A theoretical expression for the slow part (the nonlinear fluctuation part) of the pressure-strain rate is compared with simulations of anisotropic homogeneous flows. The objective is to determine the quantitative accuracy of the theory and to test its prediction that the generalized Rotta coefficient, a non-dimensionalized ratio of slow term to the Reynolds stress anisotropy, varies with direction and can be negative. Comparisons are made between theoretical and simulation values of the slow term itself and of the generalized Rotta coefficients. The implications of the comparison for two-point closure theories and for Reynolds stress modeling are pointed out

Shariff, K.

Weinstock, J.

English

NASA Technical Reports Server

N88-23108
Center for Turbulence Research
Proceedings of the Summer Program 1987
Evaluation of a Theory for Pressure-Strain Rate
By J. WEINSTOCK 1, and K. SHARIFF 2
A theoretical expression for the slow part (the non-linear fluctuation part) of the
pressure-strain rate is compared with simulations of anisotropic homogeneous flows.
The objective is to determine the quantitative accuracy of the theory and to test its
prediction that the generalized Rotta coefficient, a non-dimensionalized ratio of slow
term to the Reynolds stress anisotropy, varies with direction and can be negative.
Comparisons are made between theoretical and simulation values of the slow term
itself and of the generalized Rotta coefficients. The implications of the comparison
for two-point closure theories and for Reynolds stress modeling are pointed out.
1. Introduction and background
The slow pressure-strain rate correlation _j is a key term that occurs in Reynolds
stress modeling. Until recent years it was almost universally modeled according to
Rotta's (1951) prescription as
s= C _
_ij - q2 bij, (empirical model)
where bij =< u_u_ > -[1/3(q28ij)], u I is the fluctuating velocity along direction i,
q2 __< uiui > is twice the turbulent kinetic energy density, e is the rate of turbulent
kinetic energy dissipation, and C is an empirical constant referred to as Rotta's
constant. However, Lumley (1978) has shown that C cannot be constant and more
recently, it has been shown (Weinstock, 1981; 1982; 1985) that C is neither constant
nor the same for different directional components ij. These variations occur because
• _) depends on more than one scale of the turbulence field, and, in addition, the
scales vary with direction.
One way to account for the effect of all scales is for the slow term to be derived by
a two-point closure theory. Such a derivation has been carried out (Weinstock, 1981_
1982; 1985), a principal result of which was that _j can be expressed as an integral
over scalar energy spectra Eij(k). Standard closures such as the DIA (Kraichnan,
1959) and EDQNM (e.g., Cambon et al., 1981; Bertoglio, these proceedings) are
much more ambitious, and correspondingly complex, since they determine the en-
ergy spectrum itself. Here, Ei.i(k) is taken from simulations. It is hoped that the
relative simplicity will allow the present theory to be applied to a wider class of
flows - including those with relatively large anisotropies after suitable modeling of
1 NOAA/ERL/Aeronomy Laboratory
2 NASA Ames Research Center
t)RECEDtNG PAGE BLANK NOT FILMED
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214 J. Weinstock and K. Shaviff
the scalar spectra. This is encouraged by a "universal" character found for the
theoretical _i'j in which its dimensionless (Rotta) coefficients are very insensitive to
the shape of the energy spectrum in the small k (energy producing) region, provided
that the Reynolds number is not too small. We believe that such a universality is
crucial for predictive modeling of flows.
The theoretical _j to be tested is given by
_2_,j = -Cijebij, (No sum on i and j) (la)
):o :oCo_ = 1.08 _3) a/3 dkl dk2
eko qbo,_
1
× k_k_E(kz)[E,_,_(kl) - _E(kl)]H(k_,k2), (lb)
(k_+ k_)'/3
1
E(k) = _ [E_(k) + E,_(k) + E33(k)],
[ 2.4k2 2 (4k, k2 ) 2( 2k_ )]H(k,,ks) _ 2 k_+ k_ 0.08 ,_+ k_ 1 + _ 1 k,_+ k_ '
where the Cij are dimensionless coefficients referred to as generalized Rotta co-
efficients, Eaa(k) is the scalar spectrum for kinetic energy along direction a,
ko = (3/3) 3/2 e/q 3, and/3 is the Kolmogorov constant. The various numerical factors
arise from angular integrations of spectra in wave space. The specific angular de-
pendence of the spectra had to be modeled to make this possible. The off-diagonal
elements C12, Ca3, (723 are given elsewhere (Weinstock, 1981) and have not been
evaluated. A much simpler form of C,_ for use in Reynolds stress modeling is
a.213_.-s/3
obtained by use of the model spectrum E_ = w_,_ 0_ for ko < k < k_,, and
2/3trnt--rn-5/3
E,_ = PQ1 x Xo for k < ko, where k_ is the viscous "cut-off" wavenumber.
We refer to this E,_ as the model spectrum.
Our primary goal is to test expression (1) by comparison with computer simu-
lations. This test has also implications for standard two-point closures in general,
since such closures have in common with our closure the neglect of a two-time
fourth-order velocity cumulant.
Our article is outlined as follows: Straight forward comparisons are given in Sec.
2 where values of ¢_,_ and C,m = -@_,.,/(eb,.,,.,) obtained directly from simulations
are compared with (lb). Improvements and generalizations of the theory suggested
by simulations are in Sec. 3, and Sec. 4 contains suggestions for further simulation
tests.
2. Comparison between theory and simulation
The theory was compared with several simulations of homogeneous shear and
straining flows. Typical examples are given in Figures 1 through 3 for two cases of
homogeneous shear (S = U1,2, simulation runs C128U and C128X of Rogers, Moin
Bvaluation o] a Theory .for Pressure.Strain Rate 215
.5
"o. 0
v
v
-°5
_)
• THEORY 11
• THEORY 22
am THEORY 33
------- SIMULATION 11
......... SIMULATION 22
.... SIMULATION 33
i ! • i - ! - i • i ..-
......J
,.J" •
• • ...........in.........:'i ...............•
,.... "'il ............ -_ --_'"'"'"U" ........ ..--
2". .................... ; ............... . ....... I1" .... •
-"-k...
(a)
, I . I . i i I , I , ! ,
,T.
l.l
(b)-(d)
• INTEGRATION OF SIMULATION SPECTRA
SIMULATION
2
• i - i • o - u - i • i &-
d5
(b)
i
A
A
&
A
"t,lnl,lnl
(.I
2.5
2.0
1.5
1.0
.5
(c) _
, t . g
u . e .L.
dS
A •
R
i - ! • i . l • u - i
4
2
0
-2
-4
(d)
-6 , t
0
A •
A A
a I , I , I , I u f a I n I n I a I
10 15 5 10 15
St St
FIGURE 1. Slow pressure strain components for homogeneous shear (moderate
shear case C128U of Rogers, Moin & Reynolds, 1986). (a) Comparison of theory
with simulation for the diagonal components. (b)-(d) Generalized Rotta coefficients
as computed from the simulation data and using the simulation spectra in the
theory.
and Reynolds 1986) and for plane strain (strain directions are 2 and 3, simulation
run PXA of Lee and Reynolds 1985). For the shear cases the horizontal axis is the
total shear, St. For the plane strain case the horizontal axis is the eddy turnover
time. Figures 1 show the evolution of _ scaled on the initial dissipation rate, and
C',_,_ for run C128U, Figures 2 for run C128X and Figures 3 for plane strain run
PXA. Each graph includes simulation and theoretical values.
216 J. Weinstock and K. Shariff
.2
.1
0
-.1
-.2
(,)
• THEORY 11
• THEORY 22
B THEORY 33
SIMULATION 11
.......... SIMULATION 22
..... SIMULATION 33
• i - i " i • i - i • i
.............• ...............................=
.....=........ i, ......... ,,......... . .........u.........
• ' " i , i * l , I , I ,
y-.
¢.I
(b)-(d)
• INTEGRATION OF SIMULATION SPECTRA
SIMULATION
4
3
1
2
0
&
A •
• & •
A •
(b) *
; , i • i . t . i . * . I ,
2.5
2.0
Ne, 1.5
O
1.0
.5
• , • i "" l • I • i ' i • ,
• 6
0 14
10
§
• A
a -6
6
(c) (d!
, i , I , , * , * ' , ' • -10
2 4 6 8 10 12 0
St
• I • I • I k I • I - I •
A
A
• A
A •
l,l,l,l,l,
2 4 6 S 10
St
I i
12 14
FIGURE 2. Slow pressure strain components for homogeneous shear (high shear
case C128X of Rogers, Moin & Reynolds, 1986). a) Comparison of theory with
simulation for the diagonal components, b)-d) Generalized Rotta coefficients as
computed from the simulation data and using the simulation spectra in the theory.
(a) Dominant features of the (slow) pressure-strain rate
The simulation data show the following features:
- C_a varies between components.
- The normal components 611, 6'22, C33 each vary greatly during the simu-
lations. For example, 6'11, in plane strain run PXA, varies from -5 To +10
(in the unstrained direction). The Reynolds number defined as q4/(u_) varied
between 39.1 nad 69. Another example is that 6'22 varies from 0.6 to 2.5 in
homogeneous shear flow (C128U).
- C1] can be negative for many conditions. However, this does not imply that
Evaluation of a Theory for Pressure-Strain Rate 217
"o.
v
A
(m)
• THEORY 11
• THEORY 22
ni THEORY 33
SIMULATION 11
• SIMULATION 22• gee•o• De
.... SIMULATION 33
.06 .... ! .... i .... ! .... m .... i .... i ....
.04 • •
.02
0 "::"-_.......................................................................
-.02
(a) " •
_,04 .... l .... I .... i .... i . . . , i, , , n i, I n ,
r.)
(b)-(d)
• INTEGRATION OF SIMULATION SPECTRA
LI SIMULATION
5 .... i .... i .... _ .... i .... i .... i ....
A
10
-S
(b)
ra
.5
A
.3
.2
.1 A A
A(c)
.... I .... | .... | .... I .... I .... I . I i
•2 .4 .6 .8 1.0 1.2
teo/q _
1.4
.6
.S
.4
°_ .3'
.2
.... I .... I .... ! .... I .... I .... ! ....
A
(d) *
ol .... I .... I .... I .... I . . . i I i i , , i , , , ,
0 .2 .4 .6 .8 1.0 1.2
teo/qo2
FIGURE 3. Slow pressure strain components for homogeneous plane strain (case
PXA of Lee & Reynolds, 1985). a) Comparison of theory with simulation for the
diagonal components, b)-d) Generalized Rotta coefficients as computed from the
simulation data and using the simulation spectra in the theory.
1.4
the flow will not return to isotropy were the mean deformation to be removed
at that instant. The Lumley return to isotropy tensor, of which the pressure
strain rate is only a part, determines this.
Each of these qualitative features is predicted by the theory. There is good quan-
titative agreement of the pressure strain rate for the shear cases, discounting times
larger than St = 12 where the "box" size has an important influence. The agreement
is weaker for the case of plane strain. In the comparisons for the generalized Rotta
coefficients C_,_, the theoretical C'_,(a = 1,2,or 3) generally follows the trend of
the simulations; being small whenever the simulation C_ is small and large when-
218 J. Weinstoek and K. Shariff
ever the C_o is large, and most notably, passing through zero at the same time that
the simulation C_o (see Figure 3b) does.
During straining flows, the values of Co_, in the strained directions, were always
overpredicted by the theory. This discrepancy might be accounted for by the strong
temporal variations of b,_ which violates the present assumptions in the theory
which limit it to slow variations of boo and low mean strain-rate. Indeed, when
estimates are made to account for time-scale variation of boo by calculating the
-1 ...... :-- _1__ t _ i.lllle-Scale
...... _ ,, _._ .,a_a,_h_,_ '" ' that is used to derive (lb), the discrepancies
are reduced. However, for modeling purposes the original, unextended theory may
be sufficient.
(b) Unezpected .features of 'b_,_
- Simulations show that C,_ can be very small (much less than unity) in strain-
ing flows for a wide range of anisotropies. This smallness was unexpected, al-
though it is contained in the theory. Small Co,_ implies that intercomponent
energy transfer is a very weak process in straining flows.
- Cao can vary significantly with strong temporal variations of kinetic energy.
Surprising is the extent of difference between C11 and Cs3 in homogeneous
shear flows (Figures lb and ld).
These features are also found in the theory with small quantitative discrepancies.
3. Improvement and generalization of the theory
(i) An intriguing proposition is to derive _ij(X) ----< pS(xt)Sij(X _ +X) >, the two-
point correlation of the slow pressure-strain, from the theory. This was suggested
by Brasseur and obtained from simulations by Brasseur and Lee, Schiestel and
Rogallo (these proceedings). This correlation provides a more severe and detailed
test of closure theory than does C,_,_. It also provides a direct link between spatial
structures and Reynolds stress modeling. The theoretical &ij(x) was worked out
during the summer school, but was not compared with simulations at the time; for
example, one component of _#(x) is
1 /? /• =- dk dk. dOsinOcos(k lcosO)
_,o q
kSk_E(kb)H [ 5× ( ks + k_)4/3 sin sSE_s(k_)- _sin ?
where
8 cosS6 E_(k_)-_sin'8 E33(k_)
[ 4 k k: ]k_ =(k s+k_) 1+ 3(k2-+kZa)s]
(ii) Extend the theory to include spatial and temporal variations.
(iii) Revise the model scalar spectra E.,_(k) to account for small Reynolds num-
ber.
Evaluation of a Theory for Pressure-Strain Rate 219
As pointed out by Bertoglio the theory does not account for deformation of wave
vectors by the mean strain rate and should be modified accordingly.
(iv) Attempt to model the rapid pressure-strain rate, by a k-space closure.
(v) Compare the theoretical pressure-strain rate with a k- space model of Schiestel
(these proceedings).
4. Suggestions for future simulations
(a) Regarding the pressure-strain rate:
(i) Compare the theoretical two-point pressure-strain < pS(x')Sij(x' + x) > with
simulation.
(it) Generalize the theoretical < pSSij > to apply to channel flow, and then test
with simulations.
(iii) Using simulations, calculate the two-time fourth-moment velocity cumulant.
Determine the time scale for its decay. In particular, determine if this time scale
is shorter than the time scale for decay of second-moment correlations. Such an
ordering of time scales is basic for k-space closure theories in general, and also for
the present theory.
(b) Regarding a theory for modeling inhomogeneous flows:
I I I i !(i) Compare simulation values of < UiRjU k >, < OUiU j >, < 02U I >,
< u_u_Op/Oxk >, < Ou_Op/Oxj > with the eddy-damped quasi-Gaussian approxi-
mation (EDQG) and with a recent theory. These quantities are basic to modeling
weakly inhomogeneous and stratified flows.
(it) Verify whether or not the cumulant of
( U I I I e Ul2 12• VU )iU, iU i >=< > V < u i >. This was derived by a theory and,
if true, contradicts the quasi-Gaussian assumption for inhomogeneous flows.
acknowledgment
We are grateful to M. J. Lee, N. N. Mansour and R. S. Rogallo for invaluable
assistance with the databases, and for their interest despite their involvement in
several other CTR projects•
REFERENCES
CAMBON, C., JAENDEL, D. 8z MATHIEU, J. 1981 . J. Fluid Mech. 104,247-262.
KRAICHNAN, R. 1959 . J. Fluid Mech. 5, 497-543.
LEE, M. J. _ REYNOLDS, W. C. 1985 Numerical experiments on the structure of
homogeneous turbulence. Report No. TF-24. Mech. Engrg., Stanford Univ.
LUMLEY, J. L. 1978 . Adv. Appl. Mech. 18, 123-176.
ROGERS, M. M., MOIN, P., & REYNOLDS, W. C. 1986 The structure and mod-
eling of the hydrodynamic and passive scalar fields in homogeneous turbulent
shear flow. Report No. TF-25. Mech. Engrg., Stanford Univ.
ROTTA, J. 1951 . Z. Phys. 129, 547-572.
WEINSTOCK, J. 1981 . J. Fluid Mech. 105, 369-396.
22O
WEINSTOCK, J. 1982
WEINSTOCK, J. 1985
J. Weinstock and K. Shariff
• J. Fluid Mech. 116, 1-29.
• J. Fluid Mech. 154, 429-447.


Evaluation of a theory for pressure-strain rate

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