NASA Contractor Report 181864 

ICASE Report No. 89-43 

ICASE 

A CRITICAL COMPARISON OF TURBULENCE MODELS 

FOR HOMOGENEOUS SHEAR FLOWS IN A ROTATING 

FRAME 

Charles G. Speziale 

Thomas B. Gatski 

N. Mac Giolla Mhuiris P 

(NASA-CR-t81864) A CRITICAL COE4P~RISOI OP I8 9- 2 79 77 

TURBULEICE flODEIS FOB HOBOGBNEOUS SHEALB 

PLOUS I11 A ROTATIlG PBAiU3 Final Report 

(NASL Langley Research Center) 24 p Unclas 

CSCL 20D G3/30 0224106 

Contract No. NAS 1-1 8605 

June 1989 

Institute for Computer Applications in Science and Engineering 

NASA Langley Research Center 

Hampton, Virginia 23665-5225 

Operated by the Universities Space Research Association 

National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 23665-5225 
A CRITICAL COMPARISON OF TURBULENCE MODELS FOR 

HOMOGENEOUS SHEAR FLOWS IN A ROTATING FRAME 

Charles G. Speziale* 

ICASE, NASA Langley Research Center 

Hampton, VA 23665 

Thomas B. Gatski 

NASA Langley Research Center 

Hampton, VA 23665 

Nessan Mac Giolla Mhuiris 

Case Western Reserve University 

Cleveland, OH 44106 

ABSTRACT 

A variety of turbulence models, including five second-order closures and four twoequation models, are tested for the problem of homogeneous turbulent shear flow in a 

rotating frame. The model predictions for the time evolution of the turbulent kinetic 

energy and dissipation rate, as well as those for the equilibrium states, are compared with 

the results of physical and numerical experiments. Most of the two-equation models 

predict the same results for all rotation rates in which there is an exponential time 

growth of the turbulent kinetic energy and dissipation rate. The second-order closures 

are qualitatively superior since, consistent with physical and numerical experiments, 

they only predict this type of unstable flow for intermediate rotation rates in the range 

-0.1 5 O/S 5 0.6. For rotation rates outside this range, there is an exchange of stabilities 

with a solution whose kinetic energy and dissipation rate decay with time. Although the 

second-order closures are superior to the two-equation models, there are still problems 

with the quantitative accuracy of their predictions. 

* This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAN-18605 

and was carried out while the author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23665. 

i 
1. Introduction 

The problem of homogeneous turbulent shear flow in a rotating frame constitutes a 

non-trivial test of turbulence models since it incorporates arbitrary combinations of shear 

and rotation which can have either a stabilizing or destabilizing effect. Since the flow 

is statistically homogeneous, two-equation turbulence models and second-order closure 

models give rise to an initial value problem for a coupled set of nonlinear ordinary 

differential equations. This allows for a fairly detailed mathematical analysis of the 

models based on dynarnical systems theory and for the generation of inexpensive numerical 

solutions which have a high degree of accuracy. When these features are combined with 

the wealth of published results from physical and numerical experiments, a powerful 

tool for the testing and screening of turbulence models is obtained. This establishes the 

motivation for the present paper which constitutes an extension of the work of Speziale 

and Mac Giolla Mhuiris (1988). 

In this paper, a variety of two-equation models and second-order closure models will 

be tested for the problem of homogeneous turbulent shear flow in a rotating frame. The 

models to be considered are as follows: 

(i) the standard K - E model 

(ii) the nonlinear K - E model 

(iii) the RNG K - E model 

(iv) the Bardina K - E model 

(v) the Launder, Reece and Rodi model 

(vi) the Rotta-Kolmogorov model 

(vii) the Shih-Lumley model 

(viii) the FU, Launder and Tselepidakis model 

(ix) the RNG second-order closure model 

This represents a broad collection of both two-equation models and second-order 

closures which includes some of the older models as well as some very recent ones. 

Comparisons will be made with the physical experiments of Tavoularis and Corrsin (1981) 

and the large-eddy simulations of Bardina, Ferziger and Reynolds (1983). It will be shown 

that several of the models yield reasonably good predictions for pure shear. However, the 

predictions of all of the models degrade considerably for appreciable values of the ratio of 

the rotation rate to the shear rate, R/S. It will be shown that the two-equation models 

erroneously predict unstable flow (where the turbulent kinetic energy and dissipation rate 

grow exponentially in time) for all values of R/S. The second-order closures will be 

1 
demonstrated to be qualitatively superior to the two-equation models since they predict 

unstable flow only for an intermediate range of R/S; an exchange of stabilities occurs for 

other values of R/S so that the turbulence decays consistent with the results of physical 

and numerical experiments. Although the second-order closures are able to account for the 

stabilizing or destabilizing effect of rotations on shear flow, the quantitative accuracy of 

their predictions is rather disappointing. The most surprising finding in this regard is that 

most of the newer models do not yield improved predictions over the older second-order 

closure models which have a considerably simpler structure. 

2. The Turbulence Models to be Tested 

Three of the two-equation models (i.e., the standard K - E model, the RNG K - E 

model, and the Bardina K - E model) are of the eddy viscosity type. For these models, 

the Reynolds stress tensor is of the form (see Hanjalic and Launder 1972) 

where 

is the mean rate of strain tensor, bi is the mean velocity, K is the turbulent kinetic energy, 

and E is the turbulent dissipation rate. The constant Cp is typically taken to be 0.09 (in 

the RNG model, Cp = 0.0837). In the nonlinear K - E model of Speziale (1987), a 

quadratic correction is made to equation (1). For flows in rotating frames, the Reynolds 

stress tensor for the nonlinear K - E model is given by (see Speziale 1989) 

where CD = 1.68 and 

are the frame-indifferent Jaumann derivative, the intrinsic mean vorticity, and the local 

mean vorticity. In equation (5), Ri is the rotation rate of the reference frame relative to 

2 
an inertial framing and eijk is the permutation tensor. 

equation for the turbulent kinetic energy 

The Reynolds stress models (1)-(3) must be solved in conjunction with the transport 

(which is exact since the turbulence is homogeneous) and a modeled transport equation 

for the turbulent dissipation rate E, This modeled transport equation for E is of the general 

form (see Hanjalic and Launder 1972) 

For the standard K - E model and the nonlinear K - E model, 

CE1 = 1.45, Cc2 = 1.90 (9) 

whereas for the RNG K - E model of Yakhot and Orszag (1986) 

In the Bardina K - E model (see Bardina 1988 and Bardina, Ferziger and Reynolds 1983), 

CE1 and CE2 are functions of vij given by 

K 1- - C&l = 1.50 - 0.015- -WijWij 

E (2 

)1’2 

& (2 

K 1- - y2 C&2 = 1.83 + 0.15- -WijWij 

The second-order closure models are based on the Reynolds stress transport equation, 

which for rotating homogeneous turbulent flows takes the form 

where llij and Eij are the pressure-strain and dissipation rate correlations, respectively 

(again, eijk is the permutation tensor). In the simplified form of the Launder, Reece and 

Rodi (1975) model that will be considered in this study, the models for nij and Eij are 

as follows: 

3 

t 
where C1 and C2 are dimensionless constants that are usually taken to be 1.8 and 0.6, 

respectively. Equations (13)-( 15) are solved with the modeled dissipation rate equation 

(8) where CE1 = 1.45 and CE2 = 1.90. 

In the Rotta-Kolmogorov model (see Mellor and Herring 1973) and the RNG secondorder closure (Yakhot and Orszag 1988), the models for nij and &ij are given by 

where C1 = 3.21 and C2 = 0.224 for the Rotta-Kolmogorov model and C1 = 1.59 and 

C2 = 0.267 for the RNG second-order closure of Yakhot and Orszag (1988). Both the 

Rotta-Kolmogorov model and the RNG second-order closure are supplemented by the 

same modeled transport equation for E given in (8); in the Rotta-Kolmogorov model, 

CEl = 1.8 and CE2 = 2.0 whereas in the RNG second-order closure, CEl = 1.063 and 

CE2 = 1.72. It should be noted that the Launder, Reece and Rodi model, the RNG 

model, and the Rotta-Kolmogorov model have one characteristic in common: the modeled 

pressure-strain correlation is linear in the anisotropy tensor 

The Shih-Lumley model (1985) was constructed to satisfy realiza ity and incorporates 

more complex nonlinear anisotropic effects. The models for IIij and Eij take the form: 

4 

1 
where 

1 

a5 = -- lo (1 + 0.80) , F = 1 + 911 + 27111 (21) 

1 

p = 2 + GF (80.1 In [l + 62.4(-11+ 2.3111)]} (23) 

The turbulent dissipation rate E is obtained from the modeled transport equation (8) with 

CEl and CE2 taken as 

7 

5 CE1 = 1.44, CE2 = - + 0.49 [l - 0.331n(l - 55II)l (24) 

The FU, Launder and Tselepidakis model (1987) is one of the newest models to be 

developed by Launder and his coworkers at UMIST. This model satisfies realizability and 

in the limit of small anisotropies reduces to the simplified version of the Launder, Reece 

and Rodi model discussed earlier. Its representation for nij and &ij is as follows: 

5 
where 

C; = -6011, 7 = 0.6, T = 0.7 

The dissipation rate E is obtained from the standard modeled dissipation rate equation (8) 

with C+-l = 1.45 and CE2 = 1.90. 

For homogeneous turbulent shear flow in a rotating frame (see figure l), the mean 

velocity gradient tensor and the rotation rate of the reference frame are given by 

respectively. The nonlinear ordinary differential equations associated with each model 

are obtained in a straightforward fashion by substituting (28) into the tensor forms given 

above. These equations were solved using a Runge-Kutta-Fehlburg numerical integration 

scheme subject to the initial conditions 

(29) 

2 

3 K = KO, E = EO, ~ij = --Kobij 

at time t = 0, which corresponds to an initially isotropic turbulence. Since the transport 

equations for rij and E can be nondimensionalized into an equivalent set of transport 

equations for bij and s/SK, it follows that the solutions will only depend on the initial 

conditions through the dimensionless parameter E,/SK, (see Speziale and Mac Giolla 

Mhuiris 1988). 

3. Discussion of the Results 

The results obtained for the two-equation models will be discussed first. In figure 2, 

the time evolution of the turbulent kinetic energy K* K/Ko is shown as a function 

of dimensionless time t* = St for three rotation rates: 52/S = 0, 52/S = 0.25, and 

52/S = 0.5. The predictions of the standard and nonlinear K - E model, the Bardina 

K - E model, and the RNG K -E model are compared with the results obtained from the 

large-eddy simulations of Bardina, Ferziger and Reynolds (1983) for an initial condition 

of E~/SK~ = 0.296. It is rather disturbing to note that the newer models (i.e., the RNG 

and Bardina K - E models) yield substantially worse predictions than the standard K - E 

6 
model for pure shear as shown in figure 2a. While the RNG K - E model does well 

for the case where R/S = 0.25 (see figure 2b), it appears that this is no more than a 

coincidence since the model erroneously predicts the same results for all rotation rates 

and considerably overpredicts the growth rate of the turbulent kinetic energy for the 

R/S = 0 and R/S = 0.5 cases (see figures 2a and 2c). The Bardina K - E model does 

yield results that depend on R/S; however, the trends are in conflict with the large-eddy 

simulations. It predicts that the most energetic case occurs at R/S = 0 whereas largeeddy simulations indicate that the R/S = 0.25 case is the most energetic. Furthermore, 

in contrast to physical and numerical experiments, the Bardina K - E model does not 

predict a flow restabilization (where K and E --f 0 as t + 00) for all values of R/S that 

are sufficiently far outside of the range 0 5 R/S 5 0.5. 

The equilibrium states of each of the two-equation models are compared in Table 1 with 

the physical experiments of Tavoularis and Corrsin (1981) and the large-eddy simulations 

of Bardina, Ferziger and Reynolds (1983). It should be noted that since the LES results 

were not defiltered, they tend to considerably overpredict the anisotropy tensor and should 

only be considered for the trends that they exhibit. All the equilibrium states were 

computed with Cp = 0.09. Even though improved results are obtained when Cp is 

considered to be a function of the ratio of production to dissipation as suggested by Rodi 

(1972) (which yields (Cp), = 0.055 as used by Speziale and Mac Giolla Mhuiris 1988), 

in this study we will only show the equilibrium results corresponding to Cp = 0.09, since 

this value is almost always adopted in applications. Several observations concerning these 

results are noteworthy: (a) both the standard and nonlinear K - E models yield the same 

values for and (b12)~ which are barely in the range of the physical experiments 

for pure shear, (b) only the nonlinear K - E model is able to reproduce the trends of 

the large-eddy simulations for the normal components (bll), and (b2~)~, and (c) the 

RNG model drastically overpredicts the values of (bl2), and (SKle), in pure shear 

which accounts for the overly large growth rate of the turbulent kinetic energy shown in 

figure 2a. 

The equilibrium diagrams of (e/SK), are shown as a function of !J/S in figure 3 for 

the standard, nonlinear, RNG and Bardina K - E models. It is clear from this figure 

that all of the two-equation models considered in this study are incapable of properly 

accounting for the effects of rotations on turbulent shear flow. The standard, nonlinear, 

and RNG K - E models erroneously predict the same results for K and E independent of 

the rotation rate. While the Bardina K - E model yields equilibrium states that depend 

on R/S, they appear to be unphysical. For example, there are no stable equilibrium 

7 
solutions for rotation rates outside of the range -0.9 5 R/S 5 1.9; thus, the model does 

not predict the flow restabilization that is known to occur for most R/S outside of the 

range 0 5 R/S 5 0.5. 

The time evolution of the turbulent kinetic energy predicted by the various secondorder closure models is shown in figure 4 for the same three cases discussed above. Three 

important conclusions can be drawn from these results: (a) the Shih-Lumley model and 

Rotta-Kolmogorov model predict too small a growth rate for pure shear whereas that 

predicted by the RNG model is too large in the initial stages (see figure 4a), (b) all of 

the models yield rather bad predictions for the most energetic case of R/S = 0.25 (see 

figure 4b), and (c) the Launder, Reece and Rodi model, the Shih-Lumley model and the 

Fu, Launder and Tselepidakis model erroneously predict stable flow at R/S = 0.5 whereas 

the RNG second-order closure has much too strong a growth rate (see figure 4c). 

The equilibrium values predicted by each second-order closure is compared in Table 2 

with the physical experiments of Tavoularis and Corrsin (1981) and the large-eddy 

simulations of Bardina, Ferziger and Reynolds (1983). The performance of these models 

can be summarized as follows: (a) the Launder, Reece and Rodi model does reasonably 

well in predicting the equilibrium values for pure shear but is defective in its prediction of 

a premature restabilization at R/S x 0.39, (b) the Rotta-Kolmogorov model erroneously 

predicts the opposite extreme wherein the R/S = 0 and R/S = 0.5 case are equally 

energetic since they have the same values for (b12), and (SKI&),, and (c) the Fu, 

Launder and Tselepidakis model does the best in its prediction of the equilibrium states 

for pure shear; however, it also erroneously predicts a premature flow restabilization 

at R/S x 0.39. It should be noted that the erroneous prediction of (b12), = 0 and 

(SK/E), = 00 made by the RNG second-order closure is directly tied to its rather low 

value of CEl = 1.063. This model could become more competitive with the other secondorder closures if CEl was of the order of 30% larger. 

A characteristic equilibrium diagram for the second-order closures is shown in figure 5 

(note that all the models possess the same topological structure for (c/SK), taken as a 

function of R/S). The models have two equilibrium solutions: one where (E/SK)~ # 0 

(the elliptic region AB) for which E and K grow exponentially with time at the same rate, 

and one where (EISK), = 0 for which E and K predominantly decay with time. Hence, 

the second-order closures properly predict unstable flow only for intermediate rotation 

rates which, depending on the model, can lie anywhere in the ranges -0.11 5 Q/S 5 0.39 

to -0.09 5 R/S 5 0.61. In this intermediate band of R/S, the two solutions undergo 

an exchange of stabilities which qualitatively mimics the shear instability. Furthermore, 

8 
the second-order closures predict a universal equilibrium (i.e., one that attracts all initial 

conditions) in the unstable regime consistent with the experiments of Tavoularis and 

Corrsin (1981). 

4. Concluding Remarks 

It has been demonstrated in this study that the existing two-equation turbulence 

models of the K - E type do not properly account for the stabilizing or destabilizing 

effect of rotations on turbulent shear flow. Furthermore, the quantitative predictions of 

many of these two-equation models are rather poor for the simple case of pure shear flow. 

It appears that the nonlinear K - E model yields the best results among the two-equation 

models, however it is still highly deficient. The second-order closures are far superior 

in a qualitative sense since they do properly predict that there is unstable flow only 

for intermediate rotation rates that are not too far removed from the unstable regime 

of 0 5 R/S 5 0.5 predicted by linear stability theory. However, two major problems 

remain with the quantitative accuracy of the results of the second-order closures: (a) the 

prediction of a premature flow restabilization at R/S x 0.39, or (b) the prediction of poor 

equilibrium values for the case of pure shear, s2/S = 0. Since it is important to predict 

simple shear flows accurately in engineering applications, it is our opinion that (of the 

models considered in this study) the Fu, Launder and Tselepidakis (1987) model is on 

balance the best. However, like the other existing second-order closures, it does not yield 

accurate results when there is both shear and rotation. The development of significantly 

improved models based on invariance arguments and a dynamical systems approach will 

be the subject of a future paper. 

Acknowledgment 

One of the authors (MGM) acknowledges the support of the National Aeronautics 

and Space Administration while in residence at ICOMP, NASA Lewis Research Center, 

Cleveland, OH. 

9 
References 

Bardina, J. 1988 Turbulence modeling based on direct simulation of the Navier-Stokes 

equations. Proceedings of the 1st National Fluid Dynamics Congress, Paper No. 88- 

3747-CP, 539. 

Bardina, J., Ferziger, J. H., and Reynolds, W. C. 1983 Improved turbulence models based 

on large-eddy simulation of homogeneous, incompressible turbulent flows. Stanford 

University Technical Report TF-19. 

Fu, S., Launder, B. E., and Tselepidakis, D. P. 1987 Accommodating the effects of 

high strain rates in modelling the pressure-strain correlation. UMIST Mechanical 

Engineering Department Report TFD/87/5. 

Hanjalic, K. and Launder, B. E. 1972 A Reynolds stress model of turbulence and its 

I application to thin shear flows. J. Fluid Mech. 52, 609. 

Launder, B. E., Reece, G., and Rodi, W. 1975 Progress in the development of a Reynolds 

stress turbulence closure. J. Fluid Mech. 68, 537. 

Mellor, G. L., and Herring, H. J. 1973 A survey of mean turbulent field closure models. 

AIAA J. 11, 590. 

Rodi, W. 1972 The prediction of free turbulent boundary layers by use of a two-equation 

model of turbulence. Ph.D. Thesis, University of London. 

Shih, T-H., and Lumley, J. L. 1985 Modeling of pressure correlation terms in Reynolds 

stress and scalar flux equations. Technical Report No. FDA-85-3, Cornel1 University. 

Speziale, C. G. 1987 On nonlinear K - 1 and K - E models of turbulence. J. Fluid Mech. 

178, 459. 

Speziale, C. G. 1989 Turbulence modeling in non-inertial frames of reference. Theoret. 

Comput. Fluid Dynamics, 1, 3. 

Speziale, C. G., and Mac Giolla Mhuiris, N. 1988 On the prediction of equilibrium states 

in homogeneous turbulence. Proceedings of the 1 lth Symposium on Turbulence, 

University of Missouri-Rolla, B17:l. 

Tavoularis, S. and Corrsin, S. 1981 Experiments in nearly homogeneous turbulent shear 

flows with a uniform mean temperature gradient. Part I. J. Fluid Mech. 104, 311. 

Yakhot, V. and Orszag, S. A. 1986 Renormalization group analysis of turbulence. I. Basic 

theory. J. Sci. Comp. 1, 3. 

Yakhot, V. and Orszag, S. A. 1988 Private communication. 

10 
Equilibrium Standard 

Values K--E Model 

Nonlinear 

K-E Model 

n/s = 0.0 

bii 0 

RNG Bardina Large-Eddy 

K--E Model K--E Model ExDeriments Simulations 

nls = 0.25 

0 

0.353 

-0.252 

-0.212 

4.71 

0.050 

0.050 

-0.212 

4.71 

-0.252 

0.353 

-0.212 

4.71 

n/S = 0.5 

4.71 

0 0 0.201 0.30 

0 0 -0.147 -0.26 

-0.489 -0.247 -0.150 -0.15 

11.70 5.49 6.08 

0 0 - 0.06 

0 0 - 0.05 

-0.489 -0.218 - -0.35 

11.70 4.84 - - 

0 0 - -0.26 

0 0 - 0.25 

-0.489 -0.194 - -0.10 

11.70 4.30 - - 

I I I- I I 

Table 1. Equilibrium results for homogeneous turbulent shear flow in a rotating frame. Comparison of the predictions of the two-equation models with the Large-Eddy Simulations of Bardina, Feniger, and Reynolds 

(1983) and the experiments of Tavoularis and Corrain (1981). 

Table 2. Equilibrium results for homogeneous turbulent shear flow in a rotating frame. Comparison of the predictions of the second-order closure models with the Large-Eddy Simulations of Bardina, Ferziger, and 

Reynolds (1983) and the experiment of Tavoularis and Corrsin (1981). (LRR E Launder, Reece and 

Rodi, RK E Rotta-Kolmogorov, SL Shih-Lumley, FLT Fu, Launder and Tselepidakiis). 

11 
List of Figures 

Figure 1. Homogeneous turbulent shear flow in a rotating frame. 

Figure 2. Time evolution of the turbulent kinetic energy in rotating shear flow. Comparison of the predictions of the two-equation models with the Large-Eddy Simulations of 

Bardina, et al. (1983): (a) R/S = 0, (b) R/S = 0.25, and (c) R/S = 0.5. 

Figure 3. Equilibrium diagram for the two-equation turbulence models in rotating shear 

flow. 

Figure 4. Time evolution of the turbulent kinetic energy in rotating shear flow. Comparison of the predictions of the second-order closure models with the Large-Eddy Simulations 

of Bardina, et al. (1983): (a) R/S = 0, (b) R/S = 0.25, and (c) R/S = 0.5. 

Figure 5. Equilibrium diagram for the second-order closure models in rotating shear flow. 

12 
4.0 

3.5 

3.0 

2.5 

* = 2.0 

1.5 

1.0 

0.5 

0.0 

0 

LES 0 

Standard and Nonlinear K-E Model - 

RNG K-E Model --- 

Bardina K-E Model e.+ 

I / 

/ I 

I 

I n/s = 0.0 I 

I 

/ I 

I I' 

/ 0 

I I I I 

2 4 6 

t* 

8 10 

Figure 2. Time evolution of the turbulent kinetic energy in rotating shear flow. Comparison of the predictions of the two-equation models with the Large-Eddy 

Simulations of Bardina, et al. (1983): (a) n/S = 0, (b) 

n/S = 0.25, and (c) n/S = 0.5. 

14 
Figure 1. Homogeneous turbulent shear flow in a rotating frame. 

13 
4.0 

3.5 

3.0 

2.5 

1.5 

1.0 

0.5 

0.0 

0 2 4 

t* 

Figure 2(b). 

6 8 10 

15 
4.0 

3.5 

3.0 

2.5 

* z- 2.0 

1.5 

7.0 

0.5 

n/S = 0.5 

I 

I 

I 

I 

I 

I 

I 

I / 

1 / 

I / 

0.0 

0 2 4 6 8 70 

t' 

Figure 2(c). 

16 
8 0.20 

0.30 

Standard and Nonlinear K-E Model 

- 

0.70 - RNG K-E Model 

0.00 1 I I I I I 

-7.0 0.0 7.0 2.0 

n/s 

0.05 

0.25 

- 

I I I I I 

0.20 

?, 0.70 

w 

Bardina K-E Model 

-7.0 0.0 7.0 

vs 

2.0 

Figure 3. Equilibrium diagram for the two-equation 

turbulence models in rotating shear flow. 

17 
LES 0 FLT - - - 

LRR -- RNG - 

RK SL --- . . . . . . . . 

4. 

3. 

3. 

2. 

2. 

7. 

7. 

0. 

0. 

0 2 4 6 8 70 

t" 

Figure 4. Time evolution of the turbulent kinetic energy in rotating shear flow. Comparison of the predictions of the second-order closure models with the 

Large-Eddy Simulations of Bardina, et al. (1983): (a) 

fl/S = 0, (b) fl/S = 0.25, and (c) fl/S = 0.5. 

18 
4.0 

3.5 

3.0 

2.5 

* 2.0 

1.5 

7.0 

0.5 

0.0 

0 2 4 6 

t* 

8 10 

Figure 4( b) . 

19 
4.0 

3.5 

3.0 

2.5 

c 2.0 

k 

7.5 

7.01 

0.5 

0.0 

0 

n/s = 0.5 

4 6 

t' 

8 70 

Figure 4( c) . 

20 
0 

A B 

Figure 5. Equilibrium diagram for the second-order 

closure models in rotating shear flow. 

21 
Report Documentation Page 

1. Report No. 2. Government Accession No. 

NASA CR- 18 1864 

3. Recipient's Catalog No. 

ICASE Report NO. 89-43 I 

4. Title and Subtide 

6. Performing Organization Code 

A CRITICAL COMPARISON OF TURBULENCE MODELS 

FOR HOMOGENEOUS SHEAR FLOWS IN A ROTATING FRAME 

5. Report Date 

7. Authorls) 

C. G. Speziale 

T. B. Gatski 

N. Mac Giolla Mhuiris 

8. Performing Organization Report No. 

89-43 

Institute for Computer Applications in Science 

Mail Stop 132C, NASA Langley Research Center 

Hampton, VA 23665-5225 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Langley Research Center 

Hampton, VA 23665-5225 

and Engineering 

11. Contract or Grant No. 

NAS1-18605 

Contractor Report 

13. Type of Report and Period Covered 

14. Sponsoring Agency Code 

Langley Technical Monitor: 

Richard W. Barnwell 

17. Key Words (Suggested by Authork)) 

Seventh Symposium on Turbulent 

Shear Flows 

18. Distribution Statement 

Final Report 

16. Abstract 

19. Secsrity Classif. !of :his report) 20. Security Classif. !of !his page) 121. No. of pages 

Unclassified i 23 Un cl as s i f i 2 d 

A variety of turbulence models, including five second-order closures and four two 

equation models, are tested for the problem of homogeneous turbulent shear flow in 

a rotating frame. The model predictions for the time evolution of the turbulent 

kinetic energy and dissipation rate, as well as those for the equilibrium states, 

are compared with the results of physical and numerical experiments. Most of the 

two-equation models predict the same results for all rotation rates in which there 

is an exponential time growth of the turbulent kinetic energy and dissipation 

rate. The second-order closures are qualitatively superior since, consistent with 

physical and numerical experiments, they only predict this type of unstable flow 

for intermediate rotation rates in the range -0.1 < il / S < 0.6. For rotation 

rates outside this range, there is an exchange of stabilities with a solution 

whose kinetic energy and dissipation rate decay with time. Although the secondorder closures are superior to the two-equation models, there are still problems 

with the quantitative accuracy of their predictions. 

22. Price 

A0 3 

turbulent shear flow; rotati.ng 

frames; k- models; second-order 

closure models 

34 - Fluid Mechanics & Heat Transfer 

Unclassified - Unlimited 

I 

IASA FORM 1626 OCT 86 

NASA-Langley, 1989 

_____~ ~~