NASA-CR-176478 

19860007049 

A Reproduced Copy 

OF 

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by the 

NASA Scientific and Technical Information Facility 

FFNo 672 Aug 65 

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LANGLEY fiES!:ARCH CENTER 

LIBRARY, NASA 

1-!.'\:',1~TO~. VIRSINIA 

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NF00455 
"7.-------; (.a..l" .. ----_. ~.: 

• 

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Dep~rtmtnt of Mechanical En91neer1ng 

university of Wisconsin--Hilwaukee ,II. 

Milwaukee. W\scons'n 53201 67 (fJ 

N R&)~ . 

IMPROVEMENT OF THE SECOND- AND 

THIRD-HOHEUT MODELING OF TURBULENCE 

January 19B& 

by 

R. S. Atr.ano 

Principal Investigator 

and 

P. Goel 

Research Assistant 

Semi-Annual Progress Report on 

-A STUDY OF REYhOLDS-STR(SS CLOSURE MODELThe report documents re~(,ilrch completed during the period of July 15 througtl 

January 15. lY86 under NAStl-Lewis Re!>earch Grant NC!. flAG 3-54&. 

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ABSTRACT 

This report contains four parts of the Reynolds-str~ss closure modeling: 

1) irnprovemtnt of the k and c equations, 2) deve10pme~t of the third-moment 

transport equation, 3) formulation of the diffusion co"fficient of the~ntUIU eQuat ion by us 1ng the algebraic-stress r:-.cdel-of'· turbulence. and 4) the 

application of the Reyn~lds-stress model to a heat exchanger problem. It has 

been deoonstrated that the th1rd~~nt transport model improved the prediCtion of the triple-velocity products tn the recirculattn~ and reattaching flow 

regions in comparison w'th the existing algebra'c models for the tr1ple- . 

velocity products. Then, opt1mum values for empirical cocffic1entsare obtained for the prediction of the backward-facing step flows. Furthermore. a 

func~iona' er.prcssion is der1ved for the cotfficient of the ~mentum diffusion 

by employing the algebraic-stress model. Finally. the second-mornent closure 

is applied to a heat transfer problem. The co~utations for the flow in a 

corrugated-wall channel show that the second-moment closure improves the prediction of the heat transfer rates by 30% over the k - c model. In this 

way. the application of the Reynolds stress model to heat transfer problems 1s 

demonstrated to be prom1sin~ 

- 1 -

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I 

.. 
• 

, 

PREFACE 

This r~port outlines the progres$ of the reseArch conducted from July 

1905 to January 1986. During th\s period. the th1rd-moment closure model was 

studied as well as refining the second~~nt closure. The development of 

transport equa·'ons for the th1rd~~nts of turbulence velocity fluctuation 

has been pursued since it was observed. in the last research project period. 

that none of the existing algebraic th1rd-m~nt models can predict the levels 

of the triple velocity products correctly for the backward-facing step 'lows. 

The predi~tions by the algebraic rr~dels were particul~rly poor in the recirculating flow regions. 

In this report it is shown that the currently developed transport equations of the third-moments improve the prediction of the triple-velocity products of turbulence fluctuations in comparison with the algebraic third-moment 

~odels. 

In carrying out this project the authors are grateful for the assistance 

of Messrs. A. Bagherlee. R. Smith, T. Niess, and V. Kodali who perh'rmed a 

large amount of programming and computations. 

- ii -

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• 

TABLE OF CO~TEUTS 

ABSTRACT. 

PREFACE • • 

. . . . . . . . . . . . . . . . . . . . . . . 

. . . . . . 1 

11 

lABLE OF CONTEHTS • 

NOMENCLATURE •• 

. . . . . . . . . . . . . . . . . . . . . . ... . . . . 111 

1 

3 

6 

1. INTRODUCTION . . . . . . . . . 2. THE SECOND-MOMENT CLOSURE HODEL. . . . . . . . . . . 2.1 Preliminary Remar~s • • • • • • • • • • • • • • 6 

2.2 Improvement of the k. c and the Reynolds-Stress Equations 6 

2.3 Computed Results and Discussion • • • • 10 

2.4 Sum.arizing Remarks • • • • 14 

3. THE THIRD-HOHENT CLOSURE HODEL • • • 15 

15 

lS 

18 

21 

4. 

5. 

3.1 

3.2 

3.3 

3.4 

Preliminary Re~~rks ••••••••••••.••••••• 

Existing Models and the Transport "od~l for the Third-Homents • Co~puted Results and Discussion 

Summarizing Remarks •• 

THE ALGEBRAIC-STRESS HODEL 

· · · · · · · · · 22 

22 

23 

27 

21 

29 

31 

32 

4.1 Prel~minary Remarks 

4.2 Fomulat1on . . . . · · · · · · · · · 

· · · · · · · . . 4.3 Computed Results · · · · · · · · and Discussion 

4.3.1 Corr,puting Details 

4.3.2 External Flows. · · · · · · . . · · · · · · · · 4.3.3 Internal Flows. 

· · · · · · 4.4 Summarizing Remarks 

· · · · · · · · · · · · 

APPLICATION TO HEAT TRANSFER PROBLEMS . . . · · · · · · · 33 

5.1 Preliminary Remarks •••••. • • • • • • • 33 

5.2 Computational Method • . . • • • • • • • • • 33 

5.3 Computed ·Results and Discussion .• • • •• 35 

5.3.1 Validation Test of the Reynolds Stresses. • • • • • • 35 

5.3.2 Average Nusselt IL/mber and Sl::in Friction Coefficients 35 

5.3.3 Velocity and the ~eynolds-Stress Profiles • • • •• 36 

5.3.4. Flow 1n a Channel with Fins • • • • • • 37 

5.4 Summarizing Remarks • • • • • • • • • •• 38 

6. FINAL REMARKS AND FUTURE WORK ••••• 39 

APPENDIX--Formulation of ASH (Algebraic-Stress Hodel) • 40 

REFERENCES 44 

FIGURES .•• 46 

- i 11 -

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---. .,... .,..------.~.-.. :.: ~ . .:. .. ___ - ______ .. __ ._._.:....;~ _____ . __ .. ___ . ___ ._ .. ~_. \'fI'), 

• 

a 

Cl ,C2,CO

' 

Ck,Cp.Cp' 

Cwl ,Cw2 ,Cc' 

Ccl,Cc2·C~1· 

C.2,C+3' 

C1j 

D 

Dij 

Dr 

f 

h 

H 

NU 

p 

p 

r 

NOMEflClATU"E 

channel step height 

anisotropy 

channel width 

constants used in turbulence model 

coefficients for near-wall Reynolds stresses 

diameter of the disk 

diffusion rate of the Reynolds stresses 

diameter of the tube 

average skin friction factor 

function for wall correction 

gene-ation rate of turbulence kinetic energy, k 

generation rate of Reynolds. stresses 

local heat transfer coefficient 

step height 

secondary generation rate of Reynolds stresses 

turbulence kinetic energy (~U~) 

length of fin 

Nusselt number based on 2b 

averaged Nusselt number 

pressure fluctuation 

mean pressure 

wall heat flux 

radial coordinate 

• I' 

, ..... .. 
Re 

T 

Tb 

Tv 

u 

U 

U1N . 

Urn 

v 

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v 

x, y, z 

+ y 

r 

c 

e 

v 

p 

¢>ij 

~ubscripts 

i,j,l<.t.m 

.. __ o· _0 ___ o __ ._o.o_o ___ ~ __ _ . \!J ----------------------

Reynolds nlll"..ber based on 2b 

temperature 

bulk temperature 

wall temperature 

fluctuating velocity in x d1rect1on 

mean velocity 

inlet stream velocity 

averaged mean velocity 

fluctuating velocity in y or r direction 

mean velocity in y or r direction 

fluctuating velocity in z or e direction 

Cartesian coordinates 

dimensionless distance from wall to the first numerical 

node-point, (a k

l/2Y/v) 

constants used in EQ. (19) 

thermal diffusivity 

Kronecker delta 

energy dissipation rate 

coordinate in azimuth direction 

dynamic viscosity 

kinematic viscosity 

density 

Prandtl number 

pressure-strain correlation 

tensor notations 

- 2 -

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.. 

... 

.. 
1. INTRODUCTION 

In computations of turbulent separated shear flows. it has become more 

common to use the second-order closure of turbulence. While modern computational techniques enable us to successfully predict most of the simple turbulent shear flows. it is still difficult to accurately predict the separated 

and reattaching shear flow particularly when It is accompanied by a flow 

recirculation. This is partly because the computation should be performed 

with an elliptic approach which requires large amounts of computer time and 

storage. and partly because a turbulence model that can predict such complex 

turbulent flo~s (including flow reattachment. flow recirculation and a 

recovering boundary layer) has not yet been completely developed for universal 

usage. 

A 1~rgenumber of expprimental studies on this subject have been reported 

1n the last two decades. The Reynolds stresses in the reattachment zone were 

obtained by Etheridge and Kempl, Kim et a1. 2, Smyth3, Eaton and 

Johnston 4

• etc. Cor.monly it has bl?en shown that the -t:urbulence ener'gy level 

reaches a pr.ak value approximately one step height upstream of the reattachment point. then decays rapidly in the strearnwise direction toward the wall. 

although it decays relatively slowly along the wall in the wall vicinity 

region. This feature is in contrast with free shear flows which are not ~onstrained by the solid wall. It is also noted that the recirculating zone 

affects the reattaching shear layrr resulting in higher turbulence energy 

levels near the reattachment zone. 

Although the measurements of nigh~r-order turbulence moments are scarce 

1n literature. a few third-moment data are aVailable. The triple-velocity 

products were measured by Chandrsuda and Bradshaw5 by using a hot-wire 

- 3 -

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~==~~=--===-~_~.~-~'~~~~.-_-_~~~~. ____________ ~-~-~-'~"~-'~-'~'~"~'~'-----------------------'1:1~ 

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anemometer for a channel with a largr.r channel width of 3.5 step heights. 

Driver and Seegmiller6 used a channel with a width of 9 step heights wh~rein 

the boundary layer thickness just above the step was about 1.5 step heights 

having an inv1scid core region outside the separating layer. They ~asured 

the secondand third-moments of turbulence velocity fluctuations with a twocOmponent lOV by varying the pressure gradient in the inviscid core region. 

Through this NASA project, the authors have investigated the featur!s of . 

the second- and third-moments 1n the reattaching and recirculating flow regions. First, the existing model of the Reynolds-stress transport equations 

was modified and adjusted for the ellip~1c flows. 7 The computations' were 

m-Jde for several different cases of flow and geometry, and the 'results were 

compared witn the k - c model and the algebraic-stress model of Rodi.8 

Although the model used by Amana and Goe1 7 was the hybrid mcdel of the 

Bouss1nesq Viscosity and the Reynolds-stress equations, the prediction of the 

normal and shear stresses was improved in tha reattaching and the recovering 

region behind a s~ep. 

Meanwhlle, h was observed that the-triple-velocity products decay 

rapidly in the reattaching region which creates steep changes in the diffusion 

9-10 rates of the Reynolds stresses. In the last two reports, most of the 

existing third-moment models were tested to determine whether they could be 

employed for the prediction of such reattaching shear layers or not. As a 

result, it was recommended that a transport model for the third~~nt should 

be evolved since none of the algebraic models for the third-moments gives 

sufficiently high levels in a complex turbulent flow although the mOdels are 

satisfactory for parabolic thin shear flows. 

- 4 -

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...... 0 uw .. ,.cc '.9ISl ... ~ ... ....: ....... :~--...-. _______ . 

In this report. the ~r~ perforr~d is summarized in the following steps: 

1. Incorporation of the non-isotropic treatment for the diffusion rates 

of the k and c eQuat1on$. 

2. Determination of the empirical coefficients of the above mentioned 

diffusion rates. 

3. Development of the transport equations for the third-moments of 

turbulence. 

4. Testing the third-moment model for the reattaching shear flows 1n 

order to determine the coefficients which appear 1n the third-moment 

transport equations. 

5. Formulation of the correlation function used in the diffusion rates 

of the momentum equations. 

6. Application of the second-moment closure to a heat transfer problem. 

In the above procedure, step 5 was achieved by using the algebraic-stress 

model because of 1ts simple formulation of the Reynolds stresses. In order to 

eliminate the complexity due to the wall flows, this test was limited to only 

wakes which were crcate~ behind steps. In step 6, a periodically corrugated 

wall channel was considered. This is the basic study of a compact heat exchanger. 

- 5 -

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2. THE SECOND-MOMENT CLOSURE ~OOEL 

2.1 Prel1min~'i Remarks 

80th the turbulence energy, k, and the energy dissipation rate, c, are 

always used to define the time scale, k/c, and the length scale, k

3/2/c. 

Therefore, if the k- and c-eQuations are not functioning with the stress 

equations in the same order of treatment, the modeling is not considered to be 

complete, and thus the overall results will not be improved. To deal with the 

higher order forms of these equations, the original (unclosed) forms need to 

be employed to define the turbulence energy aud the energy dissipation rate 

and then some empirical correlations have to be reevaluated for the flows 

considered in the present study. 

The generation and dissipation rates of the k-eQuation do not require any 

empirical coefficient. On the other hand, the generation and dissipation 

rates of the c-eQuation have empirical coefficients which have been used by 

a large nu~~er of researchers for more than a decade Therefore, the key 

terms which may improve the model are the diffusion rates of the k- and 

c-eQuations that need revising for the second-moment closure. 

In the second-moment closure, the conventional Boussinesq viscosity (diffusivity) model should be replaced by the second-moment expression. In this 

process, the empirical coefficient for the diffusion rates of the k- and 

c-eQuations must be determined. However, these coefficients have not yet 

been evaluated for universal usas~. The determination of the values for these 

diffusion coefficients is attempt!!d in this section for the better prfldiction 

of the reattaching and recircula~ing flows. 

- ti -

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2.2 Improvement of k. c and the Reyno'ds-Stress Equations 

The transport equation which describes the R2ynolds-~tress variation in 

an elliptic flow field is derived as 

(1) (11 ) 

a auiu, Q 

+ a~l [v aX

1 - p(&jtU1 + &1tUj) - u1uj u1] 

(111) 

from which the turbulence kinetic energy is also derived. 

the turbulence kinetic energy is 

aU i 5 - 2v ax ax 

1 1 

( 1v) 

The definit1~n of 

There are two ways to obtain k: on~ is to solve EQ. (1) for u

2, v2, and 

w2 and substitute th! results into Eq. (2), and the other way is to formul~te the transport equation for k ~tself.When the problem is in a twodimensional plane coordinate system, the nu~~er of equations yhich need to be 

(1) 

(2) 

solved is the same either way. Here the transport equation for k is converted 

into the second-moment closure rather than solving all three normal components 

of the Reynolds stress and summing them to obtain k. This is hecause the 

k-eQuation is widely used as an indicator of turbulence levels. Thus, the 

~,-eQuation cannot be omitted when on~ is solving the tvo-eQuation model or the 

algebraic-stress model. 

Upon contracting EQ. (1) by setting j equal to i and dividing by 2, it 

yields the k-equation as follows: 

- 7 -

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aU1 uU1 -v-- axe. ax!. (3) 

(1) (11) ( 111) 

Terms (1) and (iii) on the right-hand side of Eq. (3) represent the generation 

and the dissipation rates, respectively. The dissipation can be determined by 

solving the c-equation. Term (1i), the diffusion rate, may be replaced by 

the following fonn, 

where it is assumed that the second term in the parenthesis is negligibly 

small in comparison with th~ third one in (ii) in a large-scale turbulence. 

This assumption is supported by Irwin's measurements of self-preserving wall 

jets.'l The first term in (ii) is kept in the computation in order to 

account for the viscous effect near the wall. 

To obtain the Reynolds stresses, Eq. (1) was closed as follows: 

where the pressure-strain correlation, ~ij' is divided into three parts: 

the Rotta tenn

12, the rapid term13 and the viscous term. 14 These are 

combined and given as follows: 

- a -

(4) 

( 5) 

(6) 

, 
where 

G 

. -

. -

and where fw is a function that controls the strength of the wall 

correction •. Here it is taken to be proportional to the length scale, 

k,312/c • 

The diffusion rate, Dij • 1s conventional1y apprllximated as 

for th~ computation of uiuj which is a rather rough approximation because 

the diffusion rate contains the triple-velo~ity products. A more extensive 

model for the triple-velocity pr~ducts uiujuk is given in the next 

chapter. 

The energy dissipation rate!q~ation is given as 

- 9 -

(1) 

(8) 

.. 

\ 

... .. 

.. 

~. 
'i3'm~,;'Z:-~Znt:!tf$.\"~IiiWIi(": 'Q , .b ! _am~~~~~~~~~~~~~'~' ___ 4~_~' __ ~_~~ __ '~.~~; _____ .~.i~fJ~4-----4~A~.~~'~-.... 4~.I~;4~~~~=;J'~:~~ 

---- "- . -.. .. ...... ' ......... -.. --.-.-.----.. -.---.--~---.-------------... . 

(1) (11) 

where ecl n 1.45 and ec2 • 1.92 which have been used to date by a number of researchers as was discussed in section 2.1. The generation (1) and 

the destl"Jction (ii) terms are evaluated using a direct approach. The diffusion rate is ~~dified in a similar fonm as EQ. (4): 

(9) 

(10) 

The coefficients Ck and Cc were recommended to be 0.31 and 0.15, respectively, by Pope and Whitelaw. 15 Since these values are appropriate only for 

Simple free shear flows, more extensive tests for these co~fficient~ are made 

for the computations of reattaching shear flows. The optimum values for these 

coefficients are recommended in the following subsection. 

2.3 Computed Results and Discussion 

The flow region conside~ed is given in Fig. 1. At the inlet of the flow 

field the prescribed values are given to all variables; they are ba~1cal1y 

taken from the experimental data with which the results are compare.~. Alon!} 

the top portion and the outflow sec~ion of the computational domain. the 

continuative boundary conditi~n is applied. 

- 10 -

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- .e =- --==::"' . . IS.'" 4 ,- .. ~.,..," .. 4 __ <~ •• 

At tho wall boundarhli. th~ ·wll lilw· 1s used to specHy r.iean velocities. turbulence kinet1c energ~, and the Reynolds stresses. On detenl1n1ng 

proportional constants betwe~n kana u,uj in the wall-vicintty region, w& 

- obtained tho followtna relations for u,uJ

: 

- ~dP UtUj • C'J k - (1-'1j) p ax 

where x and yare streamwist and transverse coordinates, respectively. The 

coefficients C'j used here Dr! taken fr~·Ref. 9. 

e'l • 1.21, C22 • 0.24. C'2 • 0.24. 

(11 ) 

(12) 

The dissipation rates along the wall are obtained under the ·local equilibrium 

condition·; that is. the length scale near the wall is proportional to the 

distance from the wall. 

The solution domain consists of a 60HxSH area with 52xS2 grid points. 

The grid expand$ at rates of 3S and ZS in x and y dirEctions, respectively. 

Thh system was selected after a number of grid tests ',0 produce the optimum 

grtd-independent condition for the computation of mean velocities and the 

second-moments of the turbulent velocity fluctuations. 

Figures 2 through 6 shaw the compute~ results of mean velocity ~nd the 

Reynolds stresses at several strcamwi~e locations behind the step. By using 

+ the present model for neH • 32.000. the dimensionless distance. y • froll! 

the solid wall to the first numeric; 1 node-point varies from 16 to 37. Here 

ReH denotes the Reynolds number b~sed on the step height H. The typical CPU 

time consumed for one computation tOI)K approximately 30 minutes on a UNIVAC 

1100 with about 300 iterations. It is also found that the CPU til:Je depends 

slightly on the va1ues of the coefficients Ck and C

c in EQs. (4) and 

- 11 -

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... 

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: 

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~ ... : 

.. , 

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(10). ffZlr=ly. the ccx::;Jutat10ns w1th Ck D 0.31 and Cc 

• 0.15 took about 

16~ more CPU tim2 and 50 additional iterations than in the cases of Ck • 0.1 

and C • 0.3. 

e 

As shown in Figs. 2-6. the profiles of the Reynolds stresses computed 

with the coefficients Ck • 0.31 and Cc • 0.15. which are presently 

recommended for free shear layers. are about 20% to 30% lower than the experimental data of Driver and Seegmiller6 for the nonnal stresses. and 14% lower 

for the shear stresses. As a result. the levels of the mean velocities in the 

shear layer region Are 20% higher than the ex~erimental data due to a reduction in turbulent diffusion rates. 

In order to improve the prediction of these turbulence stresses in the 

reattaching shear layer. a number of par~metric tests have been perfo~ed for 

different values of the diffusion coefficients. It was discerned that the 

smaller values for Ck and larger values for C

e g~ve better results in 

comparison with measured data. Thh is because the larger values of C

k 

cau~e the diffusion of turbulence energy to increase. thus resulting in lower 

levels of the Reynolds stresses. Similarly. the smaller values of C • in c 

turn. increase the dissipation rate of the turbulence energy which results 1n 

a reduction in the turbulence energy. 

In the present flow field an extra effect is created by the recirculating 

flow which occurs below the reattaching shear layer. The reCirculating flow 

at~.he corner of the step and the bottom wall enhances the level of turbulence 

ene:9Y in the separated flow resulting in much higher energy at the reattaching region. and this energy is transported downstream. 

After performing parametric te:ots it was found that the combination of 

Ck • 0.1 and Cc ~ 0.3 gave the best results for both the mean velocity 

and the Reynolds stresses. It is also observed in Figs. 2-6 that even the 

- 12 -

.. 

' •. .. 

" 
Bouss1nesq ~el* gives better results than the Reynolds-stress ~del ~th 

Ck • 0.31 and Cc • 0.15. The 1mprove",~nt ~de by the Reynolds-stress 

model with the new coefficients Ck • 0.1 and Cc m 0.3 is primarily attributed to the inclusion of the second~ents in the diffusion tenus which 

account for the nan-isotropic eff~cts. Table 1 compares three models 

discussed above. 

TABLE 1. Diffusion terms for k and c equations 

CIi; Cc C .. Agreement With 

Experiments 

Reynolds-stress model 

Pope and Whitelaw 0.31 0.15 Not Used Bad 

Present Study 0.1 0.3 Not Used Good 

Boussinesq Model Not Used Not Used 0.09 Fair 

The turbulence energy balance is examined by using both the presently 

determined Reynolds-stress model and the Bouss1nesq model (cases 2 i,.;d 3 in 

Tnble 1, respectively) and is shown in Figs. 7-11. Thp. variation of the 

convection term (Fig. 7) shows that both models give equally reasonable levels 

at several different locations. For the distributions of the diffusion term, 

however, the computed results do not agree well with the experimental data in 

the recirculating region, as shown at x/H • 3.9 in Fig. O. This is the reason 

*The Boussinesq model has the diffusion term in the following form: 

where ~ represents k or c, and a stands for Prandtl number for k or c. 

- 13 -

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! 

t ' 

j 

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I 

, 

! '':'' . 

~ 

.... 

I 

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why the triple-velocity products th4t represent the dIffusion of the Reynolds 

stresses are examinad by using th~ transport equations for u, u

j u

k in 

this project •. Botn the c~~uted distributions of the production (Fig. 9) and 

the dissipation (Fig. 10) are relatively 1n ~ccordance with the experimental 

data although both models always slightly overpred1ct these quantities. 

The overall balance of these terms is compared in Fig. 11. Here the 

computations are made by u$in9 the Reynolds-stress model with the presently 

recommended diffusion coefficients. It is observed that both the production 

and dissipation rates predominate in the shear layer near the step, but these 

levels decay quickly toward downstream. On the contrary, the diffusion nnd 

the convection rates stay almost at constant levels in the strcarnwise direction while they vary rapidly in the transverse direction. 

2.4 ~Jrrmariz1ng Remarks 

In this section, it has been shown that both Ck and Cr; are considerably influential 1n th~ results of turbulence qUlntities and also in the 

numerical convergence history of 1he iteration procedure. This indicates that 

the diffusion process of the turbulence energy in the reattaching shear layer 

is significant which is in contrast with the boundary layer flows. Thus, it 

is concluded that the determination of the diffusion coefficients should be 

done carefully. Probably. a more elaborate functional expression will suffice 

the requirement rather than employing the constants. 

Secondly. it beca~ clear that the third-rnoment closure should be introduced for better understanding of :he dHfusi!ln process in the flow recirculation region. This is carried out 1n the next chapter. 

- 14 -

... 

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• -- --:-::-:-.:" -.::::=-=-:.::-..::;...;...;.:.-:.~@ . 

3. THE THIRC-~OMENr CLOSURE MODEL 

3.1 Pre'irninsry R(!l'Mrks 

In closing the Reynolds-stress equation.(l). the terms of the pressurestrain (ii) and thQ diffusion (iii) have been approximated by Eqs. (6) and 

(8). respectively. The question arises whether these approximations are 

universally accepted or not. With respect to the pressure-strain correlation. 

several models have been examined in the latest report,lO and the tests for 

the best model revealed that three models by Naot et a1 •• 13 Naot et a1 •• 16 

and Launder et al.14 produced essentially the same results. In this 

chapter. attention is primarily fccussed on the diffusion term (term (iii) in 

Eq. (1». 

3.2 Existing Hodels and the Transport Hodel for the Third-Moments 

Term (iii) in Eq. (1) represents the diffusion of the Reynolds stresses 

due to molecular viscosity. pressure fluctuations, and turbulence velocity 

fluctuations. It has been shown that the diffusion due to molecular viscosity 

and pressure f1uctuations 1S negligibly small at large turbulence Reynolds 

numbers. * Thus. the triple-velocity products are the most dominant diffusive agency for the Reynolds stresses. 

It is known that the higher moments of turbulence fluctuating velocity 

vary rapidly in the reattaching shear layer towards a solid wall. S This 

affects the accuracy of the computation in the diffusive term of the Reynolds 

st~esses because it contains the third-moment velocity fluctuations. It was 

discussed in the last report'O that the triple-velocity products need to be 

*These terms appear to be small 1n most circumstances since turbulence energy budgets balance. to within experimental error, even when these are neglected. 

- 15 -

, 

... 

I 

... 

'I 

1 

- ~ 

.. 

" I 

i 
- ... nag :;;;'df'Jh-iJJi~ t..!" :-:~."!. 4~ ~ .... ----.... --. 

., .. -....... ,- -'" "'-'~ --CfJ' . ~ .. _., •••••••• __ ... ~.. ... ...... _ ..... _ ... ~ ... _. ____ •• _ ....... _ ........ _____ .. _._ ... _ ............ _ •• _~" ..... -.J __ ._. -......_~_-.-.-_-....._. • 

evaluated in a transport equation model rather than tn lin algebraic fom because the convection ~nd generation 'of the triple-velocity products have to be 

taken suitably into account according to the change in the mean strain rates. 

For this reason a full transport equatlon model of the triple-velocity products is devp.loped and used for the prediction in the reattaching shear flows. 

The complete equation is given as 

. -

(13) 

(111 ) (i v) 

where terms (i) and (11) represent the generation due to the mean ~train rate 

and the generation due to turbulence stresses, respecti~ely. Terms (iii) and 

(1v) in Eq. (13) both represent the diffusion rate of the triple-velocity 

products. Here term (iii) is correlated by assuming Gaussian form as follows: 

and term (1v) is given as: 

( 14) 

- '16 -

... 

f . ....... .. 

I 

\ 

: \ 

,,=,,' 1 
Then the final fonrn of Eq. (13) bec~~s 

In the discussion section the results of the paramet.-1c tests lire shoWil for 

optimum values for the coefficient Cp_ The term with the laminar v1scos1ty 

v 1s added to account for the v1scous effect. 

(15) 

The above ment10ned model is also compared w1th four ex1sting algebraic 

models_ These are g1ven as follows: 

(1) The model of Daly and Harlow17 

(16) 

(11) The model of Hanja11c and launder18 

(11) . 

(111) The model of Sh1r19 

(18) 

- 11 -

... 

• i.. ' - .. .., 

I 

I 

I 

~'I 
(1v) Tho model of Corr.ack et &~.20 

(19) 

where 

- 2 B'j m UiUj - 3 61jk 

(20) 

aa;j 

8 'j,k - aXk 

The coefficients are given in Table 2. 

TABLE 2. Coefficients for 01 

-B.l4xlO-3 -1.72xlO-2 -4.80xl0-2 -1.02xl0-1 

3.3 Computed Results and Discussion 

In order to establish a reliable model for the prediction of the thirdfN)ments in the reattaching shear layer, the values of the mean velocities and 

the second-moments are predetermined to solve the transport equations for 

U1Uj uk• That 1s the values of the mean velocities. uiuj ' k, and c 

computed with the method descrlbed in the preceding chapter are stored, and 

- 18 -

! <:tJ~ , 

• 

. 1 

~ 

I . 

.' 

... 

1 

f 

I 

\ 

J 

I 

1 • i i T i' \ 

I 

~ t 

• ".:. : -.I 

, 

. --.. _". 

I 

I 
or 

_--:..~~ _____ ,~====:::::::==_:,,:.= ___ ._._. ________ '''~1 

these v~lues are retrieved when the tr~nsport equations for uiujuk are 

co~uted. In this W3y, the existing r.~dels for the triple-velocity products 

are also evaluated and compared with the prescnt third-moment closure model. 

Since Eq. (15) represents the transport equations of uuu, UUV, UVV, and 

vvv in a two-dimen$ional coordinate system, and all four components ~re associated among themselves in their own equations, these four equations are 

solved iteratively. The iteration was terminated when the relative residual 

source of each equation dropped below 3X10-12• 

Figures 12-14 compare the triple-velocity product distributions for different values of the coefficient c~: 5.0, 6.0, and 10.0. The results are 

also compared with the experimental data of Chandrsuda and Bradshaw. 5 As is 

observed in this figure, the results are relatively sensitive to the value of 

Cpo As Cp becomes larger the leveis of the triple-velocity products 

decrease since this term CEQ. (14» acts as a sink of uiujuk• After a 

num~er of parametric tests it is found that the value of Cp • 5.8 gives the 

best agreement with both the experimental data of Driver and Seegmiller23 

and of Chandrsuda and Bradshaw. S This optimum value corresponds ~o an 

I 

inverse coefficient Cp • l/Cp • 0.17 which is slightly larger than the 

I 

value of Cp • 0.11 recommended for the Hanjalic and launder model (see 

Eq. (17». 

Figures 15 - 20 compare the present computations of the triple-velocity 

product profile with the experimental data of Chandrsuda and Bradshaw and of 

Driver and Seegmiller, along with the four algebraic models at se\eral different locations downstream of the step. It is observed that two peaks appear 

across the shear layer: one along the core of the separated shear laypr, the 

other near the bottom wall. Although all the computations exhibit a similar 

trend to the experimental data, the predicted levels of uiujuk are 

- 19 -

, 

, 

I 

" 

;, 

) 

( \ 

I 

) 

r 

.. , .... ' . ~ 

... 

i . I 

, 

~- : 
====~==========A~_~==~~ __________ -~"-~-~··--==~="=·~·~~·~.,,~,.~ .. ,,~ .. ______________ ~~~ 

fairly different. While the ood~l of Sh1r gives the 1GW!st levels for all 

three components, the model of Conmack et al. prov1des high levels for uvv and 

vvv but not for UUY. Unlike the model or Cormack et al •• the Daly-Harlow 

model gives high levels only for uuv and relatively low levels for both uvv 

and VVY. Among the algebraic models the results with the lfanja1ic-Launder 

model seem to be consistent showing nearly the same levels for all three compOhcnts. Th1s 1s because the Hanjalic-Launder model 1s the only one that has 

a symmetric property 1n all three directions among the algebraic correlations. 

Agreement between the present computations with the transport equat10ns 

and the experimental data is generally much better than that using the algebra1c models for two reasons. First, the coeff1cient, Cp' can be suitably 

adjusted 1n the transport equations whereas the algebraic models have been 

tested only for relatively simple shear flows and not for the reattaching 

shear layer. Second, the transport equations of uiujuk have II sym~etric 

property in all three directions; thus, the prediction with the transport 

model is better than the pr~diction with algebraic models for inhomogeneous 

flows as well as for homogenco:.:<; ones. However, tile most important point we 

should note from these results is that the location of the peak along the 

separated shear layer is predicted more accurately by using the transport 

equations than by using algebraic models. This is primarily because the 

transport model takes both the convection and the generation due to mean 

strains into account; thus, the trail of energy propagation for the 

triple~velocity products accora '; with the experimental data. 

'The results in Figs. lS-2C also suggest that the transport model ne~ds to 

be improved to give better profiles near the wall. It is shown in these 

figures that the computed levels are much higher than the ~xperimental data 

- 20 -

t 

J 

I 

, 

,I 

.. 

1 

! 

I 

I 

I 

t , . , l~. , , 

I 

I 

j 
-____ -__ ._ ... _. . '."1 ._---- ........ - .........-...-. --,-.... -----.-.---..... - .... --.. -----.-.-----..-.• - ..... - ........ - .. Ii 

ne~r the wall. In the wall adjacent region the ~del must incorporate a lowReynolds n.'l'"ber effect although the coroputat1olls in the free shear flow region 

agree well with the data. The effect of the wall boundary condition on the 

profiles of u, u

j u

k near the wall was tested by adopting several other 

models for the numerical nodes next to the wall. However. the influence of 

the wall boundary condition was negligibly small. Therefore, to 1mprove the 

prediction of the triple-velocity products in the near-wal1 region, it is 

advisable to develop a lew-Reynolds nunmer model in order to take the viscous 

effect into account. 

3.4 Summarizing Remarks 

The development of the third-moment closure has been pursued through this 

project. It is observed that the behavior of the triple-velocity products is 

completely different from that in the boundary layer flows showing rapid decay 

toward the so11d wall in the separated shear layer. 

It is noted that the transport equation mod~l dramatically improves the 

prediction of the triple-velocity products throughout the whole flow region 

when compared with the algebraic third-moment models. However, more ~xtens1ve 

tests are required for improving the transport equation model since the closing of the diffusion and pressure-stress terms of the transport equation has 

not yet been justified. Moreover, it is reco~ended that a low-Reynolds 

number near-wall model be developed for the transport equation model. 

f •. 

r j' h 

I ;\' , j : .' 

" 

., 

, 

o 

\ 

I 

- 21 -

~\ 
____ -_-._-_.-~~_ ...... __---:..'"-____ ::...=-:=:;..;.::: ______ ~_7'; .;-;-~= -:: ;:.-:. .. -~. __ .. __ •. _ .• _--_ ....... _--- .. .....-.--. · ~j - ! 

4. THE ALGEBRAIC-STRESS MODEt 

4.1 Preliminary Re~rks 

While it is important to improve the prediction of the Reynolds stresses 

in a wake region, it is equally important to develop a better expression for 

empirical coefficients for the diffusion rate of the momentum equation. This 

is because empirically dete~1ned coefficients would increase uncertainties 

and, thus the prediction becomes less accurate for more geometrically complex 

turbulent flows. 

The preceding chapters have dealt with the Reynolds-stress model. Ifowever, the algebraic-stress and the k - c models, which are the simpler 

models of turbulence, are considered in this chapter in order to investigate 

the for.;\ulation of the coefficient C by separating from many unknowns 

'" which exist in the Reynolds-stress model. If the Reynolds-stress model (RSH) 

were also used, it would be rather difficult to differentiate between the 

errors due to the RS~ and the C formulation. For futher simplification 

'" of the problem, the solid wall is completely eliminated in the t!lke region, 

thus considering only the flow behind a disk. 

The development of C is based on the algebraic-stress model (ASH) 

" which was originated by Rodi. S Although the Reynolds-stress transport model 

as described in the preceeding chapters is one of the most advanced model, the 

algebraic-stress model (ASH) improves the prediction over the k - c model. 

Probably the most advantageous point of the algebrai c-stress mod£' 1 (ASH) is 

its simplicity; the solution procedure of ~ transport equation 13 not required. The Reynolds stresses are determined by merely solving algebraic 

equations without using the line relaxation or tr~diagonal matrix solvers, and 

- 22 -

'I 

; ': 

: ~ 

! 

i 

( 

.. 

r 

1-

r 

I 

~ 

I 

1 

\ 

, 

I 

~ 

i 

I 

1 

\ 

I 1 1 

; . ': , . j , 

f't t 

: "oJ 

" I.. I ' , 

, 

I • 

{;:\ 
thus it is much more economical in conpai1son with the Reynolds-stress 

transport model. 

The above mentioned algebraic-stress model (ASH) is used to formulate the 

diffusion coefficient of the momentum equations. The model is co~pared with 

the conventional empirical constant for the computations of wakes. Since the 

formulation has to be general. lxisyrrmetric flows have been chosen so that 

three nonmal components of the stresses (u2• v

2

• and w

2

) and the shear 

stress (uv) are solved in the region behind a disk. 

4.2 Formulation 

The ASH for the axisymmetric flow can be obtained by using the propo:al 

of Rodi. S Rodi proposed that the convection-diffusion of the Ie-equation is 

proportional to that of the Reynolds-stress transport eluation such as 

ui u~ ale aU i uj --

--~ (p U ---- - D(Ie» • P Urn ax - D(U1·Uj ) Ie m aXm m 

where D(Ie) is given as . 

and where D(UiUj ) represents the diffusion rate of uiuj . The final 

form obtained is given as follo\/s: 

(21) 

(22) 

(23) 

The ASH is comprised of four algebraic equ~tions for the four Reynolds 

stresses for a two-dimensional axisymmetric flow. These arc given as follows: 

- 23 -

~' 

I 

I 

I 

r 

, 

. 

t' 

" 

, 

I 

~ 

- ,~. I 
....-~ .... "_ .. "."-- ~,,".' ..................... _-'_._--.-.. _____ ,_~ ... -.:~_.I .... ~ ___ ..... . 'ct) 

+ ~] 

The generation rate of turbulence kinetic energy· (Eq. (1» is defined as 

G • -[v (<lU + <IV) + 2 au + 2 aU + 2~] u ar ax u ax Y ar W r 

The components of the generation (EQ. (1» for Reynolds ~tresses (u2

• y2. 

- -

w2 and uv) are given as 

2 aU - <lU G11 • -2(u ax + uv ~r) 

2 3V - (IV GZ2 • -2(v ar + uv ax) 

2 V G .. -2 w - 33 r 

- 24 -

(24) 

(25) 

(2f1) 

(21) 

(28) 

· .. 

I 

• · , 

(29) 

(30) 

(31) 

~ . 
2' au 2" av - y G • -2(v -- + u -- - Uy -) 12 ar ax r 

The pressure-strain correlation as given by Eq. (6) can be written as: 

c 2' 2 ( 2 [ I: 2' 2 k) +22 • -C, k (v - 3 k) - c2 G22 - 3 G) + cwl k (v - 3 

The secondary generation rates (EQ. (7» are given as: 

2' au -- av 

H'1 • -2(u ax + uv ax> 

2 av -- au H • -2(v -- + uv --) 22 3r 3r 

2 V 

"33 - -2 w ;: 

2 au 2 av -- V U • -( u -- + v - - uv -) 12 ay ax r 

- 25 -

(32) 

.. 

(33) 

(34) 

(35) 

(3Ei) 

; 

, . 

" "'- .. 

" 

(37) 

(30) 

(39) 

(40) 

'- . 
Tho model described above is also compar~d ~1th the k - ~ model. In the 

k - c ~del the Reynolds stress?s are to~utcd by us\n9 the Douss1nesq 

viscosity concept. That is. all the Rtynolds stresses aru expressed as 

follcvs: 

P;Z • l3 pk - 2 au "t ax 

2' 2 av 

pY • 3 pk - 2 "t ar 

:2 2 V 

pw • 3 pk - 2 "t r 

-- au av 

pUY • - " (-- + --\ t ar . ax' 

Here "t is evaluated by using the followtnq equation: 

2 

"t • e" pk Ie 

In this study four different models are used out of which two are the 

k - e models and the other ~~ are ASHs. The difference in the two k - c 

models dppends upon the value given to C • \lhether it is a constant value 

" 

(41) . 

(42) 

(43) 

(44) 

(45) 

or a functional value derived from the ASn formulation. The difference in the 

two ASHs is also the same. Hence, we cnn define the foul' models as follows: 

ASH 1 --- stresses computed by ASH: 

ASi4 II -- stresses computed by ASK: 

c • 0.09 

" C .. f 

" k-c I (standard k - c, model) -- stresses computed by k - c: 

k-c II (modified k - c model) -- stresses computed by k - c: 

- ?fi -

C • 0.09 \I 

C '" f " 

. 

. : '1 .. 

... 

,I 

I 

.~ 

... 

, 
• 

_ .... ______ ._._. ___ .' ,_. __ ._ ... ____ . ____ ..... ~-,~, __ ... _. -r+,., 

(:J .~ 

, 

The function f is derived upon t4k1ng the recirculating effects du~ to 

twD-direct1onal rooan strains into account in the fomulation. The final fom 

is given as 

The details of the formulation are given in Appendix. 

The c~nstants used in the equations above ~re given in Table 3. 

TABLE 3. Constants 

Cwl 

0.1 0.125 0.015 1.5 0.4 

4.3 Comput~d Results and Discussion 

4.3.1 Computing Details 

Exploratory tests were made for different mesh sizes to 1nvestiyate an 

optimum grid independent state. Figure 22 gives mean velocity ~nd the 

Reynolds-stress profiles for differert grid sizes at x/D • 6 for the flow 

geometry in Fig. 21(a). 

(46) 

It is clear that the velocity and the Reynolds-stress profiles ~pproach 

an equilibrium state for a grid density higher than 42x42. The perc~ntage 

increase in the properties from one grid size to another is given in Table 4. 

- 27 -

,! 

t .. ' 

.. 

, 

j".- .. 

:i 

" • ~ i oJ: 'I I' 

1/ 

!j 

:t I 

¥:'"'-. 
• • 

TABLE 4. Grid test at riD Q O.S 

Mesh Sizes 32x32 to 42x42 42;:42 to 52x52 52x52 to 62x62 

U 55.0% 40.0% 5.0% 

u2 75.0% 25.0% 0.0% 

v2 66.6% 33.4% 0.0% 

oil 71.5% 28.5% O.C~ 

uv 62.5% 37.5% O.~ 

In Table 4 it is observed that there is no appreciable change in stresses 

from 52x52 to 62x62. And this is the reason why we opted for 62x62 as the 

optimum grid size for all the computations in this chapter. 

The initial values for the Reynolds stresses are given by the solutions 

of the k - c model to ensure stabl1ity in the course Clf the iteration process for the elliptic equations. The computation is ";.enninated when the 

maximum residual source of the tran5port equation fans below 0.3 percent of 

the total source of its equation. Although iteration number depends on many 

factors such as grid size. flow geometry, boundary conditions, initial 

conditions, Reynolds number, etc., the typical nurr~er ranges from 500 - 800 

for the grid size 62x62. 

The computational domain for the external flow past a disk consists of 30 

step height lengths in the x-direc~.ion and 5 step heights in the r-direct1on. 

A computation was also made by extending the length from 30 to 40 step 

heights. When the results obtained with two different geometries ~/ere compared with each other, no appreciable difference ~as observed in any properties. Thus, the geometry of 30 step height lengths is used in ~-direction for 

- 26 -

I .' \ 

\ 

, 

f 

!I 

" II 

~ 

~ ~ • " ( 

'j 

I .1 J , 

, i 

~ I: -

" 

-::-.. 

~ i ~ , ' . '1 , 

.. . "1· , ... ..i, 

'" , 

I 

I 

~' 
all the computations of the external flo~. The expansion ratio of 1.05 in 

both x- and r-d1rections is found to be the best for this flow geometry. The 

computational dc~in of the second geometry shown in Fig. 21(b) consists of a 

length of 24 step heights in x-direction and two step heights in r-d1rect1on. 

The opt1m~m expansion ratios for this case are found to be 1.05 in x and 1.00 

in r-d1rect1ons. 

4.3.2 External Flows 

Figure 23 represents the mean velocity profiles at several locations in 

the flow field beyond the body for the fk.l geometry which is shown in Fig. 

21(a). These are computed by employing k-c I. k-c II. AS" I and ASH II. 

These results are also compared with the experimental data obtained by 

canmody.21 The velocity profiles obtained with the models mentioned above 

agree with the experimental data outside the separated shear flow region but 

some Jisagreement is observed in the recirculating region which might be 

attributed to the shlrp edge of the disk used in the experimental setup caus10g a larger recircuiation region than that- predicted in the present study. 

Figures 24-21 represent the- Reynolds stresses obtained by the ASH and the 

k - c model which are compared with the experimental data. The agreement of 

the stresses obtained by ASH I with the experimental data seems to be better 

than agreement obtained by the other models. The other models have Q similar 

trend but do not agree with the experiment as closely as ASH I. The agreement 

0' all the numerical models with the experimental data is ~etter in the region 

oiltside the recirculation region. Since the Ie - c modr."IS (k-c I and k-c 

II) cannot talce the non-isotropic effect into account, the errors in the 

results obtained by the k - c models are significant, particularly in the 

recirculating region. Despite the improvement with the use of the IISM models. 

- 29 -

I 

I" 

.. 

, 

, 

I 

I , 

. 

" 

i' 

.' 
\ 

__ _ • ....,.- '.C«::";~~' ... ~ ~~--'-.:. 

" 

these models produce a second peak in the recirculating region (neDr the flow 

impingement point). which is caused by the streamwise acceleration of the flow 

near the flow impingement point. This is verified by a test in which the 

gradient of the mean velocity in x-direction (aU/ax) is omitt~d resulting 

in diminishing of this second peak. 

From the results mentioned above it is concluded that the! algebraicstress models provide better results in the whole domain of the flow geometry 

than the k - e model. rhis is evident if we observe the results around the 

flow impingement point (x/D • 3.0) as summariZed in Table 5. 

TABLE S. Percentage error for Reynolds stress at rID • 0.5 in comparison with experimental data 

uv u

2 v2 Ii 

ASH I k-e I AS" I k-c I ASH I k-c I AS!4 I k-e I 

x/O-3 30% 65% 51% 66% 30% 36% 4% 33% 

x/O-6 21% 71% 23% 61% 29% 50% 34X 53% 

x/O-9 10% 13% 10% 40% 25% 54% 18% 41% 

Figure 28 shows the turbulence kinetic energy profiles obtained by using 

the models considered in this study which are compared with the experimental 

data. These profiles also show that ASH I is better than the other models. 

Figure 29 shows the C distribution at different locations in the flow 

)I 

field and Fig. 30 shows the Gte profiles at the same locations. It is noted 

that the C profiles for ASH II and k-e II are very close to each other 

)I 

and so are the results computed with them. It is also found that there is a 

strong dependence of stress levels on the Gte distribution. If Gte is 

- 30 -

.. 

-

i -

t 

I 

tt 

, . 

~ 

! 

~ 

l 

l t I' I 

I 

\ 

I .. ... 1 . .. 

'. 

II 
- •.••..• , .. """~"":""<~.:-;'~.:';' - ,<, .-. '.- ..•.••..••• ----_..!. ':!:J ~ 

higher. then C is lower 1mproYing the Reynolds-stress levels; thus ASH I. 

I' 

which has the smallest C in most of th~ fl~_ domain, is giving th~ best 

" prediction of the Reynolds stresses aMOng the models considered. The position 

of the peak of G/e profiles shown in Fig. 30 computed by ASH I is higher 

than the peaks produced using the other models. This is b~cause ASH I creates 

only a small spreading rate. 

4.3.3 Internal Flo~s 

For the computations of internal flows only ASH I and k~e I are employed instead of using four models. Figure 31 represents the mean velocity 

prof~les in the flow direction computed by using ASH I and k-c I. These are 

compared with the experimental data of Taylor and Wh1telaw22 for DIDT = 

0.5 (Fig. 2l(b». The results obtained by using ASH I agree well with the 

experimental data and are better than those obtained using k-c I. Figures - - - 32-34 represent the three Reynolds-stress profiles (112• v

2 and ·i) 

obtained by using k-c I and ASH I which are compared with experimental 

data. 22 In these figures it is r.~so shown that ASH ~ gives better results 

for the flows in this geometry. 

Even in the internal flow geometry the second peak in the Reynolds-stress 

profiles appears in the recirculating region near the flow impingement point 

when using ASH 1. This behavior can be explained in the same manner as done 

in the case of the external flow geometry. 

The stresses shown in Figs. ~\2-34 are very small outside the mixing 1ayer 

because of negligible generation -ates. The stresses then grow slightly near 

the top wall where the mean ve1oc)ty decreases rapidly due to the viscous 

effect from the wall. It is observed that the stresses u2 and w2 are 

- 31 -

: 

.. 

• , 

... 

~. 

, 

• < 

I 

" 

.< 

~ 

\. 

/ 

, -- ;/ 

I . 

1 

I 

f 

I 

~ 

I 

l 

• 

.. 

1 

1-

, 

, 

.. 

"-

I-

: 

- , 
~1!.~'t.~~.::r~~~~:--"'::!,,:,~~- :. . ....... _. __ .~7.;-~~==-:-..:;:..:::;:.:'----'.------·---:- • '!i:.J i 

higher than y2; this feature is similar to that of th! wall boundary layer 

flows. 

Figure 35 shows the turbulence kinet1c c~ergy distribution obtained with 

both k-e I and ASH I. and they are compared with the experimental data. The 

turbulence kinetic energy shows the Mghest value near the flow impingement 

point and it decays towards downstream. Here also it is shown that the re~ 

sults computed by using ASH I agree better with the experimental data. Figure 

36 shows the G/e distribution obtained by ASH I and k-e I. The trend is 

similar to the G/e distribution for the external flo~ (see Fig. 30). 

4.4 Summarizing Remarks 

The investigation in this chapter revealed that even the algebraic stress 

model improves the prediction of the Reynolds stresses considerably when cempared with the ~ - c model. However. it is also observed that the formulation of C based on the algebraic stress model does not improve the 

'" prediction at all. This observation suggests that any model based on the 

Boussinesq viscosity concept never improves the computations of the Reynolds 

stresses. For this reason, we should discard the k - c model type of 

turbulence model and should create second- or higher-moment closure models for 

the computation of recirculating flows. 

- 32 -

I 

.. 

! . . 

i 

, 

. , 

i . 

1 

, 

i· 

i I 

J.i oJ , '. 

, 
. 

\ 

5. APPLJCATlC~ TO HEAT TRM:SFER PROBlEf.1S 

5.1 Pr~l1rninary Rel'.'.llrk~ 

The models developed in the course of th1~ project can be applied to ~~ny 

industrial problems w1thout making many modifications. One application we 

exercised recently was the computation of the flow and heat tr~nsfer rates in 

periodically corrugated wall channels whiCh appear in a corrugated-wallchannel heat exchanger. The Reynolds-stress model was primarily employed for 

the computations of the hydrodynamic turbulence variations and the heatmorrw.ntum ~nal09Y was used to evaluate th2 heat transfer rates. The results 

were compared with the computations made by using the k - c model. The main 

concern regarding the differences produced by the two different closure models 

is whether the non-isotropic effect in the corrugated wall channel is predominant or not. If it is. a significant difference will occur in the results of 

the computations. 

5.2 Computational "ethod 

The computational domain of the flow field is the region ArCOEJIHGF (in 

Fig. 37). Thh region consists of two cycles (ABCDIliGF) and the arlditional 

region (OEJI). For computations of laminar flows. only one cycle of the 

corrugated wall channel may be used with a periodic inlet/outlet condition: 

however. a two-cycle system is needed for complex t4rbulent flow computations 

because of numerical instability. In turbulent flow equations more nonlinear 

transport coefficients appear which results in a very poor histo.·~· of convergence performance in the course of the iteration process. In adeit1on. a 

two-pass procedure is employed. That is. the computation was started out with 

a constant prescribed inlet condition at AF and an outflow condition at EJ. 

With a line relaxation method, computations were iterated until relative 

- 33 -

..... 11 

i 

I 

: I 

!: " 

• I·" 

, 

.. 

: ! 

i 

, 
. ,. 

residual sources of all the trDnsport equations dec~ased below 2%. Then the 

periodic condition was activated by tran$ferring the updated values at the 

section 01 to the inlet section AF. Finally, the co~utation was tenm1nated 

when all the relative residual sources became less than 0.1%. 

At the wan boundaries. the momentum, energy and the turbulence kinetic 

energy Wlre evaluated from the ·law of the wall· v-.ile the energy dissipation 

rate was determined fr~ the ·local equ1libriurn condition.· which gives linear 

variation of the turbulence length scale from the ~olid wall. 

The average friction factor is computed as 

f .. _ --:b::--_ 6 P 

2 l pUm 

where l is the distance between corresponding planes and AP is the average 

pressure drop between these planes. 

The local Nusselt number is obtained by the following equation 

where the constant heat flux condition is employed. The bulk temperature, 

Tb, is determined as 

T _ ITIUldy 

b JIUldy 

where the integrals are to be carried over the cross-sectional area of the 

(47) 

(48) 

(49) 

channel. The absolute value of th~ velocity is taken so that the regions with 

reverse flows are also properly represented. Accordingly, the averaged 

'Iusselt number is defined as 

- 34 -

j .•• 

", 

. ! 

.. ..... 1 

/ .. 
, . 

- 2b Nu • xr I hdx (50) 

where x is the coordinate along the chnnnel wall. 

5.3 Com~uted Results and Discussion 

5.3.1 Validation Test of the Reynolds Stresses 

Due to lack of experimental data for corrugated wall channel flo~, the 

present numerical model cannot be justified for all the variables obtained •. 

In particular, turbulence quantities such as the Reynolds stre~ses need to be 

compared with some experimental data. For this reason, data for a backward 

facing ste~ flow was chosen to perform a validation test of the Reynolds 

stresses since the wall region behind a step is similar to the flow field 

along the wall AOCDE in Fig. 37. 

Figure 38 shows the computed results of the mean velocity and the 

Reynolds-stress distributions in the region behind the step. These results 

are compared with the experimental data of Driver and seegmiller. 23 Here 52 

x 52 grid points are used for the computational region of 50a x 5a. Both in 

recirculating and recovering regions, agreement between the present computations and the experimental data 1s reasonably goed (within 30%). 

5.3.2 Average Nusselt Number and Skin Friction Coefficients 

Figure 39 shows Nu/PrO. 3 as a function of Reynolds numbci. The 

experimental data of both Izumi et a1. 24 and of O'Brien and sparrow25 are 

used for comparison with the present computations. Although the geometry 

cons1dered by O'Brien and Sparrow has a bend of 120· and alb z 1.6, the levels 

of the average Nusselt number seem to be almost the same as those given by 

Izumi et al. in which a channel with a 90° bend and alb a 2 were used. 

- 35 -

\ 

, 

J 

1 

I' 

I 

\ 

! 

I 

! 

t . I 

1 

\.1 

• ! lJ 

. , 

, I 

I 
For alb • 2, the coreputations by the k - e mode1 26 and the present 

model (RSH) are also compared. It is noteworthy thdt the RSH improves the 

prediction by 30% compared with the k - ~ model. This observation is 

consistent with the case of an infinitely long channel with two bends27 in 

which it was also discerned that the prediction of the f4usselt number was 

improved by 20-30% by employing the RSH. This 1s because the turbulence 

levels 1n recirculating regions are correctly evaluated since the RSM accounts 

for non-isotropic behavior of the Reynolds stresses. 

In Fig. 39 the computations for alb • 1.5 and 1.25 are also shown. It 1s 

observed that the difference in the Nusselt number created by the change 1n 

corrugation period is minor showing only a 2-3% decrease from alb c 1.5 to 

2.0. This result is also consistent with the experiments of both Izumi et al. 

(alb -2) and O'Brien and SparrJw (alb D 1.6). 

Figure 40 shows the skin friction coefficient as a function of Reynolds 

number. Unlike the case of the Nusselt number. the skin friction rate depends 

on the corrugation period. alb. It 1s observed that the dependency of f on Re 

1s constant for both alb c 1.25 and 1.5. but for alb c 2 it decreases slightly 

as the Reynolds number increases for Re > 3000. In Fig. 40 exp~rimental data 

for a straight channel is compared with the computations. This figure shows 

that the computed curves approach the straight channel data as alb decreases 

to zero. 

5.3.3 Velocity and the Reynolds-Stress Profiles 

Figure 41 demonstrates the velocity vectors in different cl ,lnn~ls. As is 

shown. recirculating flow is created at each concave corner of t,e channel. 

The flow impingement position and the recirculating region are tiependent on 

the corrugation period, alb. 

- 3& -

, . 

I.,. .. 

" 

.~. 
Computed ~all velocities and the Reynolds stresses in the channel where 

alb. 2 and for Re • 3000 are shOim in Fig. 42. At section at the mean veloc1ty beco~~s ~4ximum near the wall AB. ~nd it decreases sharply toward the wall 

AB. This high shear str~ss is causing high levels of the stresses u2 and 

v2 in that region. If we look at the region near the opposite side of the 

channel (FG wall), the mean veloc'ty profile 1s rather smooth although the 

velocity becomes ~egat1ve n~ar the wall FG due to a recirculating flow: thus. 

the normal stresses are not as high as the shear stress uv. The levels of - - u2 and v2 interchange in the next bend section (cross-section CH) showing 

a relatively high level of v

2 and a low level of u2 because the y component of velocity is predominant over the x component mean veloci~y. At - - section H, the u2 level is higher near the wall HI, whereas the v2 level 

is high ncar the opposite side of the channel where the y component of the 

velocity is still large due to the flow deflection caused by the separation at 

the corner H. The maximum point of u2 moves to the wall DE as we ~ove 

toward downstream. 

5.3.4 Flow in a Channel with Fins 

Computations were made for flo~s 1n channels which have fins inserted at 

each bend to :ause more deflection of the flow. Local Nusselt number distributions are shown ir. Figs. 43 and 44 for alb r- 1.25 and 1.5. respectively. 

Although the general pattern of the di~~ribution is similar for both alb -

1.25 and 1~5, the distributio~ depends on the length of the fin. ~hen fins 

are inserted. the position ot the peak of the Nusselt number moves toward the 

corner C along the wall BC. This is mainly due to the expansion of the 

recirculating region downstr~Jm of the corner B with the insertion of the 

fins. A similar trend also appears near the wall D. It is commonly observed 

- 37 -

" 

, " 

; 

" 

'f 

!~ ,.1 

i 

.. 

. " ;J. 

• f. 

t, 

II 

! ~ ! .. .. 

, I 

--t,i .I 
.. (+J G:!:l"'liti'_~";:~ .... ~~-:,~ --:..: ...... :--. ._~ ...-._ .... _ ......... __ ......... __ ,<-., ... ...:.. __ .~'~ .... ~._ .. : ....... ~ ' •• -----.---••• --- r_.· -- .. .. • 

thnt the heat tr~nsfer ratel increase ftS the length of the fin 1ncrta~es up to 

Lfin/b • 0.5. However. tho c:a~e oflf1n/b • 0.75 does not dhplllY proportional increases in tht heat transfer rates. 

The dependence on the length of fins is also shown in Fig. 45 for the 

average Nusselt number. Here the average Nusselt number increases as lfin 

increases from 0 to 0.5b, but it stays at con$tant values for longer fins. 

The rate of 1ncreas~ in the Nusselt number is larger for higher Reynolds "urnbers. 

5.4 SI:!l'ITI.1 r1l1"9 Rer.-l rlts 

In this chapter, ~ saw an application of the Reynolds-stress model to a 

heat exchanger proble~. One of the most 1mpressiv~ features shown here ;s 

that the Reynolds-st~ss model improves the computation of the heat transfer 

rates by 20-30%. This is mainly because the heat tr~nsfer rates are almost 

proportional to the hydrodynamic variables, thus if the turbulence levels are 

evaluated correctly so are the heat transfer rates. 

- 38 -

'. 

j 

j 

l . 

• 
.... "'"' •• -""-:' .. ~<. -~ ..... _ ... - . -. ..-

6. FINAL REMARKS AND FUTURE WORK 

In this proJ~ct both the second- and third~mant closures have been investigated. The rr~de1s developed revealed significant improvement in the 

prediction of co~~lex turbulent shear flows that are Dccomp~nied by reattachment. recirculation, and flow deflection. However. the test~ were made only 

for each component of each model-that is. the third-moment transport fquations were solved without coupling to the second-~~ment equations. Although 

each modeling seems to be successfully carried out, there is no guarantee that 

these models perform equally well when all the transport equations nre solve~ 

simultaneously. At that stage it is anticipated that a huge amount of 

computational time and storage will be required since numbers of transport 

equations have to be solved by employing sufficiently high density of mesh 

size. Probably the problem, if any, will be numerical instability rather than 

a difficulty due to mathematical complexity. For this reason we believe it is 

extremely important to co:nplete the component study plior to consolidating all 

the models 1n one package of the co:nputer code. 

It should also be noted that there is plenty of room for improvement of 

the preselltly developed second- and third-moment models. For example. a low 

Reynolds number model coupled with a near-wall correction must be incorporated 

for the third-moment closure equations. Secondly, the development of the 

equivalent models for turbulence scalar Quantities (heat or chemical species) 

will undoubtedly advance the state~of-the-art technology in turbulence modeling for applications of many 1ndus:rial problems. 

- 39 -

I 

.. ' .. 

"" ... " 

I 

" 

-, 

'1 

'i • ,

I 

I 

I '.j 

II 

I 

:\ 

I 

~I 

... 

, 
/ 

APPENDIX 

fORMULATIon OF AS~ (Algebraic Stress "ode]) 

I 

I 

The transport equation fo~ the kinetic energy can be written as 

~ • OCk) + G - c 

The transport equation for the Reynolds stresses is given by 

where DCUiuj ) is the diffusion rate of the Reynolds stresses. Rodi8 

(A.l) 

(A.2) 

proposed that the conve~tion-diffus1on string of the k-equat1on is assumed to 

be proportional to that of the Reynolds-stress transport equation, as given in 

the equation below. 

From CA.2) and CA.3) we have 

Uiu, Ok 

k COt - D(k» • Gij - Cij + ~1j 

Now, substituting Eq. (A.') in CA.4) we have 

where ~1j is defined in EQs. (33)-(36). From the proposal of 

DalY-Harlow,17 we have. 

- 40 -

(A.3) 

(A.4) 

(A.S) 

. ; 

I 

.. 

I , 

~ / 

. , 10.. oJ 

, 

I 

.' 
". '··1 

! 

Using the ~bove result in (A.5) ~~ have 

(A.6) 

or 

(A.7) 

After rearranging, we have 

Regrouping the terms we have 

- 41 -

I i . 

, 

; . I 

., 

• A 

.' ~ 

1 

! 

I 

\ 

t .... 

, 

J 

1 

I 

, 

I 

• t : . ... ~ "'. I 

! 
Finally, the equation for Reynolds stresses is given as follows: 

By neglecting '"ail effects and the secondary mean strains. u2• v2 and uv 

are given as follows 

(l-C ) 

u2 • £ !:. 2 uy rav _ 2 au] + £ Ie 

3 c (2 + c ) ax ar 3 c 1 

2 ? Ie (1-C2) -- au av 2 

v • ~ - uv [-- - 2 --) + - k 3 c (2 + C) ar ax 3 

c 1 

substituting Eqs. (A.9) and (A.10) into (A.ll) and sir.,llfying. we get 

2 2 (l-C..,) U"'V 

(A.B) 

(A. g) 

(A.l0) 

(A.11 ) 

• - Ie c. [L + ~.] 3 . c(2 + C) ar ax c 1 

(A.12) 

After multiplying both sides by p and rearranging. we have 

- 42 -

, . 

. (tJ~ 

i 

I 

l' 

I I . 

• 

•• • 

.' ~ 

; 

i 

I 

I 

- . I 

., i.. ~ 

, 

I 

I 

~. 

i .----

I· 

~' 
: 

. -~---~",,,,,,, .. ""'.~"'- .. ~- ......... ~~~.-- .... - ..... _ .. - .... - . w~ '/ 

-- au av 2 Since Pt • -puv/(ar + ax) and Cp • Pt c/pk • we have 

C 2 ( l-C2 ) / [1 2 k

2 

( 1-C2 )2 (C 3U)2 _ 4 au 3V + C3V)2) 

~ - 3 § + C + 3 c2 § + C ar ar ax ax 

c 1 c 1 

k ( l-C2 ) V 

+ - -] 

c !!+C r 

c 1 

- 43 -

(A.13) 

(A.14) 

, 

·1 , . 

(. 

.! 

.; , .. ~ ., 

\ 

I 

I 

I , 

/. 

" 
f / 

REFERENCES 

. I 

1. Etheridge, D. H. and Ken:p, P. H., -Heasurell!!!nts of Turbulent Flow DownstrC!am of Ii Rearward-Facing Step,lI Journlll of F11J1qJ~\?chan1cs, Vol. 86, 

1978, pp. 545-56&. 

2. Kim, J., Kline, S. J. and Johnston, J. P., -Investigation of a 

Reattaching Turbulent Shear layer: Flow Over a Backward-Facing Step,- Journal of Fluids Enqineerinn, Vol. 102, 1980, pp. 302-308. 

3. Smyth, R., -Turbulent Flow Over a Plane Symrn~tric Sudden Expansion,- Journal of Fluids Engineering, Vol. 101, 1979, pp. 348-353. 

4. Eaton, J. K. and Johnston, J. P., -An Evaluation of Data for the 

Backward-Facing Step Flow,- Stanford Conferences on Complex Turbulent· 

Flows, 1980. 

5. Cha,ldrsuda, C. and Bradshaw, P., -Turbulence Structure of a Reattaching 

Mixing layer,- Journal of Fluid Mechanics, Vol. 110, 1980, pp. 171-194. 

6. Driver, D. H. and Seegmiller, H. l., -Features of a Reattaching Turbulent Shear layer in Divergent Channel Flow,- AIAA Journal, Vol. 23, 

No.2, 1985, pp. 1&3-171. 

7. Amano, R. S. and Goel, P., -Computations of Turbulent Flow Beyond Backward-Facing Steps by Using Reync.lds-Stress Closure, - AIM .10urnal. 

Vol. 23, No.9, 1985, pp. 1356-1361. 

8. Rodi, W., -The Prediction of Free Boundary layers by Use of a Two 

Equation Model of Turbulence,- Ph.D. Thesis, University of london, 

December 1972. 

9. Amano. R. S. and Goel, P., -A Study of Reynolds-Stress Closure Hodel,-

NASA-CR-174342, 1985. 

10. Amano, R. S., -A Study of the Second and Third Order Closure Models of 

Turbulence for Prediction of Separated Shear Flows,- NASA-CR-176127, 1985. 

11. Irwin, H. P. A. H., -Measurements in a Self-Preserving Plane Wall Jet in 

ft Positive Pressure Gradient,- Journ~l of Fluid Mechanics. Vol. 61, 1973, 

pp. 33-63. 

12. Rotta, J. C., -Statistische Theorie Nichthomogener Turbulenz,-

Zeitschrift fur Physik. Vol. 129. 1951, pp. 547-572. 

13. Naot, D., Shavit, A., and Wolfs~te'n, H., -Interaction 8etwe~n Components of the Turbulent Velocity Correlation Tensor Due to Pressure 

Fluctuations,- Israel Journal of Technology, Vol. 8, No.3, 1970, 

pp. 259-269. 

- 44 -

.. :tJ~ 

I 

. 

I. 

I ~ . 

I 

! 

" I i I 

t· ':. ;' , I 

.. I .. ' .1o...J ... 

,. 

I 

I , 

, 

, 

,- . 
. r i 

I I, " :01 . ' 

.. 

14. Launder, B. E., Reece, G. J. ~nd Rod1, W., 'Progress in the D~velopment 

of a Reynolds-Stress Turbulence Closure,' Journ~l of Fluid "echanics, 

Vol. 68, 1975, pp. 531-555. 

15. Pope, S. B. and Whitela~, J. H., 8The Calculation of Near-Wa~e Flows,' 

Journnl of Fluid Mechanics, Vol. 73, 1976, pp. 9-32. 

16. Naot, D., Shavit, A. ~nd Wolfshtein, H., 'Two-Point Correlation Hodel and 

the' Redistribution of Reynolds Stresses,' The Physics of Fluids, Vol. 16, 

No.6, 1973, pp. 738-743. 

17. Daly, B. J. and Harlow, F. H., 'Transport Equations in Turbulence,' The 

Physics of Fluids, Vol. 13, No. 11, 1970, pp. 2634-2649.. -

lB. Hanjalic, K. and launder, B. E., 'A Reynolds Stress Hodel of Turbulence 

and Its Application to Thin Shear Flows,' Journal of Fluid Mechanics, 

Vol. 52, Part 4, 1972, pp. 609-633. 

19. Shir, C. C., 'A Preliminary Numerical Study of Atmospheric Turbulent 

Flows in the Idealized Planetary Boundary layer,' Journal of Atmospheric Science, Vol. 30, 1973, pp. 1327-1339. 

20. Cormack, D. E., leal, l. G., and Seinfeld, J. H., 'An Evaluation of Hean 

Reynolds Stress Turbulence Models: The Triple-Velocity Correlation,' 

ASME Journal of Fluids Engineering, Vol. 100, 1918, pp. 47-54. 

21. Carmody, T., 'Establishment of the Wake Behind a Disk,a ASHE Journal of 

Basic Engineering, Vol. 86, 1964, PP. 8&9-882. 

22. Taylor, A.H.K.P. and Whitelaw, J. H., 'Velocity Characteristics in the 

Turbulent Near Wakes of Confined Axisyrrmetric Bluff Bodies,' Journal of 

Fluid Mechanics, Vol. 139, 1984, pp. 391-416. 

23. Driver, D. H. and Seegmiller. H. l., Private Communications, 1985. 

24. Izumi, R. Yamashita, H., Kaga, S. and Hiyajima, N., 'Fluid Flow and Heat 

Transfer in Corrugated Wall Channels--Experimental Study for Many Bends,' 

Proceedings of the 19th JSHE Symposium on Heat Transfer, Paper ti~. A101. 

1982. 

25. OIBrien, J. E. and Sparrow, E. H., -Corrugated-Duct Heat Transfer, ~ressure Drop, and Flow Visualization,- ASME Journal of Heat Transfer, 

Vol. 104, No.3, 1982, pp. 410-416. 

26. Amano, R. S., -A Numerical Study of laminar and Turbulent Heat Transfer 

in a Periodically Corrugat2d Wall Channel,- ASHE Journal of Heat 

Transfer, Vol. 107, tio. 3, 1985, pp. 564-569. 

27. Amana, R. S., -A Numerical S'~udy of Turbulent Heat Transfer in a Channel 

with Bends,- ASHE Paper No. 85-HI-20, 23rd National Heat Transfer 

Conference, Denver, August 1985. 

- 45 -

l 

1 

j 

J 

! 

~ . 

! ~ 

! ) .. 

i 

I I :~. 

.. 

; 

, 

j 

! -

I 

I 

I 

, , ., i.. ~ 

, 

I 

, 

.1 

I 

.;. 
t· 

, \ ,I: • 

,~~t;~-i~::~ -'.-.-~.:._~ 4 

• 

figure 1 

Figure 2 

figure 3 

Figure 4 

Figure 5 

Figure 6 

Figure 7 

Figure 8 

Figure 9 

Figure 10 

Figure 11 

Figure 12 

Figure 13 

Figure 14 

Figure 15 

figure 16 

Figure 17 

Figure 18 

Figure 19 

Figure 20 

Figure 21 

figure 22 

::igure 23 

Figure 24 

FIGUnES 

Flow domain 

Mean velocity profiles behind the step 

u2 profiles behind the step 

v2 profiles behind the step 

uy profiles beh~nd the step 

k profiles behind the step 

Convection of turbulence energy 

Diffusion of turbulence energy 

Production of turbulence energy 

Dissipation of turbulence energy 

Turbulence energy balance 

Triple-velocity products for different values of Cp 

Triple-velocity products for the different values of Cp 

Trip~e-velocity products for different values of Cp 

Tr1p1e-velocity products 

Triple-velocity products 

Triple-velocity products 

Triple-velocity products 

Triple-velocity products 

Triple-velocity products 

Flow geometries 

Grid test 

Hean velocity profiles in the flow passage for the external 

flow over a disK 

Streamwise component of Reynolds-stress distributions for the 

external flow over a disk 

- 4G -
./ 

,/ .. 

. Figure 25 

Figure 26 

Figure 27 

Figure 28 

Figure 29 

Figure 30 

Figure 31 

Figure 32 

Figure 33 

Figure 34 

Figure 35 

Figure 36 

Figure 37 

Figure 38 

Figure 39 

Figure 40 

Figure 41 

Figure 42 

Fi~urc 43 

Figure 44 

Figure 45 

/ 'r, . 

'\ 

", -~-, ... -.. ~. ott/. 

Radial ccrr.ponent of P.eynolds-stress distributions for the external flou over a disk 

Azimuth component of Reynolds-stress distributions for the external flow over a disk 

Sh~ar stress distributions for the external f1o~ over a disk 

Turbulence kinetic energy profiles for the external flow over a 

disk 

Cp prof1ies for the external flow over a disk 

Gle distributions for the e~ternal flow over a disk 

Velocity profiles in the flow passage for the internal flow 

over a cone 

Streamwise component of Reynolds-stress distribution for the 

external flow over a cone 

Radial component of Reynolds-stress distribution for the internal flow over a cone 

Azimuth component of Reynolds-stress distribution for the internal flow over a cone 

Turbulence kinetic energy distributions for the internal flow 

over a cone. 

Gle distributions for the internal flow over a cone 

Corrugated wall channels 

Velocity and the Reynolds-stress profiles in a backward facing step flow 

Aver'age Nusselt number as a function of Reynolds number 

Average skin friction coefficient as a function of Reynolds number 

Velocity vectors in channels 

Velocity and the Reynolds-stress profiles in a cO'rugated wall 

channel (alb. 2, Ra - 3000) 

local Nusselt number along the channel wall (alb ;-, 1.25, Re .. 3000) 

Local Nusselt number along the channel wall (alb g 1.5, Re z 

3000) 

Average Nusselt number as a function of the length of fin 

- 47 -

I 

" 

I 

I 

1 

I 

.. 

, 

/ 

i.! 
FLOW 

.. 

r;:~;~f;-" ~;.-

. -. 

INVISCID REGION 

----------.... 

-.... 

---- ---- VISCOUS REGION 

RECIRCULATING \ 

REGION REATTACHING 

POINT 

Figure 1 Flow domain 

r· ........ ---~ ... _ .... -........... , ..... '''-'~'''''' .... . 

REDEVELOPING 

REGION 

.. 

AI .' 

. -

. t "'->'.: 

" f ;.1 __ to _ .. 

I 

jl 

t 

r 

I 

l' 

I 

t 

I 

, 

~ 

I 

.i 

i 

~ 

J 

I t -~ 

~ 

• 

· 

~ 

J -

l 

i 

! 

~. 

t 

I 

I 

i 

~ 

l 

• . ~-~ --..... " 
L---I 

0.2 

o EXPERllUNT 10,100, .. s .. "",IIIe" 

2.0 CkoO.1,CfoO.' 

ck 00.15. Cf 00.25 

::r: CkoO•31 • Cf 00.11 --. 

:0. - - - Bou ....... q MOil.' 

1.0 

o 

o 

---_ ... -.-...... --_ .... _._ .... - .----~ ... -.-.~ .. _----...-..,.-._- ~ -. -_. 

x/H-a 0 

o o 

o o 

x/H-12 0 

o 

o 

o 

o~~~~--------~--~~----~----~~---~------~~--------------~ o 0 

Figure 2 Mean velocity profiles behind the step 

. \.'tI 

.. 

-.; 

j 

~ 

, I 

" 

... 

j 

j 

l 

i 

.; 

~ 

1 

, 

t 

a • ... 

, 

I 

I 

I 

-/
x/H-O 

0.010 

o UPUIII[NT CD, ... , • £ .. ,"'_1 -- C~' D.I. Cc 'D.~ 

-.- C~ ·D .... Ct ·D.n 

o ........ C~·D.~I. Ct .0.11 

0 ___ .......... q ..... 1 

'b 

Cbo 

o 

Figure 3 

'. 

x/H-a 

o 

o 

~JU2 IN 

~ profiles behind the step 

/ 

I 

x/H-12 

._-. CII-. 

... 

. . 
:: 

->-

Figure 4 

o o 

v2 profiles behind the step 

! 

I' 

I 

',I 

~ .' 

.. 

i 

. 

1 

I 

1 

I 

., '" J 

" I 

~. 
• 

----------------.----

::: 

-,.. 

0.005 

2.0 -- c.· •.•. Ct .I.J 

-.- C. ·o .• S. Cc ·c.n 

.... -. c.· •. , •. c; .e.u 

--- -.......... ... 

1.0 

O~I--------------------~----------~--~--------~~I~--------.~~------~ 000 

figure 5 

-u2 uvl au 

uv profiles behind the step 

• 

----.. --
"''' .. 

0.010 

o U"IIIMC'" tC._ , ..... _. 

- C.· •. '.~.O.l 

-.- C. ·G.". ~ '0.21 

~~~ •. C.' Dol'. C, '0.11 

--- o.c. .. _ ........ . 

, 

o 

o 

f;gure 6 k pre-files beh1nd the step 

o 

o 

,\ 

.. 

t 

,I. 

I 

1/ 

~ 

f\ 

, 

i 

.I 

~ . 

... 

'I 

• 

"". ~ 

" I 

I 

I 

; 

I 
2.0 

1.5 

:r: >. 1.0 

0.5 

o 

-.-'----.-----... -~--- .. ~. __ ... __ ._ .... jII....,.,..._ .... · ..... _· ..... ·• __ '_.4~_ . ..._ ........... _""_. .... _ ~ ... _w ~ ~_ ,y, ~.-

x/H- 3.11 

-0.005 0 0.005 

Figure 1 

x/H-7.11 

-0.0025 0 0.0025 

(CONVECTION) x H/UIH3 

o UP[RlloI[HT x/H -11.9 (0,,,.,, 

kegmllle" 

-STRESS MODEL 0 

-·-1l0USSIHUa 

MuOEL 

-0.002 

o 

o 

o 

0.002 

Convection of turbulence energy 

, 

• 

I '. 

~ ", 

.,.. 

.. 

· \ ~ 

. i . 

! 

1 

~ 

\ 

, 

, 

• f , 

i 

j 

.. ~ 

, : t ....... 

, 

I 

I , 

I! 

,: " 

, 

i 

I 

I 

I 

I 

L. 

, 
2.0 

o EXPERIMEHT x/H- 3.9 

111,_, , ,--smU.." 

- STRESS MODEL 

---1I0USSIHESQ MODEL 

1.5 

~ ~ 1.0 

0.5 

o 

o o 

00 

o -0.010 o 0.010 -0.0025 0 0.0025 

-('lIFFUSION) x H/UIH

3 

figure B Difrusion of turbulence energy 

x/H-l1.G 

o 

o 

o 

o 

0.002 

. __ ., 

... -, 

I 

I .," 

i" 

" 

a .. ~ 

~ 

, 

I 

I 

I /. 
• • - .-~' 

I 

• 

;.. 

2.0 

o UPERI¥ENT 

CD" .. , , t •• gmllle, I 

x/H·3.11 x/H·7.9 x/H-11.9 

- STRESS MODEL 

---IIOUSSIHESQ MODEL 

1.5 0 

0 

0 

0 . :: \ 0 \ - 1.0 i .- >-

01 

0.5 J ',. 

I 

0 I 

-0.010 0 0.010 -0.0025 0 0.0025 -O.OOZ 0 0.002 ! . 

(PRODUCTION) x HIlJ1N

3 I \ 

' .. ~ 

" I 

I 

Figure 9 Prod~ction of turbulence energy 

~. 
''. ' 

-IJ~ 

i I 

't 

."" 

• 

~-. 

2.0 . I 

o EXPEIIIMENT x/H-3.8 (D, ... , • $Mgml..." x/H-7.8 a:/H-11.D I 

I 

-STilUS MODEL ,. 

---80USSIHESQ MODEL 0 I· 

I 1.5 -- ( 0 \ 

c 0 i 

0 0 

0 0 j 

0 o ' I == 1.0 lo. 0 , , 

l' , , ,'0 

,'0 I 0 ~ 

~ 0 0 .. , 0 0 I· 0.5 '.~ 0 0 

" 0 

0 ! 

0 0 

I' . o~ !8 , 0 ) . 0 --'- -0.010 0 0.010 -0.0025 0 0.0025 -0.002 0 0.002 

'1 

(DISSIPATION) J: H/UIH

3 ,. ~ a... . 

,_. 

I 

, 

I 

• 

Figure 10 OiS5ipation of turbulence energy 
i!: 

>-

I 

I 

2.0 

1.5 

1.0 

0.5 

0 

. _______ • ____ • ________ •• _._ .. ~_. ___ ;.. ____ ~.~""' .. ,.._~ .. ""7' ...... "" 

x/H-3.9 

DIFFUSION 

CONVECTION x 

PRODUCTION + 

DISSIPATION 0 

'---- .., --, i , \ 0 + , , , \ 0 + , 

; \ ; 0 X '+ 

"' , 0 , 

, + 

\~ , I + 

0\ .... +,' 0 

x/H-7.9 

o +t"J 

o ;·~ .. O 

p. ,. 0 •• 

" , l 

, 

i 

i 

iO 

9 

q 

)-

+l , 

,~ 

, 

, ... , 

,+ 

, 

,+ 

"' 0 \ 

" ... Ifl,o x .. ,' 0 

\ i t{5 

-0.010 0 0.010 -0.0025 0 0.0025 

Figure 11 Turbulence energy balance 

-0.002 

. I 

~,._ .. ..,...n.t_ ..... ,_, .. - --'""::'" .,..... ... ~ .• ~ 

0<) 

o .0 

o ot£] 

o 

x/H-11.9 

0.002 , 1 "i ' .. ~ 

... 

J 

I 

I 

, . 

I 

, " , '/ 

I 

:,'.' '.': 
~: 

J: 

-~ 

1.5 

1.0 

0.5 

t ' 

5.0 

x/H=5.4 

o 

o 

~ ..... -::-. ----::Q: 

-'-'--": ----...... " " 

o j / . / 

O ,/ " . _ 

"",- ?"'" ......... ~-

0- ........ -- . - ...,..".- -.------

x/H=6.4 

o EXPERIMENT (Chandrs"d~ &. Bradshaw) 

/. /'", ... .... --/ . ---- C =5 0 ....... / 

I P . ./ /0 , .. - - Cp =6.0 

\ - Cp=10.0 o 

o 0.0 I 0 UUV/U'

1N 

X 10' 

". 

/'/ 

/'/ I . 

I f I . 

\ \ 

\ \ 

x/H=10.3 I x/H=12.3 

o 

o 

0 

/. // I . 

I ! 

0 " I 

, . 

0 

\ , , . 

( ) 

o 

o 

~ .' " .'¢" . , , ~ 

. ' 01 J 1 I . ' / // 

,,}6 

o 

o 

o 

~ 

• c . . : 'I 

\~, 

\ 

. , 

~-" 

. -. 

, ' 

" . '-~: 

---

Ii 

1 

~ '~ 

a 

j, 

j 

1 ./ 

. 

1 

j 

} 

'j 

l 

i 

i 

Figure 12 Triple-velocity products for, different values of Cp .-

~_·_.4_"· 

p' 

.... .ML _ _ , -'" ~ ... "- ", .. --. ' 

'\ 

'. 

, l> 

, .... - •..• -·'-4~_ •• _ G 

", '~ 
~. 

1.5 x.lH=5.4 x/H=10.3 I x/H=12.3 

5.0 . 01 0 

0 0 0 

0 /. 

,.. II, 

1.0 r ...: O~ o ,-7 o'i ,. -..-: ,/ /. ~'- I , a ... -./ '." " ..... / ....... ,.~'" I I ' . 

J: (" C{ " / I • I , , . {Ol o· \ C\\ -- ' 

\ . >- \, ". 0 \ '\ \ \ \ . 'Q " ' . . ,\ 

" '. o \~" \. 

' .......... '0:'. o \\ ....... , ~ ........ ,' .... ,~ . 0.5 I- ~ ~ 0 ... ~o 0 ~~ ... ..Q ~ .~ 

......... ·~o 0 .' 0 ~\ ". .... .... , ..... \' .\ .Q ........... , . \ ....... ... . .... \ \ \\ " ' .... . , . \ . , . , '\ " 0\ ,\ 0 I , 0 II , \ .J I . I o ;1 0 , 0 I I 

0.0 I 

0 0 0 0 

UVV JU31N X 104 

Figure 13 Triple-velocity products for the different values of Cp 

, 

.. J. 

'. \ 

\ 

. , -'. 

.. " 

t I -. ,-

I 

1 

I 

I" 

" 

.-.---~ .:., 

'. 
'. , 

1.5.- L ~/H=5.4 x/H=6.4 

5.0 0 

xIH=10.3 I x/H=12.3 

o 0 

0 0 0 l 0 

0 • 

1.0 I- \ 0 

~, 0 ' ..... ',', , .... 

. , :c . I ) 0, \ 0 \ \ 0 

-- . \ . , 

/,' 0 ~ .I "b / I /" . " ,/ " /" 0 .", 

. " ,. .- ". .~" 

0.51- 0 ~.,:~ '6" 0 ,~ 

o ;'~. ~ ... "/'. ,. . "'. "'/ ... ./ ;' ./' '" . "/ 

./' ,/ - " / 0 ,/ - " I I' j' 0 f 0 

I 

0.0 

0 0 

.\ 

~\ 0 \\ 0 -, \\ , . -\ . I 

It , I 

-, 0 il 0 " ., 

/' rj/ " ., 

0 .~ 0 ., 

0 

0 1/' 0 

/- ,! 

0 :UO 

0 II 

..L 

0 0 

VVV/U3

1N X 104 

Figure 14 T~ipl~-velocity products for different values of Cp 

v: ,-. ---......._k.. . ____ •. ~ • .l...:-. ..... /I,' •. 

., I· A. 

....... \ ---

i I 

I I - . I 

I 

I 

• I 

, 

J 

l 

i 

I 

v 

-

I -c __ 

t 

i j 

.] 

I 

) 

, . - ) 

• 

.. ..---Ca 
J: 

->-

!5. 

1.5 x/H=O.5 x/H=5.4 

5.0 

o 

1.0 t=L----· 

\-~ ... . .... \ , ' . . \ . I I 

\ ---- __ 4 'i" : / , -2 ~- -- . " 0 I " . ' ~." ..... o -EXPERIMENT.(Chandrsuda & Bradshaw) 

- TRAN~PO~T EqUATI9N,S·t 

-- CORMACK ET AL. " 9 : ., _.-'HAf;JJAlIC & -LAUN~ER 1 

-"- SHIR'~ I I Q 

0.5 

x/H=6.4 x/H=8.4 

o 

::-L 

o 

X/H=10.3/ x/H=12.3 

o 0 

. 

I 

I 

a •. 

i I @ 

. i 

---'DALY&HARLOW \ \ iO 

ooL. ~~~~~~--~------~~----~----~~--~--~ o o o 

UUV/U 3

1N X 104 

o o o 

j 

I 

I 

f 

Figure 15 Triple-velocity products 

.. ;' ~ 

.~ 4· ~1. 

i 

I 

r 

t , 

~ 

l 

, 

~ 

j. 

;". 

'-----.----......~ ..... ....... 
:r: 

->-

!) 

1.5 r I 5.0 

1.0 ()..... - "- . ::::s..;.:::"" ".,- "--:::=. .. r.._ .. -. .. __ ... -... .. 0 

0.5 

x/H=0.5 

o 

o 

, xij'j=5.4 x/H=10.31 x/H=12.3 

o 

o 

o 

0.0 ( 0 0 0 0 0 o 

iJvV/UliN X 10" 

Figure 16 Triple-velocity products 

",- ... ..,...._ .............. ,: ..... M _ ~. __ .-

__ ._._ <lj, 

, 

(. 

.. ---~ 

.... , 
1.5 r 

5.0 

1.0 t- 0 0 

J: 

-;:... 

. 0.5 

x/H=0.5 

....:~.::::..--. 

"," ./. ,,"" ........... I ....:: ~.:::: .... -. 

: /..""" 

hil 

v .. : ,I ! ,\ 0 

9 :/ : '/ 1/' .1/ 

P 

x/H=S.4 

o 

o 

){/H=6.4 

o 

o 

o 

o 

o 

x/H=8.4 I x/H=1 0.31 x/H::d 2.3 

000 

0 

~O !, 

0 i!i 0 

o 

0 

0 

o 

o 

0.0 I 0 0 0 0 0 0 

VVV/U31N X 10· 

Figure 17 Triple-velocity products 

!/ ..... -.-Af:l • ..a..r..-~, ~.I. __ ._ ........ - ... 

r ... . ~.; 

i. 

• i 

I 

I 

. t , • ! 

i . ! 

I 

• • f 

t 

i , 

I 

f 

t 

i 

l 

! , 

. 

--~ 
'VI 

....,. 

J: 

..... 

>-

1.5 x/H=4.0 x/H=6.0 

5.0 

o EXPERIMENT (Driver & Seeomiller) 

TRANSPO~TvEQUATIONS - (O<t, 

1 0 1:-- CORMACK ET ALr 0 • I ,:, ......... _ 

0.5 

_ .- HANJALIC & LAUNDER 

'\\ .. " 

: 0"', 

I i-. " ---- OAL Y & HARLOW 

_ .. _' SHIR 

/1 ~ .. "'. " ././ I \0 

// c;I / 

/ . . 

/ 90 i 

" I Q 

x/H=8.0 

°0 

o 

o 

x/H=12.0 

o 

o 

% 

en to tb cb I 0.0 I 0 0 0 0 

UUV/U31N X 104 

Figure 18 Triple-velocity products 

.... ) .... -,~!..,.,trJ.., .;u.' •. ~ L ..... 1. ... , . __ . _____ ;lL __ , .' .... _~. ______ ....... _ .. ___ .. _____ ......... _____ ........ _ to 

~ 

j 

II 

, .. -,-~ 

'--- ...... -

.' 
o 

• N 

0;-

Il 

J: 

--.. 

)( 

°0 o 0 

• 

CX) 

II 

:c: 

--.. 

>< 

o 

o 

• ~ 

II 

J: 

--.. 

X 

J~ 

°0 o 

o 

0 . .,.. 

.~;.-=.@ 

.. ~ .. V .. ··-

o 

o 

0°0 

o 

0 0 

-- LO . 

0 

H/A 

o 

o 

0 

0 . 

0 

'If 

0 

"l""'" 

)( 

:z 

... -

=> 

~ 

u 

o 

.-

IU 

> 

, 

IU 

.-

c. 

-

'" .-

IU 

l.-

:;, 

0-

-

"'-

... -~.-

I 

" 

--------.-.,~.,."\..~ 

... _-..... 

-

" 

, , 

'" ..... 

\ , 

\ 
o 

• N 

,.. 

II 

:Eo 

)( 

o 

o 

0

0

0

0 

• o o ,... 

)( 

o ~ 

o 

• ~ 

II 

J: 

....... 

)( o 

o 

J~ ~~~--------~C);-----------~~f.----~-----:C) • .,... 

H/A 

.., 

:;:) 

....... 

I~ 

\ 

\ 

.... 

~ 

v 

:;, 

-,;:J 

o 

... 

Q. 

» 

~ -u 

o 

11.1 

> 

I 

11.1 -Q. 

... 

... 

o 

N 

CI 

... 

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t:7t -u.. 

~, 

, 

.! 

J . , 

...~ , 

: .... ~ . , 

I 

I 

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--------------_--9'9----, I 

I 

I 

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I 

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~--~-.---.-D~ 

(a) 

COMPUT ATIONAL 

DOMAIN 

r-=a=--------- i 

I I 

_ J-~L :-..:.: =--==.::--=-= :- __ 1 19 

(b) 

Figure 21 Flow geometries 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I ____ J 

.. .., \ 

, . , 

. 

I' 

; 

I '. 1 

... 

1 

t 

J 

c. 

1 

i 

I 

• f 

, 

t 

[ 

1 

I 

.. ,. t 

, .... J, 

, 

I 

I' 

.! " 

.-1 

~I 
I 

i 

I 

I 

. 

. I 

i 

, 

~) 

. \ 

c -

a.. 

.... /' 

'---' 0 0.2 

UIUIH 

, 0 0.005 

2~ -2-2::12 

"2/UIN'u,/UlP~" IUIN 

'-----' 

0 0.01 u 2/U2 . IN 

if 

I 

1 0

1 ' ... '8, 

·~·U 2 

..:, IN uv/UIN 

figure 22 

'.1- ~""' •• -., .... , --.---.,,~ . 

X/D-G.O 

l 

~ \ . ' \ 

I I. 

, " u2/U2 

IN 

Grid test 

~ .,.. 

":<~ '\" . \ 

'I 

y2/U2 

IN 

Mesh Size 

62 l G2 

--- 52x52 

----- 42 x 42 

_._.- 32 x 32 

L '~ .,', . " \. " 

I , 

, \ 

\ 

l , 

\ 

) J w2/U2 

IN 

J 

! 

~ 

, 

\/ 

" 

,-

t: 
,-

~. 

c 

-... 

2 

1 

o EXPERIMENT (CarmodJ) 

ASM I 

AS'" n 

k·EI 

k·(11 

o 

o.u 0.5 

I() 

b 

0 

0 

0 

0 

0 

0 

o 

-0.5 

I 

!I 

'/ ',0 ,,- • 

" 0 } 0 

I 0 

1/ 0 /~ 

.f} o. 

0 

0.0 

Figure 23 Mean velocity profiles in the flow passage for the external 

flow over a disk 

,r.. .... """;':",,'k: ~-. .. ~ *'I. ___ ., .. Ck· ..... ~.t If" ~ _".,;;A.~;., . , 

,-__ ". _ __ -"L. .... ." ________ .. , _____ .. __ ... _. __ . _____ A 

;..-

., , 

'.I~ /' 

~- ~ .. -.. -- .. -.,~.~, .' 

• .... -- "\ \ . I 

\ 
. :.r;}) 

~ 

o 

i:: 

2 

1 .-

o EXPERIMENT ICarmody, 

ASM I 

AS'" II 

t·lI 

t·ED 

X/D °3.0 

o 

i X/D-S.O 

o 

ISO 

o 

o 

o I 0 /1 I . '0 Or.-:--"'---

0.0 

figu:-e 24 

", 

0.025 0.050 0.0 

u2/U2 

IN 

0.025 0.0 0.025 

Strea~4i~e component of Reynolds··stress distributions for the 

external flow over a disk 

~. 

r . ___ ",,I. .• !- . ... -.- .... 

I I· 

~ 

t 

• 

~ 

\ 

. 

\ \. 

\ . 

\ 
~ 

c 

-... 

0 EXPERIMENT (Carmody) 

AS'" I 

AS'" D 

k·fI 

k·fD 

I" 

II X/D-3.0 

, 

21- ," 

,t 

1 o 

o 

o 

~ rID

-

S

•

O 

11 

o 

o 

o 

,. 

~ X/D-9.O 

. 

110 /0 

10 

,\ 0 

~ 

·\0 

l\ \P 

'0 '0 I 1

10 

10 

II~ '0 

o I 0" I '/ 0' b~-""--- 0.0 0.025 0.05 0.0 

Y2/U2 

IN 

0.025 0.0 0.025 

Figure 25 Radial component of Reynolds-stress distributions for the ex- ternal flow ov~r a disk 

... .JlJ.~~ . '.; ... ;".~ •• '. ~-' " ~' 

,. 

;. 

\t' 

-~---

". 

\"'-"~- -=---
c -... 

;v. 

0 EXPERIMENT (C.,mod" 

ASM J 

ASM II 

lI-fI 

II-f 11 

2 r . 

I XIO-3.0 

~ 

~ 

1 I- \\,0 

:-.. 

~\O 

~\-. 

"~. ° '-.. ,'. 0 

hI 

=0 

, 

OL 

0.0 0.025 

h 

~ X/D-S.O 

~ 

rA 

\~ ~ ° . 

~O ,0 

, 0 

i,O 

ilo 

: j' , \ 0 

, ° J 

0.0 0.025 w2/U2 

IN 

II 

~X/D·9.0 

K 

p 

fO 

° 

, 

",~ 

~ ~ 

P !p. 

10 Ip 

/'0 

. I 

0.0 0.025 

Figure 2& Azimuth component of Reynolds-stress distributions for the external flow over a disk 

0"",' .. 

~~ 

___ ___ J1L _ _ . 

" . ._---------

b 

---.~~~,.-~-.--~ 

t-

-. -..!.:: 

.~ 

\. 

--.:. 

" 

. 

_.-=--

\, " 

--.. ' 

... ~ -.,. 

, , '-

~ 

"'" 

"'-. 
" 

~ 

~ 

a 

"i:: 

0 EXPERIMENT (Curmody) 

ASM I 

ASM 11 

II-E! 

II-Ell 

il Ii 

X/D-3.0 'Ii x/o""o.o 

ii 

21- 'I! 

I 

Ii 

I 

I 

~ 

1L ("\Ij. ,,~ 

o ! 0 b bt-----

Figure 27 

-0.025 0.0 -0.025 

uv/U2 

IN 

0.0 -0.025 0.0 

Shear stress distributions for the external flow over a disk 

~ 

,. M ,. 

, . ," ", "', ... 

r. 

1 

-I . 

f 

I " 

~ 

f 

, 

I 

f 

i 

• I 

1 

...... 

-

......... ~ ~ 

:. E ------- " .~" 
:!), 

2 

o EXPERIMENT (Carmody' 

AS'" J 

ASM II 

k'fI 

k·EII 

x/D-3.0 X/D-6.D X/i)-g.o 

o 

I 0 

~ I ~ ~ 0 j\ 0 

o ,\0, \ \ 

1 I- 1:, 0 ~'O" \ 0 ~:~, '\:~ \ '\ '",'.:,,-"':... 0 "., 0 ': () .~~, \M Y 

"'~" 0 '\ \\ 0 \,0 \ \\ ' \ 

~ . ! 

! 

--

I,!: 0 II" 0 ' b .. .., 

J

, .\ 0 ': I' 0 P . ,I : I I I 

, " 0 Ii: 0 : P !q : ; I 0 I P 

!I ! ," 0 ' 0 I' ; , I 

o 

0.0 0.025 0.050 0.0 0.025 0.0 0.025 

figure 28 

2 k/U'N 

Turbulence kinetic energy profiles for the external flow over a 

disk 

.... ,," .. '. ...... 

r ., ~, 

'\" 

~ E -------
· -"

f! 

,~~ 

---- ASM I 

~ - '- -

- ASMD Jj 

l<·fI 

- -.- k·En , I J , 

I , / 

I I , ,., 

I X/D=3.0 I X/O=6.0 ; X/O.,g.O -. 

l i ~ 

f 21- \ 

, I 

, 1 I 

\ J , 

, i 

! ---

l , h r 

~ I. I 

, , h l II 

1 

/ 1 

0 \ , -... I 

\~ .l. : '\. 1 r- I \ 

~ ~\ i\ , , 

'-

0 1 II Ir" "" 

0.2 0.0 0.1 0.2 

0.0 0.1 0.2 0.0 0.1 

CJA. 

Figure 29 Cp profiles for the external flow over a disk 

~. ! 

.. ~ . ; ~ . 

.. E 
I 

2)' 

c 

-... 

ASM I 

ASM II 

II·EI 

II·EII 

Ii X/NO 

2 Ii 

1 

00 

.:, , ~~ '\. , ...... 

. '. . ........... " "' .. '-', / . ....., ......... <. 

1/ 

2 

Figure 30 

Ix/D.SoO 

\ -~--------- ....... , _ ..... -) 

-----

.' ....... 

3 o 1 2 3 o 1 3-

G/e 

G/e distributions for the external flow over a disk 

.. ~. 

•• ~ .... ~_.!... ........... '1.... .' •••• 

,,-. ~ 

.. -\1 
0 

1.0 I 

c -... 0.5 ~ 

~ - /,j~ 

--- \ 

, 

EXPERIMENT (Taylor" Whitelaw) 

ASM I 

k·EI 

0 

0.0 

l " \ \ 

o X/D=O.4 rX/D.OO .. 

0 10 

0 I! 0 (0 

I 

0 10 

I 

IJ I 

, ./ 

0.5 

Figure 31 

0 / ,. 

" 

O~ 

/, 

I 

o Ij 

I /0 10 /0 

I 

I 

01/ ,'/ 0 if 0 '/0 

I II 

, 

Of( '1 ° '/ ° 

,10 

, i 0' 

I 

I I : '0 

0.0 0.5 0.0 0.5 0.0 0.5 0.0 0.5 

U/U1N 

Velocity profiles in the flow passage for the internal flow 

over a cone 

r· , ...... __ ., ... ~ '" •. ra. & ...... . 

L 

! I 

: I 

l"~ 

:i 

iJ 

. ~ , 

: ~ ; ~ I -, 

t ~ 

: 1 

. 1 

~ 

t 

i 

! 

1 

i 

1 

, 

4 

t 

~ 

• 

• 

~ 
0 

-.. 

;/. 

.. '''' .•.• - J ........ "- , ., . .-

0 EXPERIMENT (Tayler & Whitelaw) 

ASM I 

---- !I.E! 

1.0 Ir { 

X/D-O.99 

0 

0 L 0 

0.5l~ " , 

\ 

0 , 

, \ I 

I 

0 I I /0 I 

I 

0 / I / 0 

I 

I 

I 

I 0 

I 

I 

o I I lid I 

0.0 0.02 0.04 0.0 0.02 0.04 

'. 

( ~/D-1.99 

0 

0 

! 0 

0 

0 ~\. 0 

0 \\ 0 

I I 0 

! / 0 

0.0 0.02 0.04 

u2/U 2 

IN 

:" 

r :X/o ... 78 X/D=4.77 

0 

0 

~ 0 0 I 0 ;\ 0 

i,\ 0 \ \ 0 

~ \ 0 I \0 

10 ~ 10 

I /0 :10 

O.~ 0.02 0.04 0.0 0.02 0.04 

Figure 32 Streamwise component of Reynolds-stress distribution for the 

external flow over a cone 

::."" ... , .... i-!","") :- ._ .• _____ -"L ___ • 

........ ,-" ... ,,\ ... 4. l •• '~ 

;. 

f~~ ,-

,or:;. k: 

';'J/, ~ 

1 r: 

f' i . 

/ 

~.-

1· 

j) 

.. -

, . 

t 

) 

rfill 

~ 

.,~ 

. , 
0 

-. 

.... 

.;,,/ 

o EXPERIMENT (Taylor" Whitelaw) 

ASM I 

k·H 

1.0 ~.--~~------~----------,-------------.-~-----.-------

X/D:: 0.4 X/D= 0.99 X/D:: 1.99 X/D:: 2.78 X/D=4.77 

0 0 0 0 ~ 0 

, 

I 

, 

0.5 ~ .... ',0 " 0 0 0 1\ 0 

1 

I 

0/1 \e , 

0 0 1\ 0 

, 

I 

9/ i\ 0 ~ 0 ;\ 0 "10 

, I 

bl ! I 0 ~ \ 0 ! I 0 110 I 

I 

, I '0 

o >----l-L, ~ 

0.0 0.02 0.04 0.0 0.02 0.04 0.0 0.02 0.04 0.0 0.02 0.04 0.0 0.02 0.04 

figure 33 

~ " ..... '. 

. ~, .. ". ' ... 

V2/U~ 

Radial component of Reynolds-stress distribution for the internal flow over d cone 

{ 

I 

/ 

'," ,.'" 

I· 

\ , ',' ' 

, \ I 

I 

, 

• 

. ' 

~ . 

'f 

" 

'" ..... 

f 

j 

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: 

f 

, ' 

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I 

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,,'l 

.~ .... "'-.. . , , 

- , .. -. 
2). 

a 

-.. 

~---.,.,-----

o EXPERIMENT (lIy"" & Whllel.", 

ASM I 

.,0 

1.0ri--~~----------~----------------~~------------~--------~------

0.5 

o 

X/D .. 0.4 

o 

o 

0.0 0,02 0.04 

Figure 34 

X/D=0.99 

o 

o 

o 

o 

0.0 0.02 0.04 

o 

o 

X/D-1.99 

0 

0 

0 

0 

0 

0 

0 \ 0 

0.0 0.02 0.04 

\V21U~ 

X/D=2.7B X/D=4.77 

0 0 

0 0 

0 0 

0 0 

I 

\ 0 

r , 0 o 

I 

, ~ 0 I 0 

I 

0 !I 0 '10 I 

I 

; \ I 0 0 dO 

I 

I 

0.0 0.02 0.04 0.0 0.02 0.04 

Azimuth component of Reynolds-stress distribution for the internal flow over a cone 

' .. \ 

(' 

. ---------;....--..;.;..:;;..!-4- -"'" - . -" .' '.' '~!: ; --~ ... 

. //;/-; '- -r~~ 

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o EXPERIMENT (lar\o' , While".) 

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k·U 

1.0~--------------~--------------------------------~----------~------

X/D= 0.4 X/O=0.99 X/D -1.99 X/O-2.78 X/O-4.17 

o ~o l 0 l 0 \0 oJ ~O"'" 0 0 0 '. 0 

, , 

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, , \ 0 \\ 0 ~\ '0 

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0.0 0.02 0.04 0.0 0.02 0.04 0.0 0.02 0.04 0.0 0.02 0.04 0.0 0.02 0.04 

--.:.~ 

-- -_.' 

Figure 35 

2 k/U1N 

Turbulence kinetic energy distributions (or the internal flow 

over a cone. 

' .. .."4''- ... ,..:. .4.: . 

<,/ __ ,>"~;'{~ .. _c._ .>-;._n._, .. ' .•.. , __ " 

\ . " -~: .-'" .. -. 

H.~ ....... "" ......... 

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---------

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0.5 1.0 0.0 0.5 1.0 

GIl: 

Figure 36 G/e distributions for the internal flow over a cone 

r ___ "L .. .. ' .. " '. 

/. .-. -I .~ ... .zI'. 

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Figure 37 

~FLOW 

Corrugated wall channels 

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Figure ja 

a 0.05 -0.01 -0.01 o 

- 2 (d) uv/U1N 

Velocity and the Reynolds-stress profiles in a backward facing 

step flow 

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l EXPERIMENT 

a/b"2 j a/bn1.5 PREDICTION BY RSM 

a/b-1.25 . 

a/b"2 PREDICTION BY 

k-fMODEL 

o 

o 0 

10°;---~--------~------------+-----------~ 

102 

Re 

figure 39 . Average Nusselt number as a function of Reynolds number 

-;....-'4-.••• ~~ ••• _.o.-.. __ ••• ~,., .. , .. -,_""_~"~_~~,~",-,_"",,,,,",- ......... ,.... ~ ..... " -.-." '." • . __ .......... .,...._., ..:!l!"'.~~- ........ 

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EXPERIMENT FOR 

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alb 

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Figure 40 

Re 

Average skin friction coefficient as a function of Reynolds 

number 

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· · · · · · · · · 

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a/b=2 

Re .. 3000 

figure 42 Velocity and the Reynolds-stress profiles in a corrugated wall 

channel (alb = 2. Re = 3000) 

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80 

70 

60 

F1gure 43 

Lfin/b 

0 

~ ••••••• 0.25 

--- 0.5 

-.- 0.75 

.-. ., ............ 

• • ....... e. " ." ••••• ..... ~~ /.~... ~. ..... -_.,- . " -../~, L' • " .-

~ ..,,- . - .. ........ ~.. / •• - . e. '" "- •• _ -"~L-. ••• • " • ...,;;. .. ,. .----.... . .... " ........... " \ 

alb =1.25 

Re a 3000 

I 

C 

'-'--.,,\ 

I I 

o E 

local Nusselt number along the channel wall (alb c 1.25, Re c 

3000) 

. ---"" -'.~"" '-~' -... -- --'. -.. ' -.-- .... ------- -... 

r .! 
C") . 

o '-no 

-::J 

Z 

80 

70 

60 

or 

A 

L'ln/b 

0 

~ ••••••• 0.25 

--- 0.5 

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~ . .. ~. ...... \ . I' . :r \1 .... '. 

. 

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1,,1' • --.....,- ~. '\:.. •• - '\.... . • •• , . . .. .....:-- . .. . . . ........ ............ -. ,I. . ~ " .! • .' .. - - ••••••••••••• 'r- ...... "-/1· -. • .•• , ••• ~ .. \. ..... .. . " ._ ....... .... ~ ~. '-

I 

B C 

alb -1.5 

Re-3000 

I 

0 

Figure 44 Local Nusselt number along the channel wall (alb = 1.5. Re 

3000) 

... 

I 

E ....... 

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... . --.., 

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d 

Re=8000 

150 

.. , 

100 

cot) . 

c ... 

Q. 

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I~ Re=3000 

50 

Re .. 1000 

1 ' .. 

O+-------~--------_r--------ro 0.25 . 0.5 0.75 

Figure 45 Average Nusselt n~mber as a function of the length of fin 

I@.I',..,· l~~~ 

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I 

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End of Document