NASA Contractor Report 187611
ICASE Report No. 91-65
ICASE
DEVELOPMENT OF TURBULENCE MODELS FOR SHEAR
FLOWS BY A DOUBLE EXPANSION TECHNIQUE
V. Yakhot
S. Thangam
T. B. Gatski
S. A. Orszag
C. G. Speziale
Contract No. NAS1-18605
July 1991
Institute for Computer Applications in Science and Engineering
NASA Langley Research Center
Hampton, Virginia 23665-5225
Operated by the Universities Space Research Association
National Aeronaulic._ and
Space Administration
Langley Re_eorch Center
Hampton, Virginia 23665-5225
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DEVELOPMENT OF TURBULENCE MODELS FOR
SHEAR FLOWS BY A DOUBLE EXPANSION TECHNIQUE
V. Yakhot*, S. Thangam t_, T. B. Gatski§,
S. A. Orszag* and C. G. Speziale t
*Applied &: Computational Mathematics, Princeton University, Princeton, NJ 08544
tICASE, NASA Langley Research Center, Hampton, VA 23665
§NASA Langley Research Center, Hampton, VA 23665
ABSTRACT
Turbulence models are developed by supplementing the renormalization group (RNG)
approach of Yakhot L: Orszag with scale expansions for the Reynolds stress and production
of dissipation terms. The additional expansion parameter (r/ - Sl-(/-g) is the ratio of the
turbulent to mean strain time scale. While low-order expansions appear to provide an
adequate description for the Reynolds stress, no finite truncation of the expansion for the
production of dissipation term in powers of 7/suffices - terms of all orders must be retained.
Based on these ideas, a new two-equation model and Reynolds stress transport model are
developed for turbulent shear flows. The models are tested for homogeneous shear flow
and flow over a backward facing step. Comparisons between the model predictions and
experimental data are excellent.
tPermanent Address: Stevens Institute of Technology, Hoboken, NJ 07030.
tThis research was supported by the National Aeronautics and Space Administration under NASA
Contract No. NAS1-18605 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. Introduction
According to the Kolmogorov theory 1 of turbulence, the dynamics of velocity fluctuations
v(k, w) at the scale l = 2_r/k depend only on the mean rate of dissipation of kinetic energy
g and the length scale I. Neither the integral (L) nor dissipation (ld = 27r/kd) scales enter
the dynamics of the inertial range of scales where ld _<: l <4 L. Based on this theory, the
velocity of eddies of size I scales as gl/31a/3 so that the characteristic (turnover) time of these
eddies is
l ,.., -g_1131213"
T(1) _ v(l----)- (1)
The velocity correlation function can then be represented in the scaling form
<vi(k,w)vi(k,,j)>_k_13/3¢( w )
__113k_t3 - C(k,w) (2)
where the scaling function ¢(x) is to be determined from other considerations. The energy
spectrum E(D) is of the form
E(k) _ k 2 f C(k,_o)aw = CK-g2/ak -5/3 (3)
where C_- is the so called Kolmogorov constant. These results also yield the effective (tur-
bulent) viscosity UT at the scale l = 2rr/k:
I]T(lg ) "" VII "" _1/3]¢ -4/3 . (4)
This effective viscosity plays a profound role in turbulence modelling. For example,
it is quite reasonable to assume that the large-scale properties of the flow are governed
by effective equations of motion similar to the Navier-Stokes equations with the effects of
strong interactions between the velocity fluctuations taken into account through an effective
viscosity (see Yakhot and Orszag 2)
UT _ -_1/3L4/3. (5)
However, the strict notion of eddy viscosity requires the existence of a small parameter
I/L << 1 which is absent in turbulent flow. Still, the eddy viscosity concept proves to be
extremely useful, working much better than expected. This situation is not unique: in a
fluid close to thermal equilibrium, the molecular viscosity representation is very accurate
even when the mean free path A is not that small (A/L __ 1).
Many years before Kolmogorov's work, Osborne Reynolds realized that turbulent flow was
a random system which had to be treated using statistical methods. The work of Reynolds
was similar in concept to the work of Boltzmann, Gibbs and others who formulated the