In a recent paper, the authors compared the performance of a variety of turbulence models including the k-epsilon model and the second-order closure model based on Renormalization Group (RNG) Methods. The performance of these RNG models in homogeneous turbulent shear flow was found to be quite poor, apparently due to the value of the constant C(sub epsilon1) in the modeled dissipation rate equation which was substantially lower than its traditional value. However, recently a correction has been made in the RNG based calculation of C(sub epsilon1). It is shown that with the new value of C(sub epsilon1), the performance of the RNG k-epsilon model is substantially improved. On the other hand, while the predictions of the revised RNG second-order closure model are better, some lingering problems still remain which can be easily remedied by the addition of higher order terms

Fitzmaurice, Nessan

Gatski, Thomas B.

Speziale, Charles G.

English

NASA Technical Reports Server

NASA Contractor Report 187552
ICASE Report--No, 91-37
ICASE
AN ANALYSIS OF RNG BASED TURBULENCE MODELS
FOR HOMOGENEOUS SHEAR FLOW
Charles G. Speziale
Thomas B. Gatski
Nessan Fitzmaurice
Contract No. NAS1-I8605
April 1991
Institute for Computer Applications in Science and Engineering
NASA Langley Research Center
Hampton, Virginia 23665-5225
Operated by the Universities Space Research Association
IXl/ /X
Nalional Aeronautics and
Space Adminislralion
Langley Research Center
Hampton, Virginia 23665-5225
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https://ntrs.nasa.gov/search.jsp?R=19910014120 2020-03-19T17:36:40+00:00Z

AN ANALYSIS OF RNG BASED TURBULENCE
MODELS FOR HOMOGENEOUS SHEAR FLOW
Charles G. Speziale*
ICASE, NASA Langley Research Center
Hampton, VA 23665
Thomas B. Gatski
NASA Langley Research Center
Hampton, VA 23665
Nessan Fitzmaurice
Case Western Reserve University
Cleveland, OH 44106
ABSTRACT
In a recent paper [Phys. Fluids A2:1678-1684, 1990], the authors compared the perfor-
mance of a variety of turbulence models including the K - e model and the second-order
closure model derived by Yakhot and Orszag based on Renormalization Group (RNG) meth-
ods. The performance of these RNG models in homogeneous turbulent shear flow was found
to be quite poor, apparently due to the value of the constant Col in the modeled dissipation
rate equation which was substantially lower than its traditional value. However, recently a
correction has been made in the RNG based calculation of Cel. It is shown herein that with
the new value of Gel, the performance of the RNG K - _ model is substantially improved.
On the other hand, while the predictions of the revised RNG second-order closure model are
better, some lingering problems still remain which can be easily remedied by the addition of
higher-order terms.
*This research was supported by the National Aeronautics and Space Administration under NASA Con-
tract No. NAS1-18605 while the first author was in residence at the Institute for Computer Applications in
Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665.

A comparative study of the performance of nine independent turbulence models in ro-
tating homogeneous shear flow was recently reported by Speziale e_ aL 1 Two of the models
considered consisted of the K - e model and second-order closure model derived by Yakhot
and Orszag 2 using Renormalization Group (RNG) methods. It was rather surprising how
poorly the RNG models performed in homogeneous shear flow relative to the older, em-
pirically based models of the same general type. The origin of the deficient predictions of
the RNG models appeared to be largely due to the rather low value of the constant 6',1 in
the modeled dissipation rate equation; the RNG value of C,1 was 1.083 in contrast to the
more traditional value of 6',1 = 1.44. However, a recent re-examination of the RNG based
calculation of Gel by Yakhot and Smith 3 has led to a correction - the new value of C,1 is
1.42. Some minor changes in the values of other constants in the RNG K - e model were also
made. s In light of these changes, it would be desirable to set the record straight in regard
to what these Renormalization Group models now predict for homogeneous shear flow - a
critical test case used to evaluate the performance of models. This establishes the motivation
for the present paper.
' ' (given that i is theIn the RNG K- e model, the Reynolds stress tensor 7"ii =- uiuj u i
fluctuating velocity and an overbar represents an ensemble mean) is modeled as follows: 2'3
= . K6,j- \O=j+ (1)
1 _.--L-77_.1
where K --= ]u_-a_ is the turbulent kinetic energy, e - vc3u_/cqm# cgu_/c3z, i is the turbulent dis-
sipation rate, _i is the mean velocity, and O t, is a dimensionless constant which is calculated
to be 0.085. In homogeneous turbulence, the turbulent kinetic energy is a solution of the
transport equation
0-a_
which is exact. The turbulent dissipation rate is obtained from the RNG derived transport
equation
g 0"_i g2
= c,2-g (3)
where Col = 1.42 and C,2 = 1.68 according to the recent calculations of Yakhot and Smith. 3
These new values constitute a correction to the earlier values of C,1 = 1.063 and C,2 = 1.72
reported by Yakhot and Orszag. 2 An additional production term was also uncovered by
Yakhot and Smith3 which they were unable to close. However, an order of magnitude
analysis 3 indicated that this term is small unless there are large strain rates - a case which
will not be considered herein. Hence, we will neglect this additional term in the present
study. For the R,NG second-order closure model, the eddy viscosity model (1) is replaced
with a Reynolds stress transport model of the form 1
N + (')
where Ca and C2 are constants that are calculated to be 1.59 and 2/15, respectively. Some
clarifications are needed concerning the origin of this model which has not been published
and was obtained from a private communication with V. Yakhot. We have come to learn
that this was not intended to be a final model, but rather was the result of a low-order
calculation of the pressure:strain correlation whose purpose was to merely demonstrate that
the Rotta term - with a coefficient O1 close to the well accepted value of 1.5 - could be
formally obtained from RNG. Hence, the results predicted by this preliminary model should
be judged accordingly.
In homogeneous shear flow, an initially isotropic turbulence where
2
Tij = _K06ij, e = e0 (5)
at time t = 0 is subjected to a constant shear rate S with the corresponding mean velocity
gradient tensor
(0 S 0)
0 0 0 . (6)
O-_xJ = 0 0 0
In Figure 1, the time evolution of the turbulent kinetic energy (where K* = K/Ko and
t ° = St) predicted by the new RNG K-e model is compared with the large-eddy simulation
of Bardina et aL 4 for an initial condition of eo/SKo = 0.296. The predictions of the old
version of the RNG K - e model (where C_, = 0.0837, C,1 = 1.063, and C,_ = 1.72) as well
as the standard K - e model (where C, = 0.09, C,1 - 1.44 and C_ = 1.92) are also shown
in Figure 1. It is clear from these results that the revised RNG K - e model does the best
overall job in reproducing the growth rate of the numerical experiment on homogeneous shear
flow. Analytically, it can be shown why this is the case. From a straightforward calculation,
it can be shown that the turbulent kinetic energy and dissipation rate grow exponentially in
homogeneous shear flow as follows: 1,s
K* _ exp(M*), _* _ exp(M*)
where the dimensionless growth rate _ is given by
A = (O.i- i)(C.,- i)J " (7)
Hence, the growth rate becomes singular when C,1 = 1 w a state of affairs that explains
why the old version of the RNG K - _ model, with C_1 = 1.063, overpredicted the growth
rate of the turbulent kinetic energy by such a wide margin. The new version of the RNG
K - e model predicts a growth rate of
= 0.142
which is extremely close to the range of values obtained from physical and numerical
experiments. 6'? On the other hand, the standard K - _ model predicts the somewhat high
value of % = 0.226 which explains why this model overpredicts the LES data for K* as shown
in Figure 1. A more complete set of the equilibrium values predicted by these different ver-
sions of the K - _ model will be provided later.
In Figure 2, the time evolution of the turbulent kinetic energy predicted by the revised
RNG second-order closure model is compared with the large-eddy simulation of Bardina e_
al. 4 as well as with the predictions of the earlier version of the model and the Launder, Reece,
and Rodi s (LRR) model. The new version of the RNG model does yield better predictions
than the older version of the model since the previous value of C'_1 = 1.063 was too close
to C_1 - 1 which constitutes a bifurcation point of the dissipation rate transport equation
a8 shown by Speziale. 9 However, there are still problems with the model which gives rise to
points of inflection in the time evolution of K* - a feature that makes it inferior to other
second-order closure models such as the Launder, Reece, and Rodi model. The origin of
this problem appears to be tied to the modeling of the pressure strain correlation. In the
Launder, Reece, and Rodi model, the pressure strain correlation IIii =- IY(c%I/Oz i + c3u_lOzi)
is modeled as follows
2
H,j = -2Cl_b,j + 2C, K'_, + C3K (b_-_jk + b,k',_ - -_b_'ktSij)
+C4K(bikWjl, + bj_W'ik)
(8)
where
= \o j + ' : (9)
b,, = (r,, - 2 KS, j) /2K (10)
(in the simplified form of the Launder, Reece, and Rodi model, C1 = 1.8, C2 = 0.4 and
C3 = C4 = 1.2). This model satisfies two important consistency conditions: (a) the constant
6'2 is equal to 2/5 - a result that follows from simple symmetry conditions for H_j as well
as from Rapid Distortion Theory, a and (b) it represents a formal expansion of H_j to O(b)
in the anisotropy tensor. On the other hand, for this preliminary RNG second-order closure
model we have
where C1 = 1.59 and C2 = k15"
Hij = -2Clcb_j + 202KT_j (11)
This model is not complete to O(b) in the rapid pressure-
strain term and violates the important symmetry constraint of C2 = 2/5. The fixed points
that the resulting nonlinear ODE's for these second-order closure models give rise to in
homogeneous shear flow are of the focus type. 5 Significant deviations of (72 from a value
of 2/5 excites the imaginary parts associated with these fixed points, thus inducing inertial
oscillations which are unphysical for the case of pure shear flow.
An overview of the performance of the models can be gleaned from Table 1 which com-
pares the predicted equilibrium values with the most recent experimental data of Tavoularis
and Karnik 7 for homogeneous shear flow (this data constitutes a mean over the stronger
shear rate cases). Here (.)_ denotes the equilibrium value obtained in the limit as _ --_ oo.
Several observations concerning Table 1 are noteworthy:
(a) The revised RNG K- e model yields substantially better results than the old version
of the model and is, on balance, better than the standard K - e model. This appears to
explain why the models performed as they did in Figure 1 relative to the LES results.
(b) The only deficiency in the predictions of the new RNC K-e model for homogeneous shear
flow are in the values of the normal components of the anisotropy tensor - a shortcoming of
any model based on an isotropic eddy viscosity. However, the RNG based anisotropic eddy
viscosity model of Rubinstein and Barton 1° - which predicts (b_)=_ = 0.260 and (b2_)_ =
4
-0.196 for the normal anisotropies in homogeneous shear flow - alleviates this deficiency to
a large extent.
(c) The RNG second-order closure model does perform somewhat better with the new value
of C,1 (the model now predicts a weak exponential time growth of K" whereas the old
version of the model predicted a power law growth, with (SK/e)oo = co, due to the close
proximity of Col to the bifurcation point C,1 = 1). However, this preliminary model still
performs weakly in comparison to the more commonly used second-order closures such as
the Launder, Reece, and Rodi model. The deficiency in this model is traced to the rapid
part of the pressure-strain correlation which is O(1) instead of O(b) in the &nisotropy tensor.
In fact, the deviation of 6'2 from 0.4 to 2/15 results from the model trying to compensate
for the truncated O(b) terms n (interestingly enough, if C2 is set to 0.4 in Eq. (11), the
predictions of the model deteriorate substantially). Hence, we have little doubt that if the
RNG based calculation is extended to include the O(b) terms, the resulting model would
perform quite well in comparison to other second-order closures.
In conclusion, with the revised coefficients proposed by Yakhot and Smith s, the R.N(]
K - e model now performs well in homogeneous shear flow - particularly when the R.NG
based anisotropic eddy viscosity of Rubinstein and Barton t° is used. The RNG second-order
closure model needs further development, however. It would appear that an extension of
the rapid pressure-strain correlation to include terms of O(b) would resolve the remaining
deficiency in this model. Consequently, our current assessment of RNG based turbulence
models is now more optimistic than reported earlier.
REFERENCES
1C. G. Speziale, T. B. Gatski, and N. Mac Giolla Mhuiris, Phys. Fluids A 2, 1678 (1990).
2V. Yakhot and S. A. Orszag, J. Sci. Comput. 1, 3 (1986).
3V. Yakhot and L. M. Smith, Phys. Fluids A, submitted for publication.
4j. Bardina, J. H. Ferziger, and W. C. Reynolds, Stanford University Technical Report TF-19
(1983).
5C. G. Speziale and N. Mac Giolla Mhuiris, J. Fluid Mech. 209, 591 (1989).
aM. M. Rogers, P. Moin, and W. C. Reynolds, Stanford University Technical Report TF-P5
(1986).
7S. Tavoularis and U. Karnik, J. Fluid Mech. 204, 457 (1989).
SB. E. Launder, G. Reece, and W. Rodi, J. Fluid Mech. 68, 537 (1975).
gC. G. Speziale, Proc. Whither Turbulence Workshop, Lect. Notes Phys., ed. J. L. Lurnley
(Springer, New York, 1990), Vol. 357, p. 490.
l°R. Rubinstein and J. M. Barton, Phys. Fluids A 2, 1472 (1990).
11V. Yakhot, Private Communication (1991).
New RNG
K - _ Model
Old RNG
K - e Model
Standard
K - _ Model
New RNG
Second-Order
Closure
Old RNG
Second-Order
Closure
LRR Model
Experimental
Data
(b11) 
0
0
0.489
0.533
0.193
0.21
-0.185
-0.489
-0.217
-0.091
-0.185
-0.16
0
-0.244
-0.267
-0.096
-0.13
4.38
11.70
4.82
8.94
Co
5.65
4.8
Table 1. Comparison of the equilibrium values of the various models with the experimental
data of Tavoularis and Karnik 7 on homogeneous shear flow.
7
LIST OF FIGURES
Figure 1. Time evolution of the turbulent kinetic energy in homogeneous shear flow:
New RNG K - c model; ... 01d RNG K - e model; - - - Standard K - _ model;
o Large-Eddy Simulation. 4
Figure 2. Time evolution of the turbulent kinetic energy in homogeneous shear flow:
-- New RNG Second'0rder Closure Model; ... Old RNG Second-Order Closure Model;
-- - Launder, Reece, and l_odi Model; o Large-Eddy Simulation. 4
E
I
E
u
=
qE
=
=
8
4.0
3.5
3.0
2.5
_2.0
1.5
1.0
0.5
0.0
LES o
Standard K-¢ Model
Old RNG K- ¢ Modet
Nezu RNG K-¢ Model
• I
I
; I
' /
,' /
' //
," /
/
,' /
' /
, /
/ ,l
," //
•" //
." //
." 11./"
I I I I
0 2 4 6 8 10
_
Figure 1. Time evolution of the turbulent kinetic energy in homogeneous shear flow:
New RNG K - ¢ model; ... Old RNG K - _ model; - - - Staoudard K - e model;
o Large-Eddy Simulation. 4
4.0
LES 0
LRR Model
Old RNG Model
New RNG Model
,o.
3.5
3.0
2.5
_2.0
4b
_<
o
o
," J
J
7
 oooj1.5 ..'"•_ 01,1
1 0_o°_ _
/
/
/
/
/
/
/
/
/
/
/
/
/
0.5 -
0.0 i L l i
0 2 4 6 8 10
t
E
It
,E
Figure 2. Time evolution of the turbulent kinetic energy in homogeneous shear flow:
New RNG Second-Order Closure Model; ..- Old RNG Second-Order Closure Model;
- - - Launder, Reece, and l:todi Model; o Large-Eddy Simulation. 4
10


Report Documentation Page
Na r,.Onal_IY_r'aul( s ,-Ind
,"_Pace t_rnpn_slrabon
1. Report No.
NASA CR-187552
ICASE Report No. 91-37
2, Government Accession No.
4. Title and Subtitle
AN ANALYSIS OF RNG BASED TURBULENCE MODELS FOR HOMO-
GENEOUS SHEAR FLOW
7. Author(s)
Charles G. Speziale
Thomas B. Gatski
Nessan Fitzmaurice
9. Performing Organization Name and Address
Institute for Computer Applications in Science
and Engineering
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
3. Recipient's Catalog No.
5. Repo_ Date
April 1991
6, Performing Organization Code
8. Performing Organization Report Nol
91-37
10. Work Unit No.
505-90-52-01
11. Contract or Grant No.
NASI-18605
13, Ty_ ofRepo_andPeriodCovered
Contractor Report
14. Sponsoring Agency Code
15, Supplementary Notes
Langley Technical Monitor:
Michael F. Card
To be submitted to Physics of Fluids
A.
Final Report
16. Abstract
In a recent paper [Phys. Fluids A2:1678-1684, 1990], the authors compared the
performance of a variety of turbulence models including the K - E model and the
second-order closure model derived by Yakhot and Orszag based on Renormalization
Group (RNG) methods. The performance of these RNG models in homogeneous turbulent
shear flow was found to be quite poor, apparently due to the value of the constant
Cel in the modeled dissipation rate equation which was substantially lower than its
traditional value. However, recently a correction has been made in the RNG based
calculation of Cel . It is shown herein that with the new value of CEI, the per-
formance of the RNG K - e model is substantially improved. On the other hand, while
the predictions of the revised RNG second-order closure model are better, some
lingering problems still remain which can be easily remedied by the addition of
higher-order terms.
17. Key Words (Suggested by Author(s))
Renormalization Group; K-E model;
Second-order Closure Models
19. Security Classif. (of this report)
Unclassified
18. Distribution Statement
34 - Fluid Mechanics and Heat Transfer
Unclassified - Unlimited
20. Securi_' Cla_if. (of this page) 21. No, of pages
Unclassified 12
22. Price
A03
NASA FORM 1626 OCT 86
NASA-Langley,1991
=z z _
= -


An analysis of RNG based turbulence models for homogeneous shear flow

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