NASA Technical Memorandum 105954 _

ICOMP-92-24; CMOTT-92-12

Critical Assessment of Reynolds

Stress Turbulence Models Using

Homogeneous Flows

Aamir Shabbir and Tsan-ttsing Shih

Institute for Computational Mechanics in l'ropul,gion

Lewis Research Center

Cleveland, Ohio

Prepared for the 31 st Aerospace Sciences Meeting

sponsored by the American Institute of Aeronautics and Astronautics

__. Reno, Nevada, January ! 1-14, 1993

IW SA

(NASA-TM-105954) CRITICAL

ASSESSMENT OF REYNOLDS STRESS

TURBULENCE MODELS USING HOMOGENEOUS

FLOWS (NASA) 31 p

G3134

N93-16703

Unclas

0140771
T
Critical Assessment of Reynolds Stress Turbulence Models

Using Homogeneous Flows

Aamir Shabbir and Tsan-Hsing Shih

Institute for Computational Mechanics in Propulsion

Center for Modeling of Turbulence and Transition

NASA Lewis Research Center,

Cleveland, OH 44135

ABSTRACT

In modeling the rapid part of the pressure correlation term in the Reynolds stress

transport equations, extensive use has been made of its exact properties which were first

suggested by Rotta 1. These, for example, have been employed in obtaining the widely

used LRR model. Some recent proposals have dropped one of these properties to obtain

new models. We demonstrate, by computing some simple homogeneous flows, that doing

so does not lead to any significant improvements over the LRR model and it is not the

right direction in improving the performance of existing models. The reason for this, in our

opinion, is that violation of one of the exact properties can not bring in any new physics

into the model. We compute thirteen homogeneous flows using LRR (with a recalibrated

rapid term constant), IP and SSG models. The flows computed include the flow through

axisymmetric contraction; axisymmetric expansion; distortion by plane strain; and homogeneous shear flows with and without rotation. Results show that for most cases a properly

calibrated LRR model, which is the most general representation for a model linear in the

anisotropic tensor, performs either better or as good as the other two models of the same

level.
1. INTRODUCTION

Modeling of the pressure strain correlation is one of the central issues in the development of Reynolds stress closure models. Most of the popularly used models have utilized

the suggestions of Rotta 1 in developing models for this correlation. From the Poisson

equation for the fluctuating pressure he obtained the following expression for the pressure

correlation

(1)

(2)

where the superscript S refers to the so called slow part and the superscript R refers to

the rapid part of the pressure correlation 2'3'4. If we restrict ourselves to the homogeneous

turbulence with constant mean gradients (only such flows are considered in this study)

then Up,q can be moved outside the integral and one has to model only what else is left

inside the integral. For brevity we designate

Rotta showed that the fourth rank tensor Iq_pj has the following properties which are exact.

Iqipj = Iiq_,j = Iqijp, symmetry property

Iqipl = 0, incompressibility,

Iqijj = uqui, normalization property (4)

Launder et al. 2 utilized these constraints in obtaining the so called Launder, Reece and

Rodi (LRR) model. They first assumed the most general form of _rqipj which is linear in the

anisotropic tensor bij = (_--_/q_ - 6ij/3). Then application of the symmetry constraint

reduced the number of unknown constants to five. Further application of the incompressibility and normalization constraints reduced the number of unknown constants to one

2
which must be determined empirically. Recent study of Speziale et al. 5 has proposed a

model of Iqipj which is linear in the anisotropic tensor bij and satisfies all of the above constraints except the normalization constraint. This model, therefore, involves four empirical

constants. If the normalization constraint is applied to the SSG model then it reduces to

the LRR model. So the most general rapid term model which is linear in the anisotropic

tensor and satisfies the above mentioned constraints is the LRR model. In our opinion

the violation of the normalization property of the rapid term correlation does not bring

in any new physics to the model and, therefore, is not the right direction in improving

the performance of the currently used models. Direct comparisons of rapid term model

with DNS data e showed that the LRR model performs better than SSG model. In this

paper it will be further demonstrated, through computations of homogeneous flows, that

the properly calibrated LRR model can perform as good as SSG and IP models or even

better.

The thirteen homogeneous flows that are numerically computed include flow through

axisymmetric contractions and expansions; distortion by plane strain; homogenous shear

flows with and without rotation of coordinate axis. In the light of the objective of this

study we concentrate only on those models which use a linear (or quasi-linear) model for

the rapid term. This, therefore, includes the Launder ,Reece and Rodi 2 (LRR) model,

the isotropization of production (IP) model 2, and the Speziale, Sarkar and Gatski 5 (SSG)

model. In the next section governing equations and the models tested are given for homogeneous flows. Then a brief description is given of the flows computed followed by the

results and conclusions of the study.

2. GOVERNING EQUATIONS AND MODELS

For an incompressible homogeneous fl0w in a rotating frame the Reynolds stress equation can be written as

D

-_uiuj = - (u-7_ Uj,_ + _ Ui,k) - 2(u-7-,-,_e,.r,kjf't,.n + Uj_e,nki_,,r,)

+ lp(ui,_ + uj,i) - 2vui,kui, k

P

(3)

3
where fire is the rotation rate vector of the coordinates relative to the inertial frame. The

model for the pressure strain correlation (denoted by IIij) and a modeled dissipation rate

tensor are needed to complete the second order closure. Usually, the deviatoric part of

dissipation tensor is combined with the pressure strain, and a dissipation rate (i.e. e)

equation is needed.

The form of the dissipation rate equation used by all the models tested here is given

by

DtDe_ C_l-_u-_Ui,k - C_2_ (4)

where k = uiui/2; C_1 and CE2 are model coemcients. For LRR and IP models we use

Cd = 1.44 and C¢2 = 1.92 whereas for SSG model CE1 = 1.44 and C_2 = 1.83.

LRR MODEL

The LRR model for the pressure correlation term is given by

IIij = - C1 ebij

2

+ 0.8kS_ + (18C_11+ 12)k(b_SCk + b¢kS_k - -sb,_,,S,,,,,&j)

+ (20- 14C_)k(bikWj_ +bjkwik) (5) 11

where bij = (_/q2 _ 61j/3), Sij = (Ui,j + Uj,i)/2 and Wij = (Uij - Uj,i)/2 + emjlft,,_.

The model constants of the LRR model have evolved to slightly different values than those

originally recommended by LRR 2. The value of the Rotta constant C1 (in the return to

isotropy term) used in the present study is 3.6 This value was used, among others, by

Gibson and Launder T. The value of the rapid term constant C_ (is 0.4 in the original

LRR model) used in this study is 0.55, which is slightly higher than the value of 0.5

recommended by Morris s. Therefore, in this paper LRR model will mean equation (5)

with C1 = 3.6 and C'_' --- 0.55. It was found that the value of 0.55 led to improvement in

the performance of LRR model for all the flows tested here. (The improvements were slight

for the irrotationally strained flows but were significant for the homogeneous shear flows.)
It should be mentioned that originally LRR obtained the value of this constant from the

experiments of Champagne et a12 which were later redone under higher shear rate (see e.g.

Tavoularis and Corrsin 1°) to obtain more consistent results and these later experiments

have been used by several researchers 11 as a standard benchmark for calibrating model

....C0n_tants_ A slightly different value of rapid constant used here is merely a reflection of

this fact.

The LRR model is linear in Reynolds stress and satisfies the constraints of incompressibility, normalization and symmetry. It does not satisfy the realizability principle (

which implies that the non-negative quantities such as u s should remain non-negative, see

Shih and Lumley 4 for details). It should be further noted that the LRR model is the most

general model for the rapid term which is tensoriaUy linear in the anisotropic tensor and

satisfies the above mentioned three basic constraints.

IP MODEL

This model uses the same slow term model as the LRR but for the rapid term keeps

only the G2(Pij - 2P6ij/3) term of the LRR model. When re-written in terms of bij, Wij,

and Sij this model is given by the following expression.

2

IIij =- Clebij + C22k[(bikSjk + bjkSik -- -_bmnSmnSij)

2

+ + bi W k) + S,j] (6)

where C1 = 3.6 and C2 = 0.6 were recommended by Gibson and Launder 7.

SSG MODEL

The SSG model for the pressure correlation term is given by

1 bm,,b,.,,.,Sij) + (C3 - C73II_/_)kSij u,j = - + c;p)b,j + c2 (b,kbk -

2 ,5

+ C4k(bikSjk + bjkSik - -_bm,tS_,., ij + Csk(bikWji + bj_Wik) (7)

where the definitions of bii, Sij and Wij are the same as given earlier; P = -u-i_Ui,j and
[I = bijbij. This model involves seven empirical constants and their values are given by

C1 = 3.4, C_ = 1.80, C2 = 4.2

4

C3 = _, C_ = 1.30, C4 = 1.25, Cs = 0.40

The rapid part of the SSG model satisfies the constraints of incompressibility and symmetry

but it does not satisfy the normalization property. If the constraint of normalization was

imposed, the rapid part ofssG model would reduce to LRR model.

FLOWS COMPUTED

A total of thirteen flows are computed in the present study. These include two axisymmetric contractions (DNS)_2; two axisymmetric expansions (DNS)_2; four distortions

by plane strain (DNS)12; two homogeneous shear flows (one DNS 13 and one experiment_°);

and three homogeneous shear flows with rotation (LES) 14. For the flow through the axisymmetric contraction the mean velocity gradient is given by

)

_- 0 0

1 ov 0 Ui,i = 0 2 o,

1 ou

0 0 2 oz :_ : _ :

where au is positive. For the flow through axisymmetric expansion the mean veIoc]ty

gradient tensor is the same as above except that ou is negative. For distortion by plane

strain flow the mean velocity gradient tensor is given by

U_,j= 0 o__y_v 0

0 av

81/

where ay

_-y is negative. For the homogeneous shear flow the mean velocity gradient tensor

is given by

0 o__Uv O)

Ui,j = 0 °0_ 0

0 0 0

For the homogeneous shear flow in a rotating frame the mean velocity gradient tensor is

same as above but in addition it has a non-zero rotation rate which for the cases considered

here is given by fli = (0, 0, fl) where gt is constant and represents the rotation rate of the

6
non-inertial frame of referencerelative to the inertial frame of reference.In all thesecases

the mean velocity gradients are constant. All these flows constitute an initial value problem

and were numerically solved using a fourth order Runge-Kutta finite difference method.

3. RESULTS

Figure 1 compares the development of Reynolds stresses computed using these three

models in a flow through axisymmetric contraction with the DNS data. (Following Lee

et al. 12 all the stresses will be plotted as a function of the normalized time St.) Here we

show a typical case of 5' = 10.00 (Sko/_o = 5.573, case AXL). All the models deviate from

the DNS data. However, LRR model gives slightly better results than the SSG model for

both the components while IP model performs the worst. Figure 2 shows a comparison

for a much higher strain rate case (,5' = 100.00, Sko/_o = 55.73, case AXM). We note that

now all the models deviate even more from the DNS data. We also observe that the LRR

model gives better results than the SSG model.

Figures 3 and 4 show a similar comparison for flow through axisymmetric expansion for

two different strain rates 12. For the smaller strain rate flow (S = 0.717, Sko/_o = .408, case

EXO) SSG model reproduces the u 2 development quite well while both IP and LRR models

under-predict it. For the v 2 component all the models give similar results. Therefore, for

tMs low strain rate flow SSG model is better than the other two models. For the flow

with higher strain rate (S = 7.17, Sko/_o = 4.08, case EXP) the LRR model is in excellent

agreement with the DNS data for both the components while both IP and SSG models

show over-prediction So for this flow LRR model works the best.

Now we show comparisons for the distortion of turbulence by plane strain for four

cases of differing strain rates 12. We start from the smaller strain rate case. Figure 5

compares the evolution of the three non-zero Reynolds stress components for the flow with

strain rate S = 0.65 (Sko/eo = 0.577, case PXA). We observe that all the models predict

u'3- component reasonably with LRR model slightly better than the other two models. For

the v 2 component SSG model is closest to the data where as both LRR and IP models

7
slightly under-predict it. For the w s component all the three models give about the similar

results and predict it reasonably. Figure 6 shows the comparison for the case of S =- 1.3

(Sko/eo = 1.154, case PXB). The _- and w 2 components are reproduced better by the LRR

model. However, for the v s component SSG model gives better prediction than LRR and

IP models. The comparisons for the third case, with strain rate S = 2.6 (Sko/eo = 2.309,

case PXC), are shown in figure 7. For u _- component all the models under-predict the DNS

data. LRR model is slightly better than the SSG model. IP model is the worst of the

three. For v 2 component IP model works the best. LRR model slightly under-predicts v-_

while SSG over-predicts it. The third component w"-'_is over-predlcted by all the models

with LRR model being better than the other two. Figure 8 shows the similar comparisons

for the highest strain rate case (S = 25.0, Sok/eo = 22.227, case PXE). All the three

models under-predict the u s component. IP model is the worst of the three models. LRR

model gives slightly better result than the SSG model for this stress component. For v--7

component LRR model is the best and SSG model is the worst of the three. For the w---_-

component all the three models over-predict the DNS data with LRR model being closest

to the data. From the above four plane strain flow comparisons, we note that on the overall

LRR model works better than the other two models.

Now we turn our attention to homogeneous shear flow. First we show the comparisons

with the DNS data _3 for the shear rate S = 56.56 (Sko/eo = 2.36, case c128W). Figure 9

s_hows that both--LR--R and SSG models preform equally quite well. On the other hand IP

model is off the data for all the components.

Figure 10 shows the same comparison with the homogeneous shear flow experiment

of Tavoularis and Corrsin 1° (S = 46.8, Sko/eo = 6.46). (Following Tavoularis and Corrsin

all the Reynolds stresses shown here are normalized by mean velocity along the centerllne

of their wind tunnel and the streamwise distance is normalized by the height of the wind

tunnel.) For the u s component LRR model gives the best result whereas SSG and IP

models over-predict it. For the v s component also the LRR works the best. SSG model

slightly over-predicts the data whereas IP model is off by a larger margin. For the w--T
component both SSG and IP models reproduce the data very well whereas LRR model

over-predicts the data. For the shear stress component LRR performs very well whereas

SSG model over-predicts the data and IP model is off the data by a larger margin. So

for this experiment LRR model has better overall performance than the other two models.

Finally, we discuss the evolution of q2 for the case of rotating homogeneous shear flow.

Since no experimental or DNS data is available for this flow the comparisons will be made

(for two eases) with the LES of Bardina et al. TM. Bardina is has pointed out that in this

ease we should be careful in interpreting the comparisons and should use the LES data

for judging the model performance in a qualitative fashion. In all the cases shown here

the initial conditions corresponded to isotropic turbulence with eo/Sko = 0.298. Figure

11 shows the comparisons for the three eases of different Rosby numbers (= f_/S). For

f_/S = .25 we note that all the three models significantly under-predict the LE$ results

for q2 ; SSG being closest to the LES data and the LRR being the furthest. Qualitatively

all the three models reproduce the LES trends. For the case of I']/S = 0.50 SSG is in

excellent agreement with the LES results. Both IP and LRR give identical results and give

a smaller value of q2 than the LES. For the third case of f_/S = 1.0, all the three models

give identical results. Since no LES results are available for this case the only purpose of

showing the results is to see how the three models compare with each other.

4. CONCLUSIONS

Results were shown from numerical computation of various homogeneous turbulent

flows using three different turbulence models. All of these models use a linear (or quasilinear) model for the rapid part of the pressure strain term. The rapid term constant in the

LRR model was recalibrated to a slightly different value then the one originally proposed

by LRR. Based on their overall performance it is found that the recalibrated LRR model

works as good as the SSG and IP models. For the irrotational flows the differences the

performance of the LRR model improved over the other two models as the strain rate

increased. For the simple homogeneous shear flow experiment LRR model worked better

than the SSG model. For the DNS of the same flow both performed equally good. For

9
the rotating homogeneous shear flows both SSG and LRR model showed trends similar

to those shown by LES with SSG performing closer to LES. It is worth noting that SSG

model has seven empirical constants as compared to two in LRR model and stiU, on the

overall, it does not perform better than LRR model. Part of the reason for this may be

due to the fact that the SSG model does not satisfy the normalization constraint where

as LRR model does. For a model of the rapid pressure strain term which is tensorially

linear in the anisotropic tensor and satisfies a11 the three basic constraints LRR is the most

general model. The comparisons shown in this study demonstrate that the model obtained

by relaxing normalization property of the pressure correlation term does not lead to any

significant improvement over the LRR model and, therefore, this is not the right direction

in improving the performance of the current models.

10
REFERENCES

I Rotta, J.C. "Statistische Theorie Nichthomogener Turbulenz 1. Z. Phys., 1951, 129,

547-572.

2 Launder, B.E., Reece and W. Rodi. "Progress in the Development of Reynolds-stress

Turbulence Closure", J. Fluid Mech., 1975,68, 537-566.

s Lumley, J.L.,"Computational Modeling of Turbulent Flows". Adv. Appl. Mech.,

1978, 18, 123-176.

4 Shih, T.-H. and J.L. Lumley,"Modeling of Pressure Correlation Terms in Reynolds

Stress and Scalar Flux Equations". CorneU Rep. No. FDA-85-3, 1985, Sibley School of

Mech. and Aero. Engg., CorneU University, Ithaca, NY.

s Speziale, C.G., S. Sarkar and T.B. Gatski, "Modeling the Pressure-Strain Correlation

of Turbulence; An Invariant Dynamical Systems Approach". J. Fluid Mech., 1991, 227,

245-272.

e Shih, T.-H., and J.L. Lumley, "A Critical Comparison of Second Order Closures

with Direct Numerical Simulation of Homogeneous Turbulence", NASA TM 105351, 1992.

Fluid Mech., 1976, 8, 183-208.

7 Gibson, and B.E. Launder, "Ground Effects on Pressure in the Atmospheric Boundary Layer". JFM, 1978, 86, 491-511.

s Morris, P.J., "Modeling the Pressure Redistribution Terms". Phys. Fluids, 1984,

27, 7, 1620-1623.

9 Champagne, F.H. , V.G. Harris and S. Corrsin, "Experiments in Nearly Homogeneous Shear Flow", JFM, 1970, 41, 81-139.

10 Tavoularis, S and S. Corrsin, "Experiments in Nearly Homogeneous Turbulent Shear

Flow with a Uniform Mean Temperature Gradient", Part 1. 1981, JFM, 104, 311-347.

11 Taulbee, D.B. "Engineering Turbulence Models", Advances in Turbulence, ed. W.K.

George and R. Arndt, Hemisphere Publishing Corp.

12 Lee, M. and W.C. Reynolds, "Numerical Experiments on the Structure of Homogeneous Turbulence". Stanford Univ. Report No. TF-24, 1985.

la Rogers, M, P. Moin and W.C. Reynolds, " The Structure and Modeling of the

Hydrodynamic and Passive Scalar Fields in Homogeneous Turbulent Shear Flow", 1986,

Stanford Univ. Report No. TF-25.

14 Baxdina, J., J.H. Ferziger and W.C. Reynolds, "Improved Turbulence Models Based

on Large-Eddy Simulation of Homogeneous, Incompressible Turbulent Flows". Stanford

Univ. Report TF-19, 1983.

lSBardina, J., (1992) Private communication.

II
0.20

0.15-

U 2

0.I0-

0.05-

0.00

0.00

, I , ,

• DNS

LRR

..... IP

-"- - - SSG

0.50 : 1.00 : 1.50 2.00

St

0.20 i

0.15 I.

?]2

0.10-

0.05.

1 • I I

U ////

0.00 ' ' ' l ' ' ' i

0.00 0.50 1.00

St

• DNS

LRR

..... IF

.... SSG

1.50 2.00

Figure 1. Comparison of the models for flow through axisymmetHc contraction with the

DNS data of of Lee and Reynolds (1985), case AXL, S=10.0.

12
0.10

0.08-

0.06-

0.04

0.02

0.00

0.00

l . , , _ I .... I

• DNS

-- LRR

s

m

m

0.50 1.00 1.50

St

2.00,

0"30 I

o.25-I0.20-

V 2

0.15-

0.10-

0.05-

0.00

o.oo

_'"" • DNS

LRR

..... Ip

.... SSG

0.50 ::-: 1.00 1.50 2.003

St =

Figure 2. Comparison of the models for flow through axisymmetric contraction with the

DNS data of of Lee and Reynolds (1985), case AXM, S=100.0.

13
• DNS

LRR

0.( ..... IP

.... SSG

0.06 _1! ___i:r_z___-: ....

0.02

2100

0.02

[] DNS

--- LRR

..... IP

.... SSG

0.01

0.00 0.50 1.00 1.50 2.00

St

Figure 3. Comparison of the models for flow through axisymmetric expansion with the

DNS data of of Lee and Reynolds (1985), case EXO, S=0.717.

14
0.30

I

0.00 t '

0.00

LRR

..... IP

.... SSG

, ' ' i .... I ' ' I • ' '

0.50 1.00 1.50 2.00

, I I , 1 , ,/ • ]

/ ./ l

/ /

/

/ J

/" s. /

_, / /.

/ /"

/ d''

j s'.f

_._

St

0.30

0.20-

0.10-

• _ .... I .... I ,

• DNS

LRR

"_'_° IF

.... SSG

0•00 .... l ' i ' • • 1-- ......

0.00 0.50 i .00 i .50 2.00)

St

Figure 4. Comparison of the models for flow through a.-dsymmetric expansion with the

DNS data of of Lee and Reynolds (1985), case EXP, S=7.17.

15
• DNS

LRR

..... IP

.... SSG

U 2

0.01

0.0_

0.00 0.50 1.00 1.50 2.00

St

o.os 

J

0.02-

0.01 -

• DNS

_- LRR

..... Ip

.... SSG

0.00 ................ , ! , , , - , ,

0.00 0.50 1.00 1.50 2.00

St

Figure 5. Comparison of the models for distortion by plane strain with the DNS data of

Lee and Reynolds (1985), case PXA, S=0.65.

16
0.01

0.00

0.00

• DNS

LRR

.... IP

.... SSG

0.50 1.00 1.50

St

2.00

Figure 5 (continued from last page)

17
0.10

0.08-

0.06-

I __1

, , • , •

0"02 I

0.00, • • • i .... i

0.00 0.50 1.00

St

• I , , ,

m DNS

-- LRR

.... SSG

• /

• _:

-_ - _ _. _. .-_-. = -_- Z. 7. T---. ---

, ,_ L

1.50 2.00

0.10

0.08-

.... I , , I , I .

°v2JL.... I

0.04 - -- "-" -

0.02-

0.00-

0.00

• DNS

LRR

" .... IP

.... SSG

.... i .... i ..... i " ' "

0..50 1.00 1.50 2.00

St

Figure 6. Comparison of the models for distortion by plane strain with the DNS data of

Lee and Reynolds (1985), case PXB, S=1.3.

18
0.10 i .... . I , , , ! .... =I ,

• DNS

-- LRR

0.08- - .... IP -

.... SSG

006-

• _ . _ ................. _- --_._-

• • .... o

0.00 0.50 1.00 1.50 2.00

St

Figure 6 (continued from last page)

19
0.10

0.08

, ,, , , , I ,- , , • ' .... I ....

• DNS

LRR

.... IP

.... SSG

0.06-

0.02-

0.00 .... i .... i .... t ....

0.00 0.50 1.00 1.50

St

2.00

0.I0

0.08-

0.0___66-

132

0.041

0.02-

0.00-

0.00

.... I .... ..I .... 11 , /, , .

/_ /,/ //

,#/ / /'/

,S _' #."

/"_. C ,4

• DNS

-- LRR

..... IP

.... SSG

...... i .... I .... I .....

0.50 1.00 1.50 2.00

St

Figure 7. Compariso n of the models for distortion by plane strain with the DNS data of

Lee and Reynolds (1985), Case PXC, S--2,6.

2O
0.10

0.08-

0.06-

J • , J ° .... , J , • •

• DNS

LRR

..... IP

.... SSG

W 2

0.04 -I

__- _ • _. _ L_... "

0.02- "- "--.: -'---'-" = ';

0.00 .... _ .... _ .... I ....

0.00 0.50 1.00 1.50

St

Figure 7 (continued from last page)

2.00

21
0.05

0.00

0.00

0.0!

• DNS

LRR

..... IP

.... SSG

1,00 1.50

• DNS

_--- LRR

..... Ip

.... SSG

0.00

0.00 0.50 1.00 1.50 2.00

St

Figure 8. Comparison of the models for distortion by plane strain with the DNS data of

Lee and Reynolds (1985), case PXE, S=25.0

22
0.20

0.15-

0.10-

0.05

0.00

0.00

I .

II DNS

LRR

..... IP

.... SSG

.... i .... i .... ; ....

0.50 1.00 1.50

St

2.00

Figure 8 (continued from last page)

23
:200.00

150.00-

100.00-

50.00-

0.00-

0.00

.... ! .... I ,

• DNS

LRR

.... " IP

.... SSG

-'--"--'-_1 ......... 1 .... !....... ! -_-__

5.00 10.00 15.00 20.00 25.00 30.00

St

200.00 .,., t . ,

150.00-

100.00-

• DNS

LRR

..... Ip

.... SSG

I , • , I , 1 • , | , ,

/

I'

I'

I

I

I"

I"

/"

/"

/'

/"

/"

/ •

0.00 5.00 10.00 15.00 20.00 25.00 30.00

St

Figure 9. Comparison of the models for homogeneous shear flow with the DNS data of

Rogers et al. (1986), case C128W, S=56.568.

24
200.00

150.00-

W 2

.... I .... I .... t .... I .... I , , !

• DNS i"

/ LRR

I"

.... " IP

I"

.... SSG ,.

/

/

100.00 t ,///W

[ ,'/.,,

5o.oo-1: , "/-"

"_ -t7 0.00 • ". " .'_-t ....

0.00 5.00 10.00 15.00 20.00 25.00 30.OOJ

St

0.00 ._ .

%

UV "

%

- 100.- ',

%

I

I

- 150.- [] DNS ',

LRR

..... IP

.... SSG

-200. • • • ' _ .... _ .... _ .... i .... _ .......

0.00 5.00 10.00 20.00 25.00

-50.00-

15.00

St

30.00

Figure 9 (continued from last page)

25
40.00

30.O0-

u 2 0 4 u-_ xl

20.00-

10.00-

0.00-

7.00

, I , I I ,

• EXPERIMENT

.f

LRR 7"

..... Ip /

SSG

' I ' I ' I .... !

8.00 9.00 10.00 11.00

L

12.00

20.00 '

t 5.00-

V 2

•u2 x 10 4

10.00-

5.oo-

, _ -° . I

• EXPERIMENT

-- LRR

.... ' IP

.... SSG

I - , !

/

/

/

I

,s" f

0.00 ' _ ' i • ; '

7.00 8.00 9.00 10.00 11.00

x/H

z

|

12.00

Figure 10. Comparison of the models for homogeneous shear flow with the data Tavoularis

and Corrsin (1981).

26
40.00

30.O0-

w 2 xl 0 4

20.00-

I0.00-

0.00

7.00

I I

• EXPERIMENT

LRR

.... SSG

I , I

I ' I ' I ' I

8.00 9.00 10.00 11.00

x/H

12.00

20.00

15.00-

--_-_ X 10 4

10.00-

5.00-

I I

• EXPERIMENT

LRR

°--°_" rP

.... SSG

/

0.00- _. f , t "

7.00 8.00 9.00 10.00

=/H

I

11.00 12.00

Figure 10 (continued from last page)

27
5.00-

4.00-

3.00-

2.00-

1.00-_

TT_ t I .... _ , , , I .....

/

£Z/S = .25 ,,"

¢ /

/ /"

/

¢ /

/ /

o ," .. / 4

o ." .." / 1

o __

n _"-5 >/ 0 LES

.... SSG

0.00-

0.00 2.00 4.00 6.00 8.00 10.00

St,

-j

qo=

5.00

4.00-

3.00-

2.00-

t .00_

.... 4---. , . , -b.., , , . I . . ,,, , I ....

n/S = o.5o o LES

LRR

,i._. IF

.... SSG

0.00

0.00

.... ! ..... ___,.__L. __ .... i ....

2.00 4.00 6.00 8.00 1

St "

_ L 2.

).00

Figure 1I. Comparison of the models for {he rotating homogeneous-shear flow with the

large eddy simulation of B&rdlhk et al. (1983).

28
q_

5.00

4.00-

3.00-

2.00-

1.00-

0.00-

0.00 2.00

_IS = 1.0

4.00

St

6.00 8.00

LRR

.... SSG

10.00

Fisure 11 (continued from last page)

29
Form Approved

REPORT DOCUMENTATION PAGE OMBNo.0Z04-0188

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1. AGENCY USE ONLY (Leave blank) 12. REPORT DATE i3. REPORT TYPE AND DATES COVERED '"

December 1992

4. TITLE AND SUBTITLE

Critical Assessment of Reynolds Stress Turbulence Models Using

Homogeneous Flows

6. AUTHOR(S)

Aamir Shabbir and Tsan-Hsing Shih

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Lewis Re,arch Center

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAMES(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, D.C. 20546-0001

11. SUPPLEMENTARY NOTES

Technical Memorandum

5. FUNDING NUMBERS

WU-505-62-21

8. PERFORMING ORGANIZATION

REPORT NUMBER

E-7474

_'1'0. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM- 105954

ICOMP-92-24

CMOTT-92-12

Prepared for the 3]st Aerospace Sciences Meeting sponsored by the American Institute of Aeronautics and Astronautics,

Reno, Nevada, January 11-14, 1993.

12!. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified - Unlimited

Subjcc! Catcgory 34

13. ABSTRACT (Maximum 200 words)

In ntodcling the rapid part of the pressure correlation term in the Reynolds stress transport equations, cxlcnsivc use has

been made of its exact properties which were first suggested by Rolla. These, for example, have bccn employed in obtaining the widely used LRR model. Some recent proposals have dropped one of these properties to obtain new models.

We demonstrate, by computing some simple homogeneous flows, that doing so does not lead to any significant improvements over the LRR model and it is not the right direction in improving the performance of existing models. The reason

for this, in our opinion, is that violation of one of the exact properties can not bring in any new physics into the model.

We compute thirteen homogeneous flows using LRR (with a recalibrated rapid term constant), IP and SSG models. The

flows computed include the flow through axisymmetric contraction; axisymmetric expansion; distortion by plane strain;

and homogeneous shear flows with and without rotation. Results show that for most cases a properly calibrated LRR

model, which is the most general representation h)r a model linear in the anisolropic tensor, performs either better or as

good as the other two models of the same level.

14. SUBJECT TERMS

Turbulence, Reynolds stress models

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OF REPORT

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Unchtssificd

NSN 7540-01-280-5500

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OF ABSTRACT

Unclassified

15. NUMBER OF PAGES

30

"i6. PRICE CODE

..... j

A03

20. LIMITATION OF ABSTRACT

Standard Vorm 298 (Roy. 2 89)

Prescribed by ANSI Sld Z39 i8

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-r

g