L 

1 

NASA Contractor Report 178420 

ICASE REPORT NO. 87-73 

ICASE 

ON THE SUBGRID-SCALE MODELING 

OF COMPRESSIBLE TURBULENCE 

C. G. Speziale 

G. Erlebacher 

T. A. Zang 

M. Y. Hussaini 

Contract No. NAS1-18107 

December 1987 

[NASA-CR-178420) ON THE SUBGRIPSCALE N88- 7 4309 MODELING OF COHPRESSXBLE TURBULEWCE Pinal 

Report [IASA) 14 p CSCL 200 

Unclas 

G3/34 01 17081 

INSTITUTE FOR COMPUTER APPLICATIONS W SCIENCE AND ENGUEZRING 

NA!U Langley Research Center, Hampton, Virginia 23665 

Operated by the Universities Space Research Association 
, 

ON TEE SWGRIIFSCALE MODELING OF COMPBESSIBLE l"RBULENCE 

C. G. Speziale 

ICASE, NASA Langley Research Center 

Hampton, VA 23665 

G. Erlebacher and T. A. Zang 

NASA Langley Research Center 

Hampton, VA 23665 

M. Y. Hussaini 

ICASE, NASA Langley Research Center 

Hampton, VA 23665 

ABSTRACT 

A subgrid-scale model recently derived by Yoshizawa' for use in the 

large-eddy simulation of compressible turbulent flows is examined from a fundamental theoretical and computational standpoint. It is demonstrated that 

this model, which is only applicable to compressible turbulent flows in the 

limit of small density fluctuations, correlates somewhat poorly with the results o€ direct numerical simulations of compressible isotropic turbulence at 

low Mach numbers. An alternative model, based on Favre-filtered fields, is 

suggested which appears to reduce these limitations. 

Research was supported under the National Aeronautics and Space Administration 

under NASA Contract No. NAS1-18107 while the authors were in residence at the 

Institute for Computer Applications in Science and Engineering (ICASE), NASA 

Langley Research Center, Hampton, VA 23665. 

i 
In a recent article, Yoshizawal developed sugrid-scale models for 

possible future use in the large-eddy simulation of compressible turbulent 

flows. These models were obtained by making use of a multiscale version of 

Kraichnan's DIA f~rmalism.~,~ While the study conducted by Yoshizawa is quite 

interesting, it will be pointed out in this work that the specific subgridscale stress model derived therein is limited to slightly compressible flows 

(i.e., to flows with small density fluctuations) and also correlates somewhat 

poorly with direct numerical simulations of isotropic compressible turbulence 

at low Mach numbers. This poor correlation arises from Yoshizawa's formulation of a Smagorinsky-type model which neglects entirely the effects of momentum exchanges between the small and large scales that are accounted for in the 

subgrid-scale Leonard and Cross stress terms. These inaccuracies can be reduced by a new subgrid-scale stress model that was recently derived by the 

 author^.^ The purpose of this work is to provide a comparison of these two 

alternative approaches. Subgrid-scale models are needed for the large-eddy 

simulation of compressible turbulence with important applications to high 

speed aerodynamic flows. 

In the large-eddy simulation of compressible turbulent flows, any field 

quantity F in the flow domain D can be decomposed as follows 

F = ? + F" 

where 

= G(x NN - z,A)F(z)d3z N 

D 

is the filtered part and F" is the subgrid scale part which accounts for 

1 the scales that are not resolved by the computational grid A. Yoshizawa 
-2- 

developed subgrid scale models (based on the filtering procedure (1)) by making use of Kraichnan’s DIA formalism generalized to the compressible regime 

with a multiscale expansion. It was assumed therein that the length and time 

scales of the fluctuating fields were small compared with those of the mean 

fields. Such an assumption restricts the applicability of the models to flows 

with small density fluctuations (i.e., it is equivalent to making a small compressibility expansion). A Smagorinsky model for the subgrid scale stress was 

obtained to the first order when a state of isotropy was assumed for the zeroorder solution. Comparable subgrid scale models of the gradient transport 

type were derived, to the first order, for the remaining subgrid scale correlations (e.g., the subgrid scale heat flux). 

As an alternative to this approach, in line with the more traditional 

analyses of compressible turbulent flows, a Favre filter can be defined as 

follows : 

P 

where p is the mass density. This gives rise to the alternative decomposition 

F=?+F’ (4) 

where F’ is the subgrid scale part of F based on Favre filtering. The 

Favre filtered equations of motion for an ideal gas are given by 4 

- + - a- (PVk) = 0 

at a% 
-3- 

aTkR 

- 

a- a -N 

at 

- (PVk) + - (pv v ) = - * + - + - 

a x, 

kt axk ax, a xll 

a aT a ‘k +- arc (K T) - 

where Vk is the velocity vector, p Z pRT is the thermodynamic pressure, 

T is the temperature, and R is the ideal gas constant. In (6) - (7), 

2 avk av, 

= - 7j ppJ 6k, + p(- + -) kR ax, axk u 

w Tv -NN 

w NN -N N - - p ( VkVR - VkV, + Viva + v VI + vivi) TkR kR 

2 2 avk av, av, 

ax, axk axk @ = - 3 p(pz) + p(- + -) - 

-NNN N N T - yv 

Q, Cpp(vkT - vkT + v’T k + vkTc + v’T’) k 

are the molecular stress tensor, the subgrid scale stress tensor, the viscous 

dissipation, and the subgrid scale heat flux, respectively, whereas p is 

the dynamic viscosity and K is the thermal conductivity. 

In order to achieve closure of the equations of motion (5) - (71, subgrid 

scale models for T~ and Qk must be provided. We recently proposed the 

following models 4 
-4- 

where 

-N - N mm - 2 1/2 N (% = - p(vkvR - vkvR) + 2CRpA 11, 

S 

'kR 

N -" m(J 2 1/2 a'? 

Qk CpP [ (VkT - vkT) - - cR A I1 -1 N S axk PrT 

N 1 aGk R a: 

'k~ 7 ax, axk 

= (-+-) 

PrT is the turbulent Prandtl number (which assumes a value of approximately 

1/2), and CR and CI are dimensionless constants which were found to assume 

the values of 0.012 and 0.0066, respectively, by correlating with the results 

of direct numerical simulations of compressible isotropic turbulence. This 

model may be viewed as a compressible analogue of the linear combination model 

of Bardina, et al. . The subgrid scale stress model (12) was found to correlate well with the results of direct numerical simulations of compressible 

isotropic turbulence for average Mach numbers (for average 

Mach numbers there can be localized supersonic regions in this 

flow4). Correlation coefficients in the range of 80-95% were obtained at the 

tensor level independent of the average Mach number (see Table 1). These 

direct simulations of isotropic turbulence were performed using Fourier collocation methods on a 963 grid at a Reynolds number of ReA 2 40 based on 

the Taylor microscale. The initial energy spectrum was chosen to match that 

measured by Comte-Bellot and Corrsin , and the results (including those to be 

5 

0 - < Mo - < 0.6 

Mo - > 0.4 

6 
- 5- 

shown later in Fig. 1) were then non-dimensionalized with respect to a 

reference length scale and velocity scale of 10/r cm and 1 cm/sec, 

respectively (the physical properties of air at room temperature were used for 

the reference density and viscosity). The rms density fluctuations 

f <~'~>'/~/<p> for Mo = 0.1, Mo = 0.4, and Mo = 0.6 were 0.0034, rms 

0.0358, and 0.0739 respectively, where <*> denotes the spatial average. 

In contrast to these compressible subgrid-scale models based on Favrefiltering, Yoshizawa' represents the subgrid scale Reynolds stress 

by the Smagorinsky model (derived formally to the first-order using a twov;C'v;' 

scale DIA method; see equation (81) of Yoshizawa 1 ) 

where 

1 

vc *VI e k k 

v= * A2(2 3 3 )1'2 

e 'uu2 mn mn 

and cuu2 is a dimensionless constant that was found to assume a value of 

0.16. However, by taking the trace of (16) it follows that 

N v.2 = 0 

which is strictly true only for incompressible turbulence. Hence, as alluded 

to earlier, the subgrid scale model (16) of Yoshizawa' is only appropriate for 

weakly compressible turbulent flows (this limitation arises since 
- 6- 

a small compressibility expansion was employed which requires the density 

fluctuations to be small.) There is an additional problem with the 

Smagorinsky model (16) even when it is restricted to flows with small density 

changes as pointed out by Bardina, et al.5 This model neglects the kinematic 

Leonard stress and Cross stress Ckfi given by 

(the total kinematic subgrid scale stress 'kg : Lkfi + CkR + where 

Rkll : vi;-v;- is the kinematic Reynolds stress). For flows at low Mach 

numbers 

between Favre filtered fields and ordinary filtered fields is negligible. 

Hence, at low &ch numbers, the model derived by Yoshizawa' would correspond 

to equation (12) with the scale similarity part (i.e., the first term on the 

right-hand side of (12)) neglected. In particular, it would correspond to the 

model 

Mo 5 0.1, density changes are extremely small and the difference 

- 2 1/2 N 1 = 2CRpA 11- (Sk2 - 7 zmmSu) S 

D'kfi 

for the deviatoric part D~k4, of the subgrid scale stress tensor. The 

computations conducted by Erlebacher, et al.4 indicate that such a model correlates poorly with the results of direct numerical simulations of compressible isotropic turbulence at low Mach numbers. To be specific, correlations at 

the tensor level (i.e., correlations of exact versus modeled subgrid scale 

stress tensors) for MO = 0.1 were only of the order of 30% as compared to 

the approximate value of 85% for the more complete model given by (12) (see 

Table 1 and 2). This improvement in the case of compressible flows is 
I 

I 

-7- 

comparable to the improvement obtained by the linear combination model for incompressible flows (see Bardina, et a~~). 

A distinguishing feature of compressible turbulence models is the existence of an isotropic subgrid scale stress which cannot simply be absorbed into 

the pressure term as is customarily done for incompressible flows. The isotropic part of Yoshizawa's model, in the limit of low Mach numbers for which 

the difference between Favre and ordinary filters are negligible, is given by 

which differs from the Erlebacher, et ala4 model by the absence of the isotropic part of the Leonard and Cross stress contributions. However, this 

model correlates even more poorly than the deviatoric part of the subgrid scale stress tensor when compared with the results of direct numerical 

simulations of isotropic compressible turbulence. A scatterplot of the 

modeled (Eq. (22)) versus exact isotropic subgrid-scale stress I~kk/3 is 

shown in Figure 1 which was obtained from a 963 direct simulation of isotropic compressible turbulence at MO = 0.1. The correlation coefficient for 

this case is 15%. This correlation improves to the 85% level when the Leonard 

and Cross stresses are in~luded.~ It must be kept in mind however, that direct numerical simulations (Erlebacher, et ala4) indicate that the isotropic 

Reynolds stress affects the correlations by at most a few percent for MO < 

0.4. 

D~kR 

In conclusion, it has been pointed out that the Smagorinsky-type subgrid 

scale model developed by Yoshizawa' from the DIA for compressible large-eddy 

simulations is not applicable to turbulent flows with large density fluctua- 
-8- 

tions and correlates somewhat poorly with the results of direct numerical 

simulations of compressible isotropic turbulence at small Mach numbers. In 

order to overcome these limitations, we believe that models should be based on 

Favre filtered fields (where no restrictions are placed on the density fluctuations) and the Leonard and Cross stresses must be accounted for. The compressible subgrid scale models recently developed by the authors4 appear to be 

more general in this regard and are currently under study in a research program involving the large-eddy simulation of compressible turbulent shear 

flows 0 
- 9- 

REFERENCES 

1. A. Yoshizawa, Phys. Fluids, Vol. 29, 2152 (1986). 

2. R. H. Kraichnan, Phys. Fluids, Vol. 7, 1048 (1964). 

3. A. Yoshizawa and K. Horiuti, J. Phys. SOC. Japan, Vol. 54, 2834 (1985). 

4. G. Erlebacher, M. Ye Hussaini, C. G. Speziale, and T. A. Zang, ICASE 

Report No. 87-20 (1987). 

5. J. Bardina, J. H. Ferziger, and W. C. Reynolds, Report No. TF-19, Department of Mechanical Engineering, Stanford University (1983). 

6. G. Comte-Bellot and S. Corrsin, J. Fluid Mech., Vol. 48, 273 (1971). 

i 
-1 0- 

t 

D 

OD 

V 

S 

Mo = 0.0 

93 82 93 81 

80 85 79 84 

46 72 46 71 

56 73 56 74 

Mo = 0.6 

L+C+R C+R L+C+R C+R 

Table 1. Comparison of correlation coefficients of the exact subgrid scale 

tained ,from a 963 direct simulation of compressible isotropic turbulence 

(D 5 diagonal tensor level, OD 5 off-diagonal tensor level, V f vector level, 

-N - N NN -N 

N N 

- - cv 

- = -~(v’v + v v’), RkR -~v‘v‘ kR kR k R’ s 5 scalar level, \R = -0(vkvR - vkvR>, CkR 

Mo G Average Mach number over the computational domain). 

Table 2. Comparison of correlation coefficients between the exact and the 

modeled deviatoric subgrid scale Reynolds stress (given by Eq. (21)) obtained 

from a 96’ direct simulation of compressible isotropic turbulence 

m 

= +(VV - - VY’ 6 )). kR 3 mm kR 
-1 1- 

i . . .-- IFigure 1. Scatterplot of the isotropic modeled1 versus the exact subgrid 

scale stress obtained from a 963 direct simulation of compressible 

isotropic turbulence at 

I~kk/3 

% = 0.1. 
Report Documentation Page 

1. Report No. 

NASA CR- 178420 

ICASE Report No. 87-73 

2. Government Accession No. 

I 

4. Title and Subtitle 

ON THE SUBGRID-SCALE MODELING OF COMPRESSIBLE 

IS page) 

I ed 13 

21. No. of pages 

TURBULENCE 

22. Price 

A0 2 

7. Authork) 

C. G. Speziale, G. Erlebacher, T. A. Zang, 

and M. Y. Hussaini 

19. Security Classif. (of this report) 

Uncl as si f i ed 

9. Performing Organization Name and Address 

Institute for Computer Applications in Science 

20. Security Classif. (of t 

Uncl as si : 

and Engineering 

Mail Stop 132C, NASA Langley Research Center 

Hampton, VA 23665-5225 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Admini stration 

Langley Research Center 

Hampton, VA 23665-5225 

3. Recipient's Catalog No. 

5. Report Date 

December 1987 

6. Performing Organization Code 

8. Performing Organization Report No. 

87-73 

10. Work Unit No. 

505-90-21-01 

11. Contract or Grant No. 

NAS1-18107 

13. Type of Report and Period Covered 

Contractor Report 

14. Sponsoring hgency Code 

5. Supplementa Notes 

Langley qechnical Monitor: 

Richard W. Barnwell Submitted to Physics of Fluids 

Final Report 

6. Abstract 

A subgrid-scale model recently derived by Yoshizawa for use in the large-eddy 

imulation of compressible turbulent flows is examined from a fundamental theoetical and computational standpoint. It is demonstrated that this model, which 

s only applicable to compressible turbulent flows in the limit of small density 

luctuations, correlates somewhat poorly with the results of direct numerical 

imulations of compressible isotropic turbulence at low Mach numbers. An alterative model, based on Favre-fil tered fields, is suggested which appears to reduce 

hese 1 imi ta ti ons . 

17. Key Words (Suggested by AuthorM) irect simulation, subgrid scale 

odeling, compressible homogeneous 

urbul ence 

18. Distribution Statement 

34- Fluid Mechanics and Heat 

Transfer 

Unclassified - unlimited 

NASA FORM 1626 OCT 86 NASA-Langley, 1988