach provides theoretical support for

conventionally derived models and sometimes suggests theoretically derived forms

for the empirical elements, whether constants or functions, which appear in these

models.

We have applied RNG methods to both the eddy viscosity and Reynolds stress

transport equation models. In addition to the k — s model proposed by Yakhot

and Orszag43 , it is possible to obtain constitutive relations for Reynolds stress

and heat fluxes (Rubinstein and Barton 4,6 which are special cases of the general

results Egs.(2.1.3) and (2.1.7). By applying the perturbation theories of Yakhot

and Orszag43 to the relevant correlations, expansions in powers of the mean velocity

gradient are obtained for the stresses and heat fluxes; quadratic truncation of the

series leads to a stress model Eq.(2.1.3) with constant a 4i as, a7 and a heat flux

model Eq.(2.1.7) with constant a 2 , a3 in which the constants are in good agreement

with empirically selected values. The forms derived are also consistent with the

DIA analysis of Yoshizawa 3 5

The RNG method also provides a formulation for closing the Reynolds stress

transport equation (Rubinstein and Barton 14). Perturbative evaluation of the correlations 11" and Hq^ leads to series expansions in powers of the mean velocity

gradient. These series can be consolidated, or "resumed" using the known perturbation series for the Reynolds stresses by methods analogous to Pade approximation.

Systematic lowest order summation leads to a Reynolds stress transport equation

with a form identical to the LRR model equation and with constants in reasonable agreement with empirically chosen values. Higher order resummation leading

to nonlinear models of the type described in Sec. 3.5 remains an open possibility.

The possibility of such resummation in the context of DIA has been discussed by

Yoshizawa45,46

Recent work has focussed on nonequilibrium time dependent relations between the

Reynolds stress and the mean flow derived from a simplification of the DIA theory

of shear turbulence. In this theory, shear turbulence is modeled by a non-Markovian

eddy damping acting against the mean shear. The RNG and DIA Reynolds stress

transport models and the LRR model all assume Markovian damping; as in the

t Professor, University of California, Berkeley, CA
18 Tsan-Hsing Shih

molecular theory of transport coefficients, Markovian damping describes long time

behavior and is incorrect at short times. The most important consequence of nonMarkovian damping is a strong suppression of eddy damping at short times. This

leads to closer agreement between the present theory and rapid distortion theory

at short time. this is important in modeling oscillating shear flows: recent work of

Mankbadi47 shows that RDT based models best predict such flows. In transient

homogeneous shear flow at high strain rates, the LRR model predicts rapid onset

of eddy damping leading to excessive growth of turbulence kinetic energy at short

times. The suppression of eddy damping at short times in the present model should

lead to improved predictions for this flow as well.

Another consequence of this theory is a stress model Eq.(2.1.3) in which the coefficients a2, ... are functions of the mean strain rate. This theory can be described as

RDT with a modified total strain determined by the response function of the DIA

theory of isotropic turbulence. The introduction of a phenomenological modified total strain has often been advocated in the RDT literature to improve the agreement

between RDT and shear flow data; here the modified total strain is deduced as a

consequence of the theory. In the special but important case of simple shear flow

in which 8Ui 18xj = SSi1 6j2 , the result can be formulated in terms of Eq.(2.1.3) in

which, for example, a2 = a2 (Sk/e) and the function a3 is found exactly from RDT.

There are analogous results for the coefficients a4i as, a7 ; in simple shear flow, the

remaining terms in Eq.(2.1.3) identically vanish. Extension of this theory to other

mean shear tensors depends on the tabulation of the corresponding RDT solution.

The researchers involved with the subject in this section are R. Rubinstein and

A. Yoshizawa.t

5. Numerical Simulation

To obtain a better understanding of the effect of compressibility and rotation

on turbulence, numerical simulations of compressible homogeneous shear flows and

rotational flows are carried out. The effects of compressibility and rotation on the

energy spectrum and energy cascade between turbulent eddies has been analyzed

(Hsu and Shih48 ). These simulations support the idea of the multiple-scale model

for nonequilibrium compressible turbulent flows (W. Liou and Shih 37).

Another numerical simulation is the transition subjected to the free stream large

disturbances. The objective of this simulation is to obtain some insight into the

transition physics and to provide data base for bypass transition modeling. Based

on the assumption that the transition process is mainly controlled by large scale

motions, we use a high accuracy finite difference Navior-Stokes solver with course

grids to simulate the large scale motions of transition. A preliminary calculation of

bypass transition was carried out. Various statistics of the calculated flow field are

under examination.

The Researchers involved with the subject in this section are A. Hsu, C. Liout,

Z. Yang, A. Shabbir, T.-H. Shih.

t Professor, Tokyo University, Japan

t Professor, University of Colorado, Denver, CO
Research Activities at CMOTT 19

References

1 Shih, T.-H. and Lumley, J., "Remarks on turbulent constitutive relations," NASA

TM 106116 (1993).

a Johansson, A., Private communication.

3 Yoshizawa, A., Phys. Fluids, 27, 1377-1387 (1984).

4 Rubinstein R. and Barton, J. M., "Nonlinear Reynolds stress models and the

renormalization group," Phys. Fluids 2, 1472-1476 (1990).

5 A. Yoshizawa, J. Fluid Mech. 195, 541 (1988).

6 Rubinstein, R. and Barton, J. M., "Renormalization group analysis of anisotropic

diffusion in turbulent shear flows," Phys. Fluids 3, 415-421 (1991).

7 Lumley, J. L., "Some comments on turbulence," Phys. Fluids, 4 (1992).

8 Tennekes, H. and Lumley, J. L., A First Course in Turbulence Massachusetts

Institute of Technology, 1972.

9 Shih, T.-H., Liou, W. W., Shabbir, A., Yang, Z., and Zhu, J.," A vorticity dynamics based model for the turbulent dissipation," NASA TM 107177 (1993).

10 Launder, B.E., Reece, G.J., and Rodi, W. , " Progress in the development of a

Reynolds-stress turbulence closure," J. Fluid Mech. 68, 537-566 (1975).

11 Baldwin B. S. and Barth, J. T., "A one-equation turbulence transport model for

high Reynolds number wall-bounded flows," NASA TM 102847 (1990).

11 Spalart, P. R. and Allmaras, S. R., "A one-equation turbulence model for aerodynamic flows," AIAA-92-0439 (1992).

13 Launder, B. E. and Spalding, D. B., "The numerical computation of turbulent

flow," Computer Methods in Applied Mechanics and Engineering 3, 269-289

(1974).

14 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).

15 Chien, K.-Y., "Predictions of channel and boundary-layer flows with a lowReynolds-number turbulence model," AIAA Journal, 20, 33-38, (1982).

11 Shih, T.-H. and Lumley, J. L., "Kolmogorov behavior of near-wall turbulence and

its application in turbulence modeling," NASA TM 105663 (1992).

17 Yang, Z. and Shih, T. -H., "A new time scale based k — E Model for Near Wall

Turbulence," AIAA J. 31, 1191-1198 (1993).

18 Yang, Z. and Shih, T. -H., NASA TM to appear.

19 Yang, Z. and Shih, T. -H., "A Galliean and tensorial invariant k — E model for

near wall turbulence," AIAA 24th Fluid Dynamics Conference 93-3105, July 6-9

1993/Orlando, FL.
20 Tsan-Hsing Shih

20 Rodi, W., "Turbulence models and their application in hydraulics — A state of

the art review," Book Publication of the International Association for Hydraulics

Research, Delft, Netherlands (1980).

21 Taulbee, D. B., "An improved algebraic Reynolds stress model and corresponding

nonlinear stress model," Phys. Fluids 4, 2555 (1992).

22 S.B. Pope, "A more general effective-viscosity hypothesis," J. Fluid Mech., 72,

331 (1975)

23 Shih, T.-H., Zhu, J., and Lumley, J. L., "A realizable Reynolds stress algebraic

equation model," NASA TM 105993 (1992).

24 Zhu, J. and Shih, T.-H., "Computations of confined coflowing jets with three

turbulence models," AIAA paper 93-3120 (1993).

25 Zhu, J. and Shih, T.-H., "A numerical study of confined turbulent jets, " NASA

TM 106197 (1993).

26 Shih, T.-H., Shabbir, A., and Lumley, J. L., "Advances in modeling the pressure

correlation term in the second moment equations," NASA TM 104413 (1991).

27 Shih, T.-H. and Lumley, J.L., "Critical comparison of second order closures with

direct numerical simulation of homogeneous turbulence," AIAA J., 31, 663-671

(1993)

28 Daly, B.J. and Harlow, F.H., "Transport equations of turbulence," Phys. Fluid,

13, 2634 (1970).

29 Lumley, J.L., "Computational modeling of turbulent flows," Adv. Appl. Mech.,

18, 123 (1978).

30 Shih, T.-H. and Lumley, J. L., "Second-order modeling of boundary-free turbulent

shear flows," AIAA Journa1,30, 1553-1561 (1992).

31 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).

32 Reynolds, W. C., "Effect of rotation on homogeneous turbulence," Tenth Australian Fluid Mechanics Conference, December 11-15, 1989.

33 Lee, M.J. and Reynolds, W.C., " Numerical experiments on structure of homogeneous turbulence," Dept. Mech. Engng. Rep. TF-24, Stanford University:

Stanford California (1985).

34 Hallback, M., "Development of Reynolds stress closures of homogeneous turbulence through physical and numerical experiments," Doctoral Thesis, Royal

Institute of Technology, Stockholm, 1993.

35 Hanjalic, K., Launder, B.E. and Schiestel, R., "Multiple-time-scale concepts in

turbulent transport equation," Turbulent Shear Flow 2 (1980).

31 Duncan, B., Liou, W. W. and Shih, T.-H., "A multiple-scale turbulent model for

incompressible flow," AIAA 93-0086 (1993).
Research Activities at CMOTT 21

37 Liou, W. W. and Shih, T. H., "A multiple-scale model for compressible turbulent

flows," NASA TM 106072 (1993).

38 Launder, B. E. and Sharma, B. I., "Application of the energy-dissipation model

of turbulence to the calculation of a flow near a spinning disk," Letters in Heat

and Mass Transfer, 1, 131-138 (1974).

39 Yang, Z. and Shih, T.-H., "A k-e model for turbulent and transitional boundary

layer," Near Wall Turbulent Flows, Elsevier, 165-175 (1993).

4o S.B. Pope, "PDF methods for turbulent Reactive Flows," Prog. Energy Combust.

Sci., 11, 119-192 (1985).

41 Hsu, A. T., "Study of hydrogen diffusion flames using pdf turbulence model,"

AIAA paper 91-1780 (1991).

42 Hsu, A. T., Tsai, Y. L. T., and Raju, M. S., "A PDF approach for compressible

turbulent reacting flows," AIAA-93-0087 (1993).

43 V. Yakhot and S.A. Orszag, J. Sci. Comput. 1, 3 (1986).

44 Rubinstein, R. and Barton, J. M., "Renormalizatin group analysis of the Reynolds

stress transport equation," Phys. Fluids, 4, 1759-1766 (1992).

45 Yoshizawa, A., "Derivation of a model Reynolds stress transport equation using

the renormalization of the eddy-viscosity type representation," Phys. Fluids, 5,

707-715 (1993).

4s Yoshizawa, A., "Bridging between eddy-viscosity type and second order models

using a two-scale DIA," 9th Symposium Turbulent Shear Flows Kyoto Japan,

1993.

47 Mankbadi, R. R. and Liu, J. T. C., "Near-wall response in turbulent shear flows

subjected to imposed unsteadiness," J. Fluid Mech., 238, 55-71 (1992).

48 Hsu, A. T. and Shih, T.-H., "Effect of the Coriolis force on compressible turbulence," 9th Symposium on Turbulent Shear Flows," Kyoto, Japan, 1993.
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Center for Modeling of Turbulence and Transition 23

Research Briefs - 1993

Eddy Viscosity Models for

Turbulent and Transitional Flows

Z. Yang

1. Motivation and Objective

The purpose of this research is to develop eddy viscosity models for the complex flows of engineering interest. The models are required to be able to handle

the complex flow situations and the model calculations are to be computationally

robust.

2. Work Accomplished

In the past year, a number of projects have been conducted in the area of eddy

viscosity modeling. In addition, the linear marginal stability of the trailing line

vortex has been investigated. In the following, each project will be reported briefly.

2.1 A Galilean and Tensorial Invariant k —c Model for Near Wall Turbulence

In turbulence modeling, the k-E model is the most widely used model in engineering calculations. The standard k — c model' ,2 was devised for high Reynolds

number turbulent flows and is traditionally used in conjunction with wall functions

when 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 Launder 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 of the pre-1985 models was made by Patel et al.'. More

recently proposed models are found in Lang and Shih'. Three major deficiencies

can be pointed out about existing k — c 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 rate equation at the 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 is often the case for a real flow

situation. Patel et al.' put as the first criterion the ability of the near wall models

to be able to predict turbulent free shear flows. Third, the variable y+ was used in

the damping function fl, of the eddy viscosity formula. Since the definition of y+

involves uT , the friction velocity, any model containing y+ can not be used in flows

with separation. In addition, y + may not be well defined for flows with complex

geometry.

In an earlier paper by the authors 6, a time scale based k — c model for near

wall turbulence was proposed. In this model, 01' was chosen as the turbulent
24 Z. Yang

velocity scale. The time scale was bounded from below by the Kolmogorov time

scale. When this time scale is used to reformulate the dissipation rate equation,

there is no singularity at the wall and the introduction of a pseudo-dissipation rate

is avoided. The model constants were exactly the same as those in the standard

k — E model, which ensures the correct performance of the model far from the wall.

The damping function was proposed as a function of Ry = ^lvZy instead of y+.

Thus, the model can be used for flows with separation and reattachment.

However, the Ry dependence in the damping function makes the model coordinate

dependent. It also creates some ambiguity when the model is used for complex

geometries, for example, a corner flow. A similar problem also exists in most k — E

models. In the present study, this deficiency is overcome by introducing a new

parameter R in the damping function. The parameter is defined as

R 

Sv

(1)

where S is the modulus of the strain rate tensor of the mean velocity field.

The parameter R, defined above, is expressed in terms of the local variables of

the field and is thus coordinate independent. The physical meaning of R can be

explained as follows,

_ k _ k2 Sk

R Sv Ev E , (2)

R is then the ratio of two important physical parameters: the turbulent Reynolds

number and the time scale ratio of the turbulence to the mean flow. The variation

of R with the wall distance y+ for the case of turbulent channel flow at Re, = 180

is shown in figure 1. The gradual and monotonic increase of R with y + in the near

wall region makes R an ideal candidate for constructing the damping function.

In the present study, the form of the damping function is chosen as

fl, = [1 — exp(—a 1 R — a2 R2 — a3 R3 )] 1/2 (3)

where a 1 = 3 x 10-4 , a3 = 6 x 10- 5 , a5 = 2 x 10-6 . These constants are obtained

by calibrating the model prediction for the turbulent channel flow at Re, = 180

against the direct numerical simulation data of Kim et al.7

By introducing the parameter R, the y dependence is removed. Now, the model

is free from the three deficiencies mentioned above, and is Galilean and tensorial

invariant. This allows the model to be used in more complicated flow situations,

i.e., flows with separation. Since all the quantities in the proposed model are given

in the local variables, the model is very suitable for a general purposed CFD code

with unstructured grid.

Turbulent channel flows at different Reynolds numbers and turbulent boundary

layers with zero pressure gradient, favorable pressure gradient, and adverse pressure gradient were calculated using the present model. The model calculations were

compared with the available data from the experiments and direct numerical simulations. At low Reynolds number, the comparison between the direct numerical
Eddy Viscosity Models for Turbulent and Transitional Flows 25

simulation data and the present model is found to be excellent. At higher Reynolds

numbers, the velocity profiles are well predicted in all cases. However, the predicted

skin friction does not respond adequately with the pressure gradient. All the other

existing k — E models suffer from the same deficiency, as pointed out by Wilcox$.

The results of the calculations and the details of the model can be found in Yang

and Shih9.

2.2 On the Wall Functions for Two -Equation Turbulence Models

Near the wall, the turbulence quantities (especially the dissipation rate) change

rapidly. Very fine grids are needed to resolve such a rapid variation. This would

lead to a substantial increase in the overall number of grid points. For example,

for two dimensional boundary layer flows, more than half of the total grid points

are located in the near wall region. In addition to the increase in the number of

grid points, the fine grid spacing near the wall means that the grids will be severely

stretched, leading to what is commonly known as the numerical stiffness problem

associated with the dissipation rate equation.

The use of the wall functions which provide the boundary conditions at the log

layer rather than at the wall obviates all the difficulties mentioned above. However, existing wall functions were proposed based on the behavior of the turbulent

boundary layer over a flat plate at zero pressure gradient and it is known that they

are not universally valid.

In the present study, the validity of the wall functions is investigated for the

case of turbulent boundary layers with pressure gradient. We will shown that the

standard wall functions presented in Ref. 1 and Ref. 2 do not give an adequate

response to the pressure gradient. A new set of wall functions are derived. Rather

than using the properties of the turbulence quantities in the equilibrium turbulent

boundary at zero pressure gradient, the new wall functions are asymptotic solutions

of the governing equations (Reynolds averaged Navier-Stokes equations, the equation for the turbulent kinetic energy, and the definition of the eddy viscosity in the

framework of two-equation turbulence models) as the log layer is approached.

The assumptions involved in deriving such a new set of wall functions are: 1) the

advection term is negligible compared with the turbulent transport term in both

the mean momentum equation and the equation for turbulent kinetic energy, 2)

turbulent eddy viscosity is much larger than the molecular viscosity, 3) the mean

velocity profile follows the law of wall.

These assumptions are all met in the log layer in a high Reynolds number turbulent boundary layer with pressure gradient as long as the boundary layer remains

attached. The direct numerical simulation data of Spalart 10 shows that the advection term is at least two orders of magnitude smaller than the turbulent transport

term in the log layer. For high Reynolds number turbulent flows, the eddy viscosity

is also much larger than the molecular viscosity in the log layer. Coles" found

that the law of wall expression for the mean velocity remains valid for turbulent

boundary layers with different pressure gradients.
26 Z. Yang

Under these conditions, the new wall functions are

U = U,[ 1 In y+ +C], (4)

_ 1 1 dP

k Cμ2 [ P dx y + UT], (5)

_ UT 1 dP 2

E Ky[p dxy +UT],

where n = 0.41 is the von Karman constant, C = 5.0, and C,, = 0.09. A body fitted

coordinate system has been used in the above equations with x in the streamwise

direction and y in the direction normal to the wall. It is worth noting that when the

pressure gradient term is dropped, the new wall functions recover to the standard

wall functions.

The performances of both the standard wall functions and the new wall functions

were tested for the turbulent boundary layers with different pressure gradients. It is

known that the k — w model of Wilcox" integrated down to the wall gives good skin

friction coefficient predictions for turbulent boundary layers with different pressure

gradients. The comparison of the results from the k — w model integrated to the

wall and the results from the k — w model using the wall functions would then pin

down the effect of the wall functions.

The results for the flat plate boundary layer at zero pressure gradient is shown in

figure 2. (In this case, the new wall functions reduce to the standard wall functions.)

It is seen that the results from the k — w model integrated to the wall and the results

from the k — w model with wall functions are equally accurate when compared with

the experiment data of Wieghardt and Willmann 13 . Thus, for the zero pressure gradient boundary layer, the standard wall functions give good results. However, the

situation is quite different when the pressure gradient is not zero. Figure 3 shows

the distribution of the skin friction coefficient for Herring and Norbury flow 14 which

is a turbulent boundary layer under a favorable pressure gradient. Figure 4 shows

the skin friction distribution for Samuel and Joubert flow 15 in which the turbulent

boundary layer develops under an increasingly adverse pressure gradient. The standard wall functions fail to respond adequately to the pressure gradient, causing the

predicted skin friction to be lower than the experiment in the favorable pressure

gradient case and higher in the adverse pressure gradient case. In comparison, the

new wall functions give results that are of the same order of accuracy as the k — w

model integrated down to the wall. Thus, the capacity of the new wall functions to

capture the effect of the pressure gradient is established.

The wall functions were then used in conjunction with the standard k — E model

for Herring and Norbury flow and for Samuel and Joubert flow. It is known that the

standard k — E model with the standard wall functions poorly predicts the effect of

the pressure gradient. The comparison of the results from the standard k — E model

with the standard wall functions and the results from the standard k — E model

with the new wall functions, and the comparison of the model predictions with the

(6)

—j
Eddy Viscosity Models for Turbulent and Transitional Flows 27

experiment would tell us whether the poor performance comes from the standard

k — e model, or the wall function used, or both. The model calculations are shown

in figure 5 and figure 6, respectively. The use of the new wall functions improves

substantially the model prediction for both the favorable pressure gradient case and

the adverse pressure gradient case. However, the errors are still quite pronounced.

This indicates that the standard k — E model needs improvement in order to fully

capture the effect of the pressure gradient. This is the topic of the following two

projects.

The derivation of the wall functions and more extensive calculations using the

wall functions can be found in Yang and Shihls

2.3 A Vorticity Dynamics Based Model for the Dissipation Rate Equation

A new dissipation rate equation is proposed based on the dynamic equation for

the vorticity fluctuation. This newly proposed model equation has the advantage

over the standard E equation in that it is based on the first principle rather than

an equation made-up in analogy with the equation for the turbulent kinetic energy.

Since the production of the dissipation rate in the new equation is always positive,

the model is expected to be more robust for complex flow calculations when used

in conjunction with Reynolds stress models. When used in conjunction with the k

equation, the overall performance is found to be better than that of the standard

k — E equation. The model is given by Shih in this year's Research Brief. The details

of model development and model performance can be found in Shih et al.17.

2.4 Flow Inhomogeneity and the Dissipation Rate Equation

The dissipation rate equation is the weakest link in both the k — E and the second

order closure models. The standard E equation was made up in analogy with the

equation for turbulent kinetic energy. Recently, Shih et al. 17 proposed a model

equation for the dissipation rate based on the dynamic equation for the fluctuating

vorticity. Both the standard E equation and the model equation of Shih et al. are

homogeneous models in the sense that except for the transport term, the model

form remains the same for both homogeneous flow and inhomogeneous flow. Since

the dissipation rate represents the energy flux from the large eddies to the small

eddies in the eddy cascade, and the energy containing large eddies are sensitive to

the inhomogeneity, it can be expected that the flow inhomogeneity should enter the

modeled dissipation rate equation.

The exact dissipation rate equation is

2v \

E,t + UjE,j = VE ,jj - v(ui,kui.kuj),j P (p,ku?,k),7

- 2v ((ui, kuj, k ) + (Uk,i'U'k,i))UiJ - 2v(ujui,k)Ui,kj (7)

- 21/(ui,kUi,kUiJ) - 2v2(Ui,kjui,kj)

where ui , p represent the fluctuation field and Ui represents the mean field. It is

seen that in addition to the transport terms, the flow inhomogeneity also contributes

to the production and destruction in equation (7). However, the exact dissipation
28 Z. Yang

rate equation is very complex, containing many new unknown terms, and is very

difficult to model directly. Here, we use the invariant theory approach instead. The

invariant theory is a powerful tool to find out the possible functional form once the

independent physical variables are known. It has been widely used in continuum

mechanics. It was also used for turbulent flows by Lumley 18 , Pope 19 , Shih and

Lumley20 to find the general constitutive relations for turbulent flows.

In the present analysis, we assume that the inhomogeneity will produce another

term in the production/destruction of the modeled dissipation rate equation, which

now reads

—

2

aa(^ [(V + VT )

 aC^E 1 + Cj,SE — C2e _+4), (8) Dt Vxi i QE Vxg J

where the last term -PI represents the contribution of the inhomogeneity. In the

above equation, the inhomogeneity contribution was added to the homogeneous

model of Shih et al.17.

We use VS to measure the inhomogeneity of the mean field, where S is the size of

the strain tensor of the mean field, and Vk and VE to measure the inhomogeneity

of the turbulent field. The general form for the contribution of the inhomogeneity

to the production/destruction of the dissipation rate is

4^I = 4^ I(Vk, VE, VS; k, E, S). (9)

The homogeneous quantities (k, E, and S) enter parametrically. Since equation (9)

represents the contribution of the inhomogeneity effect, it would be zero if all the

gradient quantities are zero.

We are studying the situations when the inhomogeneity is relatively weak and

could be viewed as a perturbation about the homogeneous state. The three gradient

quantities are then related and only two of them are needed to specify the inhomogeneity. In the present study, we choose Vk and VS. Since (b I is a scalar whereas

Vk and VS are vectors, we need to form scalar quantities out of the vectors. Further, the scalar parameters should be grouped in such a way that they would have

the same dimension as other production/destruction terms in the dissipation rate

equation. There are three parameter groups altogether and the general form for the

inhomogeneity term is

44^ I = 4^ I(Vk • Vk, vt VS • Vk, vt VS • VS) (10)

where vt is the eddy viscosity used in the k — E equation and is given by

2

vt = CM 

k

E

Since we are considering flows with relatively weak inhomogeneity, we can carry

a Taylor series expansion for equation (10). To the leading order, the result is

,T), = a 1 Vk• Vk +a2 vtVS• Vk + a3 vt VS VS (11)
Eddy Viscosity Models for Turbulent and Transitional Flows 29

where al , a2 , and a3 are the coefficients from the expansion. In general, they are

functions of rl = Sk/E, which is the only invariant that can be formed from the

homogeneous quantities. In the present study, they are taken as constants.

With the inhomogeneity term given by equation (11), we have a model for the

dissipation rate equation which takes into account the effect of the flow inhomogeneity. The model constants C1E and C2E are taken the same as in Shih et al. 17 , i.e.,

C1E = 0.42 and C2E = 1.90. The model constants in the inhomogeneity terms are

taken as al = 0.1, a2 = —0.65, and a3 = 0.1. The value of o,E is changed from 1.21

in Ref. 17 to 1.4 in the present model. In the homogeneous model, the turbulent

diffusion is the only term that accounts for the effect of the inhomogeneity. Thus, it

must account for both the transport and the production due to the inhomogeneity.

In the present model, since the contribution of the inhomogeneity to the production

of the dissipation rate is accounted for explicitly, it is expected that the value of a',

be changed.

The model was then tested for turbulent boundary layer flows. Figures 7-9 show

the predicted skin friction distributions for turbulent boundary layers with zero pressure gradient (Wieghardt and Willmann flow), favorable pressure gradient (Herring

and Norbury flow), and adverse pressure gradient (Samuel and Joubert flow), respectively. The model predictions are in good agreement with the experiment in all

cases. More details of the model development and more validating calculations are

presented in a paper by Yang and Shih21.

In the works presented above, the inhomogeneity contribution was added to the

homogeneous model of Shih et al. 17 . It should be mentioned that the same inhomogeneity contribution was also added to the standard E equation and results were

found to be good too.

2.5 A Transport Equation for Eddy Viscosity

Algebraic turbulence models 21,21 have been used extensively in calculations of

aerodynamic flows. These algebraic models are easy to implement numerically and

give fairly accurate predictions for simple flows, such as that over an airfoil with an

attached boundary layer; however, they are inadequate when used for more complex

flows, such as that over an airfoil with separation. The failings of simple algebraic

models have motivated recent research into dynamical equations for eddy viscosity.

The idea of formulating an eddy viscosity transport model, first suggested by Nee

and Kovasznay24 , was recently revived by Baldwin and Barth 25 , and Spalart and

Allmaras 21 . Another motivation of working on the eddy viscosity transport model

rather than higher level models, the k — E model for example, is the observation that

near the wall the dissipation rate changes rapidly and the k — E model is numerically

stiff; while the eddy viscosity varies much more gradually and it is expected that

an equation for the eddy viscosity would be numerically better behaved.

In the models by Baldwin and Barth, and Spalart and Allmaras, wall effects were

introduced using damping functions, in which the distance to the wall y enters as

an parameter. This wall distance y is not a local property; its definition would

be ambiguous for flows in complex geometry, i.e., corner flows. Also, the damping

functions are assumed to be universal, which does not seem valid in flows with
30 Z. Yang

strong adverse pressure gradients and separation.

In the present work, we propose a new transport model for the eddy viscosity. Par

away from the wall, the model is based on the quasi-homogeneous approximation;

near the solid surface, the wall effect is introduced via an elliptic relaxation equation

first proposed by Durbin 27 in the context of the second order closure. This would

completely eliminate the damping functions. In addition, the present model is

formulated solely in terms of local variables and it is Galilean and tensorial invariant.

The model consists of a parabolic transport equation for the eddy viscosity:

2

DTvT=V(v +vT)VvT +P„— c21SIVT—c4 

L2

(12)

v

where the production term Pv is subject to the following elliptic relaxation:

LPV2 Pv — Pv = C31VVT1 2 — JSJvT . ( 13)

In these equations, the c's are model constants, the L's are length scales, and S

is the size of the strain tensor of the mean field, which is used to characterize the

gradients of the mean flow.

The length scales are given as

L2 = r 

S

2 

+ (7 ,,

vtl ) 2 (14) v

IVS1 S J

LP = min IL2V, cP max ( S

cl S I J .

(15)

The model constants used are

C2 = 0.85, c4 = 0.2, cl = 3.2, cp = 1.21 c,l, = 2,

with c3 determined by

C3 _ 1 — C2 1 — C4 (16)

IL2 1 _+K4 cm

which is obtained by using the log layer relation of the flat plate boundary layer

at zero pressure gradient. These model constants were determined on the basis of

a few points of reference: DNS of plane channel flow28 give centerline velocities of

U/UT = 18.08 at Re, = 180 and U/UT = 19.96 at Re, = 395; experiments 29 on

zero pressure-gradient boundary layers give C f = 3 x 10 -3 at Ree = 5, 000; the

spreading rate of a two-stream mixing layer based on 6 = y(0.95) — y(0.05) is 30

(1 + R)/2(1 — R)dSldx = 0.1 where R is the velocity ratio. These are points of

reference in the sense that they were targets in the selection of model constants;

they were met to varying degrees of accuracy.

A primary motivation for the present work is the current interest in turbulent

transport models for computation of complex aerodynamic flows. To test the performance of the present model for aerodynamic flows, a NA CA 4412 airfoil at 13.87°
Eddy Viscosity Models for Turbulent and Transitional Flows 31

angle of attack is simulated using the present model. This is the flow studied experimentally by Coles and Wadcock 31 . The experiment provides mean flow profiles

at x/c = 0.62, 0.675, 0.731, 0.786, 0.842, 0.897 and 0.953 on the upper surface of

the airfoil. Figure 10 shows these data along with the model calculations. x/c increases from left to right in the sequence of profiles. The model shows a premature

separation (experimentally it is near to 85% chord; the model has it at 71% chord).

However, the experimental separation bubble grows quickly and by x/c = 0.89 the

experiment and the model show the same vertical extent of the separation bubble.

The velocity profiles in figure 10 are in very much better agreement with data than

those obtained by Rogers et al.32 using the Baldwin-Lomax and Baldwin-Barth

models. The predicted pressure coefficient on the airfoil is shown in figure 11 along

with the experimental data.

In addition to this airfoil calculation, many flows of boundary layer type have

been calculated using the present model. The results of these calculations and the

details of the model development can be found in Durbin and Yang 33 , and Durbin,

Mansour, and Yang34.

2.6 Modeling of Bypass Transition

In a quiescent environment, transition is preceded by the amplification of linear

instability waves. These waves eventually break down, giving rise to turbulent

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 amplification of the linear instability waves. 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 transitional boundary layer region. Furthermore, the

performance and the life span of a turbine are directly related to the peak values of

the momentum and heat transfers both of which occur in the transitional boundary

layer region.

Priddin35 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 Scheuerer 36 , 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 Fujisawa37.

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 distinct

feature of a transitional boundary layer — intermittency. The intermittency of a

transitional boundary layer is measured by the intermittency factor which can be
32 Z. Yang

viewed as the percentage time a transitional boundary layer is in the turbulent state

due to the passing of turbulent spots.

We have proposed two eddy viscosity models for the calculation of transitional

boundary layers, which take the effect of intermittency into consideration. Both

models are based on a k — E model for near wall turbulence proposed by Yang

and Shihs . In the first model, the effect of the intermittency is introduced in the

eddy viscosity. In the second model, the effect of the intermittency is introduced

in all the terms which are generated due to the turbulence. The modifications to

the existing k — E appear in the introduction of the weighting factor in the eddy

viscosity (in the first model) or in all the turbulent terms (in the second model).

In order to close the above equations, an expression for the weighting factor is

needed. We assume the weighting factor is related to both the freestream 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. Further,

the shape factor is used 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. The details of the model

form can be found in Yang and Shih38 , and Yang and Shih39.

One of the issues in the calculation of transitional boundary layers using k — E

models is the prescription of the initial profiles for the turbulent kinetic energy and

its dissipation rate, the latter of which could not be found from the experiment

directly. An expression for the initial profiles were given in Rodi and Scheuererss

However, computations by Yang and Shih 40 which tested the effect of the initial conditions on the transition prediction found that the predicted onset of the transition

is sensitive to the initial profiles. A similar conclusion was also reached by Patankar

and Schmit41 . 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, the turbulent kinetic energy and its dissipation rate take constant profiles, the values of which are determined by the law for

the decaying turbulence.

Flat plate boundary layers with free stream turbulence levels of 3% (Case T3A)

and 6% (Case T313) respectively were calculated using the present model. These

are the benchmark cases in an ongoing project coordinated by Savill 42 , testing the

capability of turbulence models in predicting transitional flows. The variations of

skin friction coefficient c f against Re„ in these two cases are shown in figure 12 and

figure 13, respectively. Results from the experiment are 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 Launder-Sharma model performs best for transitional boundary layers. It is

clear that the present model gives a better prediction than the Launder-Sharma

model. Other features in the transitional boundary layers and the calculations of

the transitional boundary layers with other levels of freestream turbulence can be

found in Ref. 39

I
Eddy Viscosity Models for Turbulent and Transitional Flows 33

2.7 Linear Marginal Stability of a Trailing Line Vortex

The stability of the trailing line vortex (Batchelor vortex) has attracted many

researchers in the past thirty years. There are three reasons for such an interest.

First, the trailing line vortex flow is of great practical importance. Second, the

Batchelor vortex provides an archetype for swirling flows. Swirling flows are commonly found in nature and technology and their stability characteristics are of great

interest. Third, the Batchelor vortex is also used as a model of a columnar vortex

prior to its breakdown. Thus, it is hoped that the stability analysis of the Batchelor

vortex would shed some light on the phenomenon of vortex breakdown, which plays

an important role in a number of engineering flows.

The trailing line vortex is characterized by two parameters, the Reynolds number

Re and the swirling parameter q. In the linear stability analysis, the perturbations

are written in the normal mode form which is characterized by two wavenumbers,

the axial wavenumber k and the azimuthal wavenumber n. For a given pair of the

control parameters (Re, q), the stability property will also depend on wavenumbers

(k, n). We assume that in a natural environment, all perturbations are possible.

Thus, we will need to search over the (k, n) plane for a perturbation which gives the

maximum growth rate because such a perturbation dominates the flow development.

For a given trailing line vortex characterized by the pair of parameters (Re, q), if

the maximum growth flow rate is zero, the flow is said to be marginally stable.

Combinations of such (Re, q) pairs giving rise to the marginal stability form a curve

in the (Re, q) plane. This curve is called the marginal stability curve and this curve

separates the stable domain from the unstable domain in the control parameter

space. An extensive computation was carried out to generate such a curve. In the

present study, the marginal stability curve was found for Reynolds numbers up to

105 . The resulting curve is shown in figure 14.

One of the features of the marginal stability curve is that the swirling rate needed

to stabilize the flow increases with the Reynolds number and does not approach a

constant value even for Reynolds number as high as 105 . For Re = 105 , the value of

q needed to render the flow stable is larger than 6. In contrast, the asymptotic analysis by Leibovich and Stewartson43 for the inviscid perturbations of large azimuthal

wavenumbers shows that flow becomes stable for q around 1.4. For inviscid perturbations with finite values of azimuthal wavenumber, the asymptotic analysis by

Stewartson and Brown 44 show that the values of q for which the near neutral center

mode exists could be around 2.3. The recent numerical calculations by Mayer and

Powel145 show that for the inviscid perturbations, the n = —1 (in current notation)

mode has the largest unstable domain and the largest value of q is about 1.5.

The comparison between the current viscous study and the inviscid analysis by

previous researchers shows that the unstable domain of the flow (as measured by

range of q for which the flow is unstable) is larger for viscous perturbations at large

Reynolds numbers than that for the inviscid perturbations. For a value of q which is

larger than the stability bound for the inviscid perturbations, the flow is predicted

as being stable on the inviscid ground. However, the flow can be unstable to viscous

perturbations at large Reynolds numbers. As the Reynolds number is increased,
34 Z. Yang

the flow tends to be more unstable. Since the present calculations are carried out

for Reynolds number as high as 10 5 , it is expected from figure 14 that the present

viscous analysis will not approach to the inviscid results as the Reynolds number is

further increased.

For Re greater than 200, it is found that the modes giving rise to the marginal stability have an azimuthal wavenumber of n = — 1. Modes with azimuthal wavenumbers of Inj = 1 are very important in that they are the only modes that move the

fluid particles away from the axis of the vortex. It is for this reason that the Inj = 1

modes are termed `bending modes' by Leibovich, Brown, and Patel 46 . The deflection of fluid particles away from the vortex axis is one of the features of the vortex

breakdown of the spiral type, as was pointed out in Leibovich47 . Thus, our stability

analysis would provide some ingredients as found in vortex breakdown.

To see the effect of the Reynolds number on the eigenfunction, the real part of the

axial component of the eigenfunctions at Re = 104 and Re = 105 are shown in figure

15. The eigenfunctions are normalized such that the maximum value is 1. As the

Reynolds number is increased, the modes giving rise to the marginal stability moves

closer to the vortex axis. This suggests that these modes are similar to the `viscous

centers mode' found by Stewartson, Ng, and Brown 48 for the flow in a rotating

pipe. The term `viscous center mode' was used because viscosity plays an important

role for these modes even for large Reynolds number and because the modes have

nontrivial behaviors near the center of the vortex axis. The major difference between

trailing line vortex flow in the present study and flow in the rotating pipe flow is

that for the trailing line vortex flow, the present study suggests that viscous center

modes are the dominating modes near the marginal stability curve while in the

rotating pipe flow, the viscous center modes found by Stewartson, Ng, and Brown

are dominated by perturbations of the wall mode type studied by Maslowe and

Stewartson49 , and Yang and Leibovich50

More details of the linear marginal stability of the trailing line vortex could be

found in Yang and Leibovichsi

3. Future Plans

1) Turbulence modeling

The research on turbulence modeling will be carried on two fronts. The first

one is to test the eddy viscosity models we have developed for more complex flows.

The models will be eventually incorporated into an existing general purpose NavierStokes code if they pass the tests. The second direction will be the development of

Reynolds stress models, in particular for near wall flows. The eddy viscosity models

have some basic limitations which are expected to be overcome by the Reynolds

stress models.

2) Transition modeling

So far, the proposed transition models are only used for flat plate boundary layers

with zero pressure gradient. We will apply the proposed models to transitional

boundary layers with pressure gradient and curvature. We will also extend the

model to thermal boundary layers. In the theoretical aspect, we will try to derive
Eddy Viscosity Models for Turbulent and Transitional Flows 35

a set of equations from the conditional averaging point of view rather than the

Reynolds averaging point of view. This set of equations would at least serve as

a point of reference when we modify the turbulence equations for the transitional

flows.

3) Stability analysis of swirling flows

Stability analysis will be carried out for swirling flows observed in propulsion

systems. Turbulence model calculations for these swirling flows are inadequate. We

hope that the stability analysis would provide some insight on how to construct a

better turbulence model. In addition, we will be studying the mixing properties of

such flows.

4. References

' Launder, B.E. and Spalding, D.B., "The numerical computation of turbulent

flow," Computer Methods in Applied Mechanics and Engineering 3, 269 (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," Intl. J. of Heat and Mass

Transfer 16, 1119 (1973).

4 Patel, V.C., Rodi, W., and Scheuerer, G., "Turbulence models for near-wall and

low-Reynolds number flows: a review," AIAA J. 23, 1308 (1985).

5 Lang, N.J. and Shih, T.H., "A critical comparison of two equation turbulence

models," NASA TM 105237 (1991).

6 Yang, Z. and Shih, T.H., "A new time scale based k — E model for near wall

turbulence," AIAA J. 31, 1191 (1993).

7 Kim, J., Moin, P., and Moser, R., "Turbulent statistics in fully developed channel

flow at low Reynolds Number," J. Fluid Mech. 177, 133 (1987).

8 Wilcox, D.C., "Application of low Reynolds number two-equation turbulence

models to high Reynolds number flows," in Near Wall Turbulent Flows, Elsevier, 1993.

9 Yang, Z. and Shih, T.H., "A Galilean-and tensorial invariant k — E model for near

wall turbulence," AIAA Paper 93-3105 (1993).

io Spalart, P.R., "Direct simulation of a turbulent boundary layer up to Ree =

1410," J. Fluid Mech. 187, 61 (1988).

ii Coles, D. E., "The young person's guide to data," in Computation of Turbulent

Boundary Layers, V. 11, AFOSR-IFP-Stanford Conference, 1968.

12 Wilcox, D.C., "Reassessment of the scale-determining equation for advanced turbulence models," AIAA J. 26, 1299 (1988).

13 Wieghardt, K. and Willmann, W., "On the turbulent friction layer for rising

pressure," NACA TM-1314 (1951).

14 Herring, H.J., and Norbury, J.F., "Some experiments on equilibrium turbulent

boundary layers in favorable pressure gradients," J. Fluid Mech. 27, 541 (1967).
36 Z. Yang

15 Samuel, A.E., and Joubert, P.N., "A boundary layer developing in an increasingly

adverse pressure gradient," J. Fluid Mech. 66, 481 (1974).

16 Yang, Z. and Shih, T.H., "On the wall functions for two equation turbulence

models," prepared for publication.

17 Shih, T. H., Lion, W. W., Shabbir, A., Yang, Z., and Zhu, J., "Dynamics of vorticity fluctuations and the dissipation rate equation," NASA TM 105993 (1993).

18 Lumley, J. L., "Toward a turbulent constitutive relation," J. Fluid Mech., 41,

413 (1970).

19 Pope, S. B., "A more general effective-viscosity hypothesis," J. Fluid Mech., 72,

331 (1975).

20 Shih T. H. and Lumley, J. L., "Remarks on turbulent constitutive relations,"

NASA TM 106116 (1993).

21 Yang, Z. and Shih, T. H., "Flow inhomogeneity and the dissipation rate equation," In preparation.

22 Baldwin, B. S. and and Lomax, H., "Thin layer approximation and algebraic

model for separated turbulent flows," AIAA Paper 78-256 (1978).

23 Cebeci, T. and Smith, A. M. O., Analysis of Turbulent Boundary Layers, Academic Press, 1974.

24 Nee, V. W. and Kovasznay, L. S. G., "The calculation of incompressible turbulent

boundary layers by a simple theory," in Computation of Turbulent Boundary

Layers, V. I, AFOSR-IFP-Stanford Conference, 1968.

25 Baldwin, B. S. and Barth, T. J., "A one-equation turbulence transport model for

high Reynolds number wall-bounded flows," NASA TM 102847 (1990).

26 Spalart, P. R. and Allmaras, S. R., "A one-equation turbulence model for aerodynamic flows," AIAA-92-0439 (1992).

27 Durbin, P., "Near-wall turbulence closure modeling with `damping function',"

Theo. and Comp. Fluid Dynamics, 3, 1 (1991).

28 Kim, J., Private communication.

29 Purtell, L. P., Klebanoff, S., and Buckley, F. T., "Turbulent boundary layer at

low Reynolds number," Phys. Fluids 24, 802 (1981).

30 Mehta, R. D. and Westphal, R. V., "Near-field turbulence properties of single

and two-stream plane mixing layers," Experiments in Fluids 4, 257 (1986).

31 Coles, D. E. and Wadcock, A. J., "Flying hot wire study of flow past an NACA

4412 airfoil at maximum life," AIAA J., 17, 321 (1979).

32 Rogers, S. E., Wiltberger, N. L., and Kwak, D., "Efficient simulation of incompressible viscous flow over single and multi-element airfoils," AIAA Paper 92-0405

(1992).

33 Durbin, P. A. and Yang, Z., "A transport equation for eddy viscosity," in Proc.

of the 1992 Summer Program, Center for Turbulence Research, Stanford (1992).

34 Durbin, P. A., Mansour, N. N., and Yang, Z., "Eddy viscosity transport model

for turbulent flow," Submitted for publication.

35 Priddin, C.H., The Behaviour of the Turbulent Boundary Layer on a Curved

Porous Wall, PhD Thesis, Imperial College, London, 1975.
Eddy Viscosity Models for Turbulent and Transitional Flows 37

36 Rodi, W. and Scheuerer, S., "Calculation of laminar-turbulence boundary layer

transition on turbine blades," in Proc. 65th AGARD-REP Symposium on Heat

Transfer and Cooling in Gas Turbine, Bergen, Norway, 1985.

37 Fujisawa, N., "Calculation of transitional boundary layers with a refined low

Reynolds number version of a k — e model of turbulence," in Engineering Turbulence Modeling and Experiment, Elsevier, 1990.

38 Yang, Z. and Shih, T.H., "A k — E model for turbulent and transitional boundary

layers," in Near Wall Turbulent Flows, Elsevier, 1993.

39 Yang, Z. and Shih, T.H., "Modeling of transitional boundary layers," prepared

for publication.

40 Yang, Z. and Shih, T.H., "A k — c calculation of transitional boundary layers,"

in Transition and Turbulence, Sp ringer- Verlag, 1992.

41 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).

42 Savill, A.M., "Turbulence model predictions for transition under free-stream turbulence," in RAeS Transition and Boundary Layer Conference, Cambridge, England, 1991.

43 Leibovich, S. and Stewartson, K., "A sufficient condition for the instability of

columnar vortices," J. Fluid Mech., 126, 335 (1983).

44 Stewartson, K. and Brown, S. N., "Near-neutral center-modes as inviscid perturbations to a trailing line vortex," J. Fluid Mech., 156, 387 (1985).

45 Mayer, E. W. and Powell, K. G., "Viscous and inviscid instabilities of a trailing

vortex," J. Fluid Mech., 245, 91 (1992).

46 Leibovich, S., Brown, S. N., and Patel, Y., "Bending waves on inviscid columnar

vortices," J. Fluid Mech., 173, 595 (1986).

47 Leibovich, S., "Vortex stability and breakdown: Survey and extension," AIAA

J., 22, 1192 (1984).

48 Stewartson, K., Ng, T. W., and Brown, S. N., "Viscous center modes in the

stability of swirling Poiseuille flow," Phil. Trans. Roy. A., 324, 473 (1988).

49 Maslowe, S. A. and Stewartson, K., "On the linear inviscid stability of rotating

Poiseuille flow," Phys. Fluids, 25, 1517 (1982).

51 Yang, Z. and Leibovich S., "Unstable viscous wall modes in rotating pipe flow,"

AIAA Paper 91-1801 (1991).

51 Yang, Z. and Leibovich S., "On the linear marginal stability of the trailing tine

vortex," Submitted for publication.
38 Z. Yang

60.0

R

40.0

20.0

0.0 L0.0 20.0 40.0 60.0 80.0 100.0

y+

Figure 1: Variation of R for turbulent channel flow at Re, = 180.

0.0060

0

0.0050

o EXP

K-Omega model inlegroted to the wall

- - - K-Omega model with the stondord wall Function

0

u 0.0040

0

0

0.00.30

0

0

' o

0.0020

0.0 1.0 2.0 3.0 4.0 5.0

X

Figure 2: k — w prediction of the skin friction distribution for ZPG

turbulent boundary layer.
Eddy Viscosity Models for Turbulent and Transitional Flows 39

0.0060

0.0050 o EXP

K — Omego model integrated to the wall

- - - K—Omega model with the standard woll Function

-- - K— Omega model with the new woll function

Z5 0.0040

0.0030

0.0020 '

0.0 1.0 2.0 3.0 4.0 5.0

X

Figure 3: k —w prediction of the skin friction distribution for FP 

turbulent boundary layer.

0.0040

0.0030

0

0

0

0`

0

6 0.0020

0

o

o EXP

0.0010 — K—Omego model integrated to the woll

- - - K—Omega model with the standard woll Function -

- — - K—Omega model with the new wall function

00000 1

0.0 1.0 2.0 3.0 4.0

X

Figure 4: k—w prediction of the skin friction distribution for AP 

turbulent boundary layer.
40 Z. Yang

0.006

0 EXP

St. ke (with new WF)

0.005 — — - St. ke (with Clossicol WF)

u 0.004

0.003

o------ 0-- --d

0.002

0.0 1.0 2.0 3.0 4.0 5.0

x

Figure 5: Prediction of the skin friction distribution for FPG turbulent boundary layer by standard k—E model with wall functions.

0.0040

0.0030

6 00020

0.0010

o EXP

St. ke (new WF)

— — • St. ke (classical WF)

0 \\4 ^\

0\^^

0

0

0

0 \

0 \

0 \\

0 \

0 \

0.0000

0.0 1.0 2.0 3.0 4.0

X

Figure 6: Prediction of the skin friction distribution for APG turbulent boundary layer by standard k — E model with wall functions.
Eddy Viscosity Models for Turbulent and Transitional Flows 41

0.0060

0

0.0050

0

Z3 0.0040

0

0.0030

0.0020

0.0

o EXP

- Present Model

1.0 2.0 3.0 4.0 5.0 6.0

x

0

0

00

0

0

0 0

0 0 0 0

0

Figure 7: Prediction of the skin friction distribution for ZPG turbulent boundary layer by the new dissipation rate equation.
6 0.0020

0.0010

42 Z. Yang

0.0060

0.0050 o EXP

Present Model

tj 0.0040

0

0 0 0

0.0030

00020

0.0 1.0 2.0 3.0 4.0 5.0

x

Figure 8: Prediction of the skin friction distribution for FPG turbulent boundary layer by the new dissipation rate equation.

0.0040

o EXP

0.0030 Present Model

0.0000 '

0.0 1.0 2.0 3.0 4.0

X

Figure 9: Prediction of the skin friction distribution for APG turbulent boundary layer by the new dissipation rate equation.
caC)

Eddy Viscosity Models for Turbulent and Transitional Flows 43

0.10

0.08

0.06

0.04

0.02

0

t

0 0 0 0 0 0 0 1

U/U^

Figure 10: Mean velocity profiles on the suction surface.

U 0.25 0.50 0.75 1.00

x/c

Figure 11: Pressure distribution around the airfoil surface.
44 Z. Yang

0.0080

- - - Lominor

— Turbulent

0 ExP

Launder-Shormo

0.0060 — — - Model I

— Model II

u 0.0040

0.0020

0.0000 L0.0

0 - 0

I , • - _/// _~!! 

/ 0

/0

-0

200000.0 400000.0 600000.0

Re-X

Figure 12: Skin friction coefficient for T3A.

0.0080

0.0060

u 0.0040

0.0020

0

l

Laminar

— Turbulent

0 UP

. . . Launder-Shormo

— — - Model I

Model II

0.0000 L0.0 200000.0 400000.0 600000.0

Re-X

Figure 13: Skin friction coefficient for T3B.
Eddy Viscosity Models for Turbulent and Transitional Flows 45

8.0

6.0

q

4.0

2.0

0.0

0.0 20000.0 40000.0 60000.0 80000.0 100000.0

Re

Figure 14: The marginal stability curve for the trailing line vortex.

1.5

--- Re=104

Re = 105

1.0

u(r)

0.5

0.0

—0.5

\

—1.0 L

0.0 0.5 1.0

r

Figure 15: The eigenfunctions at different Reynolds numbers.
Page intentionally left blank 
Center for Modeling of Turbulence and Transition 47

Research Briefs - 1993

Testing New Two-Equation

Turbulence Models for Complex Flows

J. Zhu

1. Motivation and Objective

In the past year, several new proposals for the two-equation turbulence modeling

have been made at CMOTT. These proposals include: a functional expression of

CN,; a vorticity dynamics based model for the turbulent dissipation equation; and

a realizable Reynolds stress algebraic equation model. The objective of the present

work is to assess these proposals in complex flows of engineering importance.

2. Work Accomplished

All the proposals tested have been implemented into a conservative finite-volume

code designed for calculating incompressible elliptic flows with complex boundaries ) .

In order for the calculation to reflect the actual performance of the turbulence

models, it is crucial that the numerical diffusion be reduced to the minimum. This

is achieved via the use of second-order accurate differencing schemes and sufficiently

fine grids. Test cases include two backward-facing step flows z 3 , confined jets in a

cylindrical duct' and in a conical duct with a 5° divergence 5,6 . These test cases

have been chosen because they involve complicated flow features such as strong

adverse pressure gradient, separation and reattachment, and also because reliable

and well-documented experimental data are available, allowing the performance of

the models to be examined in detail. For comparison, the standard k — E model7

is considered in all the cases and an RNG-based k — E model' is also considered in

some of the cases.

2.1 A New Formulation of C,,

In the standard k — E model, the eddy viscosity is given by

z

vt = Cl, kCl, = 0.09

E

where the value of Cμ = 0.09 was chosen from a set of experiments for simple flows

under the equilibrium condition. Several authors have found that using a nonconstant CN, may significantly improve the k — c model's ability to predict certain

flows in which the equilibrium condition is not satisfied. On the basis of correlated

experimental data for thin shear layers, Rodi 9 related Cμ to a function of the average value of Plc (P is the production of the turbulent kinetic energy). Ljuboja

and Rodi 10 derived a CN, formula from the simplified algebraic stress model and

successfully used it to predict wall jets. Leschziller and Rodi" proposed a function

for C,y which takes into account the effect of streamline curvature and obtained improved results in the calculation of annular and twin parallel jets. However, these

modifications to C,, are either of fully empirical or of preliminary nature.

(1)
48 J. Zhu

In this work 12 , the realizability principle which requires the non-negativity of

turbulent normal stresses is applied to analyze the k — c model. The analysis leads

to

C 2/3 (2) ^` A+77

where A is a positive constant and rl is the time scale ratio of the turbulence to the

mean strain rate

11 = SKI E , S = ( 2SijSi7) 112, Sij = (Ui , j + U7,i )/2

(3)

This C,, will ensure the positivity of each component of the turbulent kinetic energy

— realizability that most existing eddy-viscosity models do not satisfy. The model

validation is made on the basis of applications to the two backward-facing step

flows experimentally studied by Driver and Seegmiller 2 (DS) and Kim, Kline and

Johnston3 (KKJ). The value of the model constant A is taken as

A = 5.5 (4)

which has been found to work well for both the test cases. The present model has

been compared with the standard k — c model and with the RNG k — E model $ . The

comparison shows that the present model effectively reduces the turbulent eddyviscosity level, resulting in significant improvement over the standard K-E model.

The RNG model generally gives very similar predictions to the present model, but

overly reduces the turbulent eddy-viscosity level in the recirculation region near the

step.

2.2 A New Dissipation Equation

Although one can write the exact c equation, it is of little use to serve as a

starting point for deriving its model counterpart because of lack of knowledge about

the various correlations in this equation. Therefore, one often ignores the exact

equation and creates a model equation which has a structure similar to that of the

turbulent kinetic energy equation. Due to its highly empirical nature, such a model

equation is often considered as a weakness in existing turbulence models.

Table 1. Comparison of the reattachment point locations

Case measurement standard k — c model present model

DS 6.1 4.99 5.36

KKJ 7f 0.5 6.35 6.8

Recently, Shih et al. 13 have proposed a new c equation based on the vorticity

fluctuation equation. The proposed c equation can be applied to any level of turbulence modeling. In this work, it is applied, in conjunction with a realizable isotropic

eddy-viscosity formulation, to calculate the two backward-facing step flows (DS

case and KKJ case 3 ). Table 1 shows the comparison of the reattachment points.

The detailed comparisons for both mean and turbulent quantities are given in Shih

et al. 13 . The comparisons show that the overall performance of the present model

is better than that of the standard k — E model.
Testing New Two-Equation Turbulence Models for Complex Flows 49

2.3 A Realizable Reynolds Stress Algebraic Equation (RRSAE) Model

The standard k — c model, like many others in the algebraic equation modeling

group, uses the Boussinesq's isotropic eddy-viscosity concept which assumes that

the Reynolds stresses are proportional to the mean velocity gradients. The concept usually does well for the shear stresses in two-dimensional mean flows of the

boundary-layer type, but not for the normal stresses due to the erroneous isotropic

nature of the concept. This suggests that linear dependence on the mean velocity

gradients is insufficient and that a more general relation is needed for more complex flows. In fact, by eliminating the convection and diffusion terms in the modeled

transport equations for the Reynolds stresses, Rodi 9 developed an algebraic stress

model (ASM) in which the Reynolds stresses are calculated by algebraic expressions. Owing to its anisotropic nature, the model does perform better than the

isotropic k — c model for certain flows; a well known example is fully-developed

flow in non-circular ducts where ASM is capable of generating turbulence-driven

secondary motions while the isotropic eddy viscosity model is not. However, ASM

does not appear in a tensorial invariant form, which may limit its generality. In addition, inappropriate modeling of higher order correlations, such as pressure-strain

correlations, will also cause deficiencies of the second-order closure based ASM.

Moreover, special care needs to be taken to prevent the turbulent normal stresses

from becoming negative (Huang and Leschziner 14 ), and the numerical implementation of ASM may even be more complicated than that of its parent second-order

closure model, especially in general three dimensional flows.

There are other approaches to developing Reynolds stress algebraic equation models. For example, Yoshizawa l ' derived a relation for the turbulent stresses using a

two-scale direct interaction approximation. It contains both linear and quadratic

terms of the mean velocity gradients. A similar relation has been also derived recently by Rubinstein and Barton 16 using Yokhot and Orszag's RNG method. An

interesting point in these two methods is that the values of the model coefficients

can all be determined analytically. Speziale 17 proposed a different expression, based

on the principle of material frame-indifference, which contains the Oldroyd derivative of the mean strain rates. However, the principle of material frame-indifference

is valid only in the limit of two-dimensional incompressible turbulence, and hence

it is not an appropriate constraint for general turbulent flows. In addition, these

non-linear models are not fully realizable and have not been extensively tested.

Recently, Shih and Lumley 18 have shown that the Reynolds stress, being a second

rank tensor, can be expressed as a fourth-order polynomial of the mean velocity gradients. This is the most general stress-strain relationship within the framework of

algebraic turbulence modeling, with the linear stress-strain relation in the Boussinesq's eddy-viscosity concept being its first-order approximation. Based on this,

Shih et al. 19 have recently proposed a quadratic stress-strain relation in conjunction with the two modeled equations of k and E for practical engineering calculations. Realizability constraints have been used to derive appropriate functional

expressions for the model coefficients so that the resulting model will ensure the

positivity of individual turbulent normal stresses, an important feature that is not
50 J. Zhu

present in most existing turbulence models. Since the model appears promising as

a competitive alternative in the turbulence modeling arsenal, it is of interest to test

its performance under various complex flow situations. A few representative test

results will be presented in the following and the detailed results can be found in

the relevant reports.

2.3.1 Backward-facing step flows (Shih et al.ls)

The empirical constants in the model have been fine-tuned in the calculation of

the two backward-facing step flows (DS case and KKJ case'). The computed and

measured reattachment points are compared in Table 2. The reattachment point

is a critical parameter which has often been used to assess the overall performance

of turbulence models as well as numerical procedures. Figs. 1(a) and 1(b) show

the comparison of computed and measured static pressure coefficient Cp along the

bottom wall. In both cases, the standard k — c is seen to predict premature pressure rises, which is consistent with its underprediction of the reattachment lengths,

while the RRSAE model captures these pressure rises quite well. The comparisons

of predicted and measured turbulent stresses uu, vv and uv are shown in Figs. 2 and

3 at various x-locations. In the KKJ-case, no experimental data for the turbulent

stresses are available in the recirculation region, and the reattachment point was

found in the experiment to move forward and backward continuously around seven

step heights downstream of the step, leaving an uncertainty of f0.5 step height

for the reattachment length. This also points to some uncertainty in the measured

turbulent quantities in the recovery region. On the other hand, the experimental

data in the DS-case should be considered more reliable because of the smaller uncertainty of the reattachment location, indicating a smaller unsteadiness of the flow.

As compared with the standard k — E model results in Figs. 2 and 3, it can be seen

that the anisotropic terms increase uu while decreasing vv, leading to significant

improvements in both uu and vv results. On the other hand, the anisotropic terms

have little impact on the turbulent shear stress The improvement obtained by

the RRSAE model in Fig. 2 for uv is due to the reduction in Cv,.

Table 2. Comparison of the reattachment point locations

Case measurement standard k — E model RRSAE model

DS 6.1 4.99 5.82

KKJ 7f 0.5 6.35 7.35

2.3.2 Confined jets in a conical duct (Zhu and Shih21)

The general features of the flow considered are sketched in Fig. 4. At the entrance,

two uniform flows, a jet of high velocity and an ambient stream of low velocity, are

discharged into the duct. Due to turbulent entrainment, the jet increases its mass

flux while spreading and this causes an equal decrease in the mass flux of the ambient

stream. An adverse pressure gradient is thus set up by the decrease in the velocity of

the ambient stream which can be considered as a potential flow. When the ratio of
Testing New Two-Equation Turbulence Models for Complex Flows 51

jet to ambient velocities at the entrance is above a critical value, a recirculation zone

occurs at the duct wall downstream of the inlet plane. This is because the jet has

consumed the whole ambient flow before reaching the wall and further entrainment

must create reverse flows in order to maintain the total mass flux conservation. The

inlet flow conditions can be characterized by the Craya-Curtet number C t and the

experiment of Binder and Kian 5,6 shows that recirculation occurs when Ct <1.1 in

the conical duct with a 5° divergence. For a given geometry, recirculation as well

as adverse pressure gradients can be intensified by reducing the value of Ct at the

entrance.

The separation and reattachment points are given in Table 3. For the reattachment point, only a range of 3.4 — 3.8D o was given experimentally due to the

high unsteadiness of the recirculating bubbles. The experiment revealed that as

Ct decreased, the separation points moved upstream while the reattchment points

remained basically unchanged. The RRSAE model captures this feature well and

predicts the locations of the recirculation bubbles better than the standard k — E

model.

Table 3. Separation and reattachment points (x9 /Do , x,./Do)

Ct experiment k — E model RRSAE model

0.775 2.5, 3.43.8 1.82, 3.17 2.15, 3.79

0.59 1.5, 3.43.8 1.43, 3.22 1.45, 3.81

Fig. 5 shows the variation of pressure coefficient along the duct wall. The pressure gradient is governed by the jet entrainment, the contraction and expansion of

the flow caused by recirculating eddies as well as the geometry of the duct. The

entrainment and the divergence of the duct can only produce a maximum pressure

difference equal to pUQ /2, while the pressure difference created by the divergence of

streamlines in the downstream part of the recirculating bubble can be much larger

than pUa /2. Regarding the comparison between predictions and experiments, it

can be seen that although both models predict practically the same total pressure

rise, the RRSAE model captures the location where the pressure starts to shoot up

much better than the standard k — E model at all the Ct values.

Detailed experimental data for the turbulent stresses uu, vv and uv are available

only at Ct = 0.59. The computed and measured radial profiles of these quantities

at four downstream locations are compared in Figs. 6(a)-6(c). With regard to

the turbulent normal stresses, the experimental data are basically followed by the

results of both the RRSAE model and the standard k — c model, with the former

predicting more anisotropy than the latter. The experimental data at x/Do = 2.5

are seen to exhibit a different trend for both vu and vv, which may possibly be due

to measurement errors. The flow visualization in the experiment indicated that the

global flow pattern was highly unsteady in the presence of recirculation. With due

regard to flow complexities and measurement difficulties, the agreement between

the predictions and measurements seen in Figs. 6(a) and 6(b) should be considered
5 2 J. Zhu

as reasonably good, but it is difficult to judge which model performs better for

the turbulent normal stresses, overall. For the turbulent shear stress uv shown in

Fig. 6(c), the results obtained with the RRSAE model are clearly better than those

with the k — E model for all the locations considered. The large discrepancy seen

at x/Do = 2.5 is partially due to the underprediction of the width of the backflow

region and partially due to the experimental uncertainty, as evidenced by the fact

that in the experimental data, the change in sign of the shear stress profile occurs

much further away from the duct wall than the velocity minimum.

2.3.3 Confined jets in a cylinder (Zhu and Shih21)

This case is taken from the experiment of Barchilon and Curtet 4 . The general

flow features are similar to those in the preceding case. Recirculation occurs when

Ct <0.96. In this case, the calculation with the RNG k — E model' is also performed

for comparison.

Figs. 7(a)-7(c) show the axial mean velocity profiles at three Ct numbers. All

the three models are seen to predict very well the upstream evolution of the flow.

As for the downstream development, the results obtained with the RRSAE model

remained in good agreement with experiments, while those obtained with the other

two models deteriorated with the RNG model producing the largest discrepencies.

The separation and reattachment points of the predicted recirculating bubbles are

compared with the experimental data in Fig. 8. The experiment indicated that as

Ct decreased, the separation point moved upstream while the reattachment point

remained practically unchanged. The comparison shows that the RRSAE model

gives the best predictions for both the separation and reattachment points.

Fig. 9 shows the variation of the recirculating flow rate with x at C t =0.305 and

0.152. This is the integral of negative velocities at each cross-section. The experiment indicated that the recirculating flow rate at C t =0.152 is about 3 times larger

than that at Ct =0.305. The results of the RRSAE model are in good agreement

with the experiment while those of the standard k — E model and the RNG model

show substantial deviations from experiment. As for the maximum recirculating

flow rate which is a critical parameter to characterize the performance of combustion chambers, the RRSAE model gives the same result as the experimental data

at Ct =0.305 and a 9% overprediction at Ct =0.152 while the other two models produce larger overpredictions. It should be pointed out that results from different

measurements' for this quantity showed considerable scatter at small Ct numbers.

The results of all the three models are within the experimental scatter.

The variation of the pressure coefficient Cp along the duct wall is shown in Fig.

10. In the cylindrical duct, the evolution of the pressure is governed by the jet

entrainment as well as the contraction and expansion of the flow caused by the

recirculating bubble. The decrease in the ambient velocity induced by the entrainment gives rise to an adverse pressure gradient, while the contraction of streamlines

produces the opposite effect. These two mechanisms interact more intensely with

each other as Ct decreases, causing the pressure to vary little in the region upstream

of the center of the recirculating bubble. In the downstream part of the recirculating bubble, the deceleration of the flow sets up an adverse pressure gradient the
Testing New Two-Equation Turbulence Models for Complex Flows 53

slope of which becomes steeper as Ct decreases. Therefore, the ability to capture

the location of the recirculation center will have a direct impact on the prediction

of the pressure. The three models capture the steep pressure gradients in the same

way as they capture the ambient velocity minimums. However, for the total pressure rise, an important parameter to the designer of jet pump devices, all the three

models are seen to give the same results which are in excellent agreement with the

measurements.

2.3.4 Concluding remarks

The extensive comparisons with the experiments clearly show the superiority of

the RRSAE model over the standard k — E model in all the test cases considered,

and the improvement is achieved at an insignificant penalty to the computational

efficiency and algorithmic simplicity of the latter.

3. Future Plans

1) Further assessment of these new proposals in flows with swirl, in diffusers, in

U- and S-shaped ducts, and in a constricted tube;

2) Extension of the present code for accommodating second-order closure models.

4. References

1 Zhu, J., "FAST-21): A computer program for numerical simulation of two dimensional incompressible flows with complex boundaries," Rept. No.690, Institute

for Hydromechanics, University of Karlsruhe, (1991).

a Driver, D.M. and Seegmiller, H.L., "Features of a reattaching turbulent shear

layer in divergent channel flow," AIAA Journal 23, 63-171 (1985).

3 Kim, J., Kline, S.J. and Johnston, J.P., "Investigation of separation and reattachment of a turbulent shear layer: Flow over a backward-facing step," Rept.

MD-37, Thermosciences Div., Dept. of Mech. Eng., Stanford University (1978).

4 Barchilon, M. and Curtet, R., "Some details of the structure of an axisymmetric

confined jet with backflow," Journal of Basic Engineering 86, 777-787 (1964).

5 Binder, G. and Kian, K., "Confined jets in a diverging duct," Proceedings, Turbulent Shear Flows 4, Karlsruhe, 7.18-7.23 (1983).

6 Kian, K., "Jets confines dans un divergent," Doctoral dissertation, l'Universite

Scientifique et Medicale de Grenoble et 1'Institut National Polytechnique de Grenoble, France (1981).

7 Launder, B.E. and Spalding, D.B., "The numerical computation of turbulent

flow," Computer Methods in Applied Mechanics and Engineering 3, 269-289

(1974).

8 Speziale, C.G. and Thangam, S., "Analysis of an RNG based turbulence model

for separated flows," NASA CR-189600 (1992).

9 Rodi, W., "Turbulence models and their application in hydraulics - A state of

the art review," Book Publication of the International Association for Hydraulic

Research, Delft, the Netherlands (1980).
54 J. Zhu

to Ljuboja, M. and Rodi, W., "Calculation of turbulent wall jets with an algebraic

Reynolds stress model," Journal of Fluids Engineering 102, 350-356 (1980).

11 Leschziner, M.A. and Rodi, W., "Calculation of annular and twin parallel jets

using various discretization schemes and turbulence model variations," Journal

of Fluids Engineering 103, 352-360 (1981).

12 Zhu, J. and Shih, T.-H., "Calculations of turbulent separated flows," NASA TM

106154 (1993).

13 Shih, T.-H., Liou, W.W., Shabbir, A., Yang, Z. and Zhu, J., "A vorticity dynamics

based model for the turbulent dissipation," NASA TM 106177 (1993).

14 Huang, P.G. and Leschziner, M.A., "Stabilization of recirculating-flow computations performed with second moment closures and third order discretization,"

Proceedings of the 5th Symposium on Turbulent Shear Flows, Cornell University,

5.19-5.24 (1985).

15 Yoshizawa, A., "Statistical analysis of the deriation of the Reynolds stress from

its eddy-viscosity representation," Physics of Fluids 27, 1377-1387 (1984).

16 Rubinstein, R. and Barton, J.M., "Nonlinear Reynolds stress models and the

renormalization group," Physics of Fluids A 2, 1472-1476 (1990).

11 Speziale, C.G., "On nonlinear k — l and k — E models of turbulence," Journal of

Fluid Mechanics 178, 459-475 (1987).

18 Shih, T.-H. and Lumley, J.L., "Remarks on turbulent constitutive relations,"

NASA TM 106116 (1993).

19 Shih, T.-H., Zhu, J. and Lumley, J.L., "A realizable Reynolds stress algebraic

equation model," NASA TM 105993 (1993).

20 Zhu, J. and Shih, T.-H., "A numerical study of confined turbulent jets," NASA

TM 106197 (1993).

21 Zhu, J. and Shih, T.-H., "Computation of confined coflow jets with three turbulence models," AIAA Paper 93-3120.
0.1

a

v

0.0

—0.1

0

I

(a) DS–case

/ 1

10 20 30 40

X

Testing New Two-Equation Turbulence Models for Complex Flows 55

0.2

0.4

0.3

a 0.2

U

0.1

0.0

-0 1

V^I - (b) KKJ–case

0 10 20 30 40

X

Figure 1. Static pressure coefficient Cp along the bottom wall.

- —: Standard K-E; : RRSAE model; •: experiment
X=20

I

2

T

1

0

3

• \I ••I

2

T

1

0

3

56 J. Zhu

3

X=8

X=2

0 1 2 0 1 2– 0.5 0.0 0.5 1.0

100uu 100vv –100uv

Figure 2. Turbulent stress profiles in DS-case.

— - —: Standard K-E; : RRSAE model; •: experiment

2

T

1

0
1

0

Testing New Two-Equation Turbulence Models for Complex Flows 57

3

2 FTT^

T

1 ^

•

0

3

2

X=7.660

•

•

X=2.667

0 1 2 3 4 0 1 2 3 4– 0.5 0.0 0.5 1.0 1.5

1 00U 100vv –10Ouv

Figure 3. Turbulent stress profiles in KKJ-case.

— - —: Standard K-E; : RRSAE model; •: experiment
58 J. Zhu

potential core boundary layer recirculation region

,,,,,

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

ambient region _ _ R

r2 - -

r mixin g region

0 0 - 1

Q ^

Ui x

Ua L B

Ux Ui

Figure 4. Flow configuration and notations
5

0

-5

-10

Ct=1 .090

10

a

0

0 5

0

0

0

5

10

5

0

—5

0 1 2 3 4 5 6 7 8

x/Do

Testing New Two-Equation Turbulence Models for Complex Flows 59

10

Figure 5. Pressure coefficient along the duct wall in the BK-case

(Notation as in figure 1)
0.0

0 5 0 5

2

1OOvv/Uo

0.5

1.0

x/Do=0.625 =1.25 =1.875 \ =2.5

(b) ^, ' \ •

• \ '

5 0 0 5

60 J. Zhu

1.0

Of 

0.5

0.0

0 5 0 5 0 5 0 5

2

1OOuu/Uo

x/Do=0.625 =1 .25 --1.875 =2.5 2.

(a) •

1.0

0.5

0.0

x/Do=0.625 =1 .25 =1.875 \ , =2.5

(C \ \'

0 10 20 0 10 20 0 10 20 0 10 20

1OOOuv/20

Figure 6. Turbulent stress profiles at Ct = 0.59 in the BK-case

(Notation as in figure 1)
(c) Ct=0.152 -

L

0.5

0.0 i ra i V i r1 i-I i

0 40 80 —5 20 45 —5

U/Um

11^

I'

11^ .IJ

10 25 —5 10 25

Testing New Two-Equation Turbulence Models for Complex Flows 61

1.0

x Do=0.375 1.125 1.875 2.5

(a) Ct=0.976

0.5

0.0 1,.-1 ,i ,7i,II i ,Nn ti, i i

0 5 10 15 0 3 6 9 0

1.0

-, 0.5

o.o

0 10 20 0 5 10 15-5 0 5 10-5 0 5 10

1.0

3 6 0 3 6

LCt=0.506 f

•

Figure 7 Axial mean velocity profiles. •: Experiment;

- —: Standard K-e; — —: RNG K-e; : RRSAE.
10-0.0

a2.0

1.0

00

Ct=0.152

0 1 2

x/Do

3 4

10

5

0

—5

10

5

0

—5

10

a

0 5

0

0

0

—5

10

5

0

—5

10

5

0

—5

l ,

0.8

0.6

0.4

0.2

00

0 1 2 3 4

x/Do

Figure 8 Separation and reattachment points

(Notation as in Figure 7)

0.8

Ct=0.305

0.4

62 J. Zhu

Ct=0.976

• Ct=0.714

• Ct=0.506

• Ct=0.305

r

• Ct=0.152

0 1 2 3 4 5 6

x/Do

Figure 10 Pressure coefficient at wall

(Notation as in Figure 7)

Figure 9 Recirculating flow rate

(Notation as in Figure 7)
Center for Modeling of Turbulence and Transition 63

Research Briefs - 1991

Low Re, k - E Models and the

Backward-Facing Step Flow

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 modelling 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 involves a study of several

two-equation turbulence models for separated flow over a backward-facing step.

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. This is particularly relevant to the application of the near-wall modelling for

separated flows. It is well known that the Chien k — E model does not accurately

resolve skin friction measurements near the reattachment point of a separated shear

layer. This is attributed to the behavior of the eddy viscosity damping function.

The question remains, however, for other LR models which do not suffer from the

same singularity in the damping function. The intention of this study is to investigate several low turbulent Reynolds number (LR) models and compare against the

performance of the standard (HR) formulation.

2. Work Accomplished

The study of four LR k —E models and the standard HR form has been completed.

This work is documented in reference'. The models investigated include the Chien2,

Jones-Launder 3 , Launder-Sharma', and Shih-Lumley 5 as well as standard form with

a two layer wall function. The experimental reference of Driver and Seegmiller 6 was

chosen because of the extensive data including LDV velocity and turbulence profiles

throughout the tunnel as well as pressure and skin friction measurements along the

tunnel wall. Below I will briefly discuss the numerical methods involved in this

analysis. Next I will present a summary of the data and include a figure depicting

the most intriguing result.

2.1 Numerical Methods

The Reynolds averaged Navier-Stokes equations for incompressible flow are solved

via the pseudo- compressibilty technique of Chorin'. This permits the use of time

marching schemes and the upwind-biased finite volume formulation. Details are

given in ',a. The system is closed by modeling the Reynolds stress tensor with the

mean velocity gradient tensor and a turbulent eddy viscosity. The eddy viscosity is
64 C. J. Steffen, Jr.

modeled with turbulent velocity and length scales, -\,fk- andE33 respectively. The

transport of k and E are modeled as shown below:

v l

k,t+U;k,j — Cv+ vt k,;l = P—E + D

J^

vt E E2

E,t+Ujej — V+ = Cz fi P —Cafe +E

where production of turbulent kinetic energy is defined as P . The constants C',

, Cl ,Ca , Qk and QE are defined a priori. The terms D, E, fl, , fl and f2 are

damping functions and correction terms for the LR formulation and are necessary

to give asymptotically correct behavior if the numerical domain is extended to the

wall.

2.2 Low Ret k — c models

The four low Re t k — E models examined include the Chien, Jones-Launder,

Launder-Sharma, and Shih-Lumley. This set was chosen with two criterion in mind:

to isolate the effect of the y+ based damping function of Chien, and to examine the

effect of damping functions based solely upon the dependent variable set. As mentioned above, the former effect results in anomalous behavior due to the singularity

associated with reattachment. The latter effect leads to a very general damping

function which is advantageous for multidimentional flows. The Jones-Launder and

Launder-Sharma models are of this latter type. Finally, the Shih-Lumley model

was chosen because the damping function is based upon both dependent and independent variables but does not include the y+ dependence.

2.3 Results

The results for the five k — e models tested agree for velocity and pressure field

data. The trends are also the same for the turbulent kinetic energy and turbulent

shear stress data available'. The results I wish to call attention to are associated

with the skin friction along the step side wall. Below in figure 1, the results are

plotted for the normalized tunnel streamwise coordinate (x/H) versus the skin friction coefficient (C f ). Notice the particularly misleading results associated with

the Chien model prediction. The effect of the y + dependence can also be observed

clearly in a contour plot of the turbulent eddy viscosity'. The only appropriate

resolution given here results from the standard formulation with the two-layer wall

function.

I believe two observations can be made about these results. First, the design of

an eddy viscosity damping function ( fμ ) appears to have an important effect upon

the resolution of the wall shear stress. Secondly, the lack of a pressure gradient

effect in the damping functions for this group of low Ret k — E models is causing

a premature recovery of the near wall velocity gradient in the redeveloping channel

flow. However, the value of the redeveloping channel velocity are underpredicted

in this same region. Further investigation into the connection between adverse

pressure gradients and the design of an eddy viscosity damping function is needed

for this class of models.
Low Ret k — E Models and the Backward-Facing Step Flow 65

3. Future Plans

Currently an effort is underway to examine the effect of various turbulence models

upon the prediction of separated flow for multistage turbomachinery. This is being

conducted within the framework of the average passage technique of Adamczyk9.

Preliminary results indicate that a change from the Baldwin-Lomax 10 model to

that of Johnston-King ll may indeed improve the numerical prediction of internal

passage aerodynamics dominated by secondary flow physics.

4. References

1 Steffen, C.J., Jr., "A Critical Comparison of Several Low Reynolds Number kepsilon Turbulence Models for Flow over a Backward-Facing Step," NASA TM

106173 (1993).

a Chien, K. -Y., "Predictions of Channel and Boundary Layer Flows with a Two

Equation Model of Turbulence," AIAA Journal 20, 33-38 (1982).

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 Launder, B.E., and Sharma, B.I., "Application of the Energy-Dissipation Model

of Turbulence to the Calculation of Flow Near a Spinning Disc," Letters in Heat

and Mass Transfer 1, 131-138 (1974).

5 Shih, T.-H., and Lumley, J.L., "Kolmogorov Behavior of Near-Wall Turbulence

and Its Application in Turbulence Modeling," NASA TM 105663 (1992).

6 Driver, D.M., and Seegmiller, H.L., "Features of a Reattaching Turbulent Shear

Layer in Divergent Channel Flow," AIAA Journal 23, 163-171 (1985).

7 Chorin, A.J., "A Numerical Method for Solving Incompressible Viscous Flow

Problems," Journal of Computational Physics 2, 12-26 (1967).

8 Steffen, C.J., Jr., "An Investigation of DTNS2D for Use as an Incompressible

Turbulence Modelling Testbed," NASA TM 105593 (1993).

9 Adamczyk, J.J., "Model Equation for Simulating Flows in Multistage T irbomachinery," ASME Paper 85-GT-226, (1985).

10 Baldwin, B., and Lomax, H., "Thin Layer Approximation and Algebraic Model

for Separated Turbulent Flow," AIAA Paper 78-0257, (1978).

11 Johnson, D.A., and King, L.S., "A Mathematically Simple turbulence Closure

Model for Attatched and Separated Turbulent Boundary Layers," AIAA Journal

23, 1684-92 (1985).
0.006

0.004

0.002

0.000

-0.002

-0.004

a

0

66 C. J. Steffen, Jr.

o Driver and Seegmiller

— Jon" n nd Launder

_....... Launder and Shama

Chien

— — Shah and Lumley

- Standard KQh Re

i

i

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0

x/H

Figure 1 Friction coefficient for the stepside wall downstream of the step
Center for Modeling of Turbulence and Transition 67

Research Briefs - 1993

Modeling of Homogeneous Scalar Turbulence

Aamir Shabbir

1. Motivation and Objectives

Modeling of the scalar field of a turbulent flow has lagged the modeling of its

velocity field. The most commonly used approaches employ a two equation model

to calculate the turbulent eddy viscosity for the turbulent velocity field but assume

a constant value of the turbulent Prandtl number to calculate the turbulent scalar

diffusivity. Obviously this approach has limitations in situations where the turbulent

Prandtl number is not constant.

To overcome this shortcoming, some recent studies have proposed new models in

which transport equations for scalar variance and its dissipation rate are solved to

calculate the thermal diffusivity. These include, for instance the work of Nagano

and Kiml and Youssef, Nagano and Tagawa. 2 The later study presents an updated

and refined version of Nagano and Kim model.

This study proposes a new two equation model for the scalar turbulence in which

transport equations for the scalar variance and its dissipation rate are used to calculate the turbulent scalar flux. This study differs from the other work in the

following two respects. (1) In the above cited works, the extension of the scalar

dissipation rate is based upon the work of Newman et al. 3 who developed the production/destruction mechanisms of the thermal dissipation equation in an analogue

fashion to those of the mechanical dissipation rate equation. The model equation

proposed in the present study is based on the exact transport equation for scalar

dissipation and, its production/destruction mechanisms differ from those proposed

in the other studies. (2) The model coefficient in the the scalar flux constitutive

relation used in the present study is not a constant but is a function of the local

invariants.

Although the newly developed scalar dissipation equation is used in the present

study at the two equation level, its use is not restricted to this level of models. It

can be used at any level of modeling which requires an equation for this variable.

Therefore, in future studies at CMOTT we will be using this new model equation

at the second order level. Another ongoing effort at CMOTT is to propose models

for bypass transitional flows. The k — E model modifications for the velocity field of

such a flow have already been proposed and tested by Yang and Shih. 4 The models

proposed here will form the basis for the scalar part of such bypass transitional

flows.

2. Work Accomplished

2.1 Model for the Scalar Dissipation Rate Equation

The exact equation for the scalar dissipation, co = -yO,i B,i, is

Ee ,t + U7 Ee,; — 7E0 ,77 7(B,i B ,i uj),j —2- B,i B,jUj i — 2'y8,i u7,iO,j
68 A. Shabbir

— 27e ,i uj E) ,ij —27

0,i e,j ui,j — 2720 7ij 9

7

ij \1)

where -y is the molecular thermal diffusivity. The first term on the right hand side

of (1) represents the molecular diffusion of scalar dissipation. The second term represents the diffusion by fluctuating velocity field. Third and fourth terms represent

the production of scalar dissipation through the gradients of mean velocity and

temperature fields. First term on the second line represents the production through

mean scalar gradients. Second term on the second line is also a production term for

the scalar dissipation rate; it represents the stretching of the scalar eddies through

the fluctuating velocity gradients. The last term on the second line represents the

destruction of the scalar dissipation rate.

We consider only wall-free flows so that the effects introduced by a wall can be

avoided. First we want to do the order of magnitude analysis of (1), for a high

Reynolds number flow, to find out which terms can be ignored. For this purpose

we assume that: l is the length scale of the large eddies; B is the scale of the scalar

fluctuations; and u is the scale for velocity fluctuations. Further we estimate that:

Ui,j — u/l; ui,j , (u/l)Rt /2 ; O, i — B/l; and 79, i 9, i — (92 1l 2 )RtPr. With this

scaling the order of magnitude of each term in ( 1) can be estimated. The result is

Eg ,t + UjEB,j

z — 703u RtPr (2.a)

'Y E9 , jj

7e2u

1 3 Z. b ( )

7( e ,i 0" uj ) ,j

7BZu

^^ 1 3

RtPr (2.c)

279 i 9,j Uj i

z

— _yB 

u

13 R1/2 Pr (2.d)

2'Y e ,i uj,i E) ,j

7B2u

^' 13

R 1/2 tPr 2.e ( )

27e ,i ujo,ij

783zu Pr

1/2 /

12.J { )

279, i 9,j ui,j

z

703

u

R3/2 Pr (2.g)

272 B ,ij B,ij

z

783 u Rt 

/2 Pr (2. f )

Note that we have used the following relation to estimate 2'y9,i 9,jUj,i

z

B,ie,j __8

2

Rt( aSij + cijRt 1/2 ...........) (3)

where a and cij are coefficients of order one. (This is similar to the estimate of WjWj

made by Tennekes and Lumley'). It should also be noted that the estimate (2.e)

above is more liberal than obtained by some of the other studies.
Modeling of Homogeneous Scalar Turbulence 69

So to the leading order we find that

— 2yB,i 9,j ui,j — 2,y O lij 0 7ij (4)

Based on (2) and (4) we expect that — 2-ye,i B,j ui,j — 2'y 2 e,ij B,ij should be of Rt

order. From (2) we also conclude that, for a large Reynolds number flow, 7EB,jj,

2-yO, i B,j Uj,i and 27'B,i ujO,ij are of relatively smaller magnitude and, therefore, can

be neglected. Under these assumptions, therefore, (1) reduces to

E o,t + Uj c o,j = — 7(B )i B,i uj),j —2-ye, z uj,iO,j —278 > z B,j ui,j — 2728 Jij B)ij (5)

Modeling of 2'ye,i uj,iO,j:

First we define a tensor Oj as

Oj =

d"i uj,i

(6)

B ,k 8 >k ul,mul.m

Using this definition we can write

278 >i 21 j,i E) ,j =27-^,/ B ,k B ak ul,m ul,m, Oj E),j (7)

Modeling Oj o( O, j /-I^ (where ^D = O, k O, k ), and after some manipulation of (7)

we obtain the following result

27

EgE

8 ,1 uj,i O ,j =Ce2 Pr ^ (8)

where Pr = v/-y and CB2 is a model coefficient.

Modeling of — 2'ye,i B,j ui,j — 2720 1ij B,ij:

We define a tensor bie as

1.2 2

B ,k e ,k 3

Using this definition 27'B,i B, j ui,j can be expressed in terms of bie as

27B ,i e ,j ui.j =27B ,k B ,k ( bid + 6ij l3 )ui , j (10)

Modeling bib OC S,j/s (where sij = (ui,j+uj,i)/2, and s = V2-s7

ij sij ) and substituting

ui ,j = Si j + u1? j (where &j = (ui ,j — 2lj i)/2) we have

S'

278 , 1 B , j ui,j a 27

s

8 ,k 8>k 

aj s ij(11)
70 A. Shabbir

As was discussed earlier, each 2-yO,i 8,j ui , j and 2-y2 B, ij 9,ij is of order Rt^ 2 but their

sum is of order Rt . This suggests that the sum of these two terms can be modeled

by the following three terms

vw

_^' EBS' (k

EO IE (12)

where S =VSijA.7 and Sij = (Ui,j + Uj,i)/2. Note that the first term is the same as

the model for 21'e,i uj,iO,j obtained above. So both of these terms can be combined

together into one term.

Therefore, the final form of the modeled scalar dissipation equation becomes

EBE EgE

EB ,t + UjEB,j = — (ujEB),j +CO1 E O S + CB2 ^ — CB3 (k + V "c) (13)

where C01, C192, and CB3 are the model coefficients.

2.2 Determination of Model Coefficients

There are three model coefficients in equation (13) and these are yet to be determined. These can be determined by considering a hierarchy of flows of increasing

complexity so that each flow allows calibration of one coefficient at a time.

First we consider decaying homogeneous turbulence. Warhaft and Lumley 6 fouled

that in such a flow the time scale ratio r = 2-k 

B 

remains unchanged. Substituting Er

Dt = 0 in equation (Al2) we obtain the following relation

CB3 =C2 - 1 + r (14)

where C2 is the constant appearing in the decay term of the mechanical dissipation

equation and is 1.9 for the model used in the present study (see appendix A).

To determine the second constant CB2 in (13), we use the same approach as

used by Jones and Musonge 7 and Nagano and Kim l .1 Both of these references

used the fact that in experiments on homogenous turbulence with non-zero mean

temperature gradients (Sirivat and Warhaft 8 ) the time scale ratio approached a

constant value toward the downstream end of the flow i.e. Dt --+0 for large t.

We use this experimental finding to obtain a value for CB2 . Therefore, by letting

Dt = 0 for this experiment we obtain

CO2 = Pr2 1

/2

3

Pvt( r) 1/2

(15)

where pz,t is the cross-stream heat flux correlation coefficient. Upon substituting

the asymptotic values of quantities from the experiment of Sirivat and Warhaft in

the above equation we obtain CO2 = 1.28. (Experiment suggests a value of 1.4 but

in computations a value of 1.28 gives a better overall agreement.) The last constant

to be determined in the scalar dissipation equation is CBI . For this purpose the

homogeneous shear flow experiment of Tavoularis and Corrsin 9 to obtain Col = 0.1.

L
Modeling of Homogeneous Scalar Turbulence 71

2.3 Modeling of the Heat Flux

In order to complete the model for the scalar turbulence, an expression is needed

for the heat flux tensor ujO. We use the model proposed by Shih and Lumleylo

We keep only those terms of their model which are linear in the mean temperature

gradient. i.e.

ui e =a1 k

2

(2)1/2O,i 

+k3

(2)1/2 ( a2 Ui,j + a3 U'ji) O, (16)

c r e2 r

The above model expression is the same as obtained by Yoshizawall and Rubenstien

and Barton 12 using DIA and RNG approaches respectively. Our purpose now is to

determine the model coefficients a l and a2 . We point out that the term involving

the coefficient a 3 is zero for all the flows considered in this study and, therefore, this

coefficient is left undetermined for now. In order to determine a l we hypothesize a

situation where

D uiB 

= as Dt (kg2)112

with a is a constant. Of course one can make other assumptions which could be

more general than this. However, this assumption seems to be adequate for the

homogeneous flows considered in this study and, therefore, we will be using this

as a guide in obtaining the formulation for a l . After differentiating the above

expression we obtain

Dui B 1 DO' 1 Dk

= a S Vk 22 (17)

Dt 262 Dt 2k Dt

We restrict our attention to homogeneous flows and substitute the transport equations for k, V2 uiB in the above relation. We note that in order to do so we need

a closed (modeled) equation for the heat flux. We employ the model equation of

Shih and Lumley". After substituting these transport equations in (17) and then

replacing uiuj and uiB with their constitutive relations (A3) and (A9) we obtain a

quadratic equation for the constant a 2 . The final solution of this quadratic equation

is further simplified to obtain the following expression for al.

EE CO (2k/3 — .45S /p)

al = — 0.72

e

(r 2 + 2r + 2) + —

O q2

(18) 4k2^2^ 02E^ ( r2 +2r+2)

The second constant a 2 is calibrated from the experiment of Tavoularis and Corrsin9

and is found to be about 0.024.

2.4 Application of Model to Homogeneous Flows

The present model for homogeneous flows is summarized in Appendix A. We

compute some homogeneous flows using this model to see if it can reproduce them
72 A. Shabbir

reasonably. For these homogeneous flows the model represents four ordinary differential equations for k, E, B 2 , and co and constitutes an initial value problem.

These equations were solved using a fourth order Runge-Kutta method. The assessment of the k — E model used with the present model has already been reported

in Shih et al. 14 . Therefore, we will be showing results for only the scalar field.

For comparison we will also be showing results obtained from Youssef, Nagano and

Tagawa model' (hereafter referred to as YNT model). Figures 1-7 shows results for

the homogeneous flow experiments which has a constant cross-stream temperature

gradient dOl dy (but no velocity gradients). The agreement between the present

model and experiment is reasonable for most of the cases, and better than the YNT

model in all the cases. We do note, however, that the present model overpredicts

the measured temperature variance for the case shown in figure 2 and underpredicts

the same quantity for the case shown in figure 4. The model performs poorly for

the case shown in figure 7. In this case the experiment shows that the temperature

variance first decreases until about xlM = 70 and then increases. Prediction of

this kind of behavior is beyond the scope of the kind of algebraic models presented

here for the heat flux (i.e. equation 18). The heat flux model given by equation

(18) can not produce a decrease in its level. Thus the production term —ui 9O,i in

the temperature variance equation also increases instead of decreasing. This leads

to the continuous increase in the temperature variance level. The correct way to

handle the kind of experimental behaviour shown in figure 7 is to use an evolution

equation for the heat flux rather than the algebraic expressions like (18).

The results for the homogeneous shear flow which has constant cross-stream temperature and velocity gradients are shown in figure 8. Both the temperature variance

and the thermal dissipation rate predicted by the current model are in reasonable

agreement with the experiment of Tavoularis and Corrsin'. On the other hand YNT

model overpredicts the experiment.

3. Future Plans

The model proposed here has been tested for simple homogeneous flows. We will

extend the model to wall bounded flows, such as a heated flat plate.

4. References

1 Nagano,Y. and C. Kim. "A two equation model for heat transport in wall turbulent shear flows", J. Heat Transfer,110, 583-589, (1988).

' Youssef, M.S., Y. Nagano and M. Tagawa. "A two-equation heat transfer model

for predicting turbulent thermal fields under arbitrary wall thermal conditions",

Int. J. Heat Mass Transfer, 35, 11, 3095-3104, (1992).

s Newman, G.R., B.E. Launder and J.L. Lumley. "Modeling the behaviour of

homogeneous scalar turbulence", J. Fluid Mech., 111, 217-232, (1981).

4 Yang, Z. and T.H. Shih. "A k — E model for turbulent and transitional boundary layers", Proceedings of the International conference on Near-Wall Turbulent

Flows, Edited by R.M. So, C.G. Speziale and B.E. Launder, 165-175, (1993).
Modeling of Homogeneous Scalar Turbulence 73

5 Tennekes, H., and J.L. Lumley. "A First Course in Turbulence". The MIT Press,

(1972).

6 Warhaft, Z., and J.L. Lumley. "An experimental study of the decay of temperature fluctuations in grid-generated turbulence", J. Fluid Mech., 88, 659-684,

(1978).

7 Jones, W.P., and P. Musonge. "Closure of the Reynolds stress and scalar flux

equations", Phys. Fluids, 31, 3589-3604, (1988).

s Sirivat, A., and Z. Warhaft. "The effect of a passive cross-stream temperature

gradient on the evolution of temperature variance and heat flux in grid turbulence", J. Fluid Mech., 128, 326-346, (1983).

9 Tavoularis, S., and S. Corrsin. "Experiments in nearly homogeneous turbulent

shear flow with a uniform mean temperature gradient", Part 1, J. Fluid Mech.,

104, 311-347, (1981).

10 Shih, T.-H., and J.L. Lumley. "Remarks on turbulent constitutive relations",

NASA TM 106116, (1993).

11 Yoshizawa, A. "Statistical modelling of passive-scalar diffusion in turbulent shear

flows", J. Fluid Mech., 195, 541-555, (1988).

12 Rubinstein, R., and J.M. Barton. "Renormalization group analysis of anisotropic

diffusion in turbulent shear flows", Phy. Fluids A, 3, 415-421, (1991).

13 Shih T.-H., and J.L. Lumley. "Influence of timescale ratio on scalar flux relaxation: modelling Sirivat & Warhaft's homogeneous passive scalar fluctuations",

J. Fluid Mech., 162, 211-22, (1986).

14 Shih, T.-H., W.W. Liou, A. Shabbir, Z. Yang and J. Zhu. "A vorticity dynamics

based model for the turbulent dissipation", NASA TM 106177, (1993).
74 A. Shabbir

Appendix A

The model used in the present study is summarized below for homogeneous flows.

(Note that the k — E model used is that of Shih et al. 14 )

Uj ax = — uix ax; — e (Al)

ae 2

Uj 

ax 

=C1 SE — C2 k

(A2)

uiuj =UT (Ui,j + Uj,i) — 

2

3 kbij (A3 )

PT 

_ k2

_ 2/3

Cμ (A4)

E '

Cμ 1.25+71+0.9^ 

where

77= k^ = kSt

S _ S,, S2, Q = 2S2 Q, (A5)

and

1 aUi auj _ 1 aUti _ aU;

SZ' 2 ( ax; + axi )' ^Z ' 2 ( ax; axi) (A6)

The constants in the e equation are: Cl = 0.42 and C2 = 1.9.

U; ae

2 _

^ = — 2zT,70 — 2E0

(A7)

U'

ax . =C

01 e0 S + CO2 

EOE^ — 

CO3 egE (A8)

Pr k

ui8 =a1 X2 ( 2 ) 1/2(9 ,i+ E2 (2)1/2(a'2Ui,j +a3 Uj, i ) 0,j (A9)

where

CEO Eg (2k/3 — .55S 02 q2/^)

al = — 0.72 4k2 ^2 

^ (r 2 + 2r + 2) + 

9 2 e^ (r2 + 2r + 2)

(A10)

and 4) = O iO i. The constants in the EB equation are:

C01 =0.1 , CO2 = 1.28, , CB3 = C2 — 1 + r (A11)

The evolution equation for the time scale ratio r =ZE a (used in section 2.2)

can be obtained by_differentiating this relation by parts and then substituting the

equations for k, E, B2 and E0 as given above. The result is

Dr uiu; aUi E 1^2 uiB a0

=(C01 S — Ci s — )r + [CB2 (Pr—) — 2 — ^r

Dt k axj CO B2 axi

+ (-2CB3 — 2 + 2C2 + 2r)—

2

(Al2)
Modeling of Homogeneous Scalar Turbulence

75

dTdy=10.3 deg C/m, U=6.3 m/s

• Expt. Data

Present Model

- - YNT Model

/

' e2' •

_ Ete

• • • • • • • •

0.0 r

1 1 1 1

20.0 60.0 x/M 100.0 140.0

Figure 1. Evolution of the normalized temperature variance and thermal dissipation

rate in the experiment of Sirivat and Warhaft (1983). The temperature variance

is normalized as 92' = 82 192 0 and the thermal dissipation is normalized as Ee =

EB For this case 92 0 = 0.0128 O C2 , u0 = 0.145515m/s, and 1 0 = 0.011937m.

e^ouo/lo

dTdy=4.48 deg C/m, U=6.3 m/s

3.0

• Expt. Data /

Present Model

- - 1'N fI Model

• 92`

E'B

0.0 1

20.0 60.0 X/M 100.0 140.0

Figure 2. Evolution of the normalized temperature variance and thermal dissipation

rate in the experiment of Sirivat and Warhaft (1983). The temperature variance

is normalized as 9 2' = 82 /92 0 and the thermal dissipation is normalized as EB =

Ea

. For this case 9 2 0 = 0.002287 0 C2 , u0 = 0.145515m/s, and 10 = 0.011937m.

e2ouo/io

3.0

2.5

2.0

1.5

1.0

0.5

2.5

2.0

1.5

1.0

0.5
• Expt. Data

— Present Model

— — 

vl' rr 1R^A ^I

•

•

E'e

• •—OL_•_•_ ♦__ -i76 A. Shabbir

3.0

2.5

2.0

1.5

1.0.

0.5

00

dTdy=3.68 deg C/m, U=6.3 m/s

• Expt. Data

Present Model

— — YNf Model i

i

V2

EeB

• • • • .

20.0 60.0 X/M 100.0 140.0

Figure 3. Evolution of the normalized temperature variance and thermal dissipation

rate in the experiment of Sirivat and Warhaft (1983). The temperature variance

is normalized as B et = 62 /62 o and the thermal dissipation is normalized as EB =

e^ov,o/[/^o For this case 0 2 o = 0.001705 ° C2 , Uo = 0.145515m/s, and to = 0.011937m.

dTdy=8.1 deg C/m, U=3.4 m/s

3.0

2.5

2.0

1.5

1.0

0.5

0.0 C I

i 1 1

20.0 60.0 X/M 100.0 140.0

Figure 4. Evolution of the normalized temperature variance and thermal dissipation

rate in the experiment of Sirivat and Warhaft ( 1983). The temperature variance

is normalized as 6 2' = 62 /62 o and the thermal dissipation is normalized as EB =

EB . For this case 62 0 = 0.009059 0 C2 , uo = 0.074701m/s, and to = 0.011071m.

9-20u0/I0
77

1.5

Modeling of Homogeneous Scalar Turbulence

dTdy=2.24 deg C/m, U=3.4 m/s

• Expt. Data

Present Model

— — YNI' Model

2.0

—r

92

1.0

0.5 E^e

0.0 l

20.0 60.0 X/M 100.0 140.0

Figure 5. Evolution of the normalized temperature variance and thermal dissipation

rate in the experiment of Sirivat and Warhaft (1983). The temperature variance

is normalized as B2' = 92 /B2 o and the thermal dissipation is normalized as c' =

EB For this case 82 0 = 0.000924 °C2 , uo = 0.074701m/s, and to = 0.011071m.

B20UO/1O

dTdy=1.81 deg C/m, U=3.4 m/s

3.0

• Expt. Data

2.5 Present Model

— — YNI' Model i

2.0

i

• 02r

1.5

0

1.0

CO

I

0.5

00

20.0 60.0 X/M 100.0 140.0

Figure 6. Evolution of the normalized temperature variance and thermal dissipation rate in the experiment of Sirivat and Warhaft (1983). The temperature

variance is normalized as 8 2 ' = 82 /92 o and the thermal dissipation is normalized

as E = 02OUO/10. For this case 9 2 0 = 0.0004471 o C2 , uo = 0.074701m/s, and

1 0 = 0.011071m.
78 A. Shabbir

3.0

2.5

2.0

1.5

1.0

0.5

00

dTdy=i .78deg C/m, U=3.4m/s

/

• Expt. Data

Present Model

— — YNT Model

/

^ —re2

•

69

20.0 60.0 x/M 100.0 140.0

Figure 7. Evolution of the normalized temperature variance and thermal dissipation rate in the experiment of Sirivat and Warhaft (1983). The temperature

variance is normalized as 02 r = V2/020 and the thermal dissipation is normalized

as er =

Ea . 

e For this case 82 0 = 0.0004955 °C2 , uo = 0.074701m/s, and

e2UUO /10

to = 0.011071m.

2.5

2.0

C114 1.5

N

1.0

• Expt.

—^ Present tilodel

— — YNT \lodel

i

0.5 l 1 1 1 1 1

6.0 7.0 8.0 9.0 10.0 11.0 12.0

x/H

Figure 8a. Evolution of the normalized temperature variance in the experiment

of Tavoularis and Corrsin (1981). The temperature variance is normalized with

02 ref = 0.01j 'C2.
Modeling of Homogeneous Scalar Turbulence 79

0.2

N CIO

U

0.1

CD

w

I

I

I I ^T —^

0 Expt_

— Present Model i

— — YNT Model

i

i

i

i

i

0.0 L

6.0 7.0 8.0 9.0 10.0 11.0 12.0

x/H

Figure 8b. Evolution of the thermal dissipation rate in the experiment of Tavoularis

and Corrsin (1981).
Page intentionally left blank 
Center for Modeling of Turbulence and Transition 81

Research Briefs - 1993

A Multiple-Scale Turbulence

Model for Incompressible Flow

B. S. Duncan

1. Motivation and Objective

In turbulent flows, the mean flow performs deformation work which transfers

energy from the mean flow to the large-scale turbulent eddies. The turbulent kinetic

energy contained within these large eddies is passed to the smaller eddies by vortex

stretching. Once the energy has been passed into eddies near the Kolmogorov scale,

it is then dissipated by the molecular viscosity. This process can be thought of as

a turbulent kinetic energy cascade'. In other words, the turbulent kinetic energy is

passed through the wave number spectrum as it cascades from large to small eddies.

The multiple-scale turbulence model which is proposed in this study splits the

energy spectrum into low and high wave number regions. The low wave number

region contains the large-scale eddies and the high wave number region consists of

the smaller, less energetic eddies. A division of this nature models the cascade of

energy from the production region, where the energy is initially created by the mean

straining work, to the dissipation region where this energy is eventually dissipated.

This concept is illustrated in Figure 1. Although a simple two-part division of

the energy spectrum cannot fully model the cascade of energy, it can simulate the

nonequilibrium energy transfer process which is beyond the capability of all singlescale models.

This concept was incorporated into an earlier multiple-time scale turbulence

model by Hanjalic, Launder and Schiestel 2 (hereafter HL&S). Based on the same

modeling methodology that was used in the development of the "standard" k—E

model', HL&S derived four transport equations to describe the turbulent characteristics of the two regions. Consequently, HL&S formulated two transport equations

for the partitioned turbulent kinetic energies. They also developed transport equations for the rate of energy transfer between the two scales and for the rate of energy

transfer to the small-scale eddies. In this model, the coefficients were written as

functions of both the ratio of the partitioned energies and the ratio of the spectral

energy transfer rates. A term containing the mean vorticity was included in the

energy transfer rate equation to account for the increased energy transfer rates in

irrotational shear flows. Their results 2 showed fairly good agreement between the

model predictions and the experimental data for jets and boundary layers.

Kim and Chen' (hereafter K&C) developed another multiple-scale model based

on the energy partitioning idea introduced by HL&S. In this model, the transport

equations were modified to include an extra source term in both the energy transfer

rate equation and the energy dissipation rate equation. In addition, the turbulent velocity scale was characterized by the total turbulent kinetic energy. K&C

calibrated the model constants which appear in the energy transfer equations for
82 B. S. Duncan

simple, wall-bounded turbulent flow problems. This model has been used for several

boundary-layer flow problems using the appropriate near-wall corrections.

The model in this study is also based on the energy partitioning concept of

HL&S. In particular, the model coefficients are dynamically dependent upon the

partitioning of the energy spectrum. The variable nature of the coefficients adjusts

the model to different flow situations. There is no need for the extra source terms

used in K&C's model or the rotational straining term in the HL&S model. These

model coefficients have been calibrated for homogeneous shear flow and decaying

grid turbulence. The present multiple-scale turbulence model has been tested for

boundary-free shear flows. Two mixing layers at different speed ratios, a planar jet

and a round jet have been evaluated. All the computations show reasonably good

agreement with the data.

2. Work Accomplished

2.1 Model Equations

Mean Flow Equations

For incompressible turbulent flow, the ensem bled- averaged equations for continuity of mass and momentum are written as

aUi = 0

axi

and

DUi _ a aUi 1 1 aP

Dt axe v axe — uZu') p axi

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

concept, the Reynolds stress can be related to the mean strain rate and a turbulent

eddy viscosity,

—Uiuj = 

vt ( OU, + allj 2 

kbij.

axj 2

Now, the momentum equation can be written as

DUi _ a aUi lap

(v + vt) Dt ax; axe _ p axi

The turbulent eddy viscosity, vt , can be characterized by the local turbulent

kinetic energy and the local length scale of the energy containing eddies,

vt oc k2' I.

The definition used for this length scale is the primary discriminating factor of eddy

viscosity turbulence models. For instance, in a k-E model this length scale is written

in terms of the turbulent kinetic energy,k, and its dissipation rate,E,

l= k2

E
A Multiple-Scale Turbulence Model for Incompressible Flow 83

In the present model, the length scale is described by the total turbulent kinetic

energy and the rate of spectral energy transfer, i.e. l = EZ . The turbulent kinetic

P

energies and the rates of energy transfer are determined by modeled transport

equations similar to the "standard" k-E equations. These transport equations are

described in the following section.

Turbulence Equations

In this multiple-scale turbulence model, the energy spectrum has been split into a

region where the turbulent kinetic energy has been produced by interaction with the

mean flow and a region where the turbulent energy has been transferred from the

production region. This division can be graphically represented by Figure 1. Now,

kP is the kinetic energy contained within the production region and E P is the rate at

which energy is passed from the low wave number range into the high wave number

range. At the high wave number end of the spectrum, kt is the kinetic energy

contained in the smaller eddies and Et is taken to be equivalent to the dissipation

rate. The modeled transport equations are

Dt aax7 [ \U + O'k axj ] + PAP — EP P

DEP

a7 U +

Up )

ax ' L \aE /I NP8x

Dkt a

^

1/t

Dt ax , ^ L v + kt l axaka J

DEt a 1/t 1

U +

aE7 1

Dt IL ax ,^ \

or //

))

ax J

where,

Pk = vt 

aU; + aUi 1 aUi

P

axi axe ) axe

Here, PkP is the production of kinetic energy by the large scale eddies. Note that

the EP term serves as a sink in the equation for k P and a source in the kt equation.

The source and sink terms in the energy transfer rate equations (EP and Et ) are

related to those in the turbulent kinetic energy equations by their corresponding

time scales, i.e., ^ and k1.

P

These transport equations, equations (1) through (4), possess several differences

from the earlier multiple-scale models. HUS include a rotational straining term

in the EP energy transfer equation (2) to improve their model's performance in

axisymmetric flows. Based on dimensional reasoning, K&C4 include additional

terms in the EP and Et equations which are nonlinear in PkP and Ep, respectively.

The present model is the least complicated as it uses neither the rotational straining

term nor the nonlinear production terms for energy transfer rates in the model

equations.

2

+ CPl kP 

PkP — Cp2 kP

+EP — Et

Et

+ cti kt

Et6p — 

ctz kt

(1)

(2)

(3)

(4)
84 B. S. Duncan

Accounting for the multiple partitions in the turbulent kinetic energy spectrum

and the spectral energy transfer rates, the eddy viscosity in the present model is

defined as

Ut —_ Cμ 

(kt + kP)2

EP

and cN, = 0.09. This is the same relationship for eddy viscosity used by K&C. With

this formulation, the single scale eddy viscosity model will be recovered when Ep

approaches E in equilibrium turbulence. The a coefficients in equations (1) through

(4) are assumed to be the following constants,

Crkp = Orkt = 1.0 and crEP = cr t = 1.3.

The other coefficients in equations (1) through (4), namely cpl , Cp2 i ctl , and ct2,

are the modeling coefficients discussed in the following section.

2.2 Model Coefficients

The coefficients for this incompressible model have been determined from analyses

of homogeneous and decaying turbulence.

Grid Turbulence

In homogeneous decaying grid turbulence, the turbulent quantities are functions

of time only and equations (1) through (4) can be simplified to

dkP _ _EP

(5)

dt

dEP EP

2

_ _c

p2

(6)

dt kP

dkt

( 7 )

dt — 

EP — Et

dEt Et EP Et

— —

cti

ct2 (g) dt kt kt

Most of the experimental evidence suggests that the turbulent kinetic energy decays

in time and can be represented by

P _ ( l n

kPo \ o

and

kt

O — ^tO)—n

where n is the decay rate and is typically of the order 1.2. From the above kinetic

energy equations, (5) and (7), the energy transfer rates must decay as

EP

t

CPO — to
A Multiple-Scale Turbulence Model for Incompressible Flow 85

and

Et t —n-1

Eto — ( to )

Therefore, the cp2 coefficient from equation (6) can be related to the decay rate,

n+1

cp2 =

n

Manipulating the simplified transport equations, (5) through (8), and using the

above relations yields a relationship between Ct2 and the other coefficients,

Ctl + k Cpl

Ct2 kt +1

Homogeneous Shear Flow

Guidelines can be established for determining the remaining two constants, cp1

and ctl , by examining the physical behavior of homogeneous shear flows. In this

flow situation, the turbulent transport equations reduce to

dip =Pkp- Ep (11)

AP

2

dt = 

Cpl LPkp — p

cp2 

^p

(12)

dkt _ Ep — Et (13)

dt

det EtEp Et

(14) dt — ctl kt —

Ct2 kt

If we define

Pkp

a=

Et

and

E

= Ep , (16)

t

then equation (11) can be combined with equation (12) to give

EP dkp

kp dep Cpla - Cp20

Likewise, equations (13) and (14) can be combined to give the following result,

E t dkt _ 0-1

kt dEt

CO — Ct2

(9)

(10)

(15)
86 B. S. Duncan

Assuming that 6 and the percentages of the kinetic energy contained in kP and kt

remain nearly constant, then

EP dkP _ Et dk _ (a — 8) (17)

kP dEP k det cpla — cp2,3

and

Et dkt _ et dk _ 0-1 (18)

kt dEt k dEt ct1Q — ct2

From the experiments , the ratio ft dE has been deduced to be 1.065. In a similar t

experiment, Tavoularis and Corrisins found this ratio to be between 0.82 and 0.94.

Clearly, this ratio is on the order of one. For simplicity, this term, k d , has been

assumed to be unity. Therefore, the following expressions for cpl and ctl can be

found from equations (17) and (18),

Cpl = 1 -0 + cp2

a a

ctl = /^ — 1 + ct2

Q Q

Equation (10) is now

1 + cp2Q

c t2 =

Q+Q kt -1

and cp2 is defined in equation (9). The coefficients, cpl , ctl and ct2 are functions of

PtP ,

a, and k In the present model, the ratios described in equations (15) and

(16) are assumed to be the following constants

a= 

Pk- 

=2.2

Et

and

Q = LP 1.05.

Et

These constants have been calibrated considering that experimental measurements

of homogeneous shear flow suggest that the ratio 

a 

should be near two. The coefficients of the present model are summarized in Table 1.

Notice that the value of is allowed to vary as the ratio of the turbulent kinetic

P

energy in the small scales to the energy contained in the large scales changes. Since

most of the energy is contained in the large scales, this ratio should remain less

than one'. This coefficient adaptability allows the model to adjust to different flow

situations and is a unique characteristic of multiple-scale eddy viscosity models.
A Multiple-Scale Turbulence Model for Incompressible Flow 87

2.3 Results and Discussions

The present multiple-scale model has been tested for two planar mixing layers and

two jet flows. As part of these tests, the model has been compared to experimental

data and to the multiple-scale models of Hanjalic, Launder and Schiestel 2 (HL&S),

Kim and Chen' (K&C), and the "standard" k-E model. In all cases, the parabolic

solution technique is started with an initial plane and the flow field evolves as the

computations march in the axial direction. The solutions are checked to insure

that they maintain a self-preserving profile. The results of these calculations are

presented in Figures 2 through 16 and in Table 2.

2.2.1 Planar Mixing Layers

For the planar mixing layers, the flow is assumed to have a thin shear layer profile

at the interface between the still air and the jet. Zero gradient boundary conditions

for the turbulence quantities are applied at the edges of the flow field and an equally

spaced grid is used.

Speed ratio = 0.0

In Figure 2, the three multiple-scale models and the "standard" k -E model are

compared to the experimental data of Wygnanski and Fiedler ? and Patel8 . At

the high speed edge, none of the four models predicts the diffusive characteristic

indicated by the data. Away from this area, however, the present model, HL&S's

model and the "standard" k-c model all are very close to Patel's data. The data

due to Wygnanski and Fiedler is considerably more diffusive than either Patel's

data or the computations. The spreading rates predicted by the four eddy viscosity

models and the spreading rate measured by Patel are listed in Table 2. For mixing

layers, the spreading rate is defined as d y dx y l The spreading rate predictions by

the present model and the "standard" k-E model are closest to the data. K&C's

model significantly underestimates the growth rate of the mixing layer.

The tendency of K&C's model to under-predict the growth of the turbulent mixing region is further seen in Figures 3 and 4. The shapes of the kinetic energy

curves and the shear stress distributions are correct, but the peak levels are well

below the data. The "standard" k-E model, HL&S's model, and the present model

predict turbulent kinetic energy levels slightly below the data but they correctly

predict the peak levels for shear stress. Apparently, the computational shear layers

tend to shift further towards the low speed side of the flow than the experimentally

measured shear layers.

In Figure 5, the ratios of the partitioned kinetic energies and the energy transfer

rates are shown. Notice that the ratio, E , is one through the mixing layer where

P

the turbulence spectrum is in equilibrium. The fact that ^ remains much less than

P

one indicates that most of the energy is contained by the large scale eddies as stated

by Tennekes and Lumley 1.

Speed ratio = 0.3

The next case considered is a mixing layer with a speed ratio of 0.3. In Figure 6,

the mean velocity profiles predicted by all four models are in very good agreement

with the experimental data measured by Spencer et. al.'. The present model and
88 B. S. Duncan

the "standard" k-E model predict an almost identical velocity profile. HL&S's and

K&C's models under-predict the growth of the mixing region, which is indicated by

the narrower mean velocity profile and also by the lower spreading rates listed in

Table 2.

As can be seen in Figure 7, the "standard" k-E and the present kr-Ep-kt-ct

model predict the kinetic energy distribution very well. Although the models due

to HL&S and K&C yield good mean velocity predictions, they both under-predict

the peak turbulent kinetic energy level. The present model and the "standard" k-E

model yield better predictions of the turbulent kinetic energy profile. All the eddy

viscosity models in this study under-predict the peak shear stress shown in Figure

8. Here, the "standard" k-E and the present model do a better job of predicting

the shear stress profile than HL&S or K&C; however, they are both slightly low in

their predictions of the peak value.

Grid Resolution Analysis

Since the mixing layer with a speed ratio of 0.3 gives the thinnest shear layer

and the slowest growth rate, this case would be the most dependent upon grid

resolution. For all the calculations discussed thus far, there were 65 points across

the flow field. Figure 9 compares the solution obtained with 65 points to the same

calculation with 101 points across the domain. As this Figure indicates, the result

is essentially unchanged as the grid is refined.

2.2.2 Planar Jet

For the jet flow simulations, the initial plane is split with a uniform velocity and

kinetic energy profile comprising approximately half of the domain, and quiescent

air comprising the other half. Zero gradient boundary conditions are applied at

the centerline of the jet and the grid has been clustered towards the centerline to

improve the accuracy of this boundary condition. Stable, self-similar profiles could

not be obtained for the HL&S model; therefore this model has not been included

in the following comparisons.

As can be seen in Figure 10, the mean velocity profiles predicted by the present

model, K&C's model, and the "standard" k-E model lie within the scatter of the

experimental data' 1,11,12 The present multiple-scale model is particularly close to

the data of Heskestad 12 . There is very little disagreement between the predicted

spreading rates and experimental spreading rate given in Table 2. For jet flows, the

spreading rate is defined as the rate of change of the half velocity point, i.e., d

The predicted spreading rates range from 0.103 by K&C's model to 0.114 by the

"standard" k-E model. All these spreading rates are in reasonably good agreement

with the experimentally determined rates.

There is considerably more disagreement between the models for the predicted

turbulent kinetic energy levels, shown in Figure 11. Both the present model and

K&C's model predict relatively flat profiles which match the data of Heskestad12.

The "standard" k-E model is in very good agreement with the experimental data,

being especially close to Bradbury's measurements.

Both multiple-scale models and the "standard" k-E model predict closest to the

peak shear stress, shown in Figure 12, although they tend to exaggerate the width
A Multiple-Scale Turbulence Model for Incompressible Flow 89

of the jet.

2.2.3 Round Jet

Using the "standard" k—E turbulence model often gives poor predictions for the

spreading rate and flow properties of a round jet. There are several methods to

adjust the "standard" k—E model, e.g., by changing the coefficient in the dissipation

equation or by adding a vortex stretching term 13

. The "standard" k—E model predictions given in Figures 13 through 15 have applied no correction methods. Neither

the present model nor K&C's model uses a three-dimensional correction term; yet,

as shown in Figure 13, they give the best match to the experimental data of Rodi14

and Wygnanski & Fiedler15 for mean velocity.

The turbulent kinetic energy is over-predicted by the "standard" k—e model as

shown in Figure 14. Closer to the data, but still not giving good predictions, are

the present model and K&C's model. Both models predict the correct centerline

turbulent kinetic energy, however, the outer edges of the jet are over-predicted.

None of the models in this report does a good job of predicting the turbulent

kinetic energy for an axisymmetric jet.

Figure 15 shows that there is also a wide spread in the shear stress predictions.

Without adjusting the modeling coefficients or adding correction terms, the "standard" k—E model predicts shear stress levels almost 1.5 times those given by the data.

Both the present model and K&C's model are relatively close to the data compared

to the "standard" k—e model, although both models over-predict the peak level by

approximately 25 percent. The two multiple-scale models, i.e. the present model

and K&C's model, predict spreading rates which are, respectively, 17 and 24 percent

too large in comparison to the data.

Figure 16 shows the ratios of Al and E across the jet for the present multiple-scale

P P

model. Notice that the ratio of kinetic energy are much less than one, indicating

that most of the energy is in the larger scales. Looking at the ratio of energy transfer

rate, the energy transfer to the dissipative scales increases near the centerline of the

jet indicating that the flow field is not in equilibrium. The "standard" k—e model has

no means to account for the increase in energy transfer rate near the centerline of the

jet and consequently over-predicts the turbulent kinetic energy and the spreading

rate.

3. Future Work

This model will be extended to wall bounded flow using both wall functions and

near wall damping functions.
90 B. S. Duncan

cμ =0.09 n=1.2 a=2.2 3=1.05

/ k

1 !­-^ p 7i 1 1 C

)3-1+CP20 p

cP1 = CfCp2 cp2 = n ctl

= Q ct2 = 8+Q^ -1 kP

QkP = 1.0 Qk, '- 1.0 QEp - 1.3 UEL - 1.3

Table 1. Multiple-Scale Turbulence Model Coefficients

Experiment HL&S K&C present standard

kP-EP—kt —Et kP-E P—kt—Et kP—EP—kt—Et k—E Model

Planar Jet 0.11 -0.12 - 0.103 0.104 0.114

Round Jet 0.085-0.095 - 0.118 0.111 0.126

21) Mix. Layer 0.179 0.146 0.126 0.152 0.159

r=0.0

21) Mix. Layer 0.052 0.061 0.064 0.078 0.082

r=0.3

Table 2. Spreading Rate Comparisons for Free Shear Flow

4. References

1 Tennekes, H. and Lumley, J. L., A First Course in Turbulence, The MIT Press

(1972).

2 Hanjalic, K., Launder, B.E., and Schiestel, R., "Multiple-time-scale concepts in

modeling turbulent flows," Turbulent Shear Flows, Springer-Verlag (1980).

3 Jones, W.P. and Launder, B.E., "The prediction of laminarization with a twoequation model of turbulence," Journal of Heat and Mass Transfer 15, 301-314

(1972).

4 Kim, S.W. and Chen, C.P., "A multiple-time-scale turbulence model based on

variable partioning of turbulent kinetic energy spectrum," NASA CR-179222,

(1987).

5 Harris, V.G. Graham, J.A.H. and Corrsin, S., "Further experiments in nearly homogeneous turbulent shear flow," Journal of Fluid Mechanics 81, 577-612 (1977).

6 Tavoularis, S. and Corrsin, S., "Experiments in nearly homogeneous shear flow

with a uniform mean temperature gradient. part 1," Journal of Fluid Mechanics

104, (1981).

7 Wygnanski, I. and Fiedler, H. E., "The two-dimensional mixing region," Journal

of Fluid Mechanics 41, 327-361 (1970).

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

(1), 69-71, (1973).

9 Spencer, B.W and Jones, B.G., "Staticical investigation of pressure and velocity
A Multiple-Scale Turbulence Model for Incompressible Flow 91

fieldss in the turbulent two-stream mixing layer," AIAA paper 71-0613, (1991).

to Gutmark, E. and Wygnanski, I., "The planar turbulent jet," Journal of Fluid

Mechanics 73, 465-495 (1976).

11 Bradbury, L. J. S., "The stucture of the self-preserving jet," Journal of Fluid

Mechanics 23, 31-64 (1965).

12 Heskestad, G., "Hot-wire measurements in a plane turbulent jet", Journal of

Applied Mechanics 42, 721-734 (1965).

13 Pope, S. B., "An explanation of the turbulent round-jet/plane-jet anomaly",

AIAA Journal 16, (1978).

14 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).

15 Wygnanski, I. and Fiedler, H. E., "Some measurements in the self preserving jet",

Journal of Fluid Mechanics 38, 577-612.
Spectral

Density

92 B. S. Duncan

Wave Number

Figure 1. Energy cascade prosess.

e wyggnonski do Fiedler

1 .0 O 

p

P Y

^^ \ — HLkSnmodel

K&C model

^^ \ --- k-c model

Q

0.8 \\

0.61

OX

E

0.4

0.2 \

-0.20 -0.10 0.00 0.10 0.20

(Y—Y.5)/x

Figure 2. Mean velocity profile for a planar mixing layer

with a speed ratio of 0. U,,, az : maximum velocity.
A Multiple-Scale Turbulence Model for Incompressible Flow 93

a Wygnonski & Fiedler 0.035 o6 o P

resent model US model K&C model

0.030 ... k-t model

0

0.025

/"-\O

0.020 0

/

\\O

k

/ \ E

I \p 

Y 0.015 ^

0.010 I \ Q^

I ^

0.005 O/

0.000 ---- %•

-0.20 -0.10 0.00 0.10 0.20

(Y-Y.$)/x

Figure 3. Turbulent kinetic energy profile for a planar mixing

layer with a speed ratio of 0. Um. s : maximum velocity.

e Wygqnonski do Fiedler

0 PPo en

1.0 O - Hd

present model

K&C model

O 0 "-_ k-t model

I p

I

0.8 O/ Q\

/ \

I I

ry '

6 EI0.6 I

D

6 I

I 1 g o

0.4 I I \\ ^^

o i 0\

I

00 I

0.2 ^\

I

00

0 0 o

0 0.0 0 ='-

-0.20 -0.10 0.00 0.10 0.20

(Y-Y.5)/x

Figure 4. Shear stress profile for a planar mixing layer with

a speed ratio of 0. U.,,: maximum velocity.
1.0

0.9

0.8

x 0.7

O

E

0.6

0.5

0.4

0.3

94 B. S. Duncan

1.6

1.4

1.2

C 

C, 1.0 4

g 0.8

eY

0.6

0.4

k,/k,

0.2

------------------------ 00-0.20 -0.10 0.00 0.10 0.20

(Y-Y.'.')/x

Figure 5. Energy transfer rate and kinetic energy ratios for

a plane mixing layer with a speed ratio of 0.

-0.08 -0.04 0.00 0.04 0.08

(Y—Y.!0/x

Figure 6. Mean velocity profile for a planar mixing layer

with a speed ratio of .3. Umax: maximum velocity.
0.040

0.035

0.030

r 0.025

a 0.020

Y 

0.015

0.010

0.005

0.000

n Spencer

ppresent model

- - - - HLdk model

— — - K&C model

------ k -c model

G Spencer

ppresent model

- - - - HL&S model

— — - K&C model

2S ---- k-c model

O

0.4

0.2

0.0

1.4

1.2

1.0

H

a 0.6

8

0.6

A Multiple-Scale Turbulence Model for Incompressible Flow 95

-0.08 -0.04 0.00 0.04 0.08

(Y-Y.5)/x

Figure 7. Turbulent kinetic energy profile for a planar mixing

layer with a speed ratio of .3. AU = Umas — Umin•

-0.08 -0.04 0.00 0.04 0.08

(Y -Y.5)/x

Figure 8. Shear stress profile for a planar mixing layer

with a speed ratio of .3. DU = Umax — Umin -
1.0

0.9

0.8

x 0.7

0

ED

0.6

0.5

0.4

0.3

96 B. S. Duncan

-0.08 -0.04 0.00 0.04 0.08

(Y -Y.S)I x

Figure 9. Grid resolution analysis of the mean velocity profile

for a planar mixing layer with a speed ratio of .3. U M" : maximum velocity.

1.0

A rZutmork k Wygnonski

\ *HSdkatod

0.8 \ — — Print model

Ki^C model

\ ---•• k-t model

0.6 \

0.4

>a

0.2

0.0 F

0. 100 0.05 0.10 0.15 0.20 0.25

Y/x

Figure 10. Mean velocity profile for a turbulent, planar jet.

Ur„ aZ : centerline velocity.
A Multiple-Scale Turbulence Model for Incompressible Flow 97

• Gutmork & WygnonW 0 o c n .a

,

o prodbury

o Nmkestbd

0. ' 0

0 

0^ • 'O —_— mttod

r toilet

0.0

.b .•' - k-s modN

0

<>

O\'^

0.06 \ \

0.03 \

g

}j 0.04

\ b.,

\ ,

0.03 \ ,''O

'•.O

0.02 •.O

0.01

`s.

0.00[

0.00 0.05 0.10 0.15 0.20 0.23

y/n

Figure 11. Turbulent kinetic energy profile for a planar jet.

Umax : centerline velocity.

2 . 5 0A utmork t WY9nonW

^eodbury

Q,^ `,^•• C^e^G modNde^

^i 0 1 . - k- c model

2.0 l^ \ \

\O

N

\n

x •,

8 

,.0 ,

0.s

a

O

0.0

0.00 0.05 0.10 0.15 0.20 0.23

r/x

Figure 12. Shear stress profile for a planar jet.

Umax : centerline velocity.
98 B. S. Duncan

1

e Wyg nonski & Fiedler

O Rodi

present model

K&C model

k–t Model

00 ^•

o O\\^'

n 0 \

O \

O \

O ^O

O

A O

o O

0

0.0 

F

0.00 0.05 0.10 0.15

Y/x

Figure 13. Mean velocity profile for a round jet.

Umax : centerline velocity.

0 . 12a Wyggnonski do Fiedler

------- O Rodi

ppresent model

— — K&C Model

p ------ k– t Model

0.10

O

0.08

Od0 \ ^

C"i

110

x

C 4C)

D 0.06

Y

0.04 \ \

0.02

OoOo n

0.00

0.00 0.05 0.10 0.15 0.20 0.25

y/x

0.8

0.6

0

E

0.4

0.2

0.20 0.25

Figure 14. Turbulent kinetic energy profile for a round jet.

Umax: centerline velocity.
2.5

2.

0.

N

O ^'

E

S

A Multiple-Scale Turbulence Model for Incompressible Flow gg

aWgyonski & Fiedler

o Abbiss \ -'x:— resent model \ — _K&C model \ k-t model

D O \\ o \

o g\

^O

o \

O \

O

o ^

r) ,.

v

0.0

0.00 0.05 0.10 0.15 0.20 0.25

Y/x

Figure 15. Shear stress profile for a round jet.

Um,,: centerline velocity.

1.0

9,/Co

0.8

0U

4

CO 0.6

Q

Y

0.4

0.2 k3kD

0.00 0.04 0.08 0.12 0.16 0.20

Y/x

Figure 16. Dissipation and kinetic energy ratios for a round jet.
Page intentionally left blank 
Center for Modeling of Turbulence and Transition 101

Research Briefs - 1993

Modeling of Turbulent Shear Flows

William W. Lion

1. Motivation and Objective

The development of advanced propulsion systems for high speed aerospace vehicles will require accurate computational models of turbulence that can be used in

the CFD calculation of individual flow components. The immediate objective of the

research activity described here is to identify robust modeling techniques for compressible turbulent flows. The main goal is to develop second-order Reynolds-stress

models for compressible flows.

2. Work Accomplished

Several approaches are now being pursued. A two-scale eddy-viscosity model was

developed earlier 1,2 and successfully applied to the calculation of compressible shear

layers. In this reporting period, the model was further assessed in the solution of

compressible boundary layers3.

A new modeling technique that describes the large-scale coherent structures in

free shear flows using a weakly nonlinear theory is also being studied. The model was

shown to be capable of providing a deterministic description of the time-dependent

turbulent motions at the large scale in incompressible free mixing layers. To further

assess the model in describing the coherent structures generated by a more complex

flow mechanism, a simple linear analysis was performed for curved mixing layers in

the past year'. It was found that some observed characteristics of the large-scale

structures might very well be described by the simple analysis.

Also, a part of the research activity involves the development and validation of a

new model equation for the turbulent dissipation rate s . The new model dissipation

rate equation may increase the numerical stability of the second-order Reynoldsstress model in complex flow calculations. In the following, progress made on the

individual research subjects in the current reporting period is briefly reviewed. The

complete analyses are given in the related cited reports.

2.1 Validation of compressible two-scale model in boundary layers

The rational for exploring the two-scale approach in the modeling of compressible turbulent flows is described in detail in a NASA TMLS . Briefly, the observed

eddy shocklets and the modification of the energy transfer process due to flow compressibility render compressible turbulence non-equilibrium, i.e., the energy is not

transferred at the same rate between the eddies of different scales. Therefore, singlescale modeling of the non-equilibrium compressible turbulence is not sufficient. The

two-scale model approach adopted here relinquishes the assumption that the energy

is transferred from the large scale to the small scale at the same rate as it is dissipated by the molecular viscosity. This allows the eddies of different sizes to respond
102 W. W. Liou

differently to compressibility effects. Since the eddy shocklets were observed in

numerical experiments to be scaled mainly with the large eddies, compressibility effects were applied explicitly to the transport equations governing the kinetic energy

of the large scale and and the rate of energy transfer from the large scale to the

small scale. The model equations, with the boundary-layer approximation, were

written as,

PDkp=

Dt

a 

o+ ^T ) akp]

ay

+ (1 — a2Mt)AT(au)2

ay ay

a3 Mt )PEP (1)

P

P

Dt

= ay ^(μ + ^T) ay ] + Cpi k μT(ay )2 — ( CP2 — Cps Mt )P ^ (2)

EP jI ^J

P

Dt, = a Ki + ^^) a

y

DEta

P = _ — Dt ay

μT aEt Et EP

^(μ + —) —] + Ct l p—

ay

Et2

— Ct2P 1 (4)

° Et kt kt

where kP and kt denote the turbulent kinetic energy of the large and the small

scale, respectively. EP denotes the energy transfer between the different scales and

Et represents the turbulent dissipation. This model has been successfully applied in

the prediction of compressible shear layers. This includes the observed reduction

of growth rate as well as the mean velocity profile. In this reporting period, the

model was further assessed in the solution of compressible boundary layers. To

account for the effects of the wall, a compressible wall-function was introduced. For

turbulent boundary layers with adiabatic wall or cooled wall, it has been shown

that the mean velocity in the fully turbulent region near the wall can be described

by a log function, similar to the log-law of the wall for incompressible boundary

layers. The use of the wall function to bridge from the high Reynolds number region

to the wall also eliminates the need for devising near-wall damping functions and

the compressibility model can be truly put to test. The results of the assessment

of the two-scale model in compressible boundary layers are included in a NASA

TM3 . The following two figures are extracted from that report. Figure 1 shows

the comparison of the predicted mean velocity profiles with measurement for a

compressible boundary layer. The flow Mach number is 2.831 and the Reynolds

number based on the momentum thickness is 424,070. The flow was studied in

an experiment and the measurement was reviewed by Fernholz and finly. The

designated flow number is 65020101. Figure 1 shows that both the present model

and the standard k — E, with Sarkar's compressibility correction, predict well the

mean velocity profile. Figure 2 shows the comparison of the calculated skin friction

coefficients, C f , using the present model and the standard k — E model, with the

Van Driest II correlation. The Reynolds number based on the momentum thickness

is 10,000. The wall is adiabatic. There is a good agreement between the present
Modeling of Turbulent Shear Flows 103

model and the Van Driest II formula at all the Mach numbers tested. The standard

k — E model, with Sarkar's compressibility corrections, underpredicts Cf . Overall,

the present two-scale model produces good agreement with measurements in both

compressible shear layers and compressible boundary layers with the same model

coefficients, which has been shown to be not feasible using the single-scale k — e

model in conjunction with other compressibility corrections.

2.2 Linear analysis of curved mixing layers

In a plane mixing layer, it has been shown that the profile distributions of the

large-scale structure can be described by linear disturbances generated by shear

instability, or Kelvin-Helmholtz instability. In an earlier report s , the author has

further shown that a predictive turbulence model can be obtained by using a weakly

nonlinear wave theory in which the local structural characteristics are described by

linear wave disturbances and their amplitudes are determined by their kinetic energy

evolution. For a curved mixing layer, the effect of centrifugal forces is introduced.

The centrifugal force may stabilize or destabilize fluctuations, depending on the

velocity distribution of the flow. The additional, yet identifiable, complexity in the

incompressible curved mixing layer presents a good test for the applicability of the

wave model in cases where flow fluctuations can be affected by more complicated

mechanisms, such as flow compressibility. Large-scale spanwise vortical structures

have been observed in both stably and unstably curved turbulent mixing layers.

These structures appear to be less two-dimensional in an unstably curved mixing

layer than they are in a stably curved mixing layer. Stationary vortical structures

have also been observed in experiments in a unstably curved mixing layer. A simple

linear analysis was performed for curved mixing layers. We are unaware of any

previsouly reported work studying this particular problem. The governing equations

for the disturbance can be reduced to the following single equation' for the vertical

velocity fluctuation, v(y),

dzv 2aU dv (au — Q) d

y z + R dy

—[(aU — a-)(az + 02 ) + ad 

z U — 2U ,Qz dU^v = 0. (5)

dyzR aU — o, dy

Figure 3 shows the distributions of the eigenfunction, v(y), for the most unstable

modes with two different stabilizing curvatures. The shapes of the eigenfunction

for the curved cases are seen to be similar to those of a plane shear layer. With

the centrifugal forces also becoming a destabilizing factor, or when the mixing is

unstably curved, more than one linear instability mode can exist in the flow. In

addition to the spatially convecting Kelvin-Helmholtz modes, stationary modes that

assume no periodicity in the streamwise direction are also found. Figure 4 shows the

countour of the streamwise vorticity of one such mode that has the highest growth

rate among the hierarchy of instability modes found at the same spanwise number.

The vorticity pattern shows that this mode is a vortex pair counterly rotating in the

streamwise direction, similar to the Gortler vortices that are observed in a flow along
104 W. W. Liou

a concave wall (destabilizing curvature). It was also found that the number of the

vortex pair increases for the mode with lower growth rate in the same hierarchy. In

a recent experiment, spatially stationary streamwise counter-rotating vortex pairs

were observed in the near field mixing region of underexpanced jets, where the flow

was also subjected to curvature effects. These preliminary results seem to indicate

that linear analyses can be used to provide local profile description of the large-scale

structure under the influence of streamline curvature.

2.3 Assessment of a new model dissipation rate equation

The new model dissipation rate equation was derived from the equation for the

mean square of fluctuation vorticity ( or enstrophy ). Models for the dominant terms

in the equation, such as the source and sink for the production of enstrophy, were

developed by examining the dynamics of vorticity fluctuations. This first-principle

based model dissipation rate equation was applied to a variety of turbulent flows

as part of a two-equation model. The details of the two-equation model and the

results of model application are described by T.H. Shih in this report and in a NASA

TM5 . Figure 5 shows the comparison of the present model predictions of the mean

velocity profile with measurements for incompressible round jets. The k — E model

prediction is also included for comparison. The agreement between the predictions

and the measurement is better for the present model than it is for the standard

k — E model. The same is also true for the spreading rate of the jet. It should be

noted that the new dissipation rate model equation can be used in conjunction with

advanced second-order models. In fact, it is expected that the new model equation

may improve the numerical stability problem associated with the Reynolds stress

equations.

3. Future Plans

(1) Identify a number of appropriate complex compressible flows as bench-mark

test cases for model assessment and validation. Preliminary candidates include, but

are not limited to, supersonic ramp flows, transonic flow over an airfoil, and flows

about turbine blades.

(2) Implement the two-scale compressible model into Navier-Stokes CFD codes.

(3) Perform linear analyses for compressible plane and curved shear layers to

investigate the effects of compressibility on the characteristics of the coherent largescale structure.
Modeling of Turbulent Shear Flows 105

4. References

1 Duncan, B. S., Liou, W.W., and Shih, T.-H., "A Multiple-Scale Turbulence Model

for Incompressible Flow," AIAA 93-0086 (1993).

z Liou, W.W. and Shih, T.-H., "A Multiple-Scale Model for Compressible Turbulent Flows," NASA TM 106072 (1993).

s Liou, W.W. and Shih, T.-H., "Compressible Turbulent Boundary Layer Predictions Using a Multiple-Scale Model," to appear in NASA TM (1993)

4 Liou, W.W., "Linear Instability of Curved free Shear Layers," AIAA 93-3253

(1993).

5 Shih, T.-H., Liou, W.W., Shabbir, A., Yang, Z., and Zhu, J., "A Vorticity Dynamics Based Model for the Turbulent Dissipation," NASA TM 106177 (1993).

6 Liou, W.W. and Morris, P.J., "Weakly Nonlinear Models for Turbulent Mixing

in a Plane Mixing Layer," Phys. Fluids, 4, pp. 2798-2808 (1992).
106 W. W. Liou

0.12

0.10

0.08

Y (m•) 0.06

0.04

0.02

0.00

0 401_0

— - k model & Serkar et nl.

present model

measurement

65020107

?0

ii

U.

Figure 1. Comparison of mean velocity profiles.M,, = 2.831, Ree = 424, 070.

1.2

1.0

Cf 0.8

C f 0

0.6

0.4

— - k-E model & Suku et sl.

present model

^^ Van Driest II

0.2 ' I

I'

0.0 1.0 2.0 3.0 4.0 5.0 6.0

MI.

Figure 2. Comparison of skin friction coefficients. M,,, = 5.0, Ree = 10, 000
-5.0 0.0 5.0 10.0

Y

1.5

1.0

V(y) 0.e

OX

-0.5

10.0

0 8.83

Modeling of Turbulent Shear Flows 107

Figure 3. Distributions of the eigenfunction v(y) for the most unstable Kelvin-Helmholtz

modes with Q = 0.3. Stably curved.

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

Y

Z

Figure 4. Streamwise vorticity w,, of the first modes of Q = 0.7 of zero frequency. Unstably

curved.
108 W. W. Liou

1.2

1.0

e

0.8

e

eq

UI Ue 0.6 e ON

e q

0.4

e q \^

A q

A

0.2

0.0

e Wygnanski & Fielder

q Rodi

--- ske model

— present model

A

0.00 0.05 0.10 0.15 0.20 0.25

y/xFigure 5. Self-similar mean velocity profiles for a round jet.
Center for Modeling of Turbulence and Transition 109

Research Briefs - 1993

PDF Models for Compressible

Reactive Flows and DNS

A.T. Hsu

1. Motivation and Objective

1)PDF. The objective of the present work is to develop a probability density

function (pdf) turbulence model for compressible reacting flows for use with a finitevolume flow solver. The probability density function of the species mass fraction

and enthalpy are obtained by solving a pdf evolution equation using a Monte-Carlo

scheme. The pdf solution procedure is coupled with a compressible finite-volume

flow solver which provides the velocity and pressure fields. A modeled pdf equation for compressible flows, capable of capturing shock waves and suitable to the

present coupling scheme, is proposed and tested. Convergence of the combined

finite-volume Monte-Carlo solution procedure is discussed, and an averaging procedure is developed to provide smooth Monte-Carlo solutions to ensure convergence.

Two supersonic diffusion flames are studied using the proposed pdf model and the

results are compared with experimental data; marked improvements over solutions

without pdf are observed.

2) DNS. Direct numerical simulation results and theoretical analysis are presented for the effect of the Coriolis force on compressible homogeneous isotropic

turbulence. It is shown that the Coriolis force serves as a frequency modulator on

turbulence. While the Coriolis force neither creates nor destroys turbulent kinetic

energy, it redistributes energy by eliminating low frequency waves and transferring

energy to waves with a frequency of 252. The dissipation rate of turbulent kinetic

energy can be either reduced or enhanced depending on whether or not the ratio

between the rotation time scale and the Kolmogorov time scale is much greater

than one. It has been demonstrated both theoretically and numerically that the

Taylor-Proudman theorem is applicable to homogeneous turbulence only when the

time scale of rotation, defined as the inverse of the frequency of the inertial waves,

approaches the Kolmogorov time scale, and that two-dimensionalization occurs in

this regime.

2. Work Accomplished

In the past year, a pdf model for compressible reactive flows has been developed

and validated in 2D supersonic hydrogen-air flames. The analysis of compressible

homogeneous turbulence under rotation is completed.

2.1 A PDF Model for Compressible Reactive Flows

The most attractive feature of the probability density function (pdf) method is

its ability to overcome the chemistry closure problem in turbulent reacting flow

computations. Much progress has been made in both pdf theory and application
110 A. T. Hsu

during the past decade'. However, most of the previous developments of pdf models

(using a pdf evolution equation) were restricted to low speed combustion. The

objective of the present work is to develop a pdf model for high speed compressible

flows that can be coupled with existing finite-volume or finite-difference flow solvers.

High speed compressible reacting flows present some unique challenges to pdf

modeling because of the existence of shocks and strong dilatation terms. In principle, the joint pdf of velocity, pressure, temperature, and species mass fraction can

be solved using a self-contained Monte-Carlo solver, and such a scheme should be

sufficient for solving compressible reacting flow problems'. Unfortunately, a shock

capturing pdf pressure solver is currently unavailable, and it may be a while before

shock capturing schemes using pdf method would be devoloped, and a even longer

time for these schemes to mature to the extent that they could be regularly used in

industry. Thus, it appears that the only alternative currently available for immediate application of the pdf method to high speed flows calls for the decoupling of

the aerodynamics from the scalar field. Since shock capturing finite-difference and

finite-volume codes have reached a certain degree of maturity, and are widely used

in industry, it seems, at least from the standpoint of practical application, that the

use of the composition pdf together with a finite-dffierence or finite-volume code is

more beneficial.

The pdf model and the corresponding Monte-Carlo solver developed in the present

work is general enough to be applied with any existing finite-volume or finitedifference codes. To demonstrate the portability of the pdf solver, we have applied

it in conjunction with two distinctly different versions of a finite-volume code known

as RPLUS3 : The first version is the original RPLUS designed specifically for high

speed combustion problems. It solves the full compressible Navier-Stokes equations

and species transport equations using an LU scheme. Algebraic and two-equation

turbulence models are available in the code. The second version is a modified

RPLUS code developed for the modeling of turbulent reacting flows with sprays

occurring inside a Wankel engine'; sufficient changes in the numerical algorithm

and code structures have been made in this version for it to be regarded as a different code. When using a pdf algorithm to solve for chemical reactions, the species

transport equations in the finite-volume code are replaced by the compostion pdf

equation.

Since the flow field, including the mean velocity, density, pressure, turbulent

kinetic energy, etc., are provided by the finite-volume flow solver, we only need to

solve for the species mass fractions and energy (or enthalpy, temperature, etc.) in

the Monte-Carlo pdf solver. For low speed reacting flows, there are two unclosed

terms in the species and temperature joint pdf equation; namely, the turbulent

diffusion term and the molecular diffusion term. For high speed flows, depending

on the specific formulation chosen for the energy equation, many more unclosed

terms could appear'. The pros and cons of various formulations are discussed in

the present work. The formulation best suited for the present objective is studied

in detail, and a modeling procedure is proposed and tested.

In incompressible flow computations, the flow equations can essentially be solved
PDF Models for Compressible Reactive Flows and DNS 111

independently of the pdf equation since the momentum and energy equations are

decoupled; one only needs to transfer information from the flow solver to the pdf

solver, but not vice versa. In contrast, the presence of the strong pressure and

density gradients makes the coupling between flow equations and the pdf equation

very important for compressible flows. Because the finite-volume solution of the

flow field is smooth, the transfer of information from the flow solver to the MonteCarlo pdf solver is straightforward, but information transfer the other way around

presents a challenge because of the relatively large statistical error. A way of obtaining smooth averaged solutions from a Monte-Carlo pdf solver, using a relatively

small number of samples that is within the capacity of today's computing facilities,

becomes an important issue in the finite-volume/Monte-Carlo coupling process.

An issue related to the problem of coupling mentioned in the previous paragraph

is the convergence of a Monte-Carlo procedure in solving an elliptic flow problem.

Applications of pdf models to elliptic flow computations are few 6,7,8,9 . When a

Monte-Carlo solver is used to simulate elliptic flows, both the definition and the

criteria of convergence are unclear, and little discussed. Since the statistical error

in a Monte-Carlo method is often much larger than the truncation error in a finitevolume or finite-difference scheme, the convergence of a combined finite-volume

Monte-Carlo scheme is difficult to measure. Without special treatment, the solution

often appears non-convergent.

A combined ensemble and time-averaging procedure for the solution is proposed in

the present work. This averaging scheme makes the measurement of the convergence

of a Monte-Carlo scheme possible, and provides sufficiently smooth solutions to be

fed back into the finite-volume flow solver, making the coupling between Monte

Carlo and finite-volume solvers feasible.

The compressible pdf model is validated using a non-reacting supersonic flow over

a ten degree ramp, where the temperature rise across the shock is computed using

the modeled pdf equation and compared with the theoretical solution. Two cases

of supersonic hydrogen combustion are studied using the proposed model: A twodimensional supersonic wall jet and a supersonic axisymmetric jet. Results compare

well with experimental data. In both of these supersonic jet cases it was shown that

with the pdf model, the solutions are markedly improved over those obtained from

a conventional CFD combustion code. Preliminary results from 3D applications are

also reported in this paper.

2.1.1 Pdf Formulation for Compressible Flows

For compressible flows, the energy equation can be cast in a number of different

forms. Pope chose to use total enthalpy2:

PH,t + pui H

j = —p ,t — qj,j + (T.7iui),.9. (1.1)

In the above equation, as well as in the equations that follow, repeated indices imply

summation over the range of the indices. By neglecting molecular diffusion effects

and assuming steady flow, the energy equation reduces to constant total enthalpy
112 A. T. Hsu

for a given sample particle. The total enthalpy formulation provides the simplest

form of the energy equation and, if the pdf for pressure and velocities are known,

the formulation can be applied to compressible flows without introducing any new

modeled terms 2 . However, in the present case, the velocity field is obtained from

a finite-volume solver, and the pdf for the velocity field is unknown. Without the

velocity pdf, one cannot back out the temperature of each sample point from a

given total enthalpy.

Kollmann5 suggested the use of specific internal energy, which leads to a formulation with A = uj,;, the velocity divergence as an random variable. In order to

solve for A, Kollmann introduced a transport equation for the velocity divergence

into the pdf formulation. The resulting pdf equation produces a large number of

new unknown terms that need modeling. Without enough experimental or direct

numercial simulation data to establish models for these new terms, it is hard to

estimate the viability of this approach.

In the present study, considering the fact that the velocity pdf is not known in our

solution, we choose to use the specific enthalpy formulation to minimize the need

of devising new models. As we shall show in the following, the specific enthalpy

formulation allows us to use existing models from previous works on second order

closure models for compressible turbulence.

The energy equation in terms of enthalpy is:

Ph,t + Puj h ,j — — 

D

Dp — qj,j + (1.2)

where 4^ is the dissipation due to viscosity. Neglecting (D from the energy equation, the evolution equation for the joint pdf of species mass fractions and specific

enthalpy, P(Y1 , • • •, YN) h; xi , t), can be written as:

(PP),t + (P < uj > P)j + (PwjP),Y; =

— (P < u,lYi, h > P)j — (< PDjYj,iilYi, h > P),Y; — (( DpIYi, h ) P)

(1.3)

h

Where the right hand side terms represent turbulent diffusion, molecular diffusion,

and the pressure effect, respectively. All the conditional means on the right hand

side of the equation are extra unknowns and need to be modeled.

In what follows the modeling of the unknown conditional means is discussed.

Since comparing with the pdf equation for low speed flows the only new term is

the conditional mean of the material derivative of pressure, we start by deriving an

appropriate model for this term.

Using C to denote the conditions in the mean, the new term can be written as

( Dp IC

I

=< p I C > ,t + < uip, i. I C >; (1.4)

decomposing the random variables into means and fluctuations, p =< p > +p' and

ui =</ ui > +ui, we have

( Dt IC) =< p>,t+<ui><p >,i +<uip'IC >,i+<p'u"ilCi >. (1.5)

--I
PDF Models for Compressible Reactive Flows and DNS 113

To use existing models for the last two terms in the above expression, we need first

to assume that the product of the fluctuating velocity and fluctuating pressure is

statistically independent of the species mass fractions and specific enthalpy.

With this assumption, the last two unknown terms can be readily modeled: The

first of the two terms, < uip' > ,i, is the trace of the pressure-strain rate tensor in

second order closure models. Most second order closure models give the following

expression for this term 8:

< u'p' > ,i. = 0.8p < k >< ui >,i,(1.6)

where < k > is the mean turbulent kinetic energy.

The last term, < p'ui i >, is the so called pressure dilatation term in second order

closure models for compressible flows. Following Sarkar9 , we write:

< p'uL,i >= —a2 pPrMt + a3 pEMt (1.7)

where Pr is the turbulent production, E is the dissipation of turbulent kinetic energy,

Mt is the turbulent Mach number, and a l and a2 are model constants given by

Sarkar as 0.15 and 0.2, respectively. All the information needed in the above models

is available from a flow field solution using a k — E model.

The final modeled pdf equation then becomes:

( PP),t + (P < u; > P),j + (Pw,P),Y; = (Dt P,j),i + M(P) — (SpP),h . (1.8)

where the first two terms on the right hand side of the equation are the modeled

terms for turbulent diffusion and molecular mixing, of which details can be found

in refs. 12,13; the last term is the term representing the compressible effect, with a

new source term defined as

SP =< p >,t + < ui >< p >.i +0.8p < k >< ui >,i — a2 pPrMt + asPEMt , (1.9)

which can be regarded as the convection velocity of a sample particle in the hdirection in the space spanned by h and Yi's.

The new term in the pdf equation simulates the compressibility effect; in other

words, it transfers information of a compressible flow field to the pdf solution. As

a test case for the above model, a non-reacting supersonic flow over a ten degree

ramp is calculated using a 40 x 50 grid. The pdf solution of the temperature across

the oblique shock is shown in Fig. I.I. The pdf solutions are taken from several

different times in the solution procedure. Since only 100 particles per cell were

used for the calculation and no time-averaging or smoothing schemes were used,

the oscillation in the solution is expected. A step function in the figure shows the

theoretical solution of temperature for the same shock. The results show that the

new model picks up the temperature rise across the shock fairly accurately.

A word of caution must be given here: The source term SP can be very large for

high Mach number flows or flows with strong shocks. When using an explicit time
114 A. T. Hsu

marching scheme in the Monte-Carlo simulation, time step must be carefully chosen

such that the "CFL number" does not exceed one to ensure a converged solution.

2.1.2 Solution Procedure

Since the Monte-Carlo method used for solving the pdf evolution equation has

been given in a previous paper' 2, and the numerical schemes used in the finitevolume code are well documented, we will only describe those details that are important or new for the present work on compressible pdf modeling and elliptic flow

solvers.

Coupling

When coupling a pdf solver with RPLUS, the species transport equations in the

RPLUS code are no longer needed. The information we need, at each marching time

step, from the flow solver (RPLUS) includes the mean velocity, pressure, density,

and a turbulence time scale or quantities such as the turbulent kinetic energy and

dissipation rate. The species transport and chemical reactions are simulated by

solving the pdf evolution equation. At every time step, the species mass fraction

field from the pdf solution is fed back to the mean flow solver for the computation

of temperature and pressure. The Monte Carlo and finite-volume solvers are run

in parallel, and information exchange occurs at every time step until a converged

solution is obtained.

The finite-volume code requires smooth temperature and pressure fields as input,

but a Monte-Carlo solution is usually not smooth unless the number of sample particles used is extremely large. Therefore, the transfer of information from a MonteCarlo solver to a finite-volume solver could either require a prohibitive amount of

computer memory, or cause divergence in the finite-volume solution. An economical

way of obtaining smooth solutions from a Monte-Carlo solver becomes a key issue

in the coupling procedure.

Pope and coworkers (e.g., 12 ) use spline curve fitting in their parabolic flow calculations to generate smooth 1D Monte-Carlo solutions. For 2D and 3D data, splines

are difficult to apply and a different way of smoothing must be found. A combined

ensemble and time-averaging scheme, as described in the following section, is used

in the present study to solve this problem.

Convergence

The convergence criteria for a Monte-Carlo simulation of a steady elliptic flow

problem is not well defined. When we use N events to simulate a random process, it

is probable that the error is smaller than 1/vrN— , which is still a rather large number

compared to the truncation errors of a finite-volume solution. The top curve in Fig.

1.2 shows the L2 norm convergence history of the ensemble averaged solution of a

jet-in-cross-flow calculation, where the error is defined as

Error = f (< On+1 > — < on >) 2 dR (1.10)

R
PDF Models for Compressible Reactive Flows and DNS 115

where < 0 > is the ensemble mean of an unknown variable, 0, n denotes the time

step, and R the solution domain. Using ensemble average only, the error curve

levels oft' before a converged solution is obtained.

The reason for this inconsistency between the L2 error curve and actual convergence history is that the relatively large statistical error ( 1/ 500 for the present

simulation) becomes dominant before a converged solution is obtained. Another

major problem caused by the relatively large statistical error is that when the

Monte-Carlo solution is transferred to the finite-volume flow solver, the statistical

error may cause the finite-volume solution to diverge. We clearly need more sampling points; however, the computer cpu time and memory requirements become

prohibitive when we further increase the number of samples.

For steady flow problems, a combined ensemble and time-averaging is used to

overcome this problem: Time averaging is performed over a set of ensemble-averaged

solutions to obtain a smoother solution. This averaging procedure is performed

continuously at every time step; for instance, supposing that we are averaging over

m time steps, at the nth time step, the information from time step n—m is discarded,

and the ensemble averaged solution from the current time step is added to the time

averaging process. A similar time averaging method has been used by Correa and

Pope9.

The curves in Fig. 1.2 shows the convergence history from averaging over 1 to

1000 iteration steps. The large jumps of the curves are caused by abandoning the

sample from the initial condition; for instance, for the error curve of averaging over

1000 time steps, the jump occurs at N = 1000, where N is the number of iteration.

This type of large excursion in the error only happens when N = Naverage, and

does not appear again at larger iterations. It appears that increasing the number

of time steps averaged by approximately three fold cause the error to reduce an

order of magnitude. The use of 100 time steps is sufficient in obtaining a smooth

solution for the purpose of transferring information to the finite-volume flow solver.

Our sample size is 500 per cell. The use of 1000 iteration steps is equivalent to

the use of 500,000 particles per cell, a number that can not possibly be handled by

today's supercomputers. This scheme provides a powerful tool for one to obtain

huge samples with limited computer resources.

In the above example, we have assumed the time step used in the Monte-Carlo

computation to be large enough that the errors in the solutions of two consecutive

time steps are statistically independent. When time steps used are very small, this

assumption may not be valid, and the effect of time averaging is greatly diminished.

In which case, we chose to select solutions from every M time steps for the time

averaging process, and make sure that the correlation of the random error reaches

zero for a time t > MAL In some of the computations reported in this paper, we

used solutions from every ten time steps in the averaging process.

Efficiency

A flow trace analysis of the pdf solver revealed that an overwhelming portion of

the cpu time was spent in selecting particles randomly for convection and diffusion
116 A. T. Hsu

simulations; this is because the selecting process can not be vectorized effectively as

is. In the case of non-reacting flow computations, more than 95% of the cpu time

is spent on this process. The computational burden lies on the need to generate

five random vectors of length N, for a sample size of N. A novel approach was

devised to accelerate the solution procedure: At the beginning of each iteration,

five random vectors of length 2N are selected. For each cell, five random indices

are selected, starting from which the samples are selected using the given random

arrays. This removed the need to generate five random arrays for each cell, and

allowed vectorization of the procedure. With this procedure, the cpu required is

reduced by a factor of four.

2.1.3 Results and Discussion

The pdf procedure is validated using experimental data for two cases of supersonic

hydrogen diffusion flames: a supersonic hydrogen round jet and supersonic wall jet.

The solution procedures used and the results are presented in the following sections.

To demonstrate the applicability of the pdf model in general 31) flows, a 3D jet-incross flow is presented.

Supersonic Round Jet

The numerical computations for the case involving the injection of a supersonic

hydrogen jet coaxially into a high-temperature, vitiated air-stream have been performed by the code that was originally developed for the modeling of turbulent, reacting flows with sprays occurring inside of a Wankel engine, which evolved from the

RPLUS code. The finite-difference formulation is based on an Eulerian-Lagrangian

approach where the unsteady, Navier-Stokes equations for a perfect gas-mixture

with variable properties together with the standard two-equation k-E turbulence

equations are solved in generalized, Eulerian coordinates on a moving grid by making use of an implicit finite-volume, Steger-Warming flux vector splitting scheme,

and the liquid-phase equations are solved in Lagrangian coordinates. The turbulent

viscosity is computed by a compressibility correction as proposed by Villasenor et

al". A complete description of the code can be found in Ref. 4. The code has recently been modified to accommodate axisymmetric flow fields before being coupled

with the pdf solver.

Combustion is modeled by the following global, five-species, two-step model developed by Rogers and Chinitz 15 for H2 -02 combustion:

H2 + 02 kf1120H

20H + H2 kf2 2H20

(1.11)

where the forward reaction rates are given as

k fi = AiTN; exp(—Ei /RT) (1.12)
PDF Models for Compressible Reactive Flows and DNS 117

with the coefficients given as:

A1=11.4x1047

Ni = —10.

El = 20.35 x 106

A2 = 2.5 x 1064

(1.13)

N2 = — 13.

E2 = 177.82 x 106

The units of Ai is m3 /mole • sec, and the units of Ei is J/kg — mole. The backward reaction rates are calculated from the Gibbs free energy and the equilibrium

coefficients.

The global model is reported to predict the oxidation of hydrogen adequately in

flows that are not dominated by long ignition delay times. The model is tested for

pressures at 1 atm, initial mixture temperatures of 1000-2000 K, and fuel/oxidizer

equivalence ratios ranging between 0.2-2.0.

The geometry and flow conditions for the coaxial-jet case are given in Fig. 1.3.

The test conditions are taken from Evans et al i6 . This test case has been the subject

of investigation by several authors 17,18 where the flow field is modeled by different

assumptions: (1) turbulence is modeled by making use of either algebraic or two

equation k-E models, (2) combustion is modeled by either finite-rate or equilibrium

chemistry models, and/or (3) the effect of turbulence on reaction rates is modeled

by either the eddy-breakup model or the assumed probability density functions

approach to describe the species fluctuations.

The computational grid used in the flow calculations is 51x61 with the grid extending 30.0 fuel-injector diameters in the flow direction and 6.5 fuel-injector diameters in the transverse direction. The grid is stretched in such a way so that more

resolution is obtained within 1.75 fuel-injector diameters about the nonindent in

the transverse direction and also more resolution is provided near the inflow in the

axial direction. The flow computations are initiated at x/d =0.33, the nearest point

at which the measurements are made. At the inflow all the flow variables are specified. The initial conditions used in the computations are similar to those chosen by

Evans et al 16, except the gas composition of central jet, which is assumed to contain small amounts of H2 O and N2 mass fractions (=0.1) instead of pure hydrogen.

The specification of inflow conditions contains a slight degree of uncertainty since

the only available information at this location is the measured Pitot pressures. And

all the other variables are chosen approximately to fit the measured pressure data.

At the exit all the flow variables are extrapolated from the interior. The treatment

of the upper boundary is determined by applying inviscid wall conditions. Values

along the nonindent are obtained by applying zero gradient extrapolation, except

the radial velocity, which is set equal to zero.

The pdf solution is obtained by making use of 100 particles per cell. The species

mass-fraction field that is supplied to the finite-volume solver from the Monte-Carlo
118 A. T. Hsu

solver is obtained from averaging the pdf solutions over the previous 100 time steps.

The combined Monte Carlo and finite-volume solver required about 0.7 ms of cpu

time per time-step per cell on a CRAY Y-MP. The inclusion of the pdf solver is

found to increase the computational effort by a factor of 2.2. For the case examined,

less than 2000 time-steps are required for the computations to reach a converged

solution.

Figs. 1.4a-d show the radial profiles of four major species, H2 , 02 , H2 O, and

N2 , from the combined Monte Carlo/finite-volume computations compared with

experimental data at four different axial locations, x/d = 8.26, 15.5, 21.7, and 27.9,

respectively. The numerical results match fairly well with the experimental data

considering the fact that the estimated error in measurements could be as large

as 15% 14 and the uncertainty involved in specifying the inflow conditions. The

estimated error in the measurements is reportedly due to the possibility that the

gas composition might be altered because of additional reactions being taking place

in the gas sampling probes as a gas chromatograph is used to measure the species

concentrations 16 . The disagreement in the calculated results showing lower than

the experimental values for the mass fractions of H2 O, 02, and H2 at axial locations

of x/d = 21.7 and 27.9 might also be attributed to the uncertainty involved in the

specification of the inflow conditions in the jet region. However, no effort is made

in the present computations to study the effect of different inflow conditions on the

ensuing flow field.

To examine the effect of the pdf model on the predicted results, a separate set

of finite-volume computations were performed without the pdf solver, with the

chemiscal source term evaluated using the mean temperature and species mass

fractions. The assumption of equal diffusivity is made in the compuation. Combustion in these computations is also modeled by the global, 2-step combustion

model of Rogers and Chinitz. And the corresponding results are summarized in

Figs. 1.5a-d. The combined Monte Carlo and finite-volume computations seem to

predict the position of the peak H2 O mass fraction rather accurately while in the

finite-volume computations the position of the flame zone is shifted radially outward

into the high-temperature region indicating that much more 02 is consumed than

what is observed experimentally. While Monte Carlo/finite-volume computations

predict negligible amounts of oxygen within the central-jet region, the non-pdf computations predict substantial amounts of 02 within this region at axial locations

of x/d =8.26 and 15.5. The reasons for this discrepancy could be attributed to a

lack of consideration of the temperature fluctuations on the reaction rate. If the

neglected temperature fluctuations are large, the reaction rates based on the mean

gas temperatures are likely to lead to smaller oxidation rates as the calculated gas

temperatures in this region remain lower than 600 K. At succeeding axial locations,

the H2 O mass fraction grows and spreads much faster in the non-pdf solution as

compared with the pdf solution. Similar to the findings in the present non-pdf computations, most if not all the previous computations, reported a radially-outward

shift in the location of the peak H2 O mass fraction into the high-temperature region

and, also, reported that 02 is consumed more rapidly in the calculated results. The
PDF Models for Compressible Reactive Flows and DNS 119

most likely cause for this disagreement may be due to incorrect temperature dependence on some of the reaction rates, leading to rapid combustion as the temperature

rises in the flame zonela

The above results demonstrated that the use of pdf turbulence model can improve

the accuracy of combustion computations considerably.

Supersonic Wall Jet

The second test case is the experiment reported by Burrows and Kurkov 19 in

1973. This experiment has been investigated numerically by several authors 17,18

Figure 6 shows its flow configuration. The two-dimensional test section measures

35.6 cm downstream of a H2 wall jet. The jet is mixed with a vitiated air stream

and burned. A high pressure gas generator supplied Mach 2.44 vitiated air at

approximately 1270K. The composition of the air is 25.8% 0 2 , 25.6% H2 O, and

48.6% N2 by mass.

The simulation of the wall-jet is performed by coupling the pdf solver with

RPLUS3 . The Baldwin-Lomax algebraic turbulence model is used in this calculation. The turbulence diffusivity used is calculated based on the eddy viscosity

and the assumption that the turbulence Schmidt number equals one. The time

scale needed in the molecular mixing model of the pdf equation is calculated using

T = y2 /vt , where vt is the turbulent viscosity from the algebraic model, and y is

the distance from the wall.

The numerical results reported in the present paper are obtained using a 71 x 61

grid in the test section. A finer grid had been tested with little change in the solution

observed. The pdf solver uses 100 samples per cell and the solution is obtained

by averaging over 100 time steps. The cpu time required on a CRAY Y-MP is

approximately 0.7 ms per time-step per cell. The calculation of the chemical reaction

accounts for 68% of the total cpu time. The required cpu-time is proportional to

the number of samples used, but is unrelated to the number of time steps being

averaged.

The overview of the solution by the coupled Monte Carlo/finite-volume computation for the wall-jet case is depicted in Fig. 1.7a-d show the temperature and the

mass fraction of hydrogen, oxygen, and water vapor. The peak temperature in this

case is approximately 2500K. The water vapor contour shows that ignition occurs

immediately after hydrogen comes in contact with the vitiated air streams. In reality, an ignition delay was observed in the experiment. This failure to predict the

ignition point is expected because partial equilibrium is assumed in the global chemistry model used. The comparison of the numerical solution and experimental data

measured at the test-section exit is shown in Fig. 1.8a-b. The computed species

mole fractions shown in Fig. 1.8a agree very well with the experimental data. The

computed total temperature profile, compared to the experimental measurement in

Fig. 1.8b, appears to be shifted away from the wall. However, we believe that the

numerical solution for the total temperature is reasonable for the following reason:

The profile of water vapor indicates that the distance between the flame surface

(peak of water vapor) and the wall is more than 2 cm; since the vitiated air stream
120 A. T. Hsu

has larger total temperature, the peak of total temperature should be farther away

from the wall than the flame surface; yet the experimental data seem to indicate

otherwise.

In the course of this application, we noticed the following. The accuracy of the

Monte Carlo/finite-volume calculation relies on a good estimation of the turbulence

diffusivity. For the present test case, the Baldwin-Lomax model seems to be adequate. To understand the effect of the turbulence diffusivity, we experimented with

various Schmidt numbers and observed significant changes in species composition.

Also noticed is the importance of the lip above the inlet of the hydrogen jet. The

lip has a thickness of 0.076 cm. For convenience, we were tempted to ignore this

thickness in the computation. However, our experience shows that shrinking the

lip thickness to zero can result in a 0.2-0.3 cm shift of the flame surface toward the

wall. The presence of any shock wave in the flow field could also cause the flame

surface to shift. As indicated earlier, the chemistry model allows immediate ignition

at the lip. The shock wave generated by ignition can reflect from the upper wall

and strike the flame, resulting in a shift of flame surface away from the lower wall.

In reality, the ignition point is located a certain distance downstream of the lip, and

the reflected shock wave does not interact with the flame in the test section. In our

numerical simulation, a non-reflective boundary condition is used at the the upper

wall to avoid the shock wave interference.

2.1.4 Conclusions

A compressible pdf turbulence model has been developed and implemented in

a Monte-Carlo solver for two- and three-dimensional elliptic flows. A combined

ensemble and time-averaging scheme is applied, which enables one to use a relatively small sample in the Monte-Carlo computation; this averaging scheme greatly

improves the efficiency and convergence of the pdf solution and, thereby, making

large scale computations feasible. The pdf solver can be easily coupled with any

existing finite-volume solvers for compressible flows. Numerical results show that,

for chemically reacting flows, the pdf method performed consistently better than

conventional CFD methods.

2.2 Effect of the Coriolis Force on Compressible Turbulence

Rotation is an important factor in many flow phenomena in nature and in engineering. Problems that are affected strongly by rotation include flows in turbomachinery, large scale motions in the atmosphere and oceans, and galactic motions.

Turbulence is important in all these examples. In order to model turbulence in a

rotating frame, a better understanding of the effects of rotation on turbulence is

required.

Experiments studying the effect of rotation on turbulence had been carried out

by various researchers (Ibbetson & Tritton (1975), Traugott (1958), Wigeland &

Nagib (1978), Hopfinger, Browand & Gagne (1982) Jacquin , Leuchter & Geffroy

(1987)). These experimental results suggest the following two effects of rotation

on turbulence: (1) rotation hampers energy cascade in turbulence, resulting in a
PDF Models for Compressible Reactive Flows and DNS 121

decreased dissipation rate for turbulent kinetic energy (Traugott (1958), Wigeland &

Nagib (1978), Jacquin, Leuchter & Geffroy (1987)); and (2) two- dimension alization

occurs under rotation (Hopfinger, Browand & Gagne (1982) Jacquin , Leuchter &

Geffroy (1987)).

Numerical studies of incompressible turbulence in a rotating frame have been

carried out by Bardina, Ferziger & Rogallo (1985), Dang & Roy (1985), Speziale,

Mansour & Rogallo (1987), and Teissedre and Dang (1987).

Bardina et al. (1985) 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

two- dimension alization does not occur as a result of the Coriolis force. Speziale

et al. (1987) confirmed these results using a smaller Rossby number, i.e., a faster

rotation, and suggested that no Taylor-Proudman reorganization would occur for

homogeneous turbulence.

Studies other than the present one dealing with compressible flows in a rotating

frame are not known to the authors.

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 artificially

dropped from the governing equations, regardless of whether or not it is negligible.

The resulting problem is a hypothetical one, but nonetheless important.

The theoretical and numerical results presented in the following show that although the Coriolis force does not appear explicitly in the turbulent kinetic energy

equation, its effect can be significant. Anisotropy does develop as a result of the

Coriolis force. Two- dimension alization is observed from DNS data for the first time.

It is shown that the Coriolis force serves as a frequency modulator on turbulence.

While the Coriolis force neither creates nor destroys turbulent kinetic energy, it

redistributes energy by eliminating low frequency waves and transfers energy to

waves with a frequency of 252. The energy distribution in the wave space is modified

accordingly: energy is shifted and restricted to either the large scale or the small

scale, depending on the rotation rate. This shift of energy spectrum could cause

either an increase or a decrease in the dissipation rate of turbulent kinetic energy.

It has been demonstrated both theoretically and numerically that the TaylorProudman theorem is applicable to homogeneous turbulence only when the time

scale of rotation, defined as the inverse of the frequency of the inertial waves, approaches the Kolmogorov time scale, and that two-dimensionalization occurs in this

regime.

2.2.1 The Applicability of the Taylor-Proudman Theorem

There is experimental evidence that a homogeneous turbulence would undergo a

two- dimensionalization process under high rotation rates (Hopfinger, Browand and
122 A.T. Hsu

Gagne (1982), Jacquin, Leuchter and Geffroy (1987)). Lesieur (1990) used a Rossby

number expansion to prove that at the zeroth order, turbulence is two-dimensional.

However, previous numerical simulations have not predicted this phenomenon, and

Speziale, Mansour, and Rogallo (1987) argued, based on rapid distortion theory,

that the Taylor-Proudman theorem does not apply in rotating isotropic turbulence.

The original Taylor-Proudman theorem is derived under the assumption that, beside a small Rossby number, Ro, the flow is steady, see, e.g., Chandrasekhar (1961).

Turbulence is an unsteady phenomenon with a broadband frequency spectrum. We

shall show in what follows that the Taylor-Proudman theorem is valid only for a

certain flow regime.

For simplicity we start with the incompressible Navier-Stokes equation in a rotating frame:

ui,t + ujui,j = —p,i — 2EijkQjuk + vui,jj (2.1)

where p is the modified pressure. For small Ro, the convection term is negligible.

Taking the curl of the above equation (with the convection term neglected) yields

wi , t = 2Qjui,j + Uwi,jj (2.2)

where wi = Eijkuk,j . If, at the limit of a very small Ekman number, wi,t = 0, then

we obtain the standard Taylor-Proudman theorem. However, in general, wi,t :^ 0.

We start our argument by assuming that a decaying homogeneous turbulence can

be decomposed into waves of the following form

ui(x, t) = 1: E uj(k, a)exp[i(kjxj + at)] (2.3)

k a

Where k and a are the wave number and angular frequency, respectively, and i

denotes the imaginary unit (not to be confused with i and k in the subscripts).

The Taylor-Proudman theorem would be applicable if Qjui,j = 0, and with waves

of the type given by eq. (2.3), it would suffice if Q j kj = 0 for all wave numbers kj.

Substituting one wave component of (2.3) to eq. (2.2), we have

fijk akj 26 k = i2Qj kjui + ivk2 Ezjkkj46k (2.4)

which can be rearranged as

/

5Ij kj = — ( 

vk22 + ia 1

Eijkkj 

uk

(2.5)

where fZ j - Q j / Q denotes a unit vector in the direction of rotation. Note that 2Q is

the intrinsic angular frequency of inertial waves in a fluid under solid body rotation.

The Taylor-Proudman theorem is valid when

^vk2 + ia^ _ a

2 + v2k4

K 1 (2.6)

2Q 2Q 
PDF Models for Compressible Reactive Flows and DNS 123

for all the possible a and k values. Eq. (2.6) shows that the Taylor-Proudman

theorem is applicable only to low frequency waves with a2 +v2 k4 « 252, and

does not apply to all waves in turbulence in general.

In what follows we try to find a criterion for which the Taylor-Proudman theorem is valid in turbulence. The turbulence time scale is limited from below by the

Kolmogorov time scale T, and the length scale is limited from below by the Kolmogorov length scale 77. Any waves with frequencies dal > 27r/T or wave numbers

k > 27r/r7 will be damped by viscosity. Thus, noticing that T = rte /v, we can make

the following estimation on the upper limit of a 2 + v2k4/(2Q):

C

2 +v2k4

252

< 27r VI + 47r2 E.7(2.7)

where E,,is the Ekman number defined by the Kolmogorov microscales:

_ v _ 1 ( 2.8) E17 2Qn 2Q-r

2.8

which is the ratio between the rotation time scale and the Kolmogorov time scale.

From the above analysis, we see that for the Taylor-Proudman theorem to be valid

in turbulence, one needs not only to have Ro —> 0, but also E.7

0.

Eq. (2.8) does not mean that at the limit of zero viscosity, or infinite Reynolds

number, the Taylor-Proudman theorem is valid; in fact, a careful examination shows

that the opposite is true. Let Roe and Ree be the turbulence Rossby number and

turbulence Reynolds number based on the turbulent integral length scale e and

turbulent velocity scale u. Then the regular turbulence Ekman number can be

written as Et = v1QP = Roe/Ret. Thus one may have a zero Ekman number by

either letting Rot go to zero or letting Ret go to infinity. But the Ekman number

defined by the Kolmogorov microscale is different:

Rot E, _

(1) ,

— = ARotRee (2.9) Ret rl

where A is an undetermined constant. Therefore, in order to have En --+ 0, one must

have Rot go to zero faster than Re, 1/2 . For finite but small Rossby numbers, there

is no Taylor-Proudman reorganization at the limit of infinite Reynolds number, a

fact remarkably different from the case of steady flows.

Using the Taylor microscale, a, we obtain a simple relation between E,, and Roa:

2

E,, = 2QA2 ^ 7) = 15 2 E\Rea = 15' Roa (2.10)

This relation shows that the criterion for the Taylor-Proudman theorem to be valid

in turbulence is Roa « 1, and that Roe « 1 is not sufficient.
124 A. T. Hsu

Define a modified microscale Ekman number as

P77 = 2T 1 + 47r 2 E77 (2.11)

According to the above analysis, turbulence subject to rotation can be divided into

the following regimes: (1)4 » 1. Turbulence will not undergo any two- dimension alization. When

Ro --> 0, the convection term is negligible and the energy cascade process is hampered, resulting in a slowing down of energy dissipation.

(2) E,, — 1. Waves with small wave numbers and small frequencies will undergo a

two-dimension alization process. However, since the Coriolis force neither generates

nor destroys turbulent kinetic energy, yet according to eq. (2.5) the low frequency

waves are eliminated, one may conclude that the effect of the Coriolis force is to

transfer energy from low frequency waves to waves with a 2 + v2 k4 — 252. Because

of this shift in energy spectrum, one expects an increase in the dissipation rate of

turbulence kinetic energy. Because the energy containing eddies are of low frequency

and small wave numbers, the effect of rotation can be significant in this regime.

(3) E,7 « 1. The Taylor-Proudman theorem is valid for this regime since all

the the turbulence waves satisfy the inequality vfa2 + v2 k4 « 252 and the right

hand side of eq. (2.5) goes to zero for all the waves. This regime may not be

physically feasible because it requires an extremely large rotation rate. For example,

for E,, = 0.1, one needs a turbulence Rossby number Roa of the order of 10-3 , a

number difficult to reach in the laboratory or in nature.

In the experiments of Wigeland & Nagib (1978) and the numerical simulations

of Bardina et al. (1985), the minimum & is greater than 40. In the simulation of

Speziale et al. (1987), the value of E,, is about 10. For these relatively high values

of the microscale Ekman number, no two-dimensional reorganization is possible,

and none was observed.

Finally, let us look at the implications of the Taylor-Proudman theorem in isotropic

homogeneous turbulence. Assume that, in a Cartesian coordinate system, Q =

(0, 0, Q); then the Taylor-Proudman theorem requires that fluid particles along any

line in the z-direction move with the same velocity. However, because the flow

field is initially random with < u >=< v >= 0, and because there is no preferred

x — y plane in the flow field, the only way for the Taylor-Proudman theorem to be

satisfied is for the fluctuation velocities in the x— and y—directions to go to zero.

2.2.2 Turbulence Equations for Rotating Fluids

The governing equations for compressible flows subjected to a constant rotation

are

P,t + (PuAj = 0

( Pui),t + (Pujui),j = —P,i + rij,j - PE ijk E klm. QjQ l xm - 2PEijkQjuk (2.12)

(Pe ),t + (Puj e ),j = —puj,j + ui, jTij - (kT j ),j

p = pRT
PDF Models for Compressible Reactive Flows and DNS 125

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 effect of the Coriolis force alone, and to compare results

with incompressible flow simulations, we artificially drop 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. Nevertheless, it allows one to identify the effects of the Coriolis force without

ambiguity.

The Reynolds stress equation for a homogeneous turbulence without mean flow

is

1

(uiuj ),t = —uk (uiuj ),k — P(uip ,j + ujp, i) + uiTjk,k (2.13)

+2IjTik,k — 2EilmS2 l 2G ,n 26j — 2EjlmQlumui

The equation for the turbulent kinetic energy, for a constant v, is

1 v

k,t = 2uiuiuj,j — 2Gip ,i — vui ui — —ui.iuj,j (2.14) 2 ^ ,j 3

We define an incompressible dissipation rate as E i = vui, jui, j and a compressible

dissipation rate as c c - 3 ui . iuj, j. The equation for incompressible dissipation is

EZt = —2vui

,juk,jui,k + vui,jui,juk k — 2vui,jp,ij

a

2 z (2.15)

—2v ui,jkui,jk — 3 v ui ijuk kj

We notice that rotation does not appear explicitly in the above equations. However, this does not mean that rotation has no effect on turbulent kinetic energy

and dissipation. A careful analysis shows that rotation affects most of the triple

correlations in the above equations. For example, the first term on the right hand

side of the energy equation is governed by

(uzuzu9,At = —Ej1"AUiuiu.,,,,j + OT (2.16)

where OT stand for "Other Terms" unrelated to rotation. The triple correlations

in the dissipation equation are governed by

(uz,7uz,7ul,l),t = —2clmn,QTnun,juj,jui j + OT (2.17)
126 A. T. Hsu

(u1,lu7,lui , 7 ) ' t —

 —ZEjImQl un^.,l ui,jui l + OT (2.18)

The last two triple correlations represent the production of dissipation. For turbulence under rotation, the change in dissipation rate is caused mainly by these triple

correlations.

2.2.3 Inertial Waves and Anisotropy

It is known that the Navier-Stokes equations for incompressible fluid subjected to

solid body rotation admit wave solutions, known as inertial waves (Chandrasekhar,

1961). For homogeneous, anisotropic turbulence, the existence of inertial waves is

manifested in Reynolds stresses. In order to explain our numerical results later, we

derive the wave solutions for the Reynolds stresses here, starting with the Reynolds

stress equations for incompressible homogeneous turbulence, assuming the rotation

to be in the X3-direction:

dui — 4QYl u2 + OT (2.19)

dt

due _ —4Q 1 u2 + OT (2.20)

dt

duiu2 

= 2Q 2 — u

dt

1) + OT (2.21)

In the following analysis, we neglect the terms unrelated to rotation (OT). Taking

time derivatives of eqs. (2.19) and (2.20), and applying eq. (2.21), we have

d2

dt 2

 (u2 — ui) _ — 16SZ 2 (u2 — U2 )

( 2.22)

It is easy to see that a solution of the above equation is

u2 — ui = A exp(i4Qt) (2.23)

where A is an undetermined constant. With OT neglected, eqs. (2.19) and (2.20)

give

dt( u

2

+ ui) = 0 (2.24)

or

u2 + ui= C (2.25)

From this equation and eq. (2.23), we obtain the following solutions:

ui = 2 exp(i4Qt ) + 2

( 2.24)

A C

u2 = - 2 exp ( i4Qt) + 2(2.25)

ul u2 = 2 exp [i(4Qt — 2 ), (2.26)
PDF Models for Compressible Reactive Flows and DNS 127

The frequency of the wave solutions for the Reynolds stresses is twice that of the

inertial waves, which is expected. We note that if the turbulence is strictly isotropic,

i.e., u2 —01 = 0, then the waves will not appear in the solutions of Reynolds stresses.

2.2.4 DNS Results

The governing equations solved in this study are those given in eq. (2.12) with

the centrifugal force neglected. An 8th order compact difference scheme is used to

solve the equations numerically. This scheme has been shown to give spectral-like

resolutions (Lele 1990), and has been successfully used in DNS (Lee, Lele, & Moin,

1991). The numerical results presented in the following are obtained using a 643

grid over a box of size 27r. The initial conditions for all the cases are the same,

and are obtained by carrying the calculation of a homogeneous, isotropic flow field

to a point where the velocity-derivative skewness reaches a constant value of about

-0.5 (Tavoularis, Bennett & Corrsin, 1978). The initial turbulence Mach number

for all the cases studied is Mt = 0.31 and the initial turbulent Reynolds number is

Rea = 12.8. The parameters of cases studied are listed in the following table.

Modified

Ekman No.

^„

Rossby No

Roa = s"

Notation

Rate

S2

00 00 0.

22 0.14 2.74

11 0.07 5.48

4.2 0.028 13.7

2.2 0.014 27.4

0.8 0.005 68.5

The modified Ekman numbers are either much larger than, or in the neighborhood

of, unity. We have not been able to compute cases for & « 1 because such cases

require extremely small time steps to resolve the rotation time scale, and are too

time consuming.

Reynolds Stresses

Figs. 2.1, 2.2, and 2.3 are the time evolution of the diagonal terms of the Reynolds

stresses for 4 = oo, 22, and 2.2, respectively. The velocity component w is parallel

to Q, and u and v are perpendicular to Q. For a relatively small rotation rate, with

E,, = 22, there is no appreciable anisotropy in terms of the Reynolds stresses. However, as E,, approaches one, as shown in Fig. 2.3, there is a distinct trend of faster

decay for the energy components in the plane perpendicular to the rotation. As the

analysis of Section 2 suggested, the only possible route of two-dimensionalization
128 A. T. Hsu

for a homogeneous turbulence is for < u > and < v > to approach zero. The

numerical results given in Fig. 2.3 support that conclusion.

The < uu > and < vv > curves in Fig. 2.3 show considerable oscillation as a

result of inertial waves. To show that the oscillation is indeed caused by the inertial

waves, we plotted < v 2 > — < u2 > versus time in Fig. 2.4, where the time is not

nondimensionalized by eddy turn over time. The angular frequency is found to be

452 from Fig. 2.4, which agrees with the theoretical analysis given in Section 4.

Turbulent Kinetic Energy

Fig. 2.5 shows three curves of the ratio between the initial turbulent kinetic

energy and energy at time t. For E,, = 22, there is a noticeable reduction in the

turbulent energy decay rate, while for E,, = 2.2, the decay rate has increased.

The decrease of the decay rate is expected: it is attributed to the hampering of

energy cascade by rotation, and has been demonstrated previously by both experiments and DNS. But the decrease is rather small compared to the incompressible

results. To see why this is the case, we try to analyze the compressible and incompressible turbulent kinetic energy separately. While actually decomposing the

energy into a compressible part and an incompressible part is very difficult, one

can estimate the trend by looking at the dilatation field and vorticity field. Since

the incompressible component of the velocity can be written as an integral of the

vorticity field, w2 , and the compressible component of the velocity can be written

as an integral of the dilatation, A, one expect the quantities < IwI 2 > and < 02 >

to be a good indicator of the incompressible and compressible energy, respectively.

These quantities are plotted in Figs. 2.6 and 2.7. The results show that while the

Coriolis force reduces the decay rate of the incompressible turbulent kinetic energy,

its effect on the compressible energy is the opposite.

A clearer picture of the effect of rotation is gained by plotting the turbulent kinetic

energy, the mean vorticity square, < l w l 2 >, and the mean velocity divergence

square, < A2 >, at t/teddy = 1 as a function of the modified Ekman number (Fig.

2.8). For the sake of comparison, the variables are normalized by their respective

values for Q = 0. The total energy curve has a maximum at E,, — 20, where the

dissipation rate of the turbulent energy reaches a minimum. For the incompressible

energy, represented by < Iw12 >, the dissipation reaches a minimum at E,, ^ 5.

The analysis given in Section 2 suggested that depending on the value of E p the

dissipation rate of turbulent kinetic energy will be either reduced or enhanced: For

4 » 1, the dissipation rate decreases with the rotation rate, and for E,, — 1, the

dissipation rate increases with the rotation rate. The numerical results presented

here agree well with that prediction.

The compressible energy always decreases as rotation rate increases, which seems

to contradict our theoretical prediction. One plausible explanation is that the compressible waves are more readily converted to non-random inertial waves. The

evidence of this can be observed from Figs. 2.6 and 2.7, where the curve for

< Do > / < A2 > at k = 2.2 shows a much stronger influence from inertial

waves than the curve for < Iwo1 2 > / < Iwl2 >. Since the mean values < . > at any
PDF Models for Compressible Reactive Flows and DNS 129

given time are obtained by averaging over the computational domain, energy contained in non-random inertial waves is not included in the mean turbulent kinetic

energy.

Dissipation Rate

Fig. 2.9 shows the dissipation rates per unit energy or E,, = oo and E,, = 0.8.

One can see that, as was predicted in Section 2, the dissipation rate has indeed

increased for E,7

Spectrum

To understand the phenomena described above, and to relate them to the analysis

given in Section 2, it is best to look at both the time and spatial spectra of the flow

field.

Fig. 2.10 shows the time spectra, at t/t,ddz, = 1, of the u-velocity, defined as

01 = J < u(t)u(t + T) > exp(ia7-)dT (2.27)

The spectra show that the Coriolis force serves as a frequency modulator for turbulence. The Coriolis force caused a considerable amount of energy to be removed

from the low frequency range, where the energy containing eddies are, and shifted

to the angular frequency of 252. Rotation prevents energy from being transferred

away from frequency 252.

When the inertial angular frequency, 252, coincides with the frequencies of the

energy containing eddies, turbulent kinetic energy is prevented from cascading to

higher frequencies. If we associate high frequencies with large wave numbers, then

this restriction reduces the amount of energy transferred to small eddies, and thus

reduces dissipation. However, when 2Q approaches the Kolmogorov time scale,

energy is transferred to small eddies, and the dissipation rate is increased.

To show that the transfer of energy to higher frequencies is accompanied by

energy transfer to smaller eddies, we plotted the one-dimensional energy spectra in

Fig. 2.11, where F21 is defined as

F'21( kl) = I f

< v(x, ()v(x

)2 

y, z) > exp(ikl r)dr (2.28)

The areas under all the curves in Fig. 2.11 are the same and equal to unity (if we

integrate from —oc to oo).

Compared with the spectrum for 52 = 0 (solid line), the curves show that for large

Ekman numbers, or small rotation rates, the energy spectrum is shifted to the left,

meaning more energy is concentrated in large eddies. Since the Reynolds numbers

are initially the same for all the cases, a decrease in the energy contained in the

small eddies means less dissipation.

As the Ekman number decreases, the energy spectra are shifted to the right, to

small eddies. In fact, the spectra for flows with rotation crosses the spectrum of

zero rotation at about E. = 4.4, suggesting that for E,, < 4.4, the dissipation rate
130 A. T. Hsu

should increase. This is consistent to both what was observed in Fig. 2.8 and what

was predicted in Section 2.

2.2.5 Conclusions

In the governing equations of turbulent kinetic energy and dissipation rate, rotation does not appear explicitly; the dissipation rate is affected by rotation through

the triple correlations in the equation. For anisotropic turbulence subjected to

solid body rotation, the solutions for the turbulent shear stresses oscillate with a

frequency equal to twice the frequency of the inertial waves.

We have shown that the Coriolis force serves as a frequency modulator on turbulence. The turbulent kinetic energy shifts from low frequencies to the frequency

of the inertial waves. The shift of energy in the frequency space is accompanied by

a shift of energy in the wave number space. Using both theoretical arguments and

DNS data, we have shown that an appropriate parameter in categorizing homogeneous turbulence subjected to solid body rotation is the modified microscale Ekman

number, E.7 . For E77

» 1, the dissipation rate of turbulent kinetic energy is reduced

by rotation due to a shift of energy to large scales. As E7 approaches one, energy

is shifted to the dissipative scales, and thus dissipation is enhanced by rotation.

Strictly speaking, the Taylor-Proudman theorem is applicable to turbulence only

when E,, « 1.

3. Future Plans

The near term goal is the application of pdf method to three-dimensional supersonic flows. In addition to 3D applications, the pdf model for compressible flows

will be further developed. Direct numercial simulation will be performed to assist

the development of new pdf models.

4. References

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Prg. Energy Combust. Sci., 1985, 11, 119-192.

2 Pope, S.B., Private communication, May, 1992.

3 Shuen, S. and Yoon, S., "Numerical Study of Chemically Reacting Flows Using a

Lower-Upper Symmetric Successive Overrelaxation Scheme." AIAA Journal, Vol

27, No 12, pp. 1752-1760, December, 1989.

4 Raju, M.S., "Heat Transfer and Performance Characteristics of a Dual-Ignition

Wankel Engine," SAE paper 920303, Feb. 1992, also accepted for publication in

The SAE Transactions.

I Kollmann, W., "The PDF Approach to Turbulent Flow." Theoret. Comput.

Fluid Dynamics, 1990, 1, 249-285.

6 Anand, M.S., Pope, S.B., and Mongia, H.C., "Pressure Algorithm for Elliptic

Flow Calculations with the PDF Method." CFD Symposium on Aeropropulsion,

NASA Lewis Research Center, April 24-26, 1990.
PDF Models for Compressible Reactive Flows and DNS 131

7 Haworth, D.C. and El Tahry, S.H., "Probability Density Function Approach for

Multidimensional Turbulent Flow Calculations with Application to in-Cylinder

Flows in Reciprocating Engines." AIAA Journal, Vol. 29, pp.208-218, 1991.

8 Roekaerts, D., "Use of a Monte Carlo PDF Mthod in a Study of the Influence of

Turbulent Fluctuations on Selectiveity in a Jet-stirred Reactor," Applied Scientific Research, Vo1.48, pp. 271-300, October, 1991.

9 Correa, S.M. and Pope, S.B., "Comparison of a Monte Carlo PDF/Finite-Volume

mean flow model with bluff-body Raman data." 24th Int'l Symp. on Combust.,

The Combustion Institute, 1992, in press.

to Shih, T-H and Lumley, J.L., "A Critical Comparison of Second Order Closures with Direct Numerical Simulation of Homogeneous Turbulence." NASA

TM 105351, November, 1991.

I1 Sarkar, S., Erlebacher, G., Hussaini, M.Y., "Compressible Homogeneous Shear:

Simulation and Modeling." NASA CR-189611, 1992.

12 Hsu, A.T., "A Study of Hydrogen Diffusion Flames Using PDF Turbulence

Model," AIAA Paper 91-1780, June, 1991.

13 Hsu, A.T. and Chen, J.Y., "A Continuous Mixing Model for PDF Simulations

and its Applications to Combusting Shear Flows." 8th Symposium on Turbulent

Shear Flows, Munich, September, 1991.

14 Pope., S.B. and Gadh, R., "Fitting Noisy Data Using Cross Validated Cubic

Smoothing Splines." Communications in Statistics, Part B, Vol.17, pp. 349-376,

1988.

15 Villasenor, R., Chen, J.-Y., and Pitz, R.W., "Modeling Ideally Expanded Supersonic Turbulent Jet Flows With Nonpremixed H2 -Air Combustion," AIAA

Journal, Vo1.30, No. 2, Feb. 1992.

17 Rogers, R.C., and Chinitz, W., "Using a Global Hydrogen-Air Combustion Model

in Turbulent Reacting Flow Calculations," AIAA Journal, Vol.21, No.4, April

1983.

18 Evans, J.S., Schexnayder, C.J., and Beach, H.L., "Application of a Two-Dimensional

Parabolic Computer Program to Prediction of Turbulent Reacting Flows," NASA

TP-1169, March 1978.

19 Eklund, D.R., Drummond, J.P., and Hassan, H.A., "Calculation of Supersonic

Turbulent Reacting Coaxial Jets," AIAA Journal, Vol.28, No. 29, Sept. 1990.

20 Evans, J.S., and Schexnayder, C.J., "Influence of Chemical Kinetics and Unmixedness on Burning in Supersonic hydrogen Flames," AIAA Journal, Vo1.18,

No.7, Feb. 1980.

2i Burrows, M. C. and Kurkov, A. P., "Analytical and Experimental Study of Supersonic Combustion of Hydrogen in a Vitiated Air Stream," TM X-2828, NASA,

1973.

zz Bardina, J., Ferziger, J.H. and Rogallo, R.S. 1985 Effect of Rotation on Isotropic

Turbulence: Computation and Modeling. J. Fluid Mech. 154, 321-336.
132 A. T. Hsu

23 Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford

Press.

24 Dang, K. and Roy, Ph., 1985 Direct and large eddy simulation of homogeneous

turbulence submitted to solid body rotation. Turbulence Shear Flows, 1985.

25 Hopfinger, E.J., Browand, F.K. and Gagne, Y., 1982 Turbulence and waves in a

rotating tank. J. Fluid Mech. 125, 505-534.

26 Ibbetson, A. and Tritton, D.J., 1975, Experiments on Turbulence in a Rotating

Fluid. J. Fluid Mech. 56, 639-672.

27 Jacquin, L. Leuchter, O. and Geffroy, P. 1987 Experimental study of homogenous

turbulence in the presence of rotation. Turbulence Shear Flow, Toulouse, France.

28 Lee, S., Lele, S.K. and Moin, P., 1991, Direct Numerical Simulation and Analysis

of Shock Turbulence Interaction. AIAA Paper 91-0523.

29 Lele, S.K., 1990, Compact Finite Difference Scheme with Spectral-like Resolution.

CTR Manuscript 107, Stanford University.

31 Lesieur, M., 1990 Turbulence in Fluids, 2nd Ed. Kluwer Academic Publishers.

31 Speziale, C.G., Mansour, N.N. and Rogallo, R.S., 1987 The Decay of Isotropic

Turbulence in a Rapidly Rotating Frame, Center for Turbulence Research, Proceedings of the Summer Program

32 Tavoularis, S., Bennett, J.C., and Corrsin, S., 1978, Velocity-derivative skewness

in small Reynolds number, nearly isotropic turbulence. J. Fluid Mech. vol. 88,

Part 1, 63-69.

33 Teissedre, C. and Dang, K., 1987 Anisotropic behavior of rotating homogeneous

turbulence by numerical simulation. AIAA paper 87-1250.

34 Traugott, S.C., 1958 Influence of Solid-body Rotation on Screen-produced Turbulence. NACA TN 4135.

35 Wigeland, R.A. and Nagib, H.M. 1978 Grid-generated Turbulence With and

Without Rotation About the Streamwise Direction. HT Fluids and Heat Transfer

Rep. R78-1, Illinois Inst. of Tech., Chicago, Illinois.
PDF Models for Compressible Reactive Flows and DNS 133

1.30

1.25

1.20

1.15

1.'0

1.05 A

I

1.00

0.95

0.90

0 10 20 30 40

X

Fig 1.1 Temperature rise through an oblique shock

predicted by the pdf turbulence model compared with

an analy tical solution.

averagea over 1 step

3 steps

10 s t e s

+

32 steps

I 100 steps

316 steps

1000 steps

0 600 1200 1800

No. of .aerations

Fig 1.2 Convergence history of the pdf solver measured by an L2 norm; combined time and ensemble

averaged solutions.

10 -2

10 -3

0

w 

10 -4

6 10-5

in

0 10-6

N

10 -1

10 -8

10 -9

10-0

D = 0.0653 m

d = 0.009525 m

Injector tin thickness = 0.0015 m

D

Lr

Macti numoer , 'i • •

Temperature, T, K .

Velocity, u, m/3

Pressure. p, MPa . . .

Masi fraction:

aye. . . . . . . . .

ao.. . . . . . . . .

ayz . . . . . . . . .

au-.0 . . . . . . . . .

Hydrogen jest ?ree stream

2.00 1.90

251 1495

2432 I 1510

0.1 0.1

1.000 0

0 0.241

0 I 0.478

0 1 0.281

1

Fig 1.3 Geometry and test conditions for the coaxial

hydrogen jet combustion.
134 A. T. Hsu

t.e

+ H2 + N2

9 A 02 .9 L 02

X H2O X H2O

8 - Q N2

8 O N2

L

I'

7L 7

O

U

10

L

^ l

2

1

a

a .2 .4 .0 .8 1.0 1.2 1,4 1.b 1.8 2.0 2.2 2.4

r- /d

Fig 1.4a Radial profiles of the predicted and measured mass fractions of major species for the pdf-CFD

solver at x/d =8.26.

O

U

L 5

N •4

N

'd

5- .7

.2

.1

a

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

r/d

Fig 1.4c Radial profiles of the predicted and measured mass fractions of major species for the pdf-CFD

solver at x/d =21.7.

1.e

9

B

7

b

7

.5

..

1a

+ H2

9 L 02

X H2O

9

p N2

1

b

J

ro

5 \\+ X

4 X

x X

L

- - -

Y\

a 2 1.0 1.2 1 A 1.b i.8 2.0 2.2 2. A

r/d

Fig 1.4b Radial profiles of the predicted and measured mass fractions of major species for the pdf-CFD

<olver at x/d =15-5.

0 0

.2 .a .6 .a l.a 1.2 t.a 1.6 1.8 2.0 2.2 2.4

r/d

Fig 1.4d Radial profiles of the predicted and measured mass fractions of major species for the pdf-CFD

solver at x/d =2T.9.
l.e

9

8

H2

02

c '420

^ v2

1.e

9

8

7

e

+ H2

0 02

X H2O

O N2

PDF Models for Compressible Reactive Flows and DNS 135

1.0

i -2

Q 02

9

X H2O

C 142 .8

.7

C

O

U

ro

L .5

N .4

N

ro

e

t.0

9

e

7

O

y 6

U

L 5

w

Ln

U1

ro

u 0

.2 .< .e 1.2 '.2 '.< t.6 1.8 2.0 2.2 2.4

/d

Fig 1.5a Radial profiles of the predicted and measured mass fractions of major species for the CFD

solver at x/d =8.26.

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.0 2.8 2.2 2.4

r/d

Fig 1-5c Radial profiles of the predicted and measured mass fractions of major species for the CFD

solver at x/d =21.7.

C

x

X

-----

n , ,•

-

f

.. < _ .c - - h i.tl 1.2 L.L L.<

Fig 1.5b Radial profiles of the predicted and measured mass fractions of major species for the CFD

solver at x/d =15.5.

U

5

U1 <

;n

ro

^ l

2

a

0 .2 < .6 1.0 +.2 1.4 1.6 1.8 2.0 !.2 !.<

r- / d

Fig 1.5d Radial profiles of the predicted and measured mass fractions of major species for the CFD

solver at x/d =27.9.

6

U

L 5

:n <

to

ro
136 A. T. Hsu

8.90 cm ; 0.476 cm total height 10.48 cm

0. 4D0 cm open height I

li 

Y

Static pressure ports

1818.3 cm---1

H Z Injection 35.6 cm

X

Fig 1.6 Flow Configuration of Burrows and Ku:kov's Experiment.

Fig 1.7a Temperature Contour of Numerical Solution Using PDF.

0.2 0.4 0.6

0.8

Fig 1.7b Hydrogen Mass Fraction Contour of Numerical Solution Using PDF.

0.05 0.: 0.15 0.2 0.25

Fig 1.7c Oxygen Mass Fraction Contour of Numerical Solution Using PDF.
0

C

0

U 0.8 0

a^ 0.6

D

>° 0.4

V)

SPECIES VOL^!AE FRACTION

o H2

02

o N2

H2O

Computed

O

° °

Fig 1.7d Water-Vapor Mass Fraction Contour of Numerical Solution Using PDF.

0.025

v 0.020

0.015

0.2 8

q 11

to $

0.005

0.000

E17=_ - - - <UU>

\ - - <vu>

\ <ww>

0.040

0.035

0.030

0.0 1.0 2.0 3.0 4.0 0.0 0.2 0.4 0.6 0.8 1.0

Y (cm) t/t",y

Fig 1.8a Composition Profile of Measured Data and Nu- Fig 2.1 Time history of Reynolds stresses for Q = 0.

merical Solution Using PDF. ( x = 35.6 cm )

0.040

0.035 F =22

H 0.030 - - - <Vu>

TOTAL 'EUPE=ATURE PROFILE w

4000 0.025-\ <wtu>

o Exoertmenta( b 0.020

3500 Computed o ^-

^, 0.015

2 3000 0.010 - - -

a 2500 000/ 

r0.005 v @00 _ o. o00

E O O O 0.0 0.2 0.4 0.6 0.8 1.0 2000 0

tAddy

6 1 500 Fig 2.2 Time history of Reynolds stresses for a moderate rotation rate of Q = 2.74.

' 000 ^ c '

500

O.0 i .i ?.0 3.0 4.0

Y

Fig 1.8b Total Temperature Profile of Measured Dat

and Numerical Solution using PDF. ( x = 35.8 cm )
2.2 - - <UU>

- - - <vv>

^ <ww>

E:J7 I\ 11

--- 

E^=22

-- 

E,=2.2 I ll

111

1"1 y

12

10

^ 8 Nd

n 6 NO

e 4

5

4

Y 3

Y 2

1

0

7.,

r

b

^y 1.

n

0 1.

ti 0.

l

0. J

U n

138 A. T. Hsu

0.040

0.035

0.030

Cn 0.025

ro 0.020

0.015

0.010

0.005 2

0 000

0.0 0.2 0.4 0.6 0.B 1.0 0.0 0.2 4 0.6 0.8 1.0

t/teddy t/t,ddy

Fig 2.3 Time history of Reynolds stresses for a high rotation Fig 2.6 The ratio of velocity divergence square, representing

rate of Q = 68.5. compressible turbulent kinetic energy.

0.010 C

0.008 7

0.006 A 6

A 0.004

0.002

N

3 5

V

V

1 0.000 4

-0.002 '3

3 v -0.004 i 2

-0.006

r

-0.008

-0 010

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 :.-t 0.6 0.8 1.0

t `teddy

Fig 2.4 Time history of < tit• > - < uu > with an angular Fig 2.7 The ratio of vorticity square, representing incoinfrequency of 452. pressible turbulent kinetic energy.

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 1.2

I/t eddy 1 /^'-,

Fig 2.8 Turbulence kinetic energy, velocity divergence

Fig 2.5 Total turbulent kinetic energy ratio. square. and vorucit:- square at t1t„1i,, = 1, normalized by their corresponding values for Q = 0.
PDF Models for Compressible Reactive Flows and DNS

E*1=m

- E,1=2.2

0.0 0.2 0.4 0.6 0.8 1.0

t/t ,,y

Fig 2.9 Dissipation rate of turbulent kinetic energy, normalized by energy.

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

00

139

0,5

0.4

0.3

0.2

0. 1

0.0

-0 1

20-0

20-5.48 20-27.4 20-54.

r^ f

I ^ I I

0 10 20 30 40 so so 70

a

Fig 2.10 Time spectra of turbulent kinetic energy.

E^1= m

E,1=11

Erl =4.4

--. E=2.2

E=0.8

0 5 10 15 20 25

K1

Fig 2.11 One-dimensional energy spectra for the How fieid at

tlt.dey _ 1.

100

10-2

10-4

10-6

10-e

10-10

10-12
Page intentionally left blank 
Center for Modeling of Turbulence and Transition 141

Research Briefs - 1993

Modeling of Turbulent, Reacting

Flows by PDF Methods

A. T. Norris

1. Motivation and Objective

The objective of this work is to further the development of Probability Density

Function (PDF) models for turbulent, reacting flows. To achieve this, work will be

performed in two different areas. The first area will involve developmental work on

an existing scalar PDF model for compressible flows. Specific areas to be worked

on are the implementation of chemical kinetics and extending the model to include

axisymmetric and 3D flows. The second area of work will involve the development

and implementation of other PDF models. Proposed topics include the development

of new models for molecular mixing and state variables such as enthalpy, and work

on parallel implementation of PDF methods. In addition, the development of a

velocity-scalar PDF method for compressible flows is proposed.

In all cases, the performance of the PDF model will be verified by application to

a variety of real flows, and comparing the predictions to the experimental data.

2. Work Proposed

In the following section, a brief background of PDF methods is presented. This

is followed by the details of the proposed work for the 1993-1994 year.

2.1 Background

Probability Density Function (PDF) methods involve solving a transport equation for the joint PDF of quantities of interest, such as velocity, dissipation, enthalpy

and composition. For all but the simplest of flows, solutions of the joint PDF transport equation are obtained by Monte Carlo schemes, where the PDF is represented

by an ensemble of particles that evolve in time by some stochastic process. The

principle advantage of this form of modeling over moment closure approaches is that

the chemical reaction is treated exactly', thus making PDF methods particularly

attractive for calculating turbulent reactive flows.

As a generalization, PDF methods can be divided into two types: The scalar PDF

method and the velocity-scalar PDF method. The scalar PDF method consists of a

modeled transport equation for the joint PDF of composition only, with the velocity

field solved by a moment closure model. This model was developed by Pope (1981)

and Dopazo and O'Brian3 (1974). This form of model was also used by Chen et all

and was extended to compressible flows by Hsu et al l . In the velocity-scalar PDF

method, the transport equation for the joint PDF of velocity and composition is

solved. This model was developed by Pope et a1 6 7 in 1986, and has the advantage

over the scalar PDF method in that convective transport is treated exactly.
142 A. T. Norris

2.2 Existing Code Development

Despite the ability of the PDF method to treat chemical reaction exactly, the

implementation of the numerical chemical kinetics in a Monte Carlo scheme is a

non-trivial matter. First, the solution of the full system of rate equations for the

thermochemistry in turbulent reacting flows requires obtaining solutions for of order 50 chemical species, governed by of order 200 stiff, non-linear rate equations.

At present this task is computationally infeasible, and so reduced mechanisms are

employed. These reduced mechanisms represent the full composition by a few representative species, typically two to five, and the reaction rates of these species are

quasi-global equations derived from the full mechanism. However even the solution

of a reduced mechanism is a computationally expensive task. In a typical PDF

calculation', there are on order 100,000 particles and 1,000 time steps, resulting in

the code performing 109 solutions of the reduced mechanism. A computationally

efficient way of reducing this task to manageable proportions is to use a look-up

table. In this table, the reduced mechanism is integrated for discrete time and

composition increments, and stored in a table. Thus integration is replaced by

interpolation, at a considerable saving in computer time. By the use of adaptive

tabulation techniques' the size of the tables can also be minimized.

To illustrate the form of a look-up table, it is useful to consider a simple function,

representing a one step, finite rate reaction of an arbitrary fuel, F, with air, A,

forming product, P;

F+A^P. (1)

Assuming equal diffusivity of all species, the rate of creation of product (reaction

rate) is just a function of the mixture fraction ^ and the mass fraction of the

product, Yp. Figure I shows a contour plot of reaction rate as a function of ^ and

Yp. Overlaid is a grid representing the adaptive look-up table used to store the

data. Reaction rates for some arbitrary composition are obtained by identifying

the cell that contains the composition, and interpolating from the four nodes.

It is proposed that the existing scalar PDF model be modified to incorporate

the use of look-up tables. Apart from an expected speed-up of the code, a library

of tables of different fuels can be generated and stored, providing users with a

more versatile tool for reactive flow calculations. In addition to the look-up table

implementation of the chemical reaction, a useful contribution would be to make

the code compatible with the CHEMKIN 10 code. CHEMKIN is a chemical kinetics

software package, designed to help incorporate gas-phase chemical reaction schemes

in fluid mechanics models. While CHEMKIN would not provide the computational

speed-up of the look-up tables, it would allow the user to quickly and simply select

any fuel-oxidizer combination required for the calculations

The extension of the scalar PDF code to allow axisymmetric and 3D calculations

is relatively trivial, code modification being the main work, and so requires no

elaboration.
Modeling of Turbulent, Reacting Flows by PDF Methods 143

2.3 Model Development

In PDF methods, the modeling of the molecular diffusion term is a region of

current research. For the case of a single scalar, 0, in constant density, homogeneous,

isotropic turbulence, the evolution equation for the scalar PDF, fo is given by:

afm = - e 

[fm < (r/p) o2 ^1^ _ >l (2) at a*

where r/p is the diffusion coefficient divided by the density, and < Q(q)lp = q > is

the expectation of some function of q, conditional upon the sample space variable

p being equal to q. Many different models have been developed for this term ',87

none of which perform entirely satisfactorily. One common assumption made in

these models is that the time scale of the mixing, Tp is half the turbulent time scale,

T = k/E, where k is the turbulent kinetic energy and E is the dissipation. While

this assumption has been shown to be reasonable for free shear flows', experiments

have shown it not to hold for more complex flows'. It is proposed to remove this

problem by providing a separate model for To, based on the scalar dissipation, co.

The expected benefits of this model are to improve the performance of the mixing

model in regions where the mixing and turbulent time scales are not proportional,

such as areas of intermittency.

Another area of potential for modeling improvement is in the area of state variables. In the compressible scalar PDF model, the evolution of the PDF of composition and specific enthalpy is calculated, while the density and velocity calculations

are input from a finite difference moment closure model. A useful development

would be to model both state variables in the PDF part of the calculation. To

achieve this, a stochastic model for density ( or another state variable, such as entropy) would need to be developed. The next logical step after this would be to

provide stochastic models for the velocity, resulting in a velocity - composition PDF

model. The advantage of this type of model is that convection is now treated

exactly, and so gradient diffusion models no longer need to be used. This is particularly important for reacting flows, where counter-gradient diffusion has been shown

to occur9.

The final area of proposed work is in parallel implementation of PDF methods.

One of the concerns with PDF methods is ensuring there are enough particles to ensure statistical accuracy, with the constraint being the amount of memory available

on a computer. For the compressible scalar PDF method of Hsu et all , a time averaging procedure is employed, however of order 100 particles per cell is still required.

However PDF methods are particularly suitable for parallel implementation as the

same program, with fewer particles, can be run on many different processors at

once, and the results averaged at the end of the runs. For full velocity-composition

PDF methods, the only change from a serial program is including the averaging

process, a simple task. Apart from very simple implementation, there is no need

for processors to communicate with each other, except at the end of the program,

making PDF parallel computing effective on distributed systems. For the scalar

PDF model, the parallel implementation of the finite difference velocity code is the
144 A. T. Norris

only complication. One solution is to employ a parallel implementation for the

scalar PDF calculations, with the mean quantities being passed to a serial velocity

calculation. This method takes advantage of the statistical accuracy of the parallel

PDF calculations, while minimizing the changes required to a serial code.

3. References

1 Pope, S.B., "PDF Methods for Turbulent Reactive Flows," Prog. Energy Combust. Sci., 11, 119-192 (1985).

2 Pope, S.B., "A Monte Carlo Method for the PDF Equations of Turbulent Reacting

Flows.," Combust. Sci. Tech., 25, 159-174 (1981).

3 Dopazo, C. & O'Brian, E.E., "An Approach to the Autoignition of a Turbulent

Mixture," Acta Astronautica, 1, 1239-1266 (1974).

4 Chen, J.Y., Kollmann, W. & Dibble, R.W., "PDF Modeling of Turbulent, NonPremixed Methane Jet Flames," Combust. Sci. Tech., 64, 315-346 (1989).

5 Hsu, A.T., Tsai, Y.L.P. & Raju, M.S., "A PDF Approach for Compressible Turbulent Reacting Flows," Proc. AIAA 31st Aerospace Sciences Meeting and Exhibit, Reno, Nevada, AIAA-93-0087 (1993).

6 Pope, S.B. & Correa, S.M., "Joint PDF Calculations of a Non-Equilibrium Turbulent Diffusion Flame," In 21st Symposium (International) on Combustion, 1341-

1348 (1986).

7 Anand, M.S. & Pope, S.B., "Calculations of Premixed Turbulent Flames by PDF

Methods," Combustion and Flame, 67(2), 127-142 (1987).

8 Norris, A.T., The Application of PDF Methods to Piloted Diffusion Flames, PhD

Thesis, Cornell University, Ithaca, 1993.

9 Pope, S.B., "Computations of Turbulent Combustion: Progress and Challenges,"

In 23rd Symposium (International) on Combustion, 591-612 (1990).

io Kee, R.J., Rupley, F.M. & Miller, J.A., "CHEMKIN-II: A Fortran Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics," Report SAND89-

8009, Sandia National Laboratories, Livermore, CA 94551
Modeling of Turbulent, Reacting Flows by PDF Methods 145

1.0

.7

y

5

.4. 

3

.2

1

0

0 .2 .3 .4 .5 .6 7 .8 .9 1.0

Fig 1: Contour plot of reaction rate as a function of ^ and Yp with the grid for

the adaptive lookup table overlaid.
Page intentionally left blank 
Center for Modeling of Turbulence and Transition 147

Research Briefs - 1993

A Time Dependent Generalization

of the Nonlinear Eddy Viscosity

Representation of Turbulence

R. Rubinstein

1. Motivation and Objective

Standard two equation turbulence models are derived in an equilibrium limit

which assumes a local Kolmogorov steady state. This assumption rules out the

possibility of correctly capturing the response of turbulence to sudden changes in

external conditions which occur when a flow is impulsively sheared or subject to

rapidly oscillating shear. Moreover, such models also do not incorporate memory

effects associated with the viscoelasticity of turbulence. Although stress transport

models do incorporate some memory effects, it will be shown that they misstate

the short time response. The goal of this work is to analyze the time dependence

of turbulence from the viewpoint of the direct interaction approximation. This is

a natural viewpoint because the direct interaction approximation is the only fully

time dependent analytical theory of turbulence now available. Models valid for

arbitrarily large strains will be introduced. These models demonstrate that both

rapid distortion theory and the Kolmogorov steady state theory are required in a

complete picture of shear turbulence.

2. Work Accomplished

The goal of this work is to derive a time dependent generalization of the nonlinear

eddy viscosity formula

2

T = 2KI — C„K (VU +VUT)

3

• Cr1 K3(VUDUT — 3VU : VUTI)

3

• CT2 K ( D U2 + VUT2 — 3 :0U VUI) (1)

3

+CT3 3(DUTVU-3DU:VUTI)

The symbols have the standard meanings: Tij = ui uj is the stress tensor, K is the

turbulence kinetic energy, e the dissipation rate, VU the mean velocity gradient, and

C,,, C,-I, CT2, CT3 are model constants. The generalization is a nonlinear viscoelastic

representation of the stress in which these constants are replaced by integrals over

the strain rate history. A typical application of such a theory would be to normal

stress effects in oscillating duct flow. It will be shown that this model predicts

time dependent response quite distinct from the time dependent response of the
148 R. Rubinstein

standard stress transport models. The theory reduces to Eq. (1) for constant strain

rates maintained for long times, but reduces to rapid distortion theory for suddenly

applied and rapidly varying strains. The implications of the model for oscillating

shear flows are discussed. This work also helps clarify the relation between rapid

distortion theory and the Kolmogorov theory of turbulence. A time dependent form

of Eq. (1) cannot be valid when the dimensionless strain rate K(Vu : Vu)' /'/E is

large. Some preliminary extensions of Eq. (1) applicable to arbitrary strain rates

will be suggested. Rapid distortion theory plays an essential role in these extensions.

2.1. Derivation of the Time Dependent Model

Eq. (1) was proposed by Yoshizawa l in order to model normal stress effects

with an explicit formula for the stress. For other discussions, see Pope , Speziale3

and Rubinstein and Barton 4 . As the derivations of this formula from analytical

theories of Yoshizawal and Rubinstein and Barton 4 suggest, Eq. (1) describes the

long-time behavior of turbulence corresponding to establishment of a Kolmogorov

steady state; application of this equation in a time-dependent problem requires

sufficiently slow changes of the strain rate that this steady state can establish itself

at each time. It should be noted that the models of Yoshizawa l and Speziale3

contain convective terms which represent effects of time dependent mean gradients.

However, these terms provide low order corrections for time dependence only; these

theories do not reduce to rapid distortion theory at very short times.

In contrast to the description by Eq. (1) of the purely viscous response of turbulence to mean strain, the viscoelastic response has been stressed by Crow'. In these

investigations, the short time response of turbulence is elastic, ?12 — KaUI /aX2 and

is determined by rapid distortion theory' , but the long time response is viscous,

T12 — vTaU1 /axe. Crow' proposed a linear viscoelastic turbulence model with these

properties, however, this model assumes purely (molecular) viscous damping, and

is therefore of limited applicability. Recently, Smith and Yakhot 7 have proposed

a more general theory of this type based on the direct interaction approximation

(DIA) of Kraichnan'. Since the DIA and its derivatives are fully time dependent

theories of the spectral dynamics of turbulence (and are, in fact, the only such

theories now available) they provide a natural basis for formulating such theories.

The DIA is a closed system of nonlinear integro differential equations for the correlation tensor Qij (k, t, s) and a response function Gij (k, t, s). A fundamental difficulty in DIA is the divergence of the response equation when Kolmogorov scaling

is assumed. Smith and Yakhot 7 propose to surmount this difficulty by introducing a regularized DIA in which the correlation equation, which is consistent with

Kolmogorov scaling, is not changed, but the response equation is suitably modified.

Previous suggestions of this sort are due to Kraichnan 9 and others. Pending a solution of this regularized DIA now in progress, Smith and Yakhot 7 suggest using

the response function for isotropic turbulence derived by Avellaneda and Majda14

for a related problem

Gij = GPij , Pij = Sij — ki kj

/k2

(2)
I A Time Dependent Generalization 149

G(k, T ) = exp { — C1E 113 k213 T[1 — exp(—c2E1/3k2/3T/2)]}

An important feature of this response function stressed by Smith and Yakhot'z is

that at short times, G — 1+0(t 2 ) as required by DIA, but G — exp(—ct) at long time

separations, as assumed (in effect) by static turbulence models. The importance of

this short time behavior will be discussed subsequently.

To derive the viscoelastic generalization of Eq. (1), we follow Leslie" and begin with the Fourier transform of the equations for the fluctuating velocity in a

homogeneous shear flow,

( d + vk2 )ui (k, t) — I iPi, .n(k) fp +q

dP dqx (3)

=k

um (p , t)un(q, t ) = Si,,,( k, t)u,(k, t)

where

Pi,nn = k m Pin + knPi,n

Sin,, _ —Ai,,, + 2k-2kikpApm + Sim k,Asra/ak,.

Aim = allikOx-m,

If Eq. (3) is formally expanded in powers of the mean strain rate about an isotropic

background state u(0),

U = u(o) + u(1) +... (4)

then DIA gives the solution for u(1),

t

u7 l) (k, t) _ f

o 

ds Gim) (k, t — s) 5,,,,n (k, s)ut° ) (k, s)

where GM = G( ° ) Pi,,,, is the response function for isotropic turbulence. If, corresponding to Eq. (4), the correlation function is written as

Q = Quo) + Q( 1 ) + .. .

then

fot

Q(1)(k, t) = ds G(o) (k, t — s)(— Air + 2 kik^k -2 Ap,.)PmQ(0) (k, t — s)

+(im) + GM (k,t — s) k, A,.na 

n 

Q",°,) (k,t — s) (5)

— QM(k, t — s) k, Arn 

as 

GM (k, t — s)

n

where (im) denotes index interchange in the immediately preceding term. Of course,

T = TM +T41)+... _ I dkQM+QM+...
150 R. Rubinstein

The behavior of this model for simple shear flow, in which aU1 119x2 = S is the only

nonzero mean velocity gradient is of particular importance. At short times, Eq. (5)

reduces to

T12 _ 4

15 KSt + O(t3)

The lowest order term agrees with rapid distortion theory. The O(t 3 ) correction

indicates a short time suppression of eddy damping which is characteristic of this

theory. In rapid distortion theory, in which eddy damping is absent altogether, the

corrections to short time behavior are of order O(t 5 ). Eq. (5) reduces at once to

the quantity evaluated by Crows if GM - 1.

To evaluate the stress at long times for constant strain, we adopt the procedures

of Yakhot and Orszag12 and assume that a universal energy spectrum E(k) exists

for k > k f , where k f is an inverse integral scale. We also assume the fluctuationdissipation relation

Q(k,T) = Q(k)G(k, I T ^)

Then

712 = v(k f)S

with

dE v(k, t) = 51 f dq O ( q , t) (5E( q ) + q dq

f

t'

O ( q , t) = JOds[G(o) ( q , s)]2

in agreement with Kraichnanio when t = oe. Explicit formulas follow by substituting the Avellaneda-Majda expression Eq. (2) for G and a Kolmogorov spectrum for

E. Note that v — k -413 as required by Kolmogorov scaling. Whereas the short time

response does not depend on the form of the spectrum, a result familiar from rapid

distortion theory, the long time response depends on the inertial range exponent

through dE/dk.

The short time behavior of this model can be contrasted to the short time behavior

of the standard Launder-Reece-Rodi 13 stress transport model, which gives for simple

shear

e 4

T12 = — CR KT12 + 

15 

SK + ... (6)

in which the terms indicated by dots are of order t 2 for short times in initially

isotropic flow. It is clear that at very short times, Eq. (6) predicts the same stress

relaxation as the equation

In

T12 (t) = G(t — s) S(s) ds

with G(t — s) = exp(—CRe(t — s)1K), so that G — 1 + O(t) for small times. It

follows that T12 = 4/15KSt + O(t 2 ) at short times. The short time behavior agrees
A Time Dependent Generalization 151

with rapid distortion theory to lowest order, but the corrections are of order 0(t2).

Thus, the stress transport models initiate eddy damping more quickly than the

present theory. This is reflected in the excessive short time growth of kinetic energy

in highly strained homogeneous shear flow. This difference is also reflected in the

response to oscillating mean shear. Suppose conditions are such that K and e can

be considered constant while S(w) = So + Sl e"'iw. Then exactly as in the theory

of linear viscoelasticity, the stress is given by the complex modulus

00

G(iw) _ f

o 

dT e'.wr G(o)(kf,T)

Of particular interest is the phase lag between stress and strain, which is zero in a

purely elastic regime. (The phase lag between stress and strain rate in 90 0 in this

case). The short time behavior G — 1 +O(t 2 ) causes a very rapid drop in the phase

lag as the frequency increases. Thus, this theory predicts a larger frequency range

for applicability of rapid distortion theory than the standard model.

This calculation helps clarify the relation between rapid distortion theory and

standard static turbulence models based, however implicitly, on a Kolmogorov

steady state. Namely, rapid distortion theory is the special case of shear flow DIA

that results from setting the response function to its short time value, G - 1 for

all times. The replacement of the DIA response function by its short time limit

has a twofold significance. First, it is consistent with the picture of rapid distortion theory as a short time theory. Second, since the DIA response function models

eddy damping, the result of nonlinear interactions, this assumption means that nonlinear interaction is ignored; neglect of nonlinear interaction in the Navier-Stokes

equations of course defines rapid distortion theory.

Calculation of the stress to second order is straightforward. The result has the

form

Q^ ) ( k, t) _ I(N)[a(N)Aip(s)Ajp(r)

1<N<6

• b(N)Aip(s)A,j(r)

• C(N)Api(s) Ajp (r) + d(N)Api(s)Apj(r)

• e(N)bijApa(s) Aap (r) + f (N) 6ij Ape(s) Ap4(r)] + (2j)

where Air = 8Ui/8x; is the mean velocity gradient, allowed to depend on time,

a (N) ,... f (N) are geometric constants, and the I(N) are integral operators

t

I(1) = J ds GO) (k, t — s) 0 dr GM (k, s — r)GM (k,t — r)Q(°)(k) J0 Jo

I(2) _

fo

 ds G( ° ) (k, t — s) fo s dr G(0) (k, s — r) dk [G(0) (k, t — r)Q(0)(k)]

I(3) _ f

o

 ds G(0) (k, t — s) fo 9 dr G(0) (k, s — r) d 22 [G(0) (k, t — r)Q(0) (k)]
152 R. Rubinstein

I(4) = t

fo ds G(o) (k, t— s) t

fo

I(5) =

fot

ds Giol (k, t — s)

t

fo 

I(6) =

fo

t

ds Giol (k, t — s)

fot

dr G(o) (k, t — r)G(o) (k, I s — r I)Q(o)(k)

dr G(0) (k, t — r) dk [G(o) (k, I s — r I)Q (o) (k))

dr G(o) (k,t — r)dk2 [G(o) (k, I s — r J)Q(o)(k))

The values of the constants are given by

(1) (2) (3) (4) (5) (6)

105a(N)27 —1 —2i2 — a —1

105b (N) 20 6 —2 6 3 —1

105c(N) —15 —15 —2 a __2 —1

105d(N)

20 —8 —2 —22 —4 —1

105e (N) 10 24 6 3 12 3

105f

(N) —4 24 6 10 12 3

The time integrations can be compared with those of Yoshizawa's theory'. As

in all derivations of this type, the constants a(N) ,... arise from integrating even

order products kl kj ,.... over spheres I k I = constant. A useful constraint on the

calculation is that it must reduce to the rapid distortion results of Maxey 6 when

G(o) - 1. An interesting reduction of this model for slowly varying strains is

suggested by the theory of nonlinear viscoelasticity, namely that the time history

integrals can be approximated by sums of Rivlin-Ericksen tensors in this case. This

replacement should lead to a model of the form proposed by Speziale 3 .

2.2. Models Valid for Large Strains

The perturbative derivation of this model restricts its applicability to weak, although possibly rapidly varying, shears. It is natural to try to derive a theory valid

for strong shear by summing this series. It proves more convenient to evaluate

the exact response function for shear flow than to evaluate the correlation function

directly. Thus, define G( ') by

Gi^1 ) = GM Spa

where S is the mean strain operator of Eq. (3). The notation abbreviates the

integral operator in

fo

t

uil) (t) = G^) ui(°i (s) ds

As above, the operator G (2) can be constructed, and by simple recursions, all G(")

for n > 2. It is at once evident from the form of this expansion that if we set
A Time Dependent Generalization 153

G( ° ) = 1, then the sum G = G(o) + GO) + • • • simply defines rapid distortion theory

for finite times and finite strain rates. The sum is explicitly known for several

standard cases including simple shear and irrotational strain. Consider for example

simple shear with S(t) = 8U1 18x2 the only nonvanishing mean velocity gradient.

The replacement of GM = 1 by a form valid at long times, say by Eq. (2) amounts

to a replacement in the formulas of RDT of the total strain

fot

a = S(r) dr

by the wavenumber dependent quantity

fnt

a(k) = G(o) (k, r) S(r) dr

This defines an "equivalent total strain," a formal device often advocated in the

RDT literature, but derived here as a consequence of an analytical theory.

It is not difficult to derive from this theory a long-time modification of Eq. (1)

valid at large strains, in which the coefficients C,,, CT1, Cr2, C, 3 are functions of

p = SKI e as suggested by Yakhot et al. 15 For example, in the case of simple shear

flow,

C,.(rl) = F( 77ICR) 1p

where T/K = F(a) is the solution of RDT for simple shear. Analogs for the normal

stress coefficients CTi (77) follow at once from RDT.

It must be stressed that this theory only applies to simple shear flow. Derivation

of a theory for three-dimensional mean velocity fields will require the solution of

RDT for these flows. A phenomenological modification of Eq. (1) has recently been

proposed by Zhu and Shih" by using realizability constraints to set the functional

dependence on rl.

We would like to conclude by outlining another approach to developing models

valid at arbitrary strain rates, now being developed in collaboration with L. Smith

and V. Yakhot. The idea is that RDT may apply not only at short times, but also to

scales for which the Kolmogorov inertial range frequency, proportional to E1/3k2/3

is smaller than the imposed strain rate S. The remaining scales are governed by

equilibrium eddy damping. This suggests a two-scale form of the response function

_ 1, k < k* G(k, t)

exp — cE113 k2/3 t, k > k*

where the cutoff scale k* is chosen so that turbulence in local energy equilibrium has

no rapidly distorted scales. It remains to determine the time of rapid distortion.

This time is finite both in homogeneous shear flow and in near wall turbulence.

In homogeneous shear flow, scales of increasing size 1/k f are continually being

created by the mean shear, but the cutoff' scale Ilk* may also increase with time;
154 R. Rubinstein

thus, turbulence at scale k is rapidly distorted from the time that it is created

until the time when k = k*. A plausible conjecture is that in fully developed

homogeneous shear flow, 1 1k* and 11k f both increase exponentially with the ratio

k*/k f constant. This theory could explain the observation that stress ratios in both

fully developed homogeneous shear flow and shear flows in local energy equilibrium

are approximately equal.

In steady near wall flows, turbulence is created in the region of high strain near the

wall but is then diffused into the equilibrium log layer l '. Thus, although the flow is

steady in time from the usual Eulerian viewpoint, from the Lagrangian viewpoint,

turbulence is highly strained only for a finite time. If we assume that imbalance

between production and dissipation is entirely the result of rapid straining, then it

is reasonable to set this time to K/K = Kl (P — e) where P = ST is production.

Preliminary comparisons with data from near wall flows and from highly strained

homogeneous shear flow are satisfactory and further development of this model is

in progress.

3. Future Plans

A two scale model of homogeneous shear flow will be developed to apply both to

fully developed flows and to rapidly sheared flows. Current models do not capture

the transient period of highly sheared flows correctly. It is hoped that the present

methodology, which incorporates the rapid distortion limit in an essential way, will

be more successful. Application of the theory to near wall turbulence will also be

attempted. This problem is more difficult than homogeneous shear flow because

the transients are due to rapid diffusion rather than to convection.

The long time constants in the nonlinear eddy viscosity representation will be

derived in this DIA framework. The introduction of the e expansion techniques of

the Yakhot-Orszag theory into DIA will be required.

4. References

1 Yoshizawa, A. "Statistical analysis of the deviation of the Reynolds stress from

its eddy viscosity representation," Physics of Fluids 27, 1377-1391 (1984).

a Pope, S. B., "A more general effective viscosity hypothesis," Journal of Fluid

Mechanics 72, 331-340 (1975).

3 Speziale, C. G., "On nonlinear K-1 and K-e models of turbulence," Journal of

Fluid Mechanics 178, 459-487 (1987).

4 Rubinstein, R. and Barton, J. M., "Nonlinear Reynolds stress models and the

renormalization group," Physics of Fluids A 2, 1472-1480 (1990).

5 Crow, S., "Viscoelastic properties of fine grained turbulence," Journal of Fluid

Mechanics 41, 81-95 (1968).

6 Maxey, M. R., "Distortion of turbulence by uniform shear," Journal of Fluid

Mechanics 124, 261-282 (1982).

7 Smith, L. M. and Yakhot, V., "Short time and long time behavior of eddy viscosity

models," Theoretical and Computational Fluid Dynamics (to appear)
A Time Dependent Generalization 155

8 Kraichnan, R. H., "The structure of turbulence at very high Reynolds number,"

Journal of Fluid Mechanics 5 459-491 (1959).

9 Kraichnan, R. H., "Kolmogorov's hypothesis and Eulerian turbulence theory,"

Physics of Fluids 7 1030-1048 (1964).

10 Kraichnan, R. H., "Eddy viscosity in two and three dimensions," Journal of the

Atmospheric Sciences 33 2315-2338 (1976).

11 Leslie, D. C., Modern developments in the theory of turbulence, Oxford (1973).

12 Yakhot, V. and Orszag, S. A., "Renormalization group analysis of turbulence,"

Journal of Scientific Computing 1 3-50 (1986).

13 Launder, B. L., Reece, G. and Rodi, W. "Progress in the development of a

Reynolds stress transport closure," Journal of Fluid Mechanics 68 537-582 (1975).

14 Avellaneda, M. and Majda, A., "Approximate and exact renormalization theories," Physics of Fluids A 4 41-63 (1992).

15 Yakhot, V., Thangam, S., Gatski, T., Orszag, S. A., and Speziale, C., "Development of turbulence models by a double expansion technique," Physics of Fluids

A 4 1510-1532 (1992).

16 Zhu, J. and Shih, T. H., "Calculation of turbulent separated flows," NASA Technical Memorandum TM-106154 (1993).

17 Yakhot, V. (private communication).
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Center for Modeling of Turbulence and Transition 157

Research Briefs - 1993

Higher-Order Accurate Osher

Schemes with Application to

Compressible Boundary Layer Stability

J. J. W. Van Der Vegt

1. Motivation and Objective

The study of boundary layer stability and transition to turbulence is one of the

classical topics in fluid mechanics. Linear and weakly non-linear theory, together

with experiments, have been successful in describing several of the important instability mechanisms in compressible boundary layers, e.g. Tollmien-Schlichting (TS)

waves, or first modes, and higher modes, which come into play at supersonic Mach

numbers, Mack15 , and theory was also successful in describing secondary instability,

Herbertlo

The complicated phenomena in the non-linear stages, leading to transition and

turbulence, however, require further understanding. Direct numerical simulations

can provide some of this information, but their application to compressible boundary layers has been hindered by many obstacles. To mention just a few, high

order accuracy is required on non-uniform grids and a severe time step limitation

is encountered due to the small grid spacing in the boundary layer when using an

explicit time integration method; whereas with implicit time integration methods it

is difficult to maintain time accuracy. Spectral methods have been very successful

in simulating incompressible flows in simple geometries, such as channel flow, e.g.

Laurien and Kleiser", but are not easily extended to more complicated geometries.

The recently popular compact finite difference schemes, Lele 13 , do not have the geometric limitations of spectral methods and have been successfully applied to mixing

layers and shock turbulence interaction, Lee et al . Unfortunately, compact finite

difference schemes and also spectral methods, cannot capture shocks and if they

appear they have to be fully resolved, which can require prohibitively small grid

spacings, Lee et al.14

There have been several attempts to use finite difference schemes for direct simulation of transition in compressible boundary layers. The most frequently used

method is the fourth order accurate version of Mac Cormack's scheme, developed

by Gottlieb and Turkel', e.g. Maestrello et al." and Bestek et aP. This method

can only achieve higher order accuracy on grids generated as the product of two

or more one-dimensional analytic transformations, limiting its applicability to relatively simple, smooth flows.

Most frequently, explicit time integration methods have been used, but for many

transitional flows the Courant- Fried richs-Lewy (CFL) time step limitation is not

necessary to maintain time accuracy. Recently Rai and Moin 21 developed a numerical scheme which solves the compressible Navier-Stokes equations using a time
158 J. J. W. Van Der Vegt

accurate upwind biased implicit method and were able to simulate bypass transition. This method alleviates the time step limitation of explicit methods, but has

as main drawback that it uses the non-conservative form of the Navier-Stokes equations and only allows grid stretching in one direction. The grid stretching, however,

does not have to be analytically defined because the higher order finite difference

approximations are generated numerically in physical space.

In this paper two alternative methods will be discussed. The first method is a

higher order accurate extension of the MUSCL scheme, originally developed by Van

Leer27 as a second order accurate extension of Godunov schemes. The scheme is

related to the multi-dimensional essentially non-oscillatory (ENO) schemes developed by Casper and Atkins4 and Harten et al. 9 . The second method is a higher

order accurate upwind biased version of the Osher scheme, which maintains its high

order accuracy on non-uniform grids. Higher order accurate Osher schemes were

also discussed by Rai 20 , but his method is only higher order accurate on a uniform

grid.

The discussion in this paper will be restricted to smooth flows, but both schemes

have been extensively tested for flows with shocks and other discontinuities in onedimensional flows. A detailed discussion of the benefits and application of these

schemes to non-smooth flows can be found in Van Der Vegt 25 . The purpose of

this paper is to investigate if these methods are accurate and efficient enough to be

used as tools for direct simulation of boundary layer instability and transition to

turbulence. The finite volume scheme has as main benefit that it is a truly multidimensional scheme, whereas the finite difference scheme uses dimensional splitting.

The finite volume scheme automatically satisfies conservation and is most closely

related to the integral formulation of the compressible Euler and Navier-Stokes

equations. It also maintains higher order accuracy at sonic points, which is not true

for the finite difference formulation. The finite volume method, however, is slightly

more costly than the finite difference scheme and requires significantly more effort

to implement. The use of an upwind scheme is beneficial for direct simulations of

compressible flow, because it automatically controls aliasing errors. There are two

types of upwind schemes, those based on flux vector splitting, e.g. Van Leer 28 and

Steger-Warming24 , and those based on a Godunov approach which use the solution

of a Riemann problem. Godunov schemes most closely mimic the physics of wave

propagation in compressible flow and have excellent shock capturing properties.

In this class of schemes the Roe and Osher approximate Riemann solvers are the

most popular, see Roe 22 and Osher and Solomon 18 . The Osher scheme has been

chosen because it has a very low numerical dissipation in boundary layers, which is

crucial for direct simulations, and it has a continuously differentiable flux, which is

important for implicit schemes. In addition it satisfies the entropy condition and

has good shock capturing properties.

In the next two sections of this paper the basic equations will be discussed and the

higher order accurate numerical schemes will be presented. The paper will conclude

with a discussion of the results of computations of boundary layer instability at

various Mach numbers.
Higher-Order Osher Schemes and Compressible Boundary Layer Stability 159

2. Work Accomplished

2.1 Navier-Stokes Equations

The Navier-Stokes equations can be considered either in integral formulation,

leading to the finite volume discretization, or in differential form, which is the basis for the finite difference discretization. The finite volume method automatically

satisfies the conservation properties of the equations but care has to be taken that

the finite difference method is in the so-called conservation form. It is otherwise

not possible to obtain a weak solution with the proper jump relations at discontinuities in the limiting case of vanishing viscosity, as demonstrated by Lax and

Wendroff12 . A detailed discussion of finite volume and finite difference methods

and their differences can be found in Vinokur29.

The integral formulation of the compressible Navier-Stokes equations is defined

as: f f / t2

J

UdV — J

UdV + J

F(U) - ndSdt = 0 (1)

Here Q(t) is the flow domain with boundary aQ(t) at time t and n the unit outward

normal vector at 352. The vector U represents the conserved variables:

(p, pu, pv, pw, e) T , with p density, u = (u, v, w) T the Cartesian velocity components, and e total energy. The matrix T, which represents the fluxes through the

surface a52(t), consists of two parts, F the inviscid flux, and V the viscous flux, with

.F=F-V. The inviscid flux contribution F has as components:

pu

Put +P

F, = puv

puw

(e + p)u

pv

puv

F2 = PV + p

pvw

( e + p)v

pw

puw

F3 = pvw (2)

Pw2 +P

(e + p)w

where the dimensionless pressure p is determined from the equation of state: p =

pc„T/(-yM2), with c„ the specific heat at constant volume, y the ratio of specific heat

at constant pressure and constant volume, M the Mach number and T temperature.

The temperature T is given by the relation: T = y(-y — 1)M2 (e — Zpu • u)/(pc„).

The viscous contribution V has as components:

0 0 0

Txx Txy Txz

V1 = Txy V2 = 

Tyy V3 - Tyz ( 3)

7*^X, 

z

T//^^y z

Tz z

IX Ny 0z
)x =uTxx + vrxy + wTxz +

Ny =uTxy + vTyy + wTyz +

Oz =UTxz + vTyz + WTxz +

(ry — I)M2 Pr ax

K aT

(ry — 1)M2 Pr ay

K aT

(-y — 1)M2 Pr az

(5)

160 J. J. W. Van Der Vegt

with the stress tensor T and the variables Nx, ay and pz defined as:

Txx =

C(2μ + A) ax 

+ A(w + az )) lRe

y

au

Txy — 

(^t(Oy

+ 

av

ax 

lRe

au

Txz = μ(az

aw 1

+ ax ) J lRe

Tyy = (2μ + av

^) ay +A( 

ax 

au 

+ Ow )) az

lRe

Tyz= 

(tt( aw

a

av

+ az )^ lRe

Ow

Tzz = C(2μ + A) az + A( ax + av )^ /Re y

K aT

(4)

Here Re represents the Reynolds number, Pr Prandtl number, r, the coefficient of

thermal conductivity and μ and \ the first and second viscosity coefficient. All

computations were done using the relation A = 3 μ, with μ given by Sutherland's

law. The non-dimensional variables are defined with respect to the freestream

velocity, density, temperature, viscosity, thermal conductivity and specific heat.

If we assume that all variables are continuously differentiable in time it is possible

to rewrite equation (1) into:

a j UdV + j P(U) • ndS = 0 (6) at^t) ast(t)

A special constraint can be derived from this expression, namely the geometric

conservation law. Inserting a uniform flow field in equation (6) we obtain:

ndS = 0 (7)

which states that the cell face OQ(t) must be closed. When dividing the total flow

field in a set of non-overlapping cells this constraint puts limitations on how to

compute the cell faces and volumes. They all have to add up to the total volume

and each cell will have to be closed; otherwise a uniform flow field will be disturbed
Higher-Order Osher Schemes and Compressible Boundary Layer Stability 161

due to contributions from the metrics. This is a non-trivial problem when deriving

higher order schemes and will be discussed in the next sections.

The differential form of the compressible Navier-Stokes equations is directly obtained from the integral formulation, equation (6), using Gauss' theorem and considering an arbitrary volume Q:

a+ V . .F(U) = o (s)

This is the conservative formulation of the compressible Navier-Stokes equations

and it is important that the discretization also can be written in conservation form.

The geometric conservation law must be satisfied and this requires great care in

dealing with the metrical coefficients in a finite difference discretization, especially

in three dimensions.

2.2 Higher Order Accurate Finite Volume Scheme

Second order accurate finite volume schemes have been around for a long time.

When extending the accuracy beyond second order several problems are encountered. It is no longer valid to equate cell averaged values with values at the cell

center and a more elaborate algorithm to reconstruct point values from cell averaged

values is required. In addition one has to compute the integrals of the fluxes over

the cell surfaces more accurately. The standard procedure of multiplying the flux

with the cell surface is only second order accurate. The geometry in a higher order

accurate finite volume method also has to be represented more accurately, especially

at the boundary. The relations for cell area and volume as given by Vinokur 29 are

no longer sufficient. They are exact for hexahedrons with straight line edges, but

not for cells with curved edges.

Until now very few attempts have been made to develop higher order accurate finite volume methods. For structured grids Harten et a1. 9 and Casper and

Atkins' developed the multi-dimensional ENO schemes and Abgrall l and Harten

and Chakravarthy8 studied these schemes for unstructered grids. Despite the fact

that the unstructured approach has more flexibility in treating complicated geometries and allows local grid refinement it was decided to use the structured grid

approach developed by Casper and Atkins'. Both CPU time and memory usage

in the unstructered schemes is so large that it is not feasible to use these methods for direct or large eddy simulations of compressible flow. The structured grid

approach, however, requires a smooth C' grid. It is possible to deal with locally

non-smooth boundaries and intersections in the finite volume approach by subdividing the grid into smooth subsections using a multi-block approach. The smoothness

requirements of the grid will limit the application of this method to fairly simple

geometries, but due to their high cost large eddy and direct simulations will be

limited to simple flows for quite a while. The finite volume method is, however,

considerably more flexible than spectral methods which require a C' grid.

The use of a structured grid makes it possible to transform the physical domain

to a simpler computational domain. Let l;, rl, and ( be the coordinates in the
162 J. J. W. Van Der Vegt

computational domain then they are related to the physical coordinates x, y, z by

the relations:

=^(x, y , z)

rl =77( x , y , z) (9)

( =(( x , y , z)

The finite volume discretization on a structured grid is obtained by dividing the

domain f2 into regular partitions Qi,j,k . Each element f2i,j, k is a hexahedron with

coordinates x i , yj and zk for the top right corner. The subdomains Qi , j, k are nonoverlapping and their sum is equal to the domain D.

The cell average Ui,j, k in a cell with index (i, j, k) is defined as:

Ui,j,k — 1 UdV (10) V01(f2 ) z,^,k

with Vol(Qi,j,k ) the volume of cell Qi,j,k. The equation for the cell average is

obtained by limiting the integration domain Q to Pi , j, k in equation (6), and is equal

to:

atUi,j,k(t) + hi,j,k (U) = 0 (11}

with the flux integral hi, j,k at the surface aQi, j,k of the cell Qi,j,k defined as:

hi,j, k (U) = 1 F(U) • ndS

Vol ( Qi,j,k) aSn ij.k (12)

1 F(U)dS

VOl ( Qi , j, k ) I-9Qj,j,,

with F the flux normal to the cell surface:

F = k1 F1 + k2 F2 + k3 F3

(13)

and n = (kl , k2 , k3 )T. This relation gives the method of lines formulation for

the cell averaged equation Ui,j,k(t). The numerical flux in a higher order finite

volume scheme now is constructed such that it approximates the exact flux at time

t = (n + 1)At up to O(hr):

Ui,j k

[Eh (At) 

UnJi j,k

= O(h') (14)

with Eh(At) the numerical solution operator. The higher order accurate finite

volume scheme therefore gives an r-th order accurate approximation to the cell

averages, not the point values as in a finite difference scheme.

The two important ingredients of a higher order accurate finite volume m ethod

are the reconstruction of the point values U(x) from the cell averaged values Ui,j,k

and the computation of the flux F at the cell face. The point values are necessary
Higher-Order Osher Schemes and Compressible Boundary Layer Stability 163

to compute the fluxes hi,j, k at the cell faces which depend on U(x) and not on

U. This is done with the reconstruction method discussed in section 3.1. The

fluxes at the cell faces are computed by considering a Riemann problem at each cell

face. This method was introduced by Godunov6 for first order accurate schemes

and extended by Van Leer27 to second order accuracy. Instead of using the exact

solution of the Riemann problem the approximate Riemann solver developed by

Osher and Solomon 18 is used, because it is less expensive than the exact solution,

but has excellent shock capturing properties and minimal dissipation in boundary

layers.

2.2.1 Reconstruction of Point Values from Cell Averages

In one dimension the most successful reconstruction technique is based on the

primitive function method, see e.g. Harten et al. 9 . This method was extended

by Harten et al.' and Casper and Atkins' to multiple dimensions using the onedimensional primitive function reconstruction technique first to compute line averages from cell averages, after which point values are computed with a second onedimensional reconstruction. It is, however, possible to define a primitive function

directly for the multi-dimensional problem without having to use one-dimensional

primitive functions.

Define the primitive function U as:

u(^, q , 0 = f < f ^ f^ U(^ /,71',S')IJ(^',77', S')I <dq'd(

(15)

Co T ?o o

then the flow field U satisfies the relation

3

U(, 77, ^) = 

I J( , ^l, 01 19006

rl, ^) (16)

with ^, rl and ( the coordinates in computational space and J the Jacobian of

the coordinate transformation. The primitive function U can be related to the cell

average U in physical space using the following relation:

Ck 77j Ci

u(Si, 77j, (k) = 

1^0 f17 0 f^o

UIJI<&7d(

k

k'

E f f f UI Jl <d77'd(

k'=ko 9'=jo i'=zo k'-1 j'-1 i'-1

j

i

f

k (17)

E JU(x, t)dV k ' =k o j'=j o i'=io Qi'.j',k/

k j i

E E Y^ VOl(Qi',j',kr)Uii,jl,kl

k ' =ko j' =7o i'=io
164 J. J. W. Van Der Vegt

This relation is the basis for the higher order finite volume scheme. It defines the

primitive function if directly at the corners of each hexahedron in computational

space (^i, rlj , (k ) and is conservative. The point values U(x) are then obtained using

equation (16).

When the flow field is smooth the following procedure can be used to compute the

pointwise data: For a surface in the plane ^ _ ^*, first approximate the left and right

states by differentiating the primitive function if with upwind biased differences in

both ^-directions with fourth order accuracy for all indices j and k. Then the 77 and

(-derivates are computed at the Gauss quadrature points and divided by the local

Jacobian to obtain the point values. This process is repeated for the planes with

rt = rt* and ( _ (* and works well for smooth flows. For flows with discontinuities

the ENO reconstruction, which tries to use only data from the smooth part of the

flow field has to be used. This will be discussed in a future paper.

Although this process is rather simple, care has to be taken to prevent loss of

accuracy due to truncation errors, because the primitive function U frequently becomes very large or small. The way to prevent this is to construct the primitive

function using only those cells around the cell with index (i, j, k) which are needed

to compute the derivatives in equation (16) and adjust the indices io,3'0 and ko for

each cell. Accuracy is further improved by combining the process of summation

and differentiation, e.g. first compute the sum with i-index, then differentiate this

result and compute the summation with j-index, and so on.

In order to preserve uniform flow it is necessary to compute the Jacobian of the

coordinate transformation at the Gauss quadrature points the same way as done

for the flow field U. This can be accomplished most easily by multiplying equation

(16) with the Jacobian and inserting a uniform flow field in equations (16-17). This

will give the Jacobian at the Gauss quadrature points with the same reconstruction

process as for U. This procedure is important because otherwise the reconstruction

process will generate small errors which can be very annoying on stretched grids.

2.2.2 Higher Order Accurate Flux Integrals

The discretization of the integral formulation of the compressible Navier-Stokes

equations (11) is completed with the approximation of the flux integrals given by

equation (12). Casper and Atkins 4 gave a detailed analysis of the accuracy required

in the reconstruction problem and the computation of the flux integrals to obtain an

accurate solution with an error of order h'. In this paper we limit the discussion to

fourth order accuracy. The use of a Gauss quadrature method is the most efficient

way to compute the flux integrals with fourth order accuracy. First the integral

over the cell boundary is split into the sum of integrals over the six cell faces:

1

6

hi,a,k(U) — Vol(Sti,^,k) ^ 1 . k 

F(U)dS (18)

l-1 ,j.

here the index l refers to one of the six faces of the hexahedron. Then the flux

integrals at each cell face can be further evaluated using the Gauss quadrature rule
Higher-Order Osher Schemes and Compressible Boundary Layer Stability 165

at each cell face and the Osher approximate Riemann solver 17,18 is used to compute

the flux at each quadrature point:

F(U)dS = 2 Fn + F R — JaFJdU JJJdS

aQ f 

N18

FL + FR — f 10FIdU JJ9m,, ldS

^n=1 I, ) 9_j

with 0S2i'j k the cell face with index l in computational space. The indices g.,,,, , , refer

to the quadrature points with index m in the cell face where the fluxes of the left

and right states FL,R and the Osher path integral are computed. The quadrature

points have coordinates (a + 1 s , z f aVdl ), assuming that the hexahedrons sides

have unit length. This relation is used by Casper and Atkins4 and requires four

calculations of the approximate Riemann flux for each cell face which significantly

increases the computing time. Harten and Chakravarthy8 suggested that only one

computation of the (approximate) Riemann flux is required to maintain accuracy in

smooth flows. Due to the fact that both the Riemann flux and also the approximate

Riemann flux according to Osher are Lipschitz continuous and JUL — URA = O(hr)

in smooth flows, it is easy to show that the third component in the integral, equation(19), can be approximated as:

Jac, IF 

IaFIdUJJ1dS = J aF^i I dUI J^, I + O(hr) (20)

I,).k r

Here the index ci refers to center of the cell face with index 1. This relation significantly reduces the computing time, while maintaining the total accuracy at the

slight expense of computing UL,R at the cell face center. In regions with discontinuities it is, however, advisable to use the (approximate) Riemann solution at all

the Gauss quadrature points.

2.3 Higher Order Accurate Finite Difference Scheme

The most difficult problem in deriving a higher order accurate finite difference

scheme is to find a way to maintain that accuracy on a non-uniform grid. In upwind

finite difference schemes, either based on flux vector splitting or using approximate

Riemann solvers, the flux is a function of more than one grid point. When deriving

the expression for the higher order differences care has to be taken to account for

the changing metrics, but this is frequently neglected. For instance the higher order

Osher scheme derived by Rai 20 is only higher order accurate on a uniform grid. In

addition to the accuracy requirement, care has to be taken that the scheme is in

conservation form and maintains uniform flow, which is a non-trivial requirement

for a finite difference scheme. The conservation property is important on physical
166 J. J. W. Van Der Vegt

grounds; the equations express conservation of mass, momentum and energy. It

is also important when dealing with discontinuities. In this paper, however, only

smooth flow fields will be considered.

Before deriving the higher order Osher scheme it is necessary to study the first

order Osher scheme in more detail . The conservative approximation to a^E using

Osher's scheme is defined as:

a2 —Ei-2) (21)

where the conservative flux is defined as:

1 t

ki+Z =2(Ei+1(Si+2)+Ei(^i+2)-

(22)

f auk+ (^i+2) dU + auE— (^i+2 )dU)

r; r;

with equivalent relations for a,,F and a( G. The symbol au represents partial differentiation with respect to U and for ease of notation the j and k indices are omitted.

In this relation Si+2 refers to the metrical coefficients which are computed at the

point with index i + z. The integrals in equation (22) are computed along a path in

phase space, Ili, and using the fact that the Riemann invariants are constant along

this path Osher was able to derive exact analytic expressions for these integrals,

see2°,22. It is important to note that although the path integral ri is from i to

i + 1, only metrical coefficients at one point must be used for consistency. As can

be seen directly from equation (22), the flux Ei+ 2

 depends on the positions with

indices i, i + 2 and i + 1. Using a Taylor series expansion with respect to both 1 and

i + 1 Rai20 was able to derive higher order conservative expressions for a^Ei . The

dependency on Z'+ 1

, however, was neglected, which reduces the order of accuracy

of the scheme on a non-uniform grid, even on mildly stretched grids. The analytical

derivations necessary to obtain Rai's higher order Osher scheme are already tedious

and taking care of the changing metrical coefficients, which would require Taylor

series expansions up to at least fourth order in three independent variables, becomes

unwieldy.

An alternative is to compute the higher order accurate flux approximations numerically. This is done using the flux-ENO scheme discussed in Van Der Vegt25,

but the stencil switching, which is part of ENO schemes, is eliminated in this paper.

As starting point a different formulation of the first order Osher scheme is used:

a^Ei = 

Jr, 1

r 

auE+ (^i- 1)dU + t J auE— (^i+ i )dU (23)

2 2

It is easy to show that both formulations are equivalent, see Osher and Solomon18.

The higher order scheme is derived using a Newton interpolation to the fluxes.

The Newton interpolation, however, cannot be directly applied to the integrals in
Higher-Order Osher Schemes and Compressible Boundary Layer Stability 167

equation (23) because of the path integrals. In order to use the Newton interpolation

we use some simple relations, which were derived by Shu and Osher 23 : the function

f(x) can always be expressed as:

f(x)

,LX

/

oz 

h(x')dx'

x— a

(24)

and using its primitive function: F(x) = ox f^ h(x')dx' the following relation is

obtained:

f (xi) = F(xi + Lx ) — F(x i — Ax ) (25)

These relations can be used to link the primitive function F to the flux integrals

df:':

dfi =f — f± 1

=F+ [xi+

2 , xi_ z ] — F + [xi— 2 , xi_ 2 ] (26)

=2F+ [xi+ 1 , xi— 1, xi— s ]

2 2 2

dfi =fi+1 — fi

=F — [xi+ ! , xi+ ] — F — [xi+ , xi_ ] (27) 2 2 2 2

_2F [xi+ z , xi+ 

z 

xiz ]

with:

dfi = J t auk-(^i+2 )dU

r

dfi = f

ri 

aU E+ ( i-2 )dU

_1

and F[xi + k , ... , xi] the k-th divided difference defined as:

F[xi+k, ... xi] = (F[xi+k, ... , x i+11 - F [xi+k-1 , ... , xi])

In the derivation of the higher order accurate scheme the specific functional form

of the functions f1 and F^: is not needed, only their divided differences.

The primitive function F is now approximated with a fifth order Newton polynomial using the divided differences defined in equations (26-27). The higher order

divided differences can be easily obtained by further extending the divided difference tables given by equations (26-27). The nodes in the Newton interpolation

are chosen such that an upwind biased scheme is obtained. This relation is then

differentiated at xi+ :

2

a£ F+ ( x i+2 ) = F+[xi+ z , xi— 2 ] + F+ [x i+2 , ... , xi— a ]

+2F+[xi+2,... xi-2] (28)

- 2F+ [xi+ 3' ... , 

xi- 5 ] 2 2
168 J. J. W. Van Der Vegt

0^F— (xi+ 1 ) =F—(xZ+ 3,xi + 1 J — F [xi+b,... ,xi 2 +l) 2 2 2 2

+ 2F- lxi+ 2 , ... , xi— 2^ (29)

+2F-(xi+2,...,xi-2^

The higher order approximation to a^tj is obtained using equation (25) and adding

the positive and negative contributions in equations (28-29):

a^Ei = 9^ F(xi+ Z ) — 9^F(xi-2 ) (30)

This relation is conservative, fourth order accurate, and maintains its higher order

accuracy on a non-uniform grid because the change in metrical coefficients is properly taken care of by means of the Newton interpolation. It satisfies the geometrical

conservation law because for each of the integrals df't, i c {i — 2, i + 21, appearing

in the divided differences, the metrics are chosen at indices i + a , i E {i — 2, i + 2 1.

The geometric conservation law then is automatically satisfied because for uniform

flow each of the integrals gives a zero contribution independent of the metrical

coefficients. The additional cost of computing the divided differences is negligible

compared to the computation of the integrals df1 and the scheme is as efficient as a

scheme with analytically derived coefficients. One additional remark must be made

about the first divided difference in equations (28-29). Their value is unknown, but

not needed, because in equation (30) only their difference is used, which is exactly

the first order contribution and given by equations (26-27).

2.4 Implementation of Higher Order Schemes

In order to obtain the high accuracy necessary for direct simulations a large

number of grid points is required. In order to efficiently run the program with such

large grids a general three-dimensional multi-block code has been written. This

gives more flexibility in managing the large memory requirements and it is easier

to generate grids for more complicated geometries. The inviscid contribution in the

compressible Navier-Stokes equations is discretized using the procedure described

in the previous sections. The viscous terms are implemented using fourth order

accurate, conservative central differences for the finite difference scheme, but the

viscous contribution is only second order accurate for the finite volume scheme. The

development of a fourth order accurate viscous discretization for the finite volume

scheme is currently in progress. Second order accurate implicit time integration

is used and in order to maintain time accuracy a Newton procedure is chosen,

equivalent to the one used by Rai and Moin21 and Rai". In this method the

equations are discretized with all the fluxes computed at the new time level n + 1.

The resulting non-linear equations are linearized using a Newton procedure and the

large system of linear equations is solved iteratively. In 20,21 the diagonal form of

approximate factorization according to Pulliam and Chaussee 19 was chosen, which

only gives a crude approximation to the solution of the linear system. In this paper

a more accurate iterative scheme based on the zebra line Gauss-Seidel method

is used and for each step in the Newton procedure the system is solved up to
Higher-Order Osher Schemes and Compressible Boundary Layer Stability 169

machine accuracy. This method is used plane by plane and fits well into the Newton

procedure.

The Newton procedure makes it possible to have an implicit spatial discretization

which is of lower accuracy than the explicit part, but because each time step the

Newton procedure is iterated time accuracy is maintained and the higher order

spatial accuracy is preserved. Usually four Newton iterations are sufficient, but for

convergence of the Newton scheme it is necessary to solve the linear system of the

implicit time integration with high accuracy.

The Newton procedure requires the computation of the Jacobian of the flux vector, which is quite cumbersome to derive, especially for the viscous terms. Currently,

both the exact Jacobian of the Osher fluxes and the approximation using the Jacobian of the Steger-Warming24 flux vector splitting are used in the implicit time

integration. The exact Jacobian of the Osher fluxes improves the convergence rate

compared with the approximate Jacobian. It is, however, approximately three times

more expensive to compute the exact Jacobian and for most cases the computing

time for both schemes is about equal. For steady flows, where the Jacobian has to

be updated only after a certain number of time steps the exact Jacobian is more

efficient.

The boundary conditions for the Osher scheme are implemented using the procedure proposed by Osher and Chakravarty 17 . In this method a Riemann initialboundary value problem is solved instead of an initial value problem, which is used

in the interior of the domain. This method is very robust and elegant, but a significant effort is required to derive all relations to compute the boundary fluxes and

Jacobians.

2.5 Results and Discussion

In this section results will be presented of two simulations of ribbon induced

boundary layer instability to demonstrate the ability of the two numerical schemes

discussed in this paper to accurately predict boundary layer instability and their

possible use for simulations of turbulent and transitional boundary layers. Although the results in this section are all two dimensional the full three-dimensional

discretization was used.

As a first step it is crucial to have extremely accurate solutions of the mean

flow, which constitute the base flow for the stability calculations. In many previous

studies an analytically defined mean flow was used, but this becomes difficult for

flows in more general geometries. The use of an analytically defined mean flow also

has as disadvantage that it does not exactly satisfy the discretized equations and

will generate numerical transients.

In Figures 1 and 2 the mean flow profiles obtained with the finite difference

scheme discussed in section 4 are plotted at 10 equally spaced stations with Rex,

the Reynolds number based on the distance from the nose of the plate, ranging from

50,000 to 320,000 versus the similarity parameter p = x Re, with Re the Reynolds

number based on plate length, x the streamwise distance from the nose of the plate

and y the normal distance. The freestream Mach number is .08. The dimensionless

normal velocity in Figure 2 is defined as: v = v Rex . At the inlet a boundary layer
170 J. J. W. Van Der Vegt

solution was specified and the grid was chosen to approximately follow the streamlines and generated with the GridGen2d package. The grid is non-orthogonal in

the interior and the normal grid spacing increases downstream. In the same plot

also the compressible Blasius solution is plotted and it is clear that all lines completely collapse for the mean flow streamwise component, Figure 1. The same is

true for the normal velocity, Figure 2, except for the asymptotic value outside the

boundary layer, which is slightly higher than the Blasius solution. This slight difference is correct because in the boundary layer solution the small displacement

of the boundary layer is not accounted for. The comparison between the theoretical and numerical solutions is remarkable considering the high Reynolds number

of the base flow. Especially the accurate solution of normal velocity component

is noteworthy, because its value is much smaller than the streamwise component

and more difficult to compute. Most tests of numerical schemes on a flat plate

boundary layer use a Reynolds number which is considerably lower and only show

the streamwise velocity component. Accurate boundary layer profiles were already

obtained with 35 points in the normal direction, but the total grid consisted of 336

x 80 points to provide sufficient accuracy for the direct simulations, discussed in the

next part. This demonstrates that the Osher scheme is considerably more accurate

in boundary layers than schemes based on flux vector splitting, Van Der Vegt26.

The mean flow profiles obtained with the finite volume scheme discussed in section

3 are similar, but the finite volume scheme turns out to be more sensitive to the

smoothness of the grid on highly stretched meshes. Care has to be taken that

the grid for the finite volume scheme is at least three times differentiable. The

sensitivity to the grid smoothness of the finite volume scheme is caused by the fact

that in the reconstruction process the cell averaged flow field U i,j,k in equation (17)

is multiplied with the cell volume. The cell volume changes much more rapidly than

the grid spacing. The finite difference scheme is less sensitive to the grid, because it

uses dimensional splitting and depends therefore only on the smoothness of the grid

in each coordinate direction. The sensitivity to the grid smoothness of the finite

volume scheme certainly needs further attention before this scheme can be used for

more general applications.

Another problem when using a high order accurate scheme to obtain steady solutions for high Reynolds number flows is the slow convergence to steady state.

Due to the minimal amount of numerical dissipation it takes a significant amount

of computing time to eliminate all transients. Convergence to steady state was

improved using an implicit scheme and CFL numbers between 1000 and 5000 were

used to obtain steady results with the fourth order accurate schemes. In order to

further speed up convergence the computations were started with a first order accurate scheme till the residue was significantly reduced, followed by the fourth order

scheme till machine accuracy was obtained.

The first simulation of boundary layer instability is a comparison with the direct

simulations of incompressible flow about a flat plate done by Fasel et a1. 5 . All

parameters were chosen as close as possible to the one used in their computations.

The free stream Mach number was .08 and the Reynolds number based on flat plate
Higher-Order Osher Schemes and Compressible Boundary Layer Stability 171

length Re was 100.000/m. The plate started at x = .5 and ended at x = 3.2 and was

extended with a buffer region with slowly increased grid spacing to x = 8.0 for the

finite difference calculations and to x = 5.0 for the finite volume calculations. The

plate has 300 grid points in streamwise direction and the buffer region consisted

of 36 points. The purpose of the buffer region was to damp out transients and

thereby minimizing reflections. Accurate non-reflecting boundary conditions for

the compressible Navier-Stokes equations are not yet available.

First a steady boundary layer solution was computed and the maximum pointwise

value of the residual was less than 10 -8 . The flow was then disturbed in a small

region by periodic suction and blowing. The suction strip started at x l = .908 and

ended at x2 = 1.13. The amplitude is given by the relation:

f, = A sin (x)(1 — cos (x)) sin (0t)

with:

x= 27r(x — x l )

X2 - xl

The amplitude A for the computations is .0001. The parameter Q is equal to 10,

which results in a frequency parameter F = 100. Here the frequency parameter is

defined as: F = Q x 106/Re.

The blowing and suction starts in the region were the boundary layer is linearly

stable. This has as benefit that transients, which occur due to the suction and

blowing, will damp out and a cleaner Tollmien-Schlichting wave is obtained. The

time trace of all the flow variables along a line which corresponds to the position

of maximum amplification was written to a file and Fourier analyzed. The Fourier

analyzed signal was then used to compute the growth rate of the TS wave. Figure 3

shows the comparison of the growth rate —a i of the streamwise disturbances with

the results of Fasel et al. 5 . A negative value of the growth rate means that the

disturbance is growing. The large initial disturbances are caused by the suction

and blowing, but after x = 1.4 the result from the finite difference scheme compares

well with that from Fasel et al. 5 , which also agree with the theoretical non-parallel

results of Gastner. The finite volume results are slightly less accurate than the finite

difference results. This is partly due to the fact that the viscous contribution in

the finite volume scheme is only second order accurate. The second simulation, a

boundary layer at M = 0.5, which is at a considerably higher Reynolds number gives

virtually identical results for both methods. The growth rates of the disturbances

of the normal velocity component, Figure 4, also compare well with the results

from Fasel et al. 5 , but the curves are shifted slightly upstream. This component

becomes unstable earlier in the simulation than that of Fasel et al. 5 . The growth

rate of the normal velocity disturbances, however, strongly depends on the vertical

position and further research is required to obtain more accurate results for this

component. Contrary to the strearnwise component there are no theoretical results

for the growth rate of the normal velocity disturbances.

The same procedure as used for the M = .08 boundary layer was used for a

simulation of a flat plate boundary layer at Mach number M = .5. The parameters were chosen equal to the calculations done by Bertolotti 2 using the linear
172 J. J. W. Van Der Vegt

Parabolic Stability Equations (PSE). This method takes the non-parallel effects of

the boundary layer into account contrary to linear stability theory. The Reynolds

number based on plate length was 500.000/m. The simulations with the finite difference scheme were done on two grids. The coarse grid has 336 x 80 grid points

and the plate started at x = .5 and ended at x = 3.2. Suction and blowing was

started at x = .725 and ended at x = .86. The second grid consisted of 436 x 80

points and the plate started at x = .5 and ended at x = 2.3. Suction and blowing

started at x = .5225 and ended at x = .6125. The buffer region ends at x = 8. for

the coarse grid and at x = 5.1 for the fine grid. The first grid has about 10 grid

points per TS wave and the fine grid has 20 point per wavelength. The frequency

parameter F was equal to 20 and the free stream temperature T,' = 206K.

This case is more difficult than the incompressible simulation because of the

higher Reynolds number and many more TS wave periods have to be covered. The

initial amplitude of the disturbances A was .0001 and Figure 5 shows a contour

plot of the pressure on the fine grid. The regular pattern of the TS waves is clearly

visible. The initial amplitude is very small and decays because the disturbance is

in the stable part of the boundary layer. More downstream the disturbance grows

very regularly, saturates and decays. The decay is partly physical and at the end

of the plate further increased by the coarsening of the grid to minimize reflections

from the outflow boundary.

Figures 6 and 7 show the spatial growth rates obtained with the finite difference

and finite volumes scheme on the fine grid and compare them with the PSE results

of Bertolotti2 . They are virtually identical till the disturbances reach the buffer

region, where the growth rate suddenly changes and the Tollmien-Schlichting wave

rapidly decays. The CPU time used for both schemes was approximately equal for

the implicit calculations, with the finite volume scheme 1.1 times more expensive

than the finite difference scheme. The approximation of the flux integral in the

finite volume method, equation (20), however, is crucial to minimize computing

time for the finite volume scheme. These plots also show that the buffer region is

quite effective in minimizing reflections from the outer wall. The simulation was

continued in all cases till the leading wave front would have travelled at least twice

the length of the domain. The sound waves, which travel faster, then would have a

chance to reflect several times through the domain, with no apparent effect on the

growth rates. The use of a buffer region is, however, not without pitfalls. One has

to be very careful to create a smooth transition with the flow domain.

The accuracy of the simulation also depends on the time step and time integration scheme used. The time integration method is a second order 3 point implicit

multi-step scheme. Four Newton iterations were used to improve time accuracy

of the implicit scheme. The residual decreased two orders of magnitude and was

approximately 5 x 10 -7 at the end of the Newton iterations. Figures 8 and 9 show

the growth rate of the normal and streamwise velocity component for simulations

with different time steps using the finite volume scheme, but all on the 436 x 80

grid. The time step At is equivalent with a CFL number 80. As can be seen from

these plots the accuracy is bounded by the time integration scheme and not by the
Higher-Order Osher Schemes and Compressible Boundary Layer Stability 173

spatial resolution. Significant improvement should be obtained by using a higher

order accurate time integration method.

The results of computations with the finite difference scheme on the coarse and

fine grid are presented in Figure 10. They show that the coarse grid simulation is

underresolved, while the fine grid simulation compares well with the PSE results

of Bertolotti 2 . It should be emphasized that it is very important to perform the

computations on different grid levels to test accuracy, especially when there are no

theoretical results available. The coarse grid results do show that the flow becomes

unstable but the growth rate is not correct and can only be checked by increasing

the resolution.

Finally, the effect of the location of suction and blowing was investigated. Figure

11. shows results of simulations with the finite difference scheme with suction and

blowing applied directly after the inflow boundary or just before the region where

the flow becomes unstable. It can be seen that the growth rates are not sensitive

to the location of suction and blowing.

To summarize the results discussed in this section it can be stated that both the

finite volume and finite difference schemes can be used for simulations of boundary

layer instability. The results for the incompressible flow were slightly better for the

finite difference scheme, but this can be attributed to the second order viscous contribution in the finite volume scheme. The biggest advantage of the finite difference

scheme is that it is less sensitive to grid stretching, but this scheme is not easily

extended to flows with sonic points, because it will lose accuracy at these points,

which is not the case for the finite volume scheme. It was also found that using a

higher order accurate time integration method will be more efficient and extreme

care has to be taken to guarantee numerical accuracy, preferably by using grids with

different resolution.

3. References

1 Abgrall, R., "On Essentially Non-Oscillatory Schemes on Unstructured Meshes:

Analysis and Implementation", ICASE Report 92-74, (1992).

2 Bertolotti, F. P., "Linear and Nonlinear Stability of Boundary Layers with Streamwise Varying Properties", Diss., Ohio State University, Columbus, Ohio, (1990).

3 Bestek, H., Thumm, A. and Fasel, H., "Direct Numerical Simulation of the ThreeDimensional Breakdown to Turbulence in Compressible Boundary Layers", Proceedings 13th International Conference on Numerical Methods in Fluid Dynamics", Rome, (1992).

4 Casper, J. and Atkins, H.L., "A Finite Volume High-Order ENO Scheme for

Two-Dimensional Hyperbolic Systems", Journal of Computational Physics, 106,

pp. 62-76, (1993).

5 Fasel, H. F., Rist, U. and Konzelmann, U., "Numerical Investigation of the ThreeDimensional Development in Boundary Layer Transition", AIAA Journal, 28, pp.

29-37, (1990).
174 J. J. W. Van Der Vegt

s Godunov, S.K., "Difference Method of Numerical Computations of Discontinuous

Solutions in Hydrodynamic Equations", Matematicheckiy Sbornik, 47, pp. 271-

306, (1959), Translated US Joint Publ. Res. Service, JPRS 7226, (1969).

7 Gottlieb, D. and Turkel, E., "Dissipative Two-Four Methods for Time Dependent

Problems", Mathematics of Computation, 30, pp. 703-723, (1976).

8 Harten, A. and Chakravarthy, S.R., "Multi-Dimensional ENO Schemes for General Geometries", ICASE Report 91-76, (1991).

9 Harten, A., Osher, S., Engquist, B. and Chakravarthy, S.R., "Uniformly High Order Accurate Essentially Non-Oscillatory Schemes III", Journal of Computational

Physics, 71, pp. 231-303, (1987).

to Herbert, Th., "Secondary Instability of Boundary Layers", Annual Review of

Fluid Mechanics, 20, pp. 487-526, (1988).

11 Laurien, E. and Kleiser, L., "Numerical Simulation of Boundary-Layer Transition

and Transition Control", Journal of Fluid Mechanics, 199, pp. 403-440, (1989).

12 Lax, P. and Wendroff, B., "Systems of Conservation Laws", Communications on

Pure and Applied Mathematics, 13, pp. 217-237, (1960).

13 Lele, S. K., "Compact Finite Difference Schemes with Spectral-like Resolution",

Journal of Computational Physics, 103, pp. 16-42, (1992).

14 Lee, S., Moin, P. and Lele, S. K., "Interaction of Isotropic Turbulence with a

Shock Wave", Thesis, Stanford University, Report TF-52 Thermosciences Division, Dept. Mech. Eng., (1992).

15 Mack, L. M., "Boundary Layer Stability Theory", in "Special Course on Stability

and Transition of Laminar Flow", AGARD Report 709, (1984).

is Maestrello, L., Bayliss, A. and Krishnan, R., "Numerical Study of Three-Dimensional

Spatial Instability of a Supersonic Flat Plate Boundary Layer", ICASE Report

89-74, (1989).

17 Osher, S. and Chakravarthy, S.R., "Upwind Schemes and Boundary Conditions

with Applications to Euler Equations in General Geometries", Journal of Computational Physics, 50, pp. 447-481, (1983).

18 Osher, S. and Solomon, F., "Upwind Difference Schemes for Hyperbolic Systems

of Conservation Laws", Mathematics of Computation, 38, pp. 339-374, (1982).

19 Pulliam, T. H. and Chaussee, D. S., "A Diagonal Form of an Implicit Approximate

Factorization Algorithm", Journal of Computational Physics, 39, (1981).

20 Rai, M. M., "Navier-Stokes Simulations of Blade-Vortex Interaction using HighOrder Accurate Upwind Schemes", AIAA paper 87-0543, (1987).

21 Rai, M. M. and Moin, P., "Direct Numerical Simulation of Transition and Turbulence in a Spatially Evolving Boundary Layer", AIAA paper 91-1607-CP, (1991).

22 Roe, P. L., "Approximate Riemann Solvers, Parameter Vectors and Difference

Schemes", Journal of Computational Physics, 43, pp. 357-372, (1981).
F

Higher-Order Osher Schemes and Compressible Boundary Layer Stability 175

23 Shu, C.-W. & Osher, S., "Efficient Implementation of Essentially Non-Oscillatory

Shock Capturing Schemes, II", Journal of Computational Physics, 83, pp. 32-78,

(1989).

24 Steger, J.L. and Warming, R.F., "Flux Vector Splitting of the Inviscid Gasdynamic Equations with Application to Finite-Difference Methods", Journal of

Computational Physics, 40, pp. 263-293, (1981).

25 Van Der Vegt, J.J.W., "ENO-Osher Schemes for Euler Equations", AIAA Paper

93-0335,(1993).

26 Van Der Vegt, J.J.W., "Assessment of Flux Vector Splitting for Viscous Compressible Flows", AIAA Paper 91-0243, (1991).

27 Van Leer, B., "Towards the Ultimate Conservative Difference Scheme V. A Second

Order Sequel to Godunov's Method", Journal of Computational Physics, 32, pp.

101-136, (1979).

28 Van Leer, B, "Flux Vector Splitting for the Euler Equations", Proc. 8th Int.

Conf. on Num. Meth. in Fluid Dyn., Berlin, Springer Verlag, (1982).

29 Vinokur, M., "An Analysis of Finite-Difference and Finite-Volume Formulations

of Conservation Laws", Journal of Computational Physics, 81, pp. 1-52, (1989).
176 J. J. W. Van Der Vegt

1.0

0.8

0.6

0.4

- - - - Compr. Blasius

0.2 Fin. difference

v v0.O 2.0 4.0 6.0 8.0 10.0 12.0 14.0

n

Figure 1. Streamwise velocity U at 10 equally spaced stations, Re x = 50.000 —

320.000, M^ _ .08, versus similarity parameter rt = 2 v"Re, compared with compressible Blasius solution (dashed line).

1.0

0.s

0.6

0.4

- - - - Compr. Blasius

0.2 Fin. difference

O.O0.O 2.0 4.0 6.0 8.0 10.0 12.0 14.0

n

Figure 2. Normal velocity V Rey at 10 equally spaced stations, Rey = 50.000 —

320.000, M^ _ .08, versus similarity parameter rt = x e, compared with compressible Blasius solution (dashed line).
Higher-Order Osher Schemes and Compressible Boundary Layer Stability 177

1.0

^---^ DNS Fasel et al. ........... DNS fin. volume

- - - - DNS fin. difference

0.5

0.0

-1.0

1.0 1.5 2.0 2.5

x

Figure 3. Spatial growth rate —a i of streamwise velocity disturbances compared

with results from Fasel et a1. 5 . M^ = .08

1.0

0.5

;

0.0

-0.5

i

i

-1.0

1.0 1.5 2.0 2.5

x

Figure 4. Spatial growth rate —ai of normal velocity disturbances compared

with results from Fasel et al. 5 . M,, = .08

-0.5
178 J. J. W. Van Der Vegt

Figure 5. Pressure contours in flat plate boundary layer, M^ = .5, initial amplitude suction and blowing at wall .0001.

3.0

2.0

1.0

0.0

-1.0

-2.0

-3-0

-4-0

0.5

PSE

--••-•--•••-- DNS fin. volume

E

- - - - DNS fin. difference€

1.0 1.5 2.0 2.5

x

Figure 6. Spatial growth rate —az of streamwise velocity disturbances using

finite volume and finite difference schemes compared with PSE results Bertolotti2,

M^=.5
Higher-Order Osher Schemes and Compressible Boundary Layer Stability 179

3.0

2.0

1.0

0.0

-1.0

-2.0

-3.0

-40

DNS fin. volume

DNS fin. difference

0.5 1.0 1.5 2.0 2.5

x

Figure 7. Spatial growth rate —a i of normal velocity disturbances using finite

volume and finite difference schemes compared with PSE results Bertolotti2, M,,, =

.5

r

^i

^--^ PSE ^

............. DNS At/4

1

- - - - DNS At/2 4'

\ — — - DNS At k

f \

1

\ p

1 I;

0.5 1.0 1.5 2.0 2.5

x

Figure 8. Spatial growth rate — ai of streamwise velocity disturbances using

finite volume scheme with different time steps, M,, = .5

3.0

2.0

1.0

0.0

-1.0

-2.0

-3.0

-40
180 J. J. W. Van Der Vegt

3.0

DNS At/4

............. DNS At/2

2.0 ---- DNS At

1.0

0.0

-1.0

-2.0

-3.0

-40

0.5 1.0 1.5 2.0 2.5

x

Figure 9. Spatial growth rate —a i of normal velocity disturbances using finite

volume scheme with different time steps, M,,,, _ .5

3.0

• - PSE

...... DNS 436x80 2.0 - - - - DNS 336x80

i

1.0

1 •.l ^

0.0

I

I

-1.0 ;I

11 I

^ 1 I

-2.0 ----------,

I I I r

...-•

I I I

I I

I

I ^

-4.0

0.5 1.0 1.5 2.0 2.5

x

Figure 10. Spatial growth rate —a i of streamwise velocity disturbances using

finite difference scheme on coarse and fine grid, M,, = .5
Higher-Order Osher Schemes and Compressible Boundary Layer Stability 181

3.0

2.0

1.0

0.0

-1.0

-2.0

-3.0

-4.00.5 1.0 1.5 2.0 2.5

x

Figure 11. Spatial growth rate —ai of streamwise velocity disturbances using

finite difference scheme for different locations wall suction-blowing, M,, _ .5
Page intentionally left blank 
Center for Modeling of Turbulence and Transition 183

Research Briefs - 1993

Calculation of Waves in Fluids Using A

High-Order Compact Difference Scheme

Sheng-Tao Yu

1. Motivation and Objective

The unsteady Euler equations are solved by a sixth-order compact difference

scheme combined with a fourth-order Runge-Kutta method. First, the accuracy

of the numerical scheme is assessed by Fourier analysis of the fully discretized,

two-dimensional Euler equations. A close form of the amplification factors and

their corresponding dispersion correlations are derived. The numerical dissipation,

dispersion, and anisotropic effects are assessed. It is found that the CFL limit for

stable calculations is about 0.8. For CFL number equal to 0.6, the smallest wave

length which is resolved without numerical damping is about 6 to 8 grid nodes. For

phase speed in the same order of acoustic waves, the corresponding time period

is resolved by about 200 to 300 time steps. Three numerical examples of waves

in fluid flows are included: (1) sound propagation in a linear shear layer posed

by Pridmore-Brown, (2) linear wave growth in a two-dimensional, compressible free

shear layer, and (3) eddy-pairing in a compressible free shear layer perturbed by two

frequencies. The results compare favorably to the analytical solutions for all three

cases. While analytical solutions of linearized flow equations are indispensible, the

adopted finite difference scheme provides comparable accuracy in simulating wave

motions in the linear regime and beyond.

2. Work Accomplished

We study the numerical characteristics when using the sixth-order compact differencing (CD6) combined with a fourth-order Runge-Kutta method (RK4) to solve

the two-dimensional Euler equations for simulating wave motions in fluid flows. For

multiple dimensional simulations, anisotropic effect in addition to the dissipation

and dispersion errors must be assessed to estimate the accuracy of the numerical

scheme. To this end, Fourier analysis of the fully discretized, two-dimensional Euler

equations must be performed.

In our previous work [2], a one-dimensional wave equation was adopted as a model

upon which Fourier analysis was performed to obtain the CFL limits and dispersion correlations. Since one can perform similarity transformation to decouple the

one-dimensional Euler equations to three scalar wave equations, this approach is

appropriate for one-dimensional calculations. On the contrary, no similarity transformation is available to decouple the two-dimensional Euler equations. Therefore,

the adoption of a two-dimensional wave equation as a model of the Euler equations

for Fourier analysis can not be justified. In this paper, we apply Fourier analysis

directly to the fully discretized, two-dimensional Euler equations. A close form
184 Sheng-Tao Yu

of the amplification factors and their corresponding dispersion correlations are derived. Consequently, the numerical accuracy in terms of dissipation, dispersion, and

anisotropic errors can be assessed.

To demonstrate the performance of the numerical method, three numerical examples are included in the present paper:

(1) Sound waves propagation in a linear shear layer. The Euler solver is applied to

simulate the problem proposed by Pridmore-Brown [4]. The wave motion is neutrally stable and the harmonic content of this flow field is limited to one frequency

mode. ZFrom the finite difference solution, the eigenfunctions are reconstructed

by Fourier transformation and compared favorably with the analytical solution.

(2) Linear wave growth in a compressible free shear layer. We first solved the compressible Rayleigh equation for the flow conditions of concern and obtain the full

spectrum of viable eigenvalues and eigenfunctions. We then arbitrarily choose

an eigenvalue of relatively long wave length and smaller growth rate. The corresponding eigenfunctions and frequency are used to perturb the prescribed mean

flow at the upstream. The growth rate of the wave in the down stream region

is then obtained from the numerical results and compared with the analytical

solution. The simulation is restricted to the linear regime, therefore, similar to

Case 1, the harmonic content of this case is limited to one Fourier mode.

(3) Nonlinear mechanisms of a compressible free shear layers. In this case, we focus on the most unstable mode for the mean flow conditions of interest. Several

perturbation schemes are applied: single frequency perturbation with small amplitude (weakly nonlinear), single frequency perturbation with relatively large

amplitude to show the eddy roll-up, and double frequencies perturbation with

the most unstable mode and its subharmonic to show the eddy pairing.

In all cases, numerical results show favorable comparison to the analytical solutions

of the Rayleigh equation in the linear regime. For nonlinear wave motions, numerical

solution show crisp resolution of the vortex roll-up and pairing.

2.1 Numerical Method

The two-dimensional Euler equations in Cartesian coordinates can be cast into a

vector form:

aQ aE aF _

at + ax + ay ^, (1)

where Q is the unknown vector and E and F are inviscid fluxes in the x and y

directions, respectively. A RK4 method is applied as the temporal discretization

and a CD6 scheme is applied to the spatial discretization.

The adopted RK4 method is originally attributed to Kutta [5] for solving ODEs.

Previously, Jameson et al. (6] used this scheme to solve the flow equations. The

algorithm is given by
Simulation of Waves in Fluid Flow 185

Q1 _ 

Qn + AtRn

Q2 _ Qn + At R1

(2)

Q3 = Qn + AtR2,

Qn+1 = Qn + 6t (Rn + 2R1 + 2R2 + R')

The superscripts 1, 2, and 3 denote intermediate steps of the RK method. RZ is

the numerical value of 9E/ax + aF/ay calculated from the flow properties at step

i. As shown in our previous paper [2], the algorithm is fourth-order accurate for

nonlinear equations and is suitable for the calculations of unsteady flows.

A CD6 scheme is used for spatially differencing the inviscid fluxes. According to

Hermite's generalization of the Taylor's series [7], one can get

ui-1 + 3ui + ui+1

_ 1

12I (u2+2 + 28ui+ 1 — 28ui-1 — ui-2) + O(Oxs),

(3)

where u could be any flow property, and the superscript ` represents the spatial

derivatives. The treatment of the boundary condition is similar to that in our

previous work [2], namely, the fourth-order compact scheme is used at locations one

grid node away from the boundary, and a third-order biased scheme is used at the

boundary.

Note that the method of characteristic (MOC) type non-reflecting boundary condition [2,8] incorporated with the one-sided biased difference is used at the computational boundary. Therefore, all numerical waves pass through the boundary without

reflection. In addition, numerical grids are stretched near boundaries to enlarge the

computational domain and enhance the non-reflection effect. As a result, lowering

the order of the spatial discretization in the vicinity of the computational boundary

is not detrimental to the numerical accuracy at interior nodes. For details of the

non-reflective boundary conditions and their implementation into the present finite

difference scheme, we refer the interested readers to Ref. [2]. The application of the

CD6 scheme with the aforementioned boundary conditions involves the inversion

of a scalar tridiagonal matrix. The inversion of the matrix incurs little penalty in

terms of CPU time.

2.2 Fourier Analysis

Fourier analysis of a two-dimensional, finite difference scheme assumes that the

solution is periodic over an infinite domain with the spatial period equal to the

computational domain (L,, and L..) in x and y directions, respectively. The computational domain is decomposed into Kx and Ky grid nodes in x and y directions,

respectively. The harmonic content of the discretized equation is limited to the

number of grid nodes used in the computational domain. A discrete solution Qi
186 Sheng-Tao Yu

at a location (i, j) and time (n) is a linear combination of Ky times Ky wave modes.

Fourier analysis is performed by substituting each wave mode of the discrete Fourier

expansion into the discretized Euler equations to calculate the amplification matrix,

G(p, q), which is defined as

n+1

G(p, q ) — p)(4)

Q"(p q) 

where Q denotes the Fourier component of the object function Q and p and q are the

wave numbers in the x and y directions, respectively. To obtain the amplification

factors of the numerical scheme, one has to diagonalize G to obtain its eigenvalues.

Since G is a 4 x 4 matrix, we expect to obtain four amplification factors. The

procedure is repeated for all wave modes, and the amplification factors over the

full spectrum of the wave numbers are obtained. In this process, we map the

function Q defined by the spatial coordinates x and y on the domain [—L,/2,L,/2] x

[ —Ly /2, Ly /2] to the wave number space [- 7r, ,Tr] for both p and q. The amplification

factors in terms of wave numbers in the first quadrant of the (p, q) plane is most

representative; solutions in other quadrants are merely alias. For this reason, the

results of our Fourier analyses are presented in the first quadrant of the (p, q) plane.

In what follows, the procedure to obtain the amplification factors of the Euler

equations discretized by the CD6-RK4 method is illustrated. First, the generalized

form of the amplification matrix G for the RK4 method, as shown in Ref. [2], is

provided by

G=I+Z+2V+6V + 24 Z4, (5)

where I is the identity matrix. Z is the Fourier component of the spatial discretization applied to the convective terms ( —aEl ax — aF/ay) in the wave number space.

To proceed, we linearize the inviscid fluxes E and F, such as

OEaE aQ — aQ

ax aQ ax

_ A 

ax ' (6)

aFaF aQ _ aQ

ay aQ ay

_ B 

ay

where A and B are the Jacobian matrices of the inviscid fluxes. Since the analysis

is linear, A and B are assumed to be constant. As shown in our previous work [2],

the discrete Fourier transformation of aQ/ax and aQ/ay is given by

F(

F (_

_ [4 sin(p) cos(p) + 56 sin(p)]Ati

ax

—Zx 12[2 cos (p) + 3]Ax'

")

_—Z

y _

[4 sin(q) cos(q) + 56 sin(g)]Ati (7)

ay 12[2 cos (q) + 3]Ax

where F(dQ/ax) is the Fourier transformation of the spatially discretized aQ/ax

and is defined as —Z^. As a result, Z can be represented as
Simulation of Waves in Fluid Flow 187

Z = AZ,: + BZy . (8)

Substitute Eqn. (8), into the amplification matrix for the RK4 method, Eqn. (5),

and G(p, q) is obtained for all wave numbers. The remaining task is to diagonalize

G.

To proceed, we perform similarity transformation of G to obtain its eigenvalues

as the numerical amplification factors of the adopted numerical scheme,

T

-1 GT =I + T-1 ZT + 

2 (

T

-1 ZT)2+

(9)

6 (T-1 ZT)3 + 24 (T-1 ZT)4.

For completeness, the matrices T -1 and T which diagonalize G are given by

1 0 0 —1/c2

0 a/ v2 Q/ v2 1/(V2-pc) 

T-1 _ 0 Q —a 0

(10)

0 —a/v'2-

—O/V2-

1/(v'2-pc)

1 0 p/(v^2_c ) p/(v'2-c)

T

_ 0 0 a/v'2- — a/v ( )

0 —a Ql v'2- —Q/v2

11

0 0 pc/v/-2-pc/v/-2-

where p is density, c is the speed of sound, a = Z./(Z2 +Zy) 1 ^ 2 , and /^ = Zy /(Zi -}

Z

2

Y

) 1/2 . The eth row of T-1 is the eth left eigenvector of G, corresponding to the

eigenvalue ge. Finally, the amplification factor 9e is

9e = 1 } '\e+tae ,+24/\e,

Q = 1 ...^4, (12)

where Ae is the nth eigenvalue of the matrix Zx A + ZyB which is also diagonalized

by T-1 and T, and are given by

A l =Zx u + Zyv,

A2 =Zxu + Zyv,

^3 =Zxu+7iyv+ C(Zx +Zy (13) )z>

A4 =Zxu+ ZY - C(Zx + Zy)2.

As a result, there are only three distinct amplification factors representing three

kinds of numerical waves propagating in the dispersive numerical medium, namely,

91,2, 93, and 94 waves. As shown in Eqns. (12) and (13), 9 1,2 represents the numerical waves simulating the flow velocity. Specifically, the velocity vector (u, v) is
188 Sheng-Tao Yu

modulated by a inner product with the vector (Z" Zy ) due to the spatial discretization. The numerical wave is further modulated by substituting the inner product

into Eqn. (12) for the RK4 method to obtain the amplification factors. Similarly,

93 and 94 waves are the numerical counterparts of the acoustic waves superimposed

on the flow motion. Unlike the flow velocity waves though, the acoustic waves propagate in all directions from a moving source and the phase speed c is modulated by

the factor (Z,",+ Zy)1 /2

It is interesting to note that the implicit schemes, such as ADI and LU methods, have four distinct amplification factors due to the fact that the approximate

factorization is introduced into the solution procedure for efficiently inverting the

coefficient matrices. Therefore, in addition to the aforementioned modulation of

the wave velocity, additional error is introduced into the numerical results and no

one-to-one comparison between the physical and numerical waves such as that of

the present finite difference method (Egns. (12) and (13)) is available.

In what follows, we use the derived amplification factors to assess the numerical

characteristics of the adopted scheme. To proceed, defining the amplification factors

in terms of frequency,

9e(p, q )

= eiw(r,$) (14)

and we can make the following interpretations:

1. Artificial Dissipation. w(p, q) is a complex number, i.e., w = a + i,Q. The imaginary part (0) represents the numerical amplification, i.e.,

9e(p, q ) = 

eiw = e—peia

(15)

I9e(p, q )I = e-0

When IgI > 1, the scheme is unstable. For calculations of unsteady flows, we want

I9I to be less than and close to unity to ensure numerical stability with minimum

artificial dissipation. We plot 

I 

g(p, q) I against p and q to illustrate the artificial

dissipation.

2. Artificial Dispersion. According to Eqn. (15), the real part of the frequency,

a(p, q), represents the artificial dispersion. Note that Fourier analysis is linear,

i.e., the analytical solution is dispersionless. We plot contours of constant a to

show phase velocities which are then used to illustrate the artificial dissipation.

3. Artificial Anisotropy. Certain direction of wave propagation will be favored by the

numerical scheme. We use various values of c, u, and v to calculate 9e i f = 2, 3, 4

according to Eqns. (12) and (13). Changes of the flow direction (u and v) allow

us to assess the anisotropy of Q1,2 . For the simulated acoustic ( q3,4 ) waves, we

turn off flow velocity to assess the isotropy of the acoustic waves.

Figure 1 shows the artificial dissipation of three different numerical waves represented by 1 -91,2 1, Ig3I and I 94 I. In order to obtain the numerical values, u, v, and
Simulation of Waves in Fluid Flow 189

c are specified. Here, u is set to be equal to v and the flow velocity is at 45 0 to

the coordinate axes. The magnitude of the flow speed is set to be equal to c. As

shown in the figure, there is negligible dissipation of 9 1,2 and 94 . On the contrary,

g3 shows significant dissipation at moderate wave numbers. Figure la is repeated

in ld in a different scale to show the dissipation effect at the moderate wave numbers. The minim of 1912,3,4 are 0.98, 0.51, and 0.99, respectively. Therefore, acoustic

waves propagating along the same direction as the flow motion suffer from the worst

dissipative error. As a result, we defined the critical CFL number by

CFL^ = max { ^Zx u + Zy v + c(Z2 + Zy) z

^Zxu+Zyv—c(Zx+Zy)2^} (16)

At

0x2 + Dy2

For this case, CFL, is 0.6. In Fig. lb (the dissipation of 93 ), there is negligible

dissipation in the low wave number region, i.e., p, q < 7r/3. This region corresponds

to waves with wave length greater than six grid nodes. Note that little dissipation

is also observed in the other three corners of the same plot. For example, for

p, q = 7r which is wave resolved by two grid nodes (or even-odd wave) in both x and

y directions, there is essentially no artificial dissipation. Later on, we shall show

that these waves are undesirable due to significant dispersive errors. Although not

shown, it is noted that for CFL, greater than 0.8, 193 (p, q)l becomes larger than

unity at certain p and q and the calculation is numerically unstable. Therefore,

the CFL, limit is 0.8 and a value between 0.5 to 0.7 is recommended for accurate

calculations.

Figure 2 shows constant a contours by which numerical phase velocities can

be interpreted. The phase velocity vectors point to the direction normal to the

contours. The distance between the consecutive contours is an indication of the

magnitude of the phase velocity; increase of the distance is a decrease of the phase

velocity and vice versa. Figure 2a shows the phase velocities of the 91,2 waves,

i.e., flow velocity waves. In this case, the flow direction is 451 . By observation,

the region in which the phase velocities are correctly simulated by the numerical

scheme is circumscribed by a dashed line. Outside this region, the phase velocities

are in erroneous directions. Similar to the dissipation error, there is little dispersive

error for p, q _< 7r/3. To assess the anisotropic effect, Fig. 2b shows the phase

velocities of the 9 1,2 waves at 22.51 to the x axis. Inside the dashed line, the waves

are well resolved. Again, in the low wave number region (p, q < 7/3), there is little

dispersive error. Figures. 2c and 2d show the phase velocities of the simulated

acoustic waves superimposed on the flow stream at 45°.

Perhaps, it is helpful to temporarily turn off the flow velocity to observe the

acoustic velocities alone as shown in Fig. 3. Theoretically, the propagation of

the acoustic waves is isotropic and is depicted by circular contours of constant a.

Again, the well resolved region is circumscribed by a dashed line; outside the dashed
190 Sheng-Tao Yu

line, acoustic waves propagate in wrong directions with erroneous magnitude. For

long wave length (at least 6 grid nodes) waves, little preference of the propagation

direction of the acoustic wave is observed; there is no evidence of the anisotropic

error. Outside the circumscribed region, however, the figure shows an increase of

separation between contours along coordinate axes to a maximum followed by a

decrease. This indicates that the acoustic waves of moderately high wave numbers

propagate too slowly along the numerical grid lines. At very high wave numbers,

the acoustic waves travel in the opposite direction of what they should. Fortunately,

for the region of p, q <7r/3, theses dispersive errors are negligible.

As discussed above, the CD6-RK4 scheme has no dissipative effect on the highwave-number waves. Nevertheless, significant dispersive and anisotropic errors are

associated with these highest-wave-number waves. Throughout the course of a computation, these high-wave-number waves keep propagating with erroneous directions

and phase speeds and eventually destroy the solution. As shown in our previous

work [2], it is appropriate to impose a small amount of high-order artificial damping

to filter out these waves, at the same time keeping the resolution at low wave modes

intact. The eighth-order artificial damping, defined as

A.D. = 8 [—ui+4 — ui-4 + 8 ( ui+3 — ui-3) (17)

- 28(ui+2 + ui-2) + 56 ( ui+l + ui-1) - 70ui]

is recommended. In the present applications, 77 = 0.003 is used.

2.3 Numerical Examples

1. Sound Wave Propagation in Linear Shear Layer

The first case is a forced two-dimensional waves propagating in a shear layer with

linear velocity profile posed by Pridmore-Brown [3]. The first-order linearized Euler

equations can be derived,

apl

+ U

+

+ Po ( o, at ax

ax ay

y

au' au'

+ U

, aU

+ v +

1 ap'

—

_ 

at ax O y Pa Ox 0, (18)

av' av' +U—+ 1 ap'

-- =0,

at Ox po Oy

where po is the mean flow density and is assumed to be a constant. To close the

equation set, the isentropic relation is used,

P = c2 P^, (19)

where c is the speed of sound of the mean flow. As proposed by Pridmore-Brown

[3], the equation set could be manipulated to obtain the equation in terms of p' and

V'
Simulation of Waves in Fluid Flow 191

1 192 p' 2 a2 p' a2 p' 2M a2 p'

C2 9t2 =(1 — M ) 19x 2 + ay c axat

(20) dM av

+ Zpoc dy 9x'

where M = U/c. The solution of p' and v' in the following form is expected,

P' = Aeit;(rx—ct) F y) ' (21)

V' = Aeik(Kx—ct)G(^, y),

where F and G are related by

dF _ ipck(1 — KM)G. (22) dG

In the above equations, k = w/c and w is the frequency of the sound wave. Substituting Eqns. (21) and (22) into Eqn. (20) and we obtain

d2 F + 2r^M' dF + k2[(1 — r^M) 2 — r^2 ]F = 0, (23) dye 1 — KM dy

where M' = dM/dy. The boundary conditions are

dF =0 at y=0 and y=L. (24)

dy

For a given frequency k = w/c, the eigenvalue r could be obtained by solving Eqns.

(23) and (24) as a boundary value problem. In Ref. [4], Pridmore-Brown solved the

eigenvalue problem by an asymptotic expansion and the final solution is presented

in the form of the complex Airy functions. In this report, we numerically solve

the equation by a finite difference method. The details of the numerical method

are beyond the scope of this report. In what follows, only a brief account of the

.numerical procedure is illustrated. First, we subdivide the computational domain

[0, L] into P grid nodes. With the aid of the second-order central difference scheme,

dF/dy and d2 F/dye are replaced by

dF N [A]f,

dy (25)

d 2 F

dye Pz^ [B]f,

where [A] and [B] are coefficient matrices of the finite difference scheme with a

dimension of P x P. The column vector f has element fi (i = 1, • • • , P) as the

discrete solution of F at grid node i. As a result of discretization, we can reformulate

Eqns. (23) and (24) to become a eigenvalue problem in terms of K such as

[A]of + r,[A] l f + K2 [A] 2 f + r,3 [A] 3 f = 0. (26)
192 Sheng-Tao Yu

Here, the coefficient matrices [A] i (i = 0, • • • , 3 ) are combinations of matrices

[A] and [B] premultiplied by coefficients as shown in Eqn. (23). Equation (26)

is a polynomial type eigenvalue problem. We used the standard QZ algorithm to

calculate the eigenvalues Ki (i = 1, • • • , P). The numerical convergence is checked

by doubling the number of grid nodes.

The Mach number of the mean flow is a linear profile from null at wall to 0.5

at the free stream. With k = 27r, the converged eigenvalue is r, = 0.842758. The

hight of the computational domain is selected to be one meter and the wave length

of the forced perturbation should be 1/n (about 1.19) meters. The length of the

computational domain is set to be 5 meters and the whole domain is subdivided

into 200 x 40 grid nodes with uniform distance between them. The magnitude

of the perturbation (A in Eqn. (21)) is equal to one hundredth. Figure 4 shows

the converged eigenfunctions F and G as functions of y. These eigenfunctions in

conjunction with the prescribed frequency k are used as the upstream perturbation

for the CFD calculation. Figure 5 shows the pressure contours of the CFD results

and the wave length is correctly simulated; the flow domain covers about 5rc (about

4.2) wave lengths. Figure 6 shows the reconstructed eigenfunctions of p' at two,

three, and four wave lengths downstream of the forced perturbation. In the figure,

the solid line is the analytical solution. For the solution at the farthest downstream

location, less than 3% of error is observed. As it can be seen, the magnitude of the

pressure fluctuation is much larger near the wall. This is because that the mean

flow distribution compresses the acoustic waves to propagate near the wall.

2. Linear Wave Growth in Compressible Free Shear Layer

The flow is defined in Fig. 7 and the fast stream is denoted by the subscript

1. The upstream perturbations (regardless of the source) are selectively amplified

by the shear layer and grow exponentially in the linear regime. Eventually, these

instability waves cause eddy roll-up and merging at the later stage (nonlinear) of

the flow development.

In this numerical example, we first discuss the mean flow solution. Then the solution of the linearized Euler equations (the Rayleigh equation) based on the given

mean flow is provided in the form of a full spectrum of eigenvalues and eigenfunctions. Among the available eigenvalues, a relatively long wave-length solution is

adopted in the present calculation and the CFD calculation is set up accordingly.

The accuracy of the CFD results is assessed by comparing to the analytical solution

in terms of growth rate and eigenfunctions.

For all solutions, flow properties of the fast stream are used as the referenced scales

and all properties are non-dimensionalized accordingly in the following equations.

In addition, the length scale is taken as 6/2 where 6 is the vorticity thickness and

is defined as

_ U1 — U2* b

(dy- )max (27)
Simulation of Waves in Fluid Flow 193

where the superscript * denotes the dimensional properties. The mean flow velocity

distribution is given by

U(y) _ 

1 + R tanh ay (28)

1+R

where R is the velocity ratio and is defined as

_ U1 — U2* R U1 + U2*'

(29)

and a is a parameter to control the thickness of the shear layer. According to

Buseman and Crocco, the Prandtl number is set to be unity, and the velocitytemperature relationship is given by

T T2+ 1 — U2 (U- U2) +2('Y- 1 ) MRU—U2)(1—U) . (30)

For the balance of the mean flow field, pressure is set to be uniform across the shear

layer and the mean flow velocity in the transverse direction V is set to be null. As

a result, all mean flow properties are defined.

Under the parallel-mean-flow assumption, the linearized disturbances have the

Fourier coefficients independent of x and t, such as

U, ( x ' y ' t ) _— Uly) i(ax I c3 t)

l )

e 31

)

P, (X , y ' t) p(y)

P'(x ' y' t) P(y)

where u(y), b(y), p(y), and p(y) are the eigenfunctions of the posed problem. For

each set of eigenfunctions, the corresponding frequency (w) and wave number (a)

are the eigenvalues.

For spatial instability problem, w is real and a is complex in which the imaginary

part (ai) is the growth rate of the instability wave. The solution of these disturbances are governed by the compressible Rayleigh equation. We use a shooting

method to solve the equation. The details of the solution procedure is out of the

scope of this paper. Figure 8 is a typical solution of the equation. In this figure,

we plot —ai against w as the full spectrum of the eigenvalues. The mean flow

conditions are given by

T1 = 300K,

Ml = 1. 5,

U2 = 0. 74,

T2 = 1.85.

In Fig. 8, we choose the case of wT = 0.10101 and ai = —0.012445 for the CFD

calculation. The chosen w and the corresponding eigenfunctions are used as the
194 Sheng-Tao Yu

imposed perturbation at the upstream boundary. Note that the eigenfunctions are

complex, therefore, the upstream perturbation is of the form,

0' _ ^'( y) cos(wt) — ^i(y ) sin(wt), (32)

where 0' could be u', v', p', or p'. Since there are four governing equations, only

four boundary conditions are needed. The magnitude of the perturbation is set to

be one thousandth. The fluctuations of other flow properties, e.g., temperature, are

automatically obtained by solving the equation set. It is interesting to note that, for

the given mean flow condition, part of the flow is subsonic. According to the MOC

type boundary condition for subsonic flow, one can only specify three boundary

conditions and let the fourth one be determined by the out-running characteristic

equation. However, in the present calculation, all four boundary conditions are

specified even for the subsonic flow due to the fact that the flow is a convective

instability problem. All flow properties at the upstream boundary are determined

by the eigenfunctions and eigenvalues; flow information carried by the out-running

characteristic matches these conditions perfectly.

The computational domain is subdivided into a 700 x 70 grid. The flow domain

along the streamwise direction covers about six wave lengths of the imposed perturbation. The numerical grid is clustered near the center to provide high resolution

of the flow field; the grid near outside boundaries is stretched to enhance the nonreflection effect. Figure 9 shows a instantaneous distribution of lu'l along the center

line of the shear layer. The x-axis is the normalized streamwise distance. The yaxis is the normalized lu'l plotted in natural log scale. As shown in the figure, the

growth rate of the wave is linear (in the log scale) and the slope is within 5% of

-ai predicted by the linear theory. Since we impose the perturbation according to

the eigenfunctions, the linear growth (in the log scale) of the instability wave starts

from the very beginning of the free shear layer; no transitional region is observed

in the vicinity of the upstream boundary.

Figure 10 shows the reconstructed eigenfunctions of u' and p' at various downstream locations compared to the prescribed eigenfunctions. Note that the eigenfunctions are normalized by the maximum. It is obvious that the CD6-RK4 finite

difference method faithfully preserves the functional shape of the original perturbation.

3. Nonlinear Mechanism of Compressible Free Shear Layer

The mean flow conditions specified in Case 2 is adopted in the present calculation. As indicated in Fig. 8, we choose the most unstable mode of the available

eigenfunctions and eigenvalue as the prescribed perturbation, i.e., W = 0.389 and

ai = —0.026067. First, a magnitude of one thousandth is used to show the linear

growth (in the natural log scale) of the instability wave. By doing so, we cover the

two particular conditions of the whole spectrum, namely, a long wave mode (Case 2)

and the most unstable mode, to assess the accuracy of the finite difference scheme
Simulation of Waves in Fluid Flow 195

in simulating linear waves. Second, we increase the magnitude of the perturbation

to show the eddy roll-up. Finally, the subharmonic of the most unstable mode is

imposed in addition to the most unstable mode. Similar to the single frequency

perturbation, the corresponding eigenfunctions of the subharmonic mode are used

in this calculation to ensure that no undesirable perturbation is introduced into the

system.

Figure 11 shows the Fourier coefficients of u' along the center line of the free

shear layer. This figure clearly shows the linear growth (in the natural log scale)

of the most unstable mode in the initial phase of the flow development. The slope

of the curve compares favorably with the analytical solution. At the later stage of

the flow development, the flow motion becomes nonlinear. The energy carried by

the most unstable mode starts being transmitted to high harmonics. As a result,

the growth of the fundamental mode stalls. Simultaneously, high harmonic modes

gain strength as the prelude of the eddy roll-up. Mathematically, the appearance

of the high harmonics is due to the convolution of the fundamental mode with itself

and its harmonics. Therefore, no subharmonic mode can be produced. Figure 12

shows the reconstructed eigenfunctions of u' and p' at 5 and 10 wave lengths of the

streamwise locations as compared to the prescribed eigenfunctions. For the streamwise location at 10 wave lengths, the flow is weakly nonlinear and the eigenfunctions

show significant deviation from the linear analytical solution.

Figure 13 shows the Fourier coefficient of the harmonic contents for flow perturbed

by the most unstable mode at a larger magnitude (one hundredth). The linear

growth of the instability wave is limited to the region in the vicinity of the upstream

boundary. At the later stage of the flow development, active eddy roll-ups are

obvious as shown in Fig. 14, in which the contours of the constant vorticity are

plotted. As shown in Fig. 13, at x = 160, a dip of Fourier coefficients is observed.

The significant modulation of the Fourier modes may be attributed to a change of

the energy transmission mechanism from the mean flow to the unsteady motions

due to the appearance of the eddy roll-up.

Finally, the two-frequency perturbation is imposed on the upstream. The magnitude of both fundamental and its subharmonic is set to be one hundredth. Figure

15 shows the distributions of the Fourier coefficients of the fundamental (w), second

harmonic (2w), and subharmonic (w12) as functions of streamwise locations. The

growth rates of both fundamental and its subharmonic waves accurately mimic the

analytical solution in the linear region. Compared to Fig. 13, Fig. 15 shows a

earlier appearance of the 2w wave due to the perturbation of the w12 wave. Again,

the fundamental and its high harmonics are modulated due to the eddy roll-up.

However, the location of the dip of the w wave occurs at x = 140. Previously, for

the case without subharmonics perturbation the dip occurs at x = 160 (see Fig.

13). As a contrast, the subharmonic mode is not affected and grows continuously.

For incompressible flows, the growth rate of the subharmonic mode shows a kink

and changes to a steeper slope when the eddy pairing appears in the shear layer.

In the present calculation, the growth of w12 wave is levelled when flow motions
196 Sheng-Tao Yu

become nonlinear. In a later stage, the growth of the W12 wave is reactivated due

to the eddy pairing. However, the slope change is much gradual compared to that

of incompressible flows. Figure 16 shows the vorticity contours of the present calculation. Compared to Fig. 14, the trend of the eddy pairing is obvious due to the

introduction of the subharmonic wave.

3. Concluding Remarks

We investigated the performance of the CD6-RK4 finite difference method in simulating linear and nonlinear wave motions in shear layers. The numerical accuracy

is assessed by the Fourier analysis and numerical examples. First, a close form of

the amplification factors and their corresponding dispersion correlations for the fully

discretized, two-dimensional Euler equations are derived. Subsequently, the numerical dissipation, dispersion, and anisotropic effects are assessed. It is found that

there are three groups of numerical waves propagating in the numerically dispersive

medium, namely, the flow velocity waves and two acoustic waves superimposed on

the flow velocity. Although, only the acoustic wave propagates in the direction of

the flow velocity suffers from significant dissipation error, all three groups of waves

suffer from dispersive errors at high wave numbers. For the present scheme, the

CFL number limit for stable and accurate calculations is about 0.8. Under the

CFL limit, dissipation, dispersive, and anisotropic artifacts are negligible for wave

numbers less than 7r/3 which corresponds to waves resolved by more than 6 grid

nodes.

Three numerical examples of waves in fluid flows are presented: (1) sound propagation in a linear shear layer posed by Pridmore-Brown, (2) wave growth in a

two-dimensional, compressible free shear layer, and (3) eddy-pairing in a compressible free shear layer perturbed by dual frequencies. The results compare favorably

to the analytical solutions for linear wave motions. For nonlinear waves, the CFD

results provide crisp resolution of the appearance of high harmonics, modulation of

wave modes, eddy roll-up, and eddy pairing.

4. References

1 Lele, S. K. "Compact Finite Difference Schemes with Spectral-Like Resolution,"

J. Comput. Phys., 103, 16-42 (1992).

2 Yu, S. T., Tsai, Y.-L. P., and Hsieh, K. C., "Runge-Kutta Methods Combined

with Compact Difference Schemes for the Unsteady Euler Equations," AIAA

Paper 92-3210 (1992). (also submitted to JCP).

3 Tsien, H. S., J. Am. Rocket Soc., 22, 139 (1952).

4 Pridmore-Brown, D. C., "Sound Propagation in a Fluid Flowing through an Attenuating Duct," J. Fluid Mech., 393-406 (1958).

5 Carnahan, B., Luther, H. A., and Wilkes, J. O., "Applied Numerical Methods,"

John Wiley & Sons, Inc., 363 (1969).

6 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).
Simulation of Waves in Fluid Flow 197

7 Colatz, L., "The Numerical Treatment of Differential Equations," Springer Verlag, 538 (1966).

8 Giles, M. B., AIAA J. 28, 12, 2050 (1990).
0

II

a

w

U

c.

l°

a^

i~

a^

U

e1-

CO N

Q

cy- N N

U ^ ^

y

Iv

II II

O .^ U V

V

Cd

•^^ II a a

•y

^ II II

2

10 12

r `..^

m cd 1^ U

50

T — a = - =

'. jaoriyt^.C198 Sheng-Tao Yu
7r

P

7r

P

P

7r

7r

4

i p

7r

4

Ir

Simulation of Waves in Fluid Flow 199

4

4

7r 7r

2a al , 2 for u = v and c = Vu-2 +v2. 2b al , 2 for tan -1 (v/u) = 22.5° and c = u

2 {- v2

2c a3 for u = v and c = u 2 + v2. 2d a4 for u = v and c = 7 -+v2.

Fig. 2 Dispersion characteristics of the CD6-RK46 scheme for CFL=0.6.
p

200 Sheng-Tao Yu

q

IT

7r

Fig. 3 Dispersion characteristics of the acoustic waves using the CD6-RK46 scheme

for CFL=0.6.
1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0

Simulation of Waves in Fluid Flow 201

0.2 0.4 0.6 0.8 1.0

P

0.0000 0.0003 0.0006

v

Fig. 4 Theoretical eigenfunction components p' and v' for acoustic waves in a duct

with linear shear.
202 Sheng-Tao Yu

C

CQ

CCd

C

O

Cd

C

U w

v1 Q

O

cCd^

.oa3

^ V

O +' U V

4r i^r O^6 cc

o

O O u1^11

C
Simulation of Waves in Fluid Flow 203

1.0 g

o

Theoretical Solultion

II 2 wave lengths. OM

Oa

0 80 3 wave lengths.

OM

0 4 wave lengths. rIM

0.6

0.4

0.2

0.0

0.2 0.4 0.6 0.8 1.0

P

Fig. 6 Reconstructed eigenfunction of p' from the CFD results using the CD6-RK4

method for acoustic waves in a duct with linear shear.
204 Sheng-Tao Yu

J

I

cli

ti

a^

a^

a^

a^

a^

a^tia0

O

N

U

a

U

cn

C^
Simulation of Waves in Fluid Flow 205

0.025

0.020

v . i 0

0.005

n ^. n

v. v0

0.0 0.2 0.4 0.6 0.8

W

Fig. 8 The spatial growth rate as a function of the angular frequency from linear

theory for the compressible free shear layer
Theoretical Envelop,

6

-10

-12

0 100 200 300

x

4

206 Sheng-Tao Yu

Fig. 9 Instantaneous distribution of lu'l along the center line of the free shear layer.
10

8

6

4

,,

0

—4

—6

—8

—IO

-..o _._ 0.4 0.6 0.8 1.J

i,^ l

207

Li C. 16MJ.^ 1.0

10

g

2

— 4

Fig. 10 Reconstructed eigenfunction components u' and p' from the CFD results for

the compressible shear layer.

J
208 Sheng-Tao Yu

-2

-3

-4

-5

—6

-7

0 40 80 120 160 200

x

Fig. 11 The Fourier coefficients of the instability waves in a compressible free shear

layer perturbed by the most unstable mode at the magnitude of one thousandth.
0.2 0.4 0.6 0.8 1.0

(ul

10

8

6

4

2

0

—2

-4

—6

—8

—10

0.0

4'9nnnilnf^nn n{' W-- o- wl-i d wIngii 209

10

8

6

L

2

n

v

—2

—L

—6

—R

— i v

0.0 0.2 0.4 0.6 0.8 1.0

Fig. 12 Reconstructed eigenfunction components u' and p' from the CFD results for

the free shear layer perturbed by the most unstable mode.
210 Sheng-Tao Yu

-2

Linear Solution

3

w

-4

_ 5 

`Nq

2w

° o°

°°°cam

° ° °

-6 °^ 3w q CIA

°°o q

-7 °°°

0 40 80 120 160 200

x

Fig. 13 Fourier coefficients of the unsteady motion in a compressible free shear layer

perturbed by the most unstable mode at a magnitude of one hundredth.
Simulation of Waves in Fluid Flow 211

.a

b

a^

a

H

a^

a

a^

^b

a^

a

O

O

7ti

a^

b

no

4. OJ

O

Uya

0

_ CO

^ C

O

U ^

4r ri

O

O

O E

O O

O

U

v^

r-.
212 Sheng-Tao Yu

-2

-3

-4

-5

-6

-7

0 40 80 120 160 200

x

Fig. 15 The Fourier coefficients of the instability waves in a compressible free shear

layer perturbed by the most unstable mode and its subharmonic.
a^

ti

a

Cd

4-^

b

c V

O

M

Simulation of Waves in Fluid Flow 213

w m

O

•^ b

G

N cd

> 'd

z

cd

O ^

U rn

4. (^

O

07 y

O O

C O

r-^

44

blo
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215

Center for Modeling of Turbulence and Transition

Research Briefs -1993

Visitors' Research Activities

A summer visitor program was initiated this year to promote collaborative research

with academia. The following describes the technical activities of the visiting researchers.

In addition, CMOTT has also initiated collaborative applied research with the CFD

groups in Rockwell Int. (headed by M. Sinder) and General Electric Co. (headed by S.

Syed) for the purpose of improving their turbulent flow calculations in propulsion system

design. This effort also helps CMOTT establish feedback channels with industry customers

and the supply/demand relationship between CMOTT and its customers.

J.-Y. Chen ( University of California at Berkeley )

Direct Numerical Simulation of Turbulent Flows with Realistic Chemistry

The main objective of this research is to study the interaction between turbulence and

realistic chemical kinetics in nonpremixed turbulent flows using Direct Numerical Solutions

(DNS). The starting point is the development of a feasible strategy of carrying out realistic

chemical kinetics in DNS without adding a significant burden to the computational task.

Such a task can be accomplished by considering specially designed fuel mixtures of H2/AR

and in a specific regime of turbulent combustion. By properly choosing the contents of

the fuel mixtures, both the flame thickness and the degree of stiffness caused by chemical

kinetics can be adjusted so that the flame will be fully resolved both in time and space.

The computational task needed for the realistic chemical kinetics can be make manageable

by using reduced chemical reaction mechanisms and efficient algorithms for evaluating

chemical kinetics source terms.

The present development includes:(1) construction of a look-up table for the source

terms for the hydrogen chemistry 2H2+02=Z2H 2 O and for the Zeldovich NO mechanism

N2 +02 =Z2NO; (2) implementation of the interpolation scheme into a high-order finite

difference DNS code for compressible homogeneoous turbulence as well as a spectral code

for incompressible flows, and (3) testing of these newly developed computer codes. This

development facilitates studies of the effects of density variation, compressibility, and the

interaction of chemistry on turbulence. Study have been conducted for homogeneous turbulence with and without a forcing scheme to maintain kinetic energy. The preliminary

results reveal that the compressibility causes the flame temperatures in either lean or rich

sides higher than the counterparts in incompressible flows. A significant departure from

chemistry equilibrium is also observed as the turbulence time scale can be smaller than the

characteristic chemistry time scale. Future development is planned to impose a constant

shear on the homogeneous turbulence.
216

Center for Modeling of Turbulence and Transition

Research Briefs -1993

P.G. Huang ( Eloret Institute/NASA Ames Research Center )

Computation of Complex Shock/Boundary Layer Interactions Using A TwoScale Turbulence Model

The present consists of two parts: implementation of a two-scale model by Liou and

Shih (1993) in to the compressible code by Huang and Coakley (1992) and testing of the

model in flows with complex shock/boundary layer interactions.

The first part of the work, i.e., the implementation of the two-scale model into the

compressible code, has been completed. The model has been tested successfully on a flat

plate flow with zero pressure gradient. for Mach numbers ranging from 0.1 to 5.

The second part of the work has been part of the work has been partially completed.

We have tested the model on Settles' 24 degree compressible corner flow, selected based on

a recent review article (Settles and Dodson (1993). The preliminary results have shown

that the two-scale model produces results similar to that of the k — e model, although the

skin friction in the recovery region is slightly higher. The outcome of the study will be

submitted to the International Conference on Flow Interaction, to be held in Hong Kong,

September 5-9, 1994.

Arne V. Johansson ( Royal Institute of Technology, Stockholm

The cooperation between CMOTT and the Royal Institute of Technology has been

started off in the area of turbulence modelling and numerical simulation of turbulence and

transition. Several common points of interest were penetrated during the two-week visit,

although the main focus was devoted to the study of general formulations of constitutive

relations, with application to turbulence modelling. The particular area of interest for such

general formulations of efforts in the direction of constructing explicit algebraic Reynolds

stress models have shown promising results. The present work of clarifying some of the

fundamentals for such relations have partly been written up during this visit. In addition

to completing this effort, continued collaboration is planned in e.g. the direct numerical

simulation area.

Chaoqun Liu ( University of Colorado at Denver)

Flow transition is one the fundamental and unsolved problems in modern fluid mechanics. The existing numerical studies are quite limited for the following reasons:

1. Most of them use temporal approach which is non-physical,

2. The codes blow up at the flow breakdown stage before transition (pre-onset simulation

only)

3. Very expensive in cpu cost (around 100 - 1000 CRAY hours fora 3-D flat plate).

A new technology was developed during my visit at ICOMP in May-August, 1993,

which led to a successful numerical simulation of the whole process of flow transition at
217

Center for Modeling of Turbulence and Transition

Research Briefs -1993

acceptable CPU cost. A fourth-order finite difference scheme on stretched and staggered grids, a frilly implicit time-marching technique, a semi-coarsening multigrid based on

the so-called approximate line-box relaxation, and a buffer domain for the outflow boundary conditions were all used high-order accuracy, good stability, and fast convergence. A

new fine-coarse-fine grid mapping technique was developed to keep the code running after

the laminar flow breaks down. A number of numerical simulations have been performed

which show good agreement with linear stability theory, secondary instability theory and

some experiments. This potentially provides a tool for direct numerical simulation of bypass transition. The cost for a typical case with 162*34*34 grid is around 2 CRAY-YMP

CPU hours for 10 T-S periods.
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219

Center for Modeling of Turbulence and Transition

Research Briefs -1993

Appendix A

Organization - 1993

Position Chart
220

Center for Modeling of Turbulence and Transition

Research Briefs -1993

ICOMP Director Dr. Louis A. Povinelli,

Deputy Chief

Internal Fluid Mechanics Division

NASA Lewis Research Center

Coordinator Dr. Meng-Sing Liou

Senior Scientist

Internal Fluid Mechanics Division

NASA Lewis Research Center

CMOTT Technical Leader Dr. Tsan-Hsing Shih

Senior Research Associate

ICOMP, NASA Lewis Research Center
221

Center for Modeling of Turbulence and Transition

Research Briefs -1993

Current Listing of Members

ICOMP

NASA LeRc

ICOMP

Sverdrup Tech., Inc.

ICOMP

Names/Term

Duncan, Beverly

7/1991 - present

Hsu, Andrew T.

5/1990 - present

Liou, William W.

11/1990 - present

Norris, Andrew

4/1993 - present

Rubinstein, Robert

7/1991 - present

Shabbir, Aamir

5/1990 - present

Shih, Tsan-Hsing

5/1990 - present

Steffen, Christopher J. Jr

10/1990 - present

Van der Vegt, Jacobus J

10/1991-8/1993

Yang, Zhigang

7/1990 - present

Yu, Sheng-Tao

3/1991 - present

Zhu, Jiang

4/1992 - present

Turbulence Modeling

Upwind Algorithms

Incompressible Flow

Two-Equation

Turbulence Models

Modeling of Bypass

Transition

Modeling of Chemical

Reacting Flows, DNS

Application of Turbulence

Models in Complex Flows

Affiliation Research Areas

Sverdrup Tech., Inc. Multiple-Scale Turbulence

Models

Sverdrup Tech., Inc. PDF Turbulence Modeling,

DNS

ICOMP Compressible Flow

Modeling, Weakly

Nonlinear Wave Models

ICOMP PDF Turbulence Modeling,

DNS

ICOMP Analytical Theories of

Turbulence

ICOMP Buoyancy Effects on

Turbulence, Turbulence

Modeling

ICOMP DNS Compressible Flows

High Order Shock

Capturing Schemes
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223

Center for Modeling of Turbulence and Transition

Research Briefs -1993

Appendix B

CMOTT Biweekly Seminars

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 are the abstract of the semainars during the reporting period.
224

^^ ninsn vT90 CENTER FOR MODELING OF

ICOMP TURBULENCE AND TRANSITION

Lewis Research Center

Ohlo Aeroepecs

7d Irtgttute \^.a

^bd NI S^

Biweekly Meeting Series (1992- 13)

Analytical Theories of Turbulence Applied to

Second Order Closures and Time-Dependent Modeling

by

Robert Rubinstein

Sverdrup Tech. Inc.

Wed., August 19, 1992

1:30-2:30 PM

Room 228, SVR Building

ABSTRACT

The general prograrn of deriving turbulence models from analytical theories like renormalization group and direct interaction approximation will

be described. Renormalization group analysis of the Reynolds stress

transport equation leads to the Lau nder- Reece- Rodi as a lowest order

approximate model with theoretically computed constants in good agreement with accepted values. However, the analysis shows that this model

is not exact and justifies its replacement either with higher order nonlinear models or with a generalization of the stress transport equation

based on a decomposition of the Reynolds stress. Application of the

direct interaction approximation to time-dependent turbulence models

will be discussed.

CONTACT: William W. Liou, PABX 3-6682
225

FOR 

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^^iwsn z CENTER FOR MODELING OF

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Lew. R--h Center m

OhioAeroepeoe

Instkute \^-d

obd Nit s^ Biweekly Meeting Series (1992- 14)

Some Practical Turbulence Modeling Options for Full

Reynolds Averaged Navier-Stokes Calculations of 3D Flows

by

Trong T. Bui

NASA Lewis Research Center

Wed., September 2, 1992

1:30-2:30 PM

Room 228, SVR Building

ABSTRACT

New turbulence modeling options currently under development for

Proteus, a general purpose compressible full Reynolds averaged NavierStokes code, are discussed. The turbulence modeling capability in the

3-D version of Proteus in the current work consists of four turbulence

models: the Baldwin-Lomax, the Baldwin-Barth, the Chien k-e, and the

Launder-Sharma k-e models. Five compressibility corrections and one

length scale correction are also available for the k-e models. Features

of the Proteus turbulence modeling package include: well documented

and easy to use turbulence modeling modules, uniform integration of

turbulence models from different classes, automatic starting options

for turbulence calculations using any turbulence model, and fully vectorized L-U solver for one- and two-equation models. Validation test

cases include the incompressible and compressible flat plate turbulent

boundary layers, turbulent developing S-duct flow, and glancing shock

wave/turbulent boundary layer interaction. Sensitivity of the turbulence

solutions with y+ computation and compressibility options are examined.

The test cases show that the highly optimized one- and two-equation

turbulence models can be used in routine 3-D Navier-Stokes computations with no significant increase in CPU time as compared with the

algebraic Baldwin- Lomax model.

CONTACT: William W. Liou, PABX 3-6682
226

,LJ^^ FOR COMA **# GT9i CENTER FOR MODELING OF

IC MP ? TURBULENCE AND TRANSITION

L..is R—mh Csnty

iP) Ohio A.rosp-* Inslhuts ^.d

obd NIS°^ Biweekly Meeting Series (1992- 15)

Calculation of Unsteady Turbulent Flow over

Oscillating Airfoil

by

S.-W. Kim, K. Zaman and J. Panda

NASA Lewis Research Center

Wed., September 16, 1992

1:30-2:30 PM

Room 228, SVR Building

ABSTRACT

A numerical method to solve unsteady turbulent flows with moving

boundaries and calculation of unsteady turbulent flow over an oscillating

NACA 0012 airfoil at Reynolds number of 44,000 are presented. The

Navier-Stokes equations defined on Lagrangian-Eulerian coordinates are

solved by a time-accurate finite volume method that incorporates an

incremental pressure equation for the conservation of mass. Tile turbulence field is described by a multiple-time-scale turbulence model. The

numerical method successfully predicts the large dynamic stall vortex

and the trailing edge vortex that are periodically generated by the oscillating airfoil. The calculated turbulence field show that the transition

from laminar to turbulent state occurs widely along the airfoil. The calculated streaklines and the ensemble-average mean velocity profiles are

in good agreement with the measured data.

CONTACT: William W. Liou, PABX 3-6682
227

/F-o C0

nun y CENTER FOR MODELING OF

ICOMP TURBULENCE AND TRANSITION

Lewin Research Center

Ohlo Aerospace

7

Institute .a

a^"d NI So\^ Biweekly Meeting Series (1992-16)

Heat Transfer in Film-Cooled Turbine Blades

by

Vijay K. Garg

NRC Senior Research Associate

Wed., September 30, 1992

1:30-2:30 PM

Room 228, SVR Building

ABSTRACT

There is a growing tendency to use higher temperatures at the inlet to a gas turbine in an effort to improve the thermal efficiency and

increase the specific power output. This calls for increasingly effective

means of cooling the turbine blades. One cooling technique presently

receiving wide application is film cooling. In order to study the effect

of film cooling on the flow and heat transfer characteristics of actual

turbine blades, Rod Chima's three-dimensional Navier-Stokes code has

been modified. The effects of film cooling have been incorporated into

the code in the form of appropriate boundary conditions at the hole

locations on the blade surface. Each hole exit (generally an ellipse) is

represented by several control volumes. This provides the code an ability to study the effect of hole shape on the film-cooling characteristics.

Different velocity and temperature profiles for the injected gas can be

specified at the hole exit. These include uniform, laminar or turbulent

(17th power-law) profiles. The code can also handle either a specified

heat flux or a variable temperature condition on the blade surface. A

periodic C-grid with nearly half a million grid points is used. Comparison

with experimental data is fair.

CONTACT: William W. Liou, PABX 3-6682
228

FOR COItA

, 0 CENTER FOR MODELING OF

ICOMP D TURBULENCE AND TRANSITION

L.N. Resew h CGnW

Ohio Atrosp^ * ^^

7

InslftW^ .d

\bd to So^^ Biweekly Meeting Series (1992-17)

Unsteady Flows, Their Impact on

Turbomachinery Blade Row Performance

by

John Adamczyk

NASA Lewis Research Center

Wed., October 14, 1992

1:30-2:30 PM

Room 228, SVR Building
229

/F'o c0

IWLSA * GT90z CENTER FOR MODELING OF

ICOMP TURBULENCE AND TRANSITION

Lewis Research Center T Ohio Aeiospxs

InslttWe -dam

Ni Biweekly Meeting Series (1992-18)

Experimental Investigation of Turbulent Supersonic

Developing Pipe Flow

by

D.O. Davis

NASA Lewis Research Center

Wed., October 28, 1992

1:30-2:30 PM

Room 228, SVR Building

ABSTRACT

Turbulent, supersonic, developing pipe flow is being investigated in the

NASA Lewis 5"X5" Supersonic Wind Tunnel. The objective of the study

is to establish a baseline dataset for the validation of turbulence measurement techniques. The resulting dataset should also be useful for turbulence model and CFD code validation. The mean flow and turbulence

field is being measured with pressure probes and hot-wire anemometry.

Preliminary results have been obtained for a Mach 3 inlet condition and

a flow development length of x/D=32 (D=2.0 inches). These results,

along with a discussion of the problems of hot-wire anemometry in a

compressible boundary layer will be presented. Input from turbulence

modelers and code developers regaarding future testing with the supersonic pipe flow facility will be solicited.

CONTACT: William W. Liou, PABX 3-6682
230

/^oR 

CoMA^

AVSA r9" CENTER FOR MODELING OF

ICOMP ' TURBULENCE AND TRANSITION

ZO Lewis Rssesrch Center m

S Ohio Ao-specs

1 InstiMe .dam

^a°bd N1 Biweekly Meeting Series (1992-19)

Validation of a Two-Scale Turbulence Model for Incompressible Shear Flows

by

Beverly S. Duncan

Sverdrup Technology Inc.

Wed., November 4, 1992

1:30-2:30 PM

Room 228, SVR Building

ABSTRACT

A two-scale eddy viscosity model has been developed which splits the

energy spectrum into a high wavenumber regime and a low wavenumber

regime. Dividing the energy spectrum into multiple regimes simplistically

emulates the cascade of energy through the turbulence spectrum. This

new model has been calibrated and tested for turbulent shear layers.

Calculations of mean and turbulent properties show good agreement to

experimental data for a plane jet, a round jet and two mixing layers.

Preliminary results for boundary layers will also be presented.

CONTACT: William W. Liou, PABX 3-6682
231

^Foa c0Mp

WSA

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Z CENTER FOR MODELING OF

ICOMP TURBULENCE AND TRANSITION

Lewis Research Center

Ohio Aerospace ,Wt. \^-d -

^bd "' s^ Biweekly Meeting Series (1992-20)

Extension of a Two-Scale Turbulence Model to

Compressible Free Shear Flows

by

William W. Liou

ICOMP

Wed., November 18, 1992

1:30-2:30 PM

Room 228, SVR Building

ABSTRACT

A two-scale model for compressible turbulent flows is described. The

model incorporates a notion that the effect of compressibility on turbulence is mainly oil energetic large eddies. The small eddies are

affected through the increased spectral energy transfer from the large

eddies due to the effect of compressibility. The turbulent eddy-viscosity

is determined by the total turbulent kinetic energy and the energy transfer rate, which is different from the energy dissipation rate. The model is

tested against high-speed mixing layers. The results agree satisfactorily

with measured data.

CONTACT: William W. Liou, PABX 3-6682
232

FOB COMA

IC MP 'C ENTER FOR MODELING OF

TURBULENCE AND TRANSITION

Lewis R-h C*Mw -

ON. Awosp 0

In0hute \^.d

obO NI Biweekly Meeting Series (1992-21)

Prediction of Surface Roughness Effects on

Turbine Aerodynamic and Meat Transfer Performance

by

Robert J. Boyle

IFMD

Wed., December 2, 1992

1:30-2:30 PM

Room 228, SVR Building

ABSTRACT

Surface roughness has been shown to affect the aerodynamic performance of turbomachinery. Tests on the SSME fuel turbine showed a

20% increase in overall loss between smooth nearly a doubling of the

blade profile loss. An approach to predicting the effects of blade surface roughness on the aerodynamic performance and heat transfer rates

is discussed. An algebraic turbulence model, which includes the effect

of surface roughness, was incorporated into a quasi three-dimensional

Navier-Stokes analysis code (RVCQ3D). Comparisons are shown for both

heat transfer and aerodynamic performance data.

CONTACT: William W. Liou, PABX 3-6682
233

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CENTER FOR MODELING OF

TURBULENCE AND TRANSITION Lewis Research C.MM m

S Ohl. Aerospace

Institute

Oyd Ni g7

Biweekly Meeting Series (1992-22)

Evaluation of the Turbulence Models in PARC

by

Nick Georgiadis

Nozzle Technology Branch

Propulsion System Division

Wed., December 16, 1992

1:30-2:30 PM

Room 228, SVR Building

ABSTRACT

The PARC2D/3D Navier-Stokes codes are used to analyze a variety

of complex turbulent propulsion flows. Both algebraic and two-equation

turbulence models are available in PARC. This presentation will compare the capabilities of PARC's turbulence models to predict several

flows including benchmark turbulent test cases (flow over a flat plate

and over a backward-facing step), the Sajben transonic diffuser flow,

and other propulsion flow cases. The algebraic turbulence models that

are investigated are the P.D. Thomas model (the standard turbulence

model in PARC which was optimized for free shear layer flows but also

calculates wall boundary layers), the Baldwin-Lomax model, and a new

combination model which uses the Modified Mixing Length (MML) for

wall boundary layers and the Thomas model for free shear layers. The

two-equation model that is also investigated is the Chien k-epsilon model

with modifications for compressibility added by Nichols. The presentation will show that there are significant differences among flow solutions

obtained with these turbulence models for even the simplest of flows.

CONTACT: William W. Liou, PABX 3-6682
234

CENTER FOR MODELING OF

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TURBULENCE AND TRANSITION ZO L..A. Rp. h C.ntw

Ohlo A.ro.p.c.

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3d 

N1 57^^

Biweekly Meeting Series (1993-1)

The Application of PDF Methods to the Modeling of

Turbulent Diffusion Flames

by

A.T. Norris

Sibley School of Mechanical and Aerospace Engineering

Cornell University, Ithaca, New York 14853.

Wed., January 20, 1992

1:30-2:30 PM

Multipurpose Room, OAI Building

ABSTRACT

The modeled transport equation for the joint probability density function (jpdf) of velocity and dissipation provides a closed set of equations

for the modeling of turbulent flows. Features of this method include

the presence of length and time scales, intermittency and the exact

treatment of convection. This approach is extended to reactive flows by

including composition, (i.e. Solving for the jpdf of velocity, dissipation

and composition.) which has the feature that reaction is treated exactly.

A particle-based Monte Carlo scheme is used to solve the transport

equation, for several different classes of flow of increasing complexity.

The simplest flow considered is a variable density plane mixing layer

while the most complex is a piloted CO/H2/N2 - Air diffusion flame

close to extinction. Details of the modeling, including molecular mixing

models and reduced thermochemical mechanisms, are discussed for each

applicable flow.

CONTACT: Aa.inir Shabbir, PAB1 3-39231
235

^j^^

NASA

*** _r9^ CENTER FOR MODELING OF

IC MP TURBULENCE AND TRANSITION

Z Lewis Research Center Z O

Ohi o

stNute a^° Biweekly Seminar Series (1993-2)

^Obd NI SOS

A Realizable Reynolds Stress Algebraic Equation Model

For Complex Flows

by

Jiang Zhu and Tsan-Hsing Shih

CMOTT, ICOMP

Wed., February 10, 1992

1:30-2:30 PM

Room 2B206, OAI Building

ABSTRACT

The invariance theory in continuum mechanics is applied to analyze

Reynolds stresses in high Reynolds number turbulent flows. The analysis leads to a turbulent constitutive relation that relates the Reynolds

stresses to the mean velocity gradients in a more general form in which

the classical isotropic eddy viscosity model is just the linear approximation of the general form. On the basis of realizability analysis, a set of

model coefficients are obtained which are functions of the time scale

ratios of the turbulence to the mean strain rate and the mean rotation

rate. These coefficients will ensure the positivity of each component

of the turbulent kinetic energy — realizability that most existing turbulence models fail to satisfy. Separated flows over backward-facing step

configurations are taken as applications. The calculations are performed

with a conservative finite-volume method. Grid-independent and numerical diffusion-free solutions are obtained by using differencing schemes

of second-order accuracy on sufficiently fine grids. The calculated results are compared in detail with the experimental data for both mean

and turbulent quantities. The comparison shows that the present proposal significantly improves the predictive capability of K — c based two

equation models.

CONTACT: Aamir Shabbir, 962-3149
236

NASA vT9 0

ICOMP CENTER FOR MODELING OF

TURBULENCE AND TRANSITION

LOW$ R—rch C.MV

OAb A.rac. ^^ ^d ^.,to.p

as Biweekly Seminar Series (1993-3)

^bd NI S7^

An Algebraic Turbulence Model for

Three-Dimensional Viscous Flows

by

R.V. Chima, P.W. Giel and R.J. Boyle

Propulsion Systems Division

Wed., February 24, 1993

1:30-2:30 PM

Multipurpose Room, OAI Building

ABSTRACT

An algebraic turbulence model is proposed for use with three dimensional Navier-Stokes analyses. It incorporates features of both the

Baldwin-Lomax and Cebeci-Smith models. The Balwin-Lomax model

uses the maximum of function f(y) to determine length and velocity

scales. An analysis of Baldwin-Lomax model shows that f(y) can have

spurious maximum close to the wall, causing numerical problems and

non-physical results. The proposed model uses the integral relations to

determine Pu, and b used in the Cebeci-Smith model. It eliminates a

constant in the Baldwin-Lomax model and determines the two remaining constants by comparison to Cebeci-Smith formulation. Pressure

gradient effects, a new wake model, and the implementation of these

features in a three-dimensional Navier-Stokes code are also described.

Results are shown for a flat plate boundary layer, an annular cascade,

and endwall heat transfer in a linear turbine cascade. The heat transfer

results agree well with experimental data which shows large variations

in endwall Stanton number contours with Reynolds number.

CONTACT: Aamir Shabbir, 962-3149
237

/J^^ *** V r9i\ NASA o

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ICOMP

Lewla Research Center

iP No Aerospace

nstRute d

d^bd 

NI

CENTER FOR MODELING OF

TURBULENCE AND TRANSITION

Biweekly Seminar Series (1993-4)

Computations of Separated Aerodynamic Flows Using

a Modified Near Wall k-E Model

by

Dr. Yuichi Matsuo

National Aerospace Laboratory

Tokyo, Japan

Thursday, March 4, 1993

2:00-3:00 PM

Multipurpose Room, OAI Building

ABSTRACT

Complex aerodynamic flows with separation were computed by using a near wall k — e model. The model is modified so that the near

wall damping functions do not include friction velocity and so that the

predicted turbulence quantities are matched to the variations from DNS

data. In order to apply the model to practical flows, an efficient and

robust flow solver was developed. The code is based on a finite difference approach where a state-of-the-art upwind scheme is used for the

convection terms and further source terms of the turbulence quantities

are treated implicitly. Some representative computations were carried

out for flows over backward facing step, a NACA0012 airfoil, a supersonic compression corner and the ONERA M6 wing. The last one is a

3D case. The results show that the present model provides satisfactory

predictability even for the complex flows with separation.

CONTACT: Aamir Shabbir, 962-3149
238

11i/LSA T9'o CENTER FOR MODELING OF I C O MP 

L^,.R.^.^,GM_ TURBULENCE AND TRANSITION

Oslo A r pr

InWhut

a

Biweekly Seminar Series ( 1993-5)

Direct Numerical Simulation of Boundary Layer

Flow Over Surface Roughness

by

Russell G. De Anna

IFMD

Wednesday, March 10, 1993

1:30-2:30 PM

Multipurpose Room, OAI Building

ABSTRACT

Results from a direct numerical simulation of transitional flow over a surface with spherical roughness elements of height k and a surface with

random roughness of maximum height fkn,, ax are presented. Periodic

boundary conditions in the streamwise and spanwise directions simulate

an infinite array of roughness elements, while a body force, designed to yield

the streamwise Blasius velocity in the absence of roughness, maintains

the flow. At kl6* = 0.72, the mean velocity field in the spherical roughness domain contains secondary flow patterns within the region below

2k, for Reynolds numbers, Uk k/v, between 90 and 225. The streamwise

vorticity at these low Reynolds numbers is simply a result of the fluid's

continuity and does not indicate rotating fluid or effects of inertia. The

spheres distort the original Blasius profile into a mildly inflected layer

containing low-momentum regions behind each sphere. These regions

engender unsteady disturbances near the wall; however, the distribution

of body force with vertical position above the wall is such that growth is

suppressed in this region. Growth does occur in the unstable layer above

the spheres where the body force is larger. The disturbance frequency

is fixed by both the mean, streamwise velocity in the most unstable

layer and the spacing between spheres; it is not the blunt-body vortexshedding frequency expected for isolated bodies. When an oscillating

component was added to the steady, Blasius body force, the response

was independent of both forcing frequency and amplitude and, once

again, depended on the mean velocity and the spacing between spheres.

CONTACT: Aamir Shabbir, 962-3149
239

^ *IWLSI^ ^9 0 CENTER FOR MODELING OF

Z T(',(lT/fU â . YTT„-,TTT T,TT/Yn A AT T) mn A ATOTrTTnAT

^ Lewls Research Csntw m

if^^ Ohlo Aefospsos ^

Biweekly Seminar Series (1993- 6)

NI

Three-Dimensional Navier-Stokes Analysis and

Redesign of an Imbedded Bellmouth Nozzle

in a Turbine Cascade Inlet Section

by

P. W. Giel, J. R. Sirbaugh, I. Lopez, J. Van Fossen

Sverdrup Tech./ARMY/NASA

Wednesday, April 7, 1993

1:30-2:30 PM

Multipurpose Room, OAI Building

ABSTRACT

A computational analysis of an imbedded bellmouth inlet was performed with the PARC code to identify and eliminate the source of

measured pitchwise flow non-uniformity in the NASA LeRC Transonic

Turbine Blade Cascade. The computational domain extended from the

beginning of a constant span section to a plane just upstream of the cascade of turbine blades. Spanwise symmetry allowed modeling of just half

of the span. The blockage and acceleration effects of the blades were

accounted for by specifying a periodic static pressure exit condition interpolated from an RVC313 code isolated blade calculation. Calculations

of the original geometry showed total pressure loss regions consistent

in strength and in location to experimental measurements. An examination of the results shows that the distortions are caused by a pair of

vortices that originate as a result of the interaction of the flow with the

imbedded bellmouth. Computations were performed for an inlet geometry which eliminated the imbedded bellmouth by bridging the region

between it and the upstream wall. This analysis indicated that eliminating the imbedded bellmouth eliminates the troublesome pair of vortices,

resulting in a flow with much greater pitchwise uniformity.

CONTACT: Aamir Shabbir, 962-3149
240

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NISA CENTER FOR MODELING OF IC MP D TURBULENCE AND TRANSITION

z LwM R....rch C~

J^^ Oh to A.ro.p.c.

InslNul. .d

r^bd NI S^ Biweekly Seminar Series (1993- 7)

A PDF Approach for Compressible Turbulent

Reacting Flows

by

Andrew Hsu

Sverdrup Tech.

Wednesday, April 21, 1993

1:30-2:30 PM

Room 2B-204, OAI Building

ABSTRACT

The objective of the present work is to develop a probability density function (pdf) turbulence model for compressible reacting flows for

use with a CFD flow solver. The probability density function of the

species mass fraction and enthalpy are obtained by solving a pdf evolution equation using a Monte-Carlo scheme. The pdfsolution procedure

is coupled with a compressible CFD flow solver which provides the velocity and pressure fields. A modeled pdf equation for compressible flows,

capable of capturing shock waves and suitable to the present coupling

scheme, is proposed and tested. Convergence of the combined finitevolume Monte-Carlo solution procedure is discussed, and an averaging

procedure is developed to provide smooth Monte-Carlo solutions to ensure convergence. Two supersonic diffusion flames are studied using

the proposed pdf model and the results are compared with experimental

data; marked improvements over CFD solutions without pdf are observed.--Preliminary applications-of pdf to 3D flows are also reported.

CONTACT: Aamir Shabbir, 962-3149
241

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ICOMP

01

Lewis Research Center m

iP Ohlo Aerospace

Institute \^d^

Obd N I 57

CENTER FOR MODELING OF

TURBULENCE AND TRANSITION

Biweekly Seminar Series (1993-8)

On the linear stability of a trailing line vortex

by

Zhigang Yang

ICOMP

Wednesday, May 5, 1993

1:30-2:30 PM

Multipurpose Room, OAI Building

ABSTRACT

The viscous linear stability of a trailing line vortex (Batchelor vortex)

is studied. The flow is characterized by two parameters, the Reynolds

number Re and the rotation rate q. The marginal stability curve which

separates the stable domain from the unstable domain was searched over

the (Re, q) plane. It is found that on the marginal stability curve, q

increases with the Reynolds number and does not approach a constant

even when the Reynolds number is as large as 10 5 . The values of q for

large Reynolds numbers are higher than the inviscid counterpart. These

findings suggest that modes giving rise to the marginal stability are

viscous and do not approach the inviscid limit as the Reynolds number

goes to infinity. These modes have an azimuthal wavenumber n — -1

when the Reynolds number is larger than 200, in contrast to n — -2 for

smaller Reynolds numbers. As the Reynolds number is increased, the

eigenfunctions of the marginal stability modes become more and more

concentrated near the axis of the vortex, suggesting that these modes

are viscous center modes in the limit of large Reynolds number.

CONTACT: Aamir Shabbir, 962-3149
242

F °R C04t

co CENTER FOR MODELING OF ICOMP y 

TURBULENCE AND TRANSITION

ZO Lwrls R—,.h Cwdw m

iP Ohlo A*ro pw* n

Inphut

"Obd N , s°^aBiweekly Seminar Series (1993-9)

Weakly Nonlinear Models for Turbulent Free Shear Flows (3)

- Linear Instability of Curved Free Shear Layers

by

William Liou

ICOMP

Wednesday, June 9, 1993

1:30-2:30 PM

Room 2B-204, OAI Building

ABSTRACT

Turbulence closure schemes based on a weakly nonlinear theory with

a description of the dominant large-scale structures as instability wave

have been applied successfully in the prediction of various plane and

axisymmetric free shear flows. In order to extend the wave model to

curved mixing layers, a linear stability anslysis has been performed. Two

mean velocity profiles that represent stably and unstably curved free

mixing layers were considered. In this presentation, results of the linear instability study for five curvature Richardson number are described.

The instability characteristics of the mixing layer were found to vary

significantly with the introduction of the curvature effects. The results

also indicate that, in a manner similar to the Urtler vortices observed in

a boundary layer along a concave wall, instability modes of spatially developing streamwise vortices pairs may appear in unstably curved mixing

layer.

CONTACT: Aamir Shabbir, 962-3149
243

z IC

NIGA

OMP CENTER FOR MODELING OF

TURBULENCE AND TRANSITION Lewis Research Center

Ohio Aerospace n

Ii^aobdI nst hule

N 1 Biweekly Seminar Series (1993 -10)

---

Development of Reduced Chemistry for CFD Applications

by

J.-Y. Chen

Department of Mechanical Engineering

University of California at Berkeley

Berkeley, California

Friday, June 11, 1993

1:30-2:30 PM

Multipurpose Room, OAI Building

ABSTRACT

Simplified chemical reaction mechanisms are often required for simulation of reacting flows to reduce computational time. Recent development of simplified mechanisms for complex combustion processes

is based on systematic reduction of the detailed chemical mechanism

rather than curve fits to limited experimental observattions. Consequently, these newly developed features are capable of capturing many

salient features of the detailed chemical kinetics. More importantly, for

CFD applications, these features of the detailed mechanism can be obtained by a small number of scalars. This presentation will be given from

the point view of CFD developers and users. The concept of systematic

reduction method will be introduced with a simple thermal NO reaction

mechanism. A step by step reduction of the detailed mechanism will

be illustrated for hydrogen and methane air combustion. Automation

of the procedures for constructing reduced reaction mechanisms by a

computer program is currently being developed. Results for hydrogen

combustion with NOX formation will be presented.

CONTACT: Aamir Shabbir, 962-3149
244

l j 

*j1/GA r9p CENTER FOR MODELING OF

ICOMP ' TURBULENCE AND TRANSITION

Laois A^rcn C*Mw

Ohio A*ropM Institute Biweekly Seminar Series (1993-11)

Some Turbulence Modeling Issues Related to Heat Transfer

in Turbomachinery

by

Ali Ameri

Turbomachinery Flow Physics Branch

Wednesday, June 16, 1993

1:30-2:30 PM

Multipurpose Room, OAI Building

ABSTRACT

Navier-Stokes calculations were carried out in order to predict the

heat transfer rates on surfaces of turbine blades. The code TRAF was

modified to handle a variety of two-equation models in addition to the

baseline Baldwin and Lomax model. The calculations were performed

efficiently by utilizing a multigrid method. The results of the calculations generally agree with the experimental measurements in the laminar

and the fully turbulent regimes. The transition process is however not

well predicted. The two-equation model results also show a very distinct

sensitivity to the assigned free stream length scale of turbulence. The

accuracy level of the results obtained using the Baldwin-Lomax model

was comparable to those obtained using the two-equation models. With

that in mind a transition model was incorporated into the code which

when used in conjunction with the B-L model produced very good results. The extension of the two-equation and the transition model to

3-d calculations will be addressed and some example cases will be considered.

CONTACT: Aamir Shabbir, 962-3149
Z CENTER FOR MODELING OF 245 PJASA

Icom.

o

TURBULENCE AND TRANSITION Z O Lawia R. arch Comm 

S^ Ohio Aerospace

Inafhut. .a

^a°bd Ni Biweekly Seminar Series (1993-12)

Modeling of Intercomponent Transfer In Reynolds Stress

Closures Of Homogeneous Turbulence

by

Arne V. Johansson

Department of Mechanics

Royal Institute of Technology

Stockholm, Sweden

Monday, June 21, 1993

1:30-2:30 PM

Multipurpose Room, OAI Building

ABSTRACT

In classical Reynolds stress closures of turbulent flows transport equations are formulated for the Reynolds stress tensor and dissipation rate.

For homogeneous turbulence there is no spatial redistribution of energy

and the modeling difficulties lie in the treatment of c-equation and the

intercomponent transfer processes. The transport equations for u,7u may

be replaced by equations for the kinetic energy, k, and stress anisotropy

tensor ai; (- uiu;/k- 26i; /3) to effectively separate the "amplitude" related

issues involved in the prediction of k and a from the relative distribution

among the components described by the normalized, traceless tensor

ail.

Intercomponent transfer in the transport equations for ai; (or u;u;) is

represented by H^') , Ii^') denoting the so called rapid and slow pressure

strain rate terms, respectively, and a term related to the effects of

anisotropic dissipation rate.

Detailed data for the individual terms obtained from direct numerical simulations and physical experiments are presented and used in the

evaluation of the modeling ideas and other aspects of intercomponent

transfer.

New experimental results will be presented where a pair of specially

built hot-wire X-probes have been used to investigate the anisotropy of

the dissipation rate in axisymmetric turbulence. The trends are to be

well described by the model proposed by Hallback et al. (Phys. Fluids

1990) both for the simulation results and for the higher Reynolds number

data obtained from the physical experiments.

For low turbulence Reynolds numbers the simulations have shown

the slow part of the pressre-strain rate to be strongly suppressed. A

simple model for the variation of Rotta constant has been shown to

capture this Reynolds number variationl.

Recently developed modeling ideas for the rapid pressure-strain rate

will also be discussed. Strong realizability an kinematic constraints have

been used to derive tensorially correct forms of the model, and the model

constants have been calibrated by use of rapid distortion theory.

CONTACT: AarrLir Shabbir, 962-3149
246

ICOMP â

CENTER FOR MODELING OF

TURBULENCE AND TRANSITION L..wls R. ,h C..t.r

Ohio A.ro.p.c.

1n.t= .dam

d NItilt s Seminar Series (1993-13)

Computation of Confined Coflowing Jets with

Three Turbulence Models

by

J. Zhu and T.-H. Shih

CMOTT/ICOMP

Wednesday, June 30, 1993

1:30-2:30 PM

Multipurpose Room, OAI Building

ABSTRACT

A numerical study of confined jets in a cylindrical duct is carried out

to examine the performance of two recently proposed turbulence models: an RNG-based K-e model and a realizable Reynolds stress algebraic

equation model. The former is of the same form as the standard K-e

model but has different model coefficients. The latter uses an explicit

quadratic stress-strain relationship to model the turbulent stresses and

is capable of ensuring the positivity of each turbulent normal stress.

The flow considered involves recirculation with unfixed separation and

reattachment points and severe adverse pressure gradients, thereby providing a valuable test of the predictive capability of the models for complex flows. Calculations are performed with a finite-volume procedure.

Numerical credibility of the solutions is ensured by using second-order

accurate differencing schemes and sufficiently fine grids. Calculations

with the standard K-c model are also made for comparison. Detailed

comparisons with experiments show that the realizable Reynolds stress

algebraic equation model consistently works better than the standard

K-e model in capturing the essential flow features, while the RNG-based

K-e model does not seem to give improvements over the standard K-e

model under the flow conditions considered.

CONTACT: Aamir Shabbir, 962-3149
247

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ICOMP Lewie R—rch Center m

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CENTER FOR MODELING OF

TURBULENCE AND TRANSITION

Biweekly Seminar Series (1993-14)

Turbulence Modeling for Compressible Flows

Part I - Modeling Mixing and Boundary Layer Flows

by

P.G. Huang

Eloret Institute/

Modeling and Experimental Validation Branch

NASA-Ames, CA

Monday, July 12, 1993

1:30-2:30 PM

Room 2B-201, OAI Building

ABSTRACT

The present seminar centers on the "dissipation-transport" equation

and its role in predicting the compressible law of the wall. First, a skin

friction and velocity profile family for compressible turbulent boundary

layers is developed. The profile family has been compared with a range

of high speed flow data with great success, including supersonic and

hypersonic experiments and a recent compressible channel flow DNS.

Predictions of the velocity profiles using standard turbulence models

have shown that the unmodified models have given rise to too small

value of von Karmam constant, K, in the log-law region. Thus, if the

models are otherwise accurate, the "wake" component is over-predicted

and the predicted skin friction is lower than the expected value. The

magnitude of the errors that results from neglecting the dependence on

density depends on the variables used to specify the length scale.

To agree with experimental values for r, in compressible boundary

layer, the apparent eddy viscosity must be increased. This "compressible effect" — which is an artifact of conventional turbulence modeling

rather than something real — is exactly opposite to that in mixing layers,

where the growth rate, and by implication the eddy viscosity, decreases

with increasing Mach number. As a consequence, recent compressibility

modifications proposed for the mixing layer has increased the errors in

predictions of flat-plate compressible boundary layer flows.

CONTACT: Aamir Shabbir, 962-3149
248

IGA CENTER FOR MODELING OF

Icom. TURBULENCE AND TRANSITION

ZO Lewis Research Center m

S^ Ohio A. We

Institute .d -

d N1 Biweekly Seminar Series (1993- 15)

Turbulence Modeling for Compressible Flows

Part II - Complex Flow Computations

by

P.G. Huang

Eloret Institute/

Modeling and Experimental Validation Branch

NASA-Ames, CA

Wednesday, July 14, 1993

1:30-2:30 PM

Room 213-204, OAI Building

ABSTRACT

Calculations of high Mach number turbulent flows have become a

major challenge in CFD in recent years. In the present seminar, attention

will be focused on some of the recent activities undergoing at NASA

Ames on predictions of flows with complex shock-wave/boundary-layer

interactions. Numerical methods to solve mean flow and turbulence

equations, including Reynolds stress transport models, will be discussed.

Comparison of low-Reynolds-number and wall-function techniques will

also be made.

The seminar presents results of calculations for a range of 2-1) and

3-13 compressible turbulent flows using both two-equation and Reynolds

stress transport models. Comparisons with the experimental data have

shown that baseline models under-predict the extent of flow separation

but over-predict the heat transfer near flow re-attachment. Modifications to the models are described which remove the above-mentioned

deficiencies.

CONTACT: Aamir Shabbir, 962-3149
249

FOR C04fo

CENTER FOR MODELING OF

ICOMP y

TURBULENCE AND TRANSITION

ZO Lewls Research Center Z

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^aObd N , s^ Biweekly Seminar Series (1993-16)

Towards the Simulation of a Full Turbofan Engine

In the Meridional Plane

by

M. Stewart

Sverdrup Technology

Wednesday, July 28, 1993

1:30-2:30 PM

Room 2B-204, OAI Building

ABSTRACT

The numerical simulation of the aerodynamics of a full jet engine is a

problem of interest in engineering research and design. Existing analysis

techniques deal with individual components and largely neglect intercomponent effects. Yet the aerodynamic performance of a jet engine

depends on these components working together efficiently.

In this simulation of a jet engine, the three-dimensional flow equations are averaged to axisymmetric flow equations, defined in the twodimensional meridional plane of an engine. The engine's meridional

plane is covered with a multiblock grid which resolves blades and other

components. The meridional plane includes external flow outside the engine and the internal ducts. The turning effects of blades, combustion,

blockage, losses and real gas effects are represented in the equations

with terms and interior conditions.

Some numerical, modeling and physical issues in this simulation will

be discussed including accounting for losses, imposing internal conditions

and the numerical stability of large multistage compressors and turbine

components.

The method will be demonstrated with two examples. Convergent

numerical solutions will be shown for the 1.15 Pressure Ratio Fan Engine

model. The applicability of these methods to commercial engines will

be demonstrated with the Energy Efficient Engine.

(Refreshments will be provided)

CONTACT: Aamir Shabbir, 962-3149
250

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ICOMP CENTER FOR MODELING OF

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Biweekly Seminar Series (1993-17)

Effect of the Coriolis Force on Compressible Turbulence

by

T.-H. Shih (ICOMP)

and

A.T. Hsu (Sverdrup Technology)

Wednesday, August 11, 1993

1:30-2:30 PM

Multipurpose Room, OAI Building

ABSTRACT

Direct numerical simulation results and theoretical analysis are presented for the effect of the Coriolis force on compressible homogeneous

isotropic turbulence. It is shown that the Coriolis force serves as a

frequency modulator on turbulence. While the Coriolis force neither

creates nor destroys turbulent kinetic energy, it redistributes energy by

eliminating low frequency waves and transferring energy to waves with

a frequency of 2Q. The dissipation rate of turbulent kinetic energy can

be either reduced or enhanced depending on whether or not the ratio between the rotation time scale and the Kolmogorov time scale is

much greater than one. It has been demonstrated both theoretically

and numerically that the Taylor-Proudman theorem is applicable to homogeneous turbulence only when the time scale of rotation, defined

as the inverse of the frequency of the inertial waves, approaches the

Kolmogorov time scale, and that two-dimensionalization occurs in this

regime.

(Refreshments at 1:15)

CONTACT: Aamir Shabbir, 962-3149
251

Center for Modeling of Turbulence and Transition

Research Briefs -1993

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 turbulence- driven

secondary flows, " 11th Australian Fluid Mechanics Conference, Hobart, Australia (Dec.

1992).

Brown, S., Leibovich, S. and Z. Fang, "On the linear instability of the Hall-Stewartson

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," AIAA 93-0086 (1993).

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 combusting 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).
252

Center for Modeling of Turbulence and Transition

Research Briefs -1993

Liou, W. W. and Shih, T.-H., "A two-scale model for compressible turbulent flows," NASA

TM 106072 (1993).

Liou, W. W. and Shih, T.-H., "Compressible turbulent boundary layer predictions using

a multiple-scale model," NASA TM (to appear) (1993).

Liou, W. W., "Linear instability of curved free shear layer," AIAA 93-3252 (1993).

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, 4, 2788-2808 (1992).

Liou,. W. W. and Morris, P. J., "The eigenvalue specturm of the Rayleigh equation for a

plane shear layer," International Journal of Numerical Methods in Fluids, 15, 1407-1415

(1992).

Liou, W. W., "A new energy transfer model for turbulent free shear flow," NASA TM

105854 (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 twoequation turbulence models ," NASA TM 105376, (1991).

Moin, P., Shih, T.-H., Driver, D. and Mansour, N., "Direct numerical simulation of a

three-dimensional turbulent boundary layer," Physics of Fluids A 2, 10, 1846-1853 (1990).

Norris, A.T., "The application of PDF methods to piloted diffusion flames," Ph.D. Thesis,

Cornell University, 1993.

Rubinstein, R. and Barton, J. M., "Renormalization group analysis of the Reynolds stress

transport equations, nonlinear Reynolds stress models and the renormalization group,"

Phys. Fluids A 4, 1759 1766 (1992).

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).
253

Center for Modeling of Turbulence and Transition

Research Briefs -1993

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., Liou, W.W., Shabbir, A., Yang, Z., and Zhu, J., "A vorticity dynamics based

model for the turbulent dissipation," NASA TM 106177 (1993).

Shih, T.-H. and Lumley, J.L., "Remarks on turbulent constitutive relations," NASA TM

106116 (1993).

Shih, T.-H. and Lumley, J.L., "Kolmogorov behavior of near-wall turbulence and its application in turbulence modeling," NASA TM 105663 (1992)

Shih, T.-H., Chen, J.-Y. and Lumley, J. L., "Second order modeling of free 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 — E 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 Turbulence, 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., "A critical comparison of several low Reynolds number k-epsilon turbulence models for flow over a backward-facing step," NASA TM 106173 (1993).

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, L. 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," AIAA Paper 93-0335 (1993)
254

Center for Modeling of Turbulence and Transition

Research Briefs -1995

Van der Vegt, J., "Assessment of flux splitting for viscous compressible flows," AIAA Paper

91-0243 (1991).

Van der Vegt, Jaap, "Overview of the Osher approximate Riemann solver for threedimensional flows," NASA TM, in print (1992).

Yang, Z. and Leibovich, S., "On the lindar marginal stability of the trailing line vortex,"

Submitted for publication.

Yang, Z. and Leibovich, S., "Unstable viscous wall modes in rotating pipe flow," AIAA

Paper 91-1801, (1991). Also submitted to Physics Fluids A for publication.

Yang, Z. and Shih, T.H., "A k — e modeling of near wall turbulence," Proc. of 4th Intl.

Sym. on CFD, Davis, CA, 1305-1310, (1991). Also available as NASA TM 105238.

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 Shih, T.H., "A Galilean and tnesorial invariant k — e model for near wall

turbulence," AIAA Paper 93-3105 (1993).

Yang, Z. and Shih, T.H., "A k — e calculation of transitional boundary layers," Proc. Near

Wall Turbulent Flows, Elsevier, 1993.

Yang, Z. and Shih, T.H., "On the wall functions for two equation turbulence models,"

prepared for publication.

Yang, Z. and Shih, T.H., "An invariant theory based model for the dissipation rate equation," In preparation.

Yang, Z. and Shih, T.H., "A k — e calculation of transitional boundary layers," Transition

and Turbulence, Springer-Verlag, 1992.

Yang, Z. and Shih, T.H. "A new time scale based k —e model for near wall turbulence,"

AIAA J. 31, 1191 (1993).

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).

Zhu, J., "FAST-2D: A computer program for numerical simulation of two dimensional incompressible flows with complex boundaries," Rept. No.690, Institute for Hydromechanics,

University of Karlsruhe (1991).

Zhu, J. and Shih, T.H., "Calculations of turbulent separated flows, NASA TM 106154

(1993).

Zhu, J. and Shih, T.H., "A numerical study of confined turbulent jets," NASA TM 106197

(1993).

Zhu, J. and Shih, T.H., "Computation of confined coflow jets with three turbulence models," AIAA Paper 93-3120.
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January 1994 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Center for Modeling of Turbulence and

Transition: Research Briefs-1993

WU-505-90-5K 6. AUTHOR(S)

William W. Liou, editor

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION

REPORT NUMBER

National Aeronautics and Space Administration

Lewis Research Center E-8202

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

National Aeronautics and Space Administration NASA TM-106383

Washington, D.C. 20546-0001 ICOMP-93-44; CMOTT-93-16

11. SUPPLEMENTARY NOTES

William W. Liou, Institute for Computational Mechanics in Propulsion and Center for Modeling of Turbulence and

Transition, 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. DISTRIBUTION CODE

Unclassified - Unlimited

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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

(CMOT'T) from June 1992 to July 1993. It is also an annual report to the Institute for Computational Mechanics in Propulsion

located at Ohio Aerospace Institute and NASA Lewis Research Center. The main objectives of the research activities at CMOTT

are to develop, validate and implement turbulence and transition models for flows of interest in propulsion systems. Currently, our

research covers eddy viscosity one- and two-equation models, Reynolds-stress algebraic equation models, Reynolds-stress

transport equation models, nonequilibrium multiple-scale models, bypass transition models, joint scalar probability density

function models and Renormalization Group Theory and Direct Interaction Approximation methods. Some numerical simulations

(LES and DNS) have also been carried out to support the development of turbulence modeling. Last year was CMOTT's third

year in operation. During this period, in addition to the above mentioned research, CMOTT has also hosted the following

programs: an eighteen-hour short course on "Turbulence—Fundamentals and Computational Modeling (Part I)" given by

CMOTT at the NASA Lewis Research Center; a productive summer visitor research program that has generated many encouraging results; collaborative programs with industry customers (e.g. P.&W. and RocketDyne) to help improve their turbulent flow

calculations for propulsion system designs; a biweekly CMOTT seminar series with speakers from within and without the NASA

Lewis Research Center, including foreign speakers. In addition, CMOTT members have been actively involved in the national

and international turbulence research activities. The current CMOTT roster and organization are listed in Appendix A. Listed in

Appendix B are the abstracts of the biweekly CMOTT seminar. Appendix C lists the papers contributed by CMOTT members.

14. SUBJECT TERMS 15. NUMBER OF PAGES

Turbulence modeling; Transition 255

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