A subgrid-scale model recently derived 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 applicable only to compressible turbulent flows in the limit of small density fluctuations, correlates somewhat poorly with the results of 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

Erlebacher, G.

Hussaini, M. Y.

Speziale, C. G.

Zang, T. A.

English

NASA Technical Reports Server

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 S U B G R I P S C A L E  N88- 7 4309 MODELING OF COHPRESSXBLE TURBULEWCE P i n a l  
Report [ I A S A )  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 
https://ntrs.nasa.gov/search.jsp?R=19880004927 2020-03-20T08:59:41+00:00Z
, 
ON TEE SWGRIIFSCALE MODELING OF COMPBESSIBLE l"RBULENCE 
C. G. Spezia le  
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 r e c e n t l y  der ived by Yoshizawa' f o r  use i n  t h e  
large-eddy s imula t ion  of compressible tu rbu len t  f lows is  examined from a fund- 
amental t h e o r e t i c a l  and computational s tandpoin t .  It is  demonstrated t h a t  
t h i s  model, which is  only app l i cab le  t o  compressible tu rbu len t  f lows i n  t h e  
l i m i t  of small d e n s i t y  f l u c t u a t i o n s ,  c o r r e l a t e s  somewhat poorly with the  re- 
s u l t s  o€ d i r e c t  numerical  s imula t ions  of compressible i s o t r o p i c  turbulence  a t  
low Mach numbers. An a l t e r n a t i v e  model, based on Favre - f i l t e r ed  f i e l d s ,  i s  
suggested which appears  t o  reduce these  l i m i t a t i o n s .  
Research was supported under t h e  Nat ional  Aeronautics and Space Adminis t ra t ion  
under NASA Contract  No. NAS1-18107 while  t he  au thors  were i n  res idence  a t  t h e  
I n s t i t u t e  f o r  Computer Appl ica t ions  i n  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 subgrid- 
scale 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 formula- 
tion of a Smagorinsky-type model which neglects entirely the effects of momen- 
tum exchanges between the small and large scales that are accounted for in the 
subgrid-scale Leonard and Cross stress terms. These inaccuracies can be re- 
duced 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 N N  - 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 mak- 
ing use of Kraichnan’s DIA formalism generalized t o  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 com- 
pressibility expansion). A Smagorinsky model for the subgrid scale stress was 
obtained to the first order when a state of isotropy was assumed for the zero- 
order solution. Comparable subgrid scale models of the gradient transport 
type were derived, to the first order, for the remaining subgrid scale corre- 
lations (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 decompo- 
sition 
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  
a t  
- ( P V k )  + - (pv v ) = - * + - + -
a x, k t  axk ax, a xll 
a a T  a ‘k + -  arc (K T) - 
where Vk is  the  v e l o c i t y  vec to r ,  p Z pRT is the  thermodynamic p res su re ,  
T is  the  temperature ,  and R i s  t h e  i d e a l  gas  cons tan t .  In  (6) - ( 7 ) ,  
2 avk av, 
= - 7j p p J  6k, + p(- + -) kR ax, axk u 
w Tv 
-N N 
w 
- N N  N N  + v V I  + v i v i )  - - p ( VkVR - VkV, + Viva k R  TkR 
2 2 avk av, av, 
ax, axk axk @ = - 3 p ( p z )  + p(- + -) - 
T - yv 
-NNN N N 
Q, Cpp(vkT - vkT + v’T k + vkTc + v’T’) k 
a r e  t h e  molecular stress t enso r ,  t h e  subgrid s c a l e  stress t enso r ,  t h e  v iscous  
d i s s i p a t i o n ,  and the  subgrid s c a l e  hea t  f l u x ,  r e spec t ive ly ,  whereas p i s  
t h e  dynamic v i s c o s i t y  and K is  t h e  thermal conduct iv i ty .  
I n  o rde r  t o  achieve c l o s u r e  of t he  equat ions  of motion ( 5 )  - (71 ,  subgr id  
scale models f o r  T~ and Qk must be provided. We r e c e n t l y  proposed t h e  
fo l lowing  models 4 
-4- 
where 
- N  - N m m  - 2 1 / 2  N 
(% = - p(vkvR - vkvR) + 2CRpA 11, 
S 
'kR 
N 
2 1 /2  a'? - "  m ( J  
Qk CpP [ (VkT - vkT) - - cR A I1 -1 
N S axk PrT 
R N 1 aGk a: 
'k~ 7 ax, axk 
= (-+-) 
PrT is t h e  tu rbu len t  P rand t l  number (which assumes a value of approximately 
1 /2 ) ,  and CR and CI a r e  dimensionless  cons t an t s  which were found t o  assume 
t h e  va lues  of 0.012 and 0.0066, r e s p e c t i v e l y ,  by c o r r e l a t i n g  wi th  t h e  r e s u l t s  
of d i r e c t  numerical  s imula t ions  of compressible i s o t r o p i c  turbulence.  This  
model may be viewed as a compressible analogue of t h e  l i n e a r  combination model 
of Bardina,  e t  a l .  . The subgrid scale stress model (12) was found t o  corre-  
l a t e  w e l l  wi th  t h e  results of d i r e c t  numerical  s imula t ions  of compressible 
i s o t r o p i c  turbulence  f o r  average Mach numbers ( f o r  average 
Mach numbers t h e r e  can be l o c a l i z e d  supersonic  reg ions  i n  t h i s  
f low4).  Cor re l a t ion  c o e f f i c i e n t s  i n  t h e  range of 80-95% were obtained a t  t h e  
t enso r  l e v e l  independent of t he  average Mach number (see Table 1). These 
d i r e c t  s imula t ions  of i s o t r o p i c  turbulence  were performed us ing  Four i e r  co l lo-  
c a t i o n  methods on a 963 g r i d  a t  a Reynolds number of ReA 2 40 based on 
t h e  Taylor microscale .  The i n i t i a l  energy spectrum was chosen t o  match t h a t  
measured by Comte-Bellot and Corrs in  , and t h e  r e s u l t s  ( inc luding  those t o  be 
5 
0 - < Mo - < 0.6 
Mo - > 0.4 
6 
- 5- 
shown la te r  i n  Fig.  1) were then non-dimensionalized wi th  r e spec t  t o  a 
r e fe rence  l e n g t h  scale and v e l o c i t y  s c a l e  of 10/r  cm and 1 cm/sec, 
r e s p e c t i v e l y  ( t h e  phys ica l  p r o p e r t i e s  of a i r  a t  room temperature  were used f o r  
t h e  re ference  d e n s i t y  and v i s c o s i t y ) .  The rms d e n s i t y  f l u c t u a t i o n s  
f < ~ ' ~ > ' / ~ / < p >  f o r  Mo = 0.1, Mo = 0.4, and Mo = 0.6 were 0.0034, rms 
0.0358, and 0.0739 r e s p e c t i v e l y ,  where <*> denotes  t h e  s p a t i a l  average. 
I n  c o n t r a s t  t o  t hese  compressible subgrid-scale  models based on Favre- 
f i l t e r i n g ,  Yoshizawa' r e p r e s e n t s  t he  subgrid scale Reynolds stress 
by the  Smagorinsky model (der ived formal ly  t o  the  f i r s t - o r d e r  using a two- 
v;C'v;' 
scale DIA method; see equat ion  (81) of Yoshizawa 1 ) 
where 
1 
vc * V I  e k k 
v =  * A2(2 3 3 )1'2 
e 'uu2 mn mn 
and cuu2 is a dimensionless  cons tan t  t h a t  was found t o  assume a va lue  of 
0.16. However, by t ak ing  the  trace of (16) it fol lows t h a t  
N v.2 = 0 
which i s  s t r i c t l y  t r u e  only  f o r  incompressible  turbulence.  Hence, as a l luded  
t o  ear l ie r ,  the  subgr id  scale model (16) of Yoshizawa' is  only appropr i a t e  f o r  
weakly compressible t u r b u l e n t  f lows ( t h i s  l i m i t a t i o n  arises s i n c e  
- 6- 
a small compress ib i l i t y  expansion was employed which r equ i r e s  t h e  d e n s i t y  
f l u c t u a t i o n s  t o  be small.) There is  an a d d i t i o n a l  problem wi th  the  
Smagorinsky model (16) even when i t  i s  r e s t r i c t e d  t o  flows with small d e n s i t y  
changes as pointed o u t  by Bardina,  e t  al.5 This  model neg lec t s  t he  kinematic  
Leonard stress and Cross stress Ckfi g iven by 
( t h e  t o t a l  kinematic  subgrid s c a l e  stress 'kg : Lkfi + CkR + where 
Rkll : vi;-v;- is t h e  kinematic  Reynolds stress). For flows at low Mach 
numbers 
between Favre f i l t e r e d  f i e l d s  and o rd ina ry  f i l t e r e d  f i e l d s  i s  n e g l i g i b l e .  
Hence, a t  low &ch numbers, the  model der ived  by Yoshizawa' would correspond 
t o  equat ion  ( 1 2 )  wi th  t h e  s c a l e  s i m i l a r i t y  p a r t  ( i . e . ,  t h e  f i r s t  term on t h e  
right-hand s i d e  of (12))  neglected.  I n  p a r t i c u l a r ,  i t  would correspond t o  t h e  
model 
Mo 5 0.1, d e n s i t y  changes a r e  extremely small and t h e  d i f f e r e n c e  
- 2 1 / 2  N 1 = 2CRpA 11- (Sk2 - 7 zmmSu)  
S 
D'kfi 
f o r  t h e  d e v i a t o r i c  p a r t  D ~ k 4 ,  of t h e  subgrid scale stress tensor .  The 
computations conducted by Er lebacher ,  e t  al.4 i n d i c a t e  t h a t  such a model cor- 
r e l a t e s  poorly with the  r e s u l t s  of d i r e c t  numerical s imula t ions  of compressi- 
b l e  i s o t r o p i c  turbulence  a t  low Mach numbers. To be s p e c i f i c ,  c o r r e l a t i o n s  a t  
t he  tensor  l e v e l  (i.e.,  c o r r e l a t i o n s  of exac t  versus  modeled subgrid s c a l e  
stress t enso r s )  f o r  MO = 0.1 were only of t he  o rde r  of 30% as compared t o  
t h e  approximate va lue  of 85% f o r  t he  more complete model given by (12) ( s e e  
Table  1 and 2) .  Th i s  improvement i n  the  case  of compressible flows i s  
I 
I 
-7- 
comparable to the improvement obtained by the linear combination model for in- 
compressible flows (see Bardina, et a ~ ~ ) .  
A distinguishing feature of compressible turbulence models is the exist- 
ence of an isotropic subgrid scale stress which cannot simply be absorbed into 
the pressure term as is customarily done for incompressible flows. The iso- 
tropic 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 iso- 
tropic part of the Leonard and Cross stress contributions. However, this 
model correlates even more poorly than the deviatoric part of the sub- 
grid 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 iso- 
tropic 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 di- 
rect 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- 
t i o n s  and c o r r e l a t e s  somewhat poorly with the  r e s u l t s  of d i r e c t  numerical  
s imula t ions  of compressible  i s o t r o p i c  turbulence  a t  small Mach numbers. I n  
o rde r  t o  overcome these  l i m i t a t i o n s ,  we be l i eve  t h a t  models should be based on 
Favre f i l t e r e d  f i e l d s  (where no r e s t r i c t i o n s  are placed on t h e  d e n s i t y  f luc tu -  
a t i o n s )  and t h e  Leonard and Cross stresses must be accounted f o r .  The com- 
p r e s s i b l e  subgrid scale models r e c e n t l y  developed by t h e  au thors4  appear t o  be 
more genera l  i n  t h i s  regard and are c u r r e n t l y  under s tudy i n  a research  pro- 
gram involving t h e  large-eddy s imula t ion  of compressible tu rbu len t  shea r  
flows 0 
- 9- 
REFERENCES 
1. A. Yoshizawa, Phys. F lu ids ,  Vol. 29, 2152 (1986). 
2. R. H. Kraichnan, Phys. F lu ids ,  Vol. 7, 1048 (1964). 
3. A. Yoshizawa and K. H o r i u t i ,  J. Phys. SOC. Japan, Vol. 54, 2834 (1985). 
4. G. Erlebacher ,  M. Ye Hussaini ,  C. G. Spez ia le ,  and T. A. Zang, ICASE 
Report No. 87-20 (1987). 
5. J. Bardina,  J. H. Fe rz ige r ,  and W. C. Reynolds, Report No. TF-19, Depart- 
ment of Mechanical Engineering, Stanford Univers i ty  (1983). 
6. G. Comte-Bellot and S. Corrs in ,  J. F lu id  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 o f  c o r r e l a t i o n  c o e f f i c i e n t s  of t h e  exact subgrid scale 
t a i n e d  ,from a 963 d i r e c t  s imula t ion  o f  compressible i s o t r o p i c  tu rbu lence  
(D 5 diagonal  t enso r  l e v e l ,  O D  5 off-diagonal t enso r  l eve l ,  V f v e c t o r  l e v e l ,  
cv - - N N - N  - N N N  - N  - = -~ (v ’v  + v v’), RkR -~v‘v‘ k R  k R  k R ’  s 5 scalar l e v e l ,  \R = -0(vkvR - vkvR>, CkR 
Mo G Average Mach number over t h e  computational domain). 
Tab le  2. Comparison o f  c o r r e l a t i o n  c o e f f i c i e n t s  between the e x a c t  and the 
modeled d e v i a t o r i c  subgrid s c a l e  Reynolds stress (g iven  by Eq. (21)) obtained 
from a 96’ d i r e c t  s imula t ion  o f  compressible i s o t r o p i c  turbulence 
m = + ( V V  - - VY’ 6 )). k R  3 m m  kR 
-1 1- 
i . . . - -  I -  
Figure 1. Scatterplot of the isotropic modeled1 versus the exact subgrid 
scale stress obtained from a 963 direct simulation of compressible 
isotropic turbulence at 
I ~ k k / 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. Spez ia le ,  G. Er lebacher ,  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 
I n s t i t u t e  f o r  Computer Appl ica t ions  i n  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 s t ra t ion  
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 qechnica l  Monitor: 
Richard W. Barnwell Submitted t o  Phys ics  of F lu ids  
F i n a l  Report 
6. Abstract 
A subgrid-scale  model r e c e n t l y  der ived by Yoshizawa f o r  use i n  t h e  large-eddy 
imula t ion  of compressible t u r b u l e n t  f lows i s  examined from a fundamental theo- 
e t i c a l  and computational s tandpoin t .  It i s  demonstrated t h a t  t h i s  model, which 
s only  app l i cab le  t o  compressible tu rbu len t  flows i n  the  l i m i t  of small d e n s i t y  
l u c t u a t i o n s ,  c o r r e l a t e s  somewhat poorly wi th  t h e  r e s u l t s  of d i r e c t  numerical  
imula t ions  of compressible i s o t r o p i c  turbulence  a t  low Mach numbers. An a l te r -  
a t i v e  model, based on Favre- f i l  t e r ed  f i e l d s ,  i s  suggested which appears  t o  reduce 
hese 1 i m i  t a  ti ons . 
17. Key Words (Suggested by AuthorM) 
i r e c t  s imula t ion ,  subgr id  s c a l e  
ode l ing ,  compressible homogeneous 
urbul  ence 
18. Distribution Statement 
3 4 -  Fluid  Mechanics and Heat 
Transfer  
Unclass i f ied  - unl imi ted  
NASA FORM 1626 OCT 86 
NASA-Langley, 1988 


On the subgrid-scale modeling of compressible turbulence

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