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OCR for page 35
RNG Modeling Techniques for Complex Turbulent Flows
S. A. Orszag and G. E. Karniadakis
Princeton University, Princeton, USA
A. Yakhoti
Ben-Gurion University of the Negev, Beesheva, Istael
V. Yakhot
Princeton University, Princeton, USA
Permanent address: Ben-Gurion University of the Negev
Beesheva, Israel
Abstract
In this paper, we combine RNG modeling tech-
niques with spectral element discretization proce-
dures to formulate an algorithm appropriate for
simulating high Reynolds number turbulent flows
in complex geometries. Three different approaches
of modeling are followed based on RNG algebraic,
differential k - c, and subgrid-scale models for the
turbulent viscosity. Results obtained for the fully
developed channel flow, and for the separated flout
in a cavity and over a backwards- facing step sug-
gest that all three formulations are suitable for tur-
bulent flow simulations. The implementation of
RNG models within the framework of a high-order
discretization scheme (i.e. spectral element meth-
ods) is essential, as it results in resolution of fine
turbulence structures even with the simple differ-
ential k - ~ model at relatively small number of
degrees of freedom.
1. RENORMALIZATION GROUP FORMULATION
Renormalization Group Theory (RNG) has proven to be a
very useful tool in theoretical physics and statistical me-
chanics to study systems with a very large number of de-
grees of freedom. The most prominent use of this approach
was in developing the theory of phase transitions of the
second kind. Although RNG theory has found its way into
fluid mechanics relatively recently, particularly in develop-
ing turbulence theory [1], it has already proved to be a
valuable research tool, and it has provided a series of inter-
esting theoretical and numerical results. A milestone for
RNG theory in fluid mechanics was the establishment of
the so-called correspondence principle, stating that in the
inertial range the behavior of the small-scale Navier-Stokes
turbulence is statistically equivalent to the modeled Navier-
Stokes equation with the additicjn of a random noise term.
This principle makes it possible to use all the formalism of
classical RNG theory.
Recently, renormalization group methods have been devel-
oped [1i, [2] to analyse a variety of turbulent flow problems.
For homogeneous turbulent flows, such important quanti-
ties as the Kolmogrov constant, Batchelor constant, tur-
bulent Prandtl number, rate of decay to isotropy, skewnes
factor, etc. have been obtained directly from this theory in
good agreement with available data. Efforts in developing
RNG methods for sub-grid (large-eddy simulation) model
constants have also been notably successful [34.
RNG methods involve systematic approximations to the
full Navier-Stokes equations that are obtained by using
perturbation theory to eliminate or decimate infinitesimal
bands of small scale modes, iterating the perturbation pro-
cedure to eliminate finite bands of modes by constructing
recursion relations for the renormalized transport coeffi-
cients, arid evaluating the parameters at a fixed point in the
lowest order of a dimensional expansion around a certain
critical dimension. The decimation procedure, when ap-
plied successively to the entire wavenumber spectrum leads
to the RNG equivalent of full closure of the Reynolds av-
eraged Navier-Stokes equations. The resulting RNG trans-
port coefficients are differential in character as opposed to
ad hoc algebraic coefficients of conventional closure meth-
ods. All constants and functions appearing in the RNG
closures are fully determined by the RNG analysis.
In essence, the RN(; method provides an analytical method
to eliminate small scales from the Navier-Stokes equations,
thus leading to a dynamically consistent description of the
large-scales. The formal process of successive elimination of
small scales together with re-scaling of the resulting equa-
tions results in a calculus for the derivation of transport
approximations in turbulent flows.
35
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The renormalization group method has been used to derive
three different types of models: an algebraic eddy viscos-
ity model, a differential k-e, and a subgrid scale turbulence
model for large eddy simulationst1~. The governing equa-
tions for the flow motion are the Navier-Stokes equations:
Bt + Vj00Vi = _ PIP + ~ ~ deli
along with the incompressibility constraint
dvi = 0 (2)
Here z, is the total viscosity defined as z' = zoo+ Z'T (the sum
of the molecular and turbulent viscosity, respectively).
RNG Algebraic Model
This model is the simplest of all, however it is not very
general as it requires the a priori postulation of the charac-
teristic integral length scale, and thus an assumption needs
to be made based on physical considerations. The eddy vis-
cosity ~ is obtained from the following relation:
~ = ',o~1 + H¢~2 ye 3~4 _ C'4~/3 (3)
where H(x) is the Heaviside function defined by H(x) = :z:
for x > 0 and H(~) = 0 otherwise, and /\ is an integral
length scale of the turbulence in the inertial range. The
constants a = 0.120 and C = 100 are derived in t13. The
mean dissipation rate ~ can be expressed entirely through
the resolvable field as
L, ~ Levi + ~Vj )2 (4)
The total viscosity is then obtained by solving an algebraic
cubic equation at every node of the computational domain
at each time step.
RNG k-e Model
This is a differential model based on differential recursion
relations discussed in t13. It is free of adjustable parame-
ters, however two additional equations are derived for the
turbulent kinetic energy k, and the rate of dissipation e.
Dk p ~ ok
Dt Ski ( 6mi )
Dt 0zi ( B~i
(5)
(6)
Here we define the production term P based on the turbu-
lent viscosity
p AT ( Levi + ~Vj)2
and the terms P. Y are defined subsequently from:
P = 0.656~/2YCP (7b)
y = 63/2y (7c)
The RNG procedure provides the additional two differ-
ential equations for Ye and ti/2 ~ which can be integrated as
follows
yul/2 ~ _ ~
~ (t3 - 1 + C)~/2 (8a)
~/7 = C\k + ilk ~ (t3— 1 + C)~/2 (8b)
Here the coefficients (c, and (°rk,~k) are known con-
stants derived from the asymptotic behavior of the model,
while the parameter ct involved in equations (5-6) is the
inverse total Prandtl number defined from the RUG alge-
braic relation (coo refers to molecular properties)
—1.3929 ~0.632~ a+2~3929 ~0.3679_ ~0 (9)
a0 - 1.3929 DO ~ 2.3929 ~
The total viscosity z' is implicitly defined from equations
(8) at each node of the computational domain. In the high
Reynolds number region where ~ ~ HOD we obtain
k2
~ = 0.0845— (10)
This high Reynolds number approximation is similar to
that commonly used in algebraic models. Thus, the al-
gebraic models contain terms like A, which diverge when
k ~ O (i.e., near wall regions or separation zones) thus
creating immense difficulties computationally. The differ-
ential relations obtained via RNG techniques, however, do
not contain such singular behavior and thus have great pm
tential for success with separating flows. The differential
relations provide definitive interpolation formulae to con-
nect low and high Reynolds regions of the flow.
RNG subgrid-scale Model
In large eddy simulations (LES) the velocity field vi is de-
composed into large scales vi and subgrid components v,.
It is the modeling of terms involved these latter compo-
nents that is crucial for accurate computations as previous
attempts using various subgrid models t4] have indicated.
The RNG subgrid model is derived by elirrunation of modes
from the interval AO A, where A = ~y(~/4 (] = 0.20) is
a dissipation cut-off limit, and the wa~revector AO = lie~r
can be expressed through the computational mesh size A.
The general RNG derived viscosity given by
(il)
can then be expressed in terms of a length scale /`, which
in LES represents the width of a suitably chosen Gaussian
(7a) filter. Thus, equation (11) reduces to an identicalequation
as in (3), where /& is given by the integral
/~4 = J ; J (k2 + p2 + q237/2 (12)
36
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Here Ai = or// (for i = 1,2,3) and /\, = 24. The above
integral can be readily evaluated by breaking it up to three
asymptotic integrals and a finite triple integral that can
be computed numerically. In cases of simple geometries
(i.e. plane channels) the integral can be evaluated through
algebraic relations [3~.
2. SPECTRAL ELEMENT METHODOLOGY
Let us denote by Q the three-dimensional computational
domain, and by ~Q the computational boundary-surface;
we can then rewrite the governing equations (1) in the form,
dvj _0pi + [3 ~z'~Vi) + Hi in Q (13)
where Hi includes all nonlinear and forcing terms (e.g. the
random force to be included in LES). The separation of
linear and nonlinear terms in the above equation leads
naturally to a mixed (explicit-implicit) time advancement
scheme. The temporal discretization proceeds by employ-
ing the splitting scheme; accuracy and relative advantages
of the scheme in the current applications are discussed in
[5~. The resulting system of equations is a system of sepa-
rately solvable elliptic equations for pressure and velocity.
The discretization of these equations in space is obtained
through the spectral element method. For simplicity we
only present the two-dimensional equations, however our
implementation is general for truly three-dimensional ge-
ometry and flow. Here we follow a 'layered' approach, ac-
cording to which the discretizations and solvers are con-
structed on the basis of a hierarchy of nested operators
proceeding from the highest to the lowest derivatives. This
approach is motivated by the fact that the highest deriva-
tives in an equation govern the continuity requirements,
conditioning, and stability of the system. Given the brevity
of the current paper we shall limit our description of spec-
tral element methods to the innermost layer, the elliptic
'kernel', which represents both the pressure equation and
the viscous corrections in equation (13~.
A typical elliptic equation for a field variable ~ can be
put in a standard Helmholtz equation form with variable
coefficients as follows
~ `p 6+ ~ >2¢ = f in Q for j = 1,2 (14)
In addition, let us assume homogeneous boundary condi-
tions ~ = 0 on Q. Equation (14) can then be further
discretized using planar spectral elements in plane x—y.
If we define Ho the standard Sobolev space that contains
functions which satisfy homogeneous boundary conditions,
and introduce testfunctions ?,6 ~ Ho, we can then write the
equivalent v~ariational~tatement of (14) as,
/ unit?/) ~ ~ ds + A2 J.n Adds = - J.n Fed (15)
The spectral element discretization corresponds to numer-
ical quadrature of the variational form (15) restricted to
the space Xh C Ho. The discrete space Xh is defined
in terms of the spectral element discretization parameters
(`K, N:, Ad), where K is the number of "spectral elements",
and N., N2 are the degrees of piecewise high-order polymo-
mials in the two directions respectively that fill the space
Xh. By selecting appropriate Gauss-Lobatto points (k and
corresponding weights ppe = pppq, equation (15) can be re-
placed by,
i=1 p=0 g=0 ~ 0=j 0~j]{Pq ~ ~ ~ ~ ~ Pm art =
K N1 N.
~ ~ ~ Pm]"[¢f]~[16~
k=1 p=0 q=0
Here Jkq is the Jacobian of the transformation from global
to local coordinates (m, y) ~ (r, I), for the two-dimensional
element k. The Jacobian is easily calculated from the par-
tial derivatives of the geometry transformation r-, r`,, so, s....
The next step in implementing (16) is the selection of a
basis which reflects the structure of the piecowise smooth
space Xh. We choose an interpolant basis with components
defined in terms of Legendre-Lagrangian interpolants, h`(rj) =
6,j. Here, rj represents local cordinate and [,j is the Kronecker-
delta symbol. It was shown in [6], [7] that such a spectral
element implementation converges spectrally fast to the ex-
act solution for a fixed number of elements K and N ~ oo,
for smooth data and solution, even in non-rectilinear ge-
ometries.
Having selected the basis we cam proceed in writing the
local to the element k spectral element approximations for
For ~k) as follows,
ok = ~m"~(r~kn(~) Vm, n ~ (O. ..., Nay, (O. ..., N2),
(17a)
where ilk n iS the local nodal value of ¢. The geometry is
also represented via similar type tonsorial products with
same-order polynomial degree, i.e.
8, Y) = ~ sm"'Ymn~hm(7~)hn~s) Vm, 7:
~ (O. ..., N'.), (O. ..., N2), (17b)
Here Ok n' yk n are the global physical coordinates of the
node mn in the k element. This isoparametric mapping
leads to a compatible pressure formulation without the
presence of spurious modes t54.
We now insert (17) into (16) and choose test functions Can,
which are non-vanishing at only one global node to arrive
at the discrete matrix system. This procedure is straight-
forward and here we cite the final matrix system,
37
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Representative terms from entire chapter:
turbulent flow
K Ni N2
~~ 1 ~~ ~~ (H~!mn+H,~Y~!mn+) 2 JO Bk Bkn)
We simulate the lid-driven cavity flow at Reynolds num-
ber R = SO,OOO with the algebraic model (equation (3~)
employed in the computation. As a characteristic length
~ needed for the model at each nodal point we consider
the distance from the point to the nearest wall which is
proportional to the characteristic size of the largest eddies
of turbulence in this flow. In figure 2 we plot the velocity
vectors of the mean turbulent flow and in figure 3 several
~ velocity profiles in the neighborhood of the right lower
corner of the cavity. It is seen that the flow undergoes sep-
aration as we move from the center of the cavity to the
corner, as expected. However, the size of the recirculation
zone as well as the strength of the reverse flow is weak due
to the effects of the increased apparent viscosity. Indeed,
direct numerical simulations on the same mesh at Reynolds
number R = 10, 000 shows regions of multiple small and
large size instantaneous eddies (figure 43; the time-a~reraged
flow however exhibits only very small recirculation zones
sirrular to the ones shown in figures 2-3.
: ; i:
Xa0.85 X.O.9O Xe0.95
BOTTOM WALL
Fig. 3 Velocity (x—component) profiles at station
x= 0.85 (a); 0.90 (b); 0.95 (c) in the
neighborhood of the downstream secondary
eddy
To examine the spatial variation of the eddy-viscosity after
a stationary state is reached we plot in figure 5a the total
viscosity at a vertical station through the center of the
cavity. We see that the distribution is smooth and peaks
at the lid where there is enhanced mixing, while it achieves
a minimum at the wall (molecular viscosity) and in the
region close to the center of the cavity where the fluid is
almost stagnant. Thus, the total viscosity increases by
almost three orders of magnitude from its molecular to its
maximum value at the moving wall. In figure Sb we plot
the total viscosity at the same position as in figure 5a but at
Reynolds number R = 25, 000. We see in this case that the
region close to the bottom wall where the fluid motion is
weaker the region of molecular viscosity is broader, whereas
the outside distribution is similar to the one in figure 5a.
r7 r ,.
t t .
~ ~ `_
\~ ~ ~
N`_
`_
`_
.
_
__
_
~ ~ __
if
l I:
_ .. , ~
Fig. 4 Instantaneous velocity vectors at
R = 10,000: The DOW is computed via direct
spectral element simulation
Lit ~ ,
~ As,
_~
_~`
_ ~
1 1 1
1 1 1
l
J 1
/'
K. ~ <.,~,~'A
~~ ~ _. _
J
in
2x10-5
0.1283E-O1
(b) /
4xlO-S ~
o.sooo X
o. Y l.ooo
I/
1 \/
0.5000 X
o. Y 1.000
Fig. 5 Total viscosity variation at a vertical station
through the cavity center
(a) R = 50,000
(b) R = 25,000
39
k—~ Modeling of Turbulent Channel Flow
We simulate first flow in a plane channel at Reynolds num-
ber R* = 1000. Here the Reynolds number is based on
the wall-shear velocity U* and the channel half width. The
computational domain extends from y+ = 8.5 to y+ =
1991.5, so that the constant shear regions next to the walls
are excluded from the simulation. Two-dimensional spec-
tral elements are employed for the discretization, (K =
12), of order N = 9, so that there are 97 collocation
points across the channel. The boundary conditions are
prescribed u, k and ~ at the y-sides of the domain, while
periodicity conditions in the streamwise direction (x) en-
sure one-dimensionality of the mean flow. The simulation
starts from an equilibrium state and integration proceeds
for a long time. Results of the mean quantities are pre-
sented in figures 6 and 7. As a next test we simulated the
same flow but with the computational domain extended
to walls. The boundary conditions at the walls are now
no-slip for velocity, zero for kinetic energy, and Neumann
/:
Ox
v
8.5258O 5 X 50.0
I .0
Fig. 6 Mean velocity and eddy-nscosity profiles at
R* = 1000 for the turbulent Dow in a
channel
2.9
K
/
Q 14
0 ·— ~ ~ '0.0014
50.0 X 50.0
8.5 Y 1991.5
-
Fig. 7 Turbulent kinetic energy (k) and dissipation
rate (~) profiles at R* = 1000 for the
turDtllent flow in a channel
condition for the dissipation rate. In order to minirruze
discretization errors and ensure accurate imposition of the
zero flux condition on ~ we increased the resolution to or-
der N = 15 per element. The results of this simulation
were identical to results displayed in figures 6 and 7, which
suggests that the zero flux condition on ~ is a meaningful
one.
k—~ Modeling of Turbulent Flow Over a
Backward—Facing Step
The geometry for this flow and the corresponding spectral
element mesh are shown in figure 8; this geometry is identi-
cal with the geometry used in experiments of Kim et al [10]
and in the computations of Avva et al [11~. The outflow
length is taken to be twenty times the step height H. A
total of K = 92 elements of order N = 9 were employed in
the discretization. The Reynolds number R* = It - ~ is 1870
(or R. = 45, 000~. In this simulation the computational
domain includes the walls, where we impose the zero flux
condition for the disipation rate e; at the outflow Neumann
conditions are specified for all field variables. The inflow
conditions match the measured profiles in the experiment
of Kim et al [10] and employed in the computations of Awa
et al t11~. Finally, the initial data are based on the results
reported in t11) and some initial perturbation.
Fig. 8 Spectral Element Mesh employed for the
simulation of flow over a backward—facing
step.
In figure 9 we. first present the results of this simulation
in the form of streamlines. In addition to the large re-
circulation zone shown in this plot eddies of smaller size
of opposite rotation appear at the step test section floor
juncture consistent with the experimental findings [123. To
the best of our knowledge no other simulation has resolved
such fine structures previously using a single or a zonal
modeling approach. Furthermore, the length of the recir-
culation zone (figure 9-10) is computed to be L = 7.3H
in close agreement with the experimental value t10~. Most
studies todate fall short of predicting the correct value due
to errors both in numerics and turbulence models. In fig-
ure 10 and 11 we plot all mean variables at a station very
close to the reattachment point. The eddy viscosity distri-
bution attains its maximum close to that point where the
turbulence intensity exhibits its extremum.
40
- - - ~
~ - ~
~ ~ ~ -
- ~
Fig. 9 Spectral Element—RNG/k—~ Simulation of
Dow over a step at R* = 1870. The curves
are mean Dow streamlines while color
represents turbulent kinetic energy
(red=max, :Uue—min). In this calculation,
no ad hoc fitted parameters or experimental
input is used. The recirculation zone length
is 7.3 step heights in agreement with the
experiments of Kim, et al (1979~. The small
vortices in the step corner are observed
experimentally too.
-Q 027
I/'/
. /-/ ~.v
13,800 X ~,800
O Y 1500
Fig. 10 Mean velocity and eddy—viscosity profiles at
R* = 1870 for the DOW described in Figure 9
30.3:j
K
j) it/' ~
i /
1/
V/
t3180°
o
Large-Eddy Simulations of Turbulent Channel Flow
In the last example we employ the RNG subgrid-scale model
to simulate the turbulent channel flow at R* = 185. This
simulation corresponds to identical conditions as the di-
rect simulation recently reported by Kim, Main and Moser
(1987) t13~. In our simulation we have used however 60
times fewer grid points. In particular, for this case we em-
ploy a global spectral discretization based on Fourier ex-
pansion in strearnwise and spanwise directions, and Cheby-
shev expansion in the inhomogeneous direction (16 x 64 x
643. The initial conditions are based on three-dimensional
Tollmien-Schlichting waves. The results of our simulation
are essentially identical with the results of the direct simu-
lation as shown in figures 12 and 13, and in close agreement
with the experiments t14~. The agreement extends also to
higher order statistics as well as to flow structure and the
streak spacing; the results of our simulations are plotted
in figure 14 as color contour plots of the fluctuating ve-
locity component at a plane close to the wall (here red
indicates low velocity; blue high velocity). The mean sep-
aration between streaks is )* ~ 90 t3] in close agreement
with the experimentally observed spacing; all previous LES
have failed to predict the correct value of streak separation
[15~.
0.027
X 13,800
Y 1500
Fig. 11 Turbulent kinetic energy (k) and dissipation
rate (~) profiles at the reattachment point
(R* = 1870) (DOW as in Figure 9)
.'
,~
Y+
_ ~
10' :~
Y+
Fig. 12 Mean Velocity Profile at R$ = 185 of the
turbulent channel DOW computed via
RNG - arge eddy simulations —: 5.0 +
2.5Iny+
41
3~0
2.5
2.0
1.0
0.5 -
O -
!~\urms
o o
:W
o
Yrms
-
20 40 60
Y+
Turbulent intensities for the Dow described
in Firure 12. Also shown are experimental
results of Kreplin and Ackerman (1979)
Contours (low velocity: red; high velocity:
blue) of near—wall fluctuations x—velocity in
a turbulent channel £10w large eddy
simulation using the renormalization
grou~based subgrid scale eddy viscosity at
R* = 185. Observe the alignment of the
£10w in the streamwise direction and the
formation of low~peed streamwise streaks.
4. CONCLUSIONS
Turbulent flows are difficult to compute because they re-
quire order R3 work and order R9/4 storage to resolve dy-
namically significant velocity fluctuations at large Reynolds
number R. These requirements become more severe in com-
plex geometry flows where a broader spectrum of significant
scales is present. Direct numerical simulation although can
provide a very faithful description of the flow is limited
severely at the present time and in the next few years by the
aforementioned computational requirements to flow simu-
lations at a Reynolds number of a few thousands. Progress
in simulating realistic flows of engineering importance can
be made if reliable and well validated modeling techniques
are combined with highly-accurate numerical methods. In
the current work, we demonstrated how RNG methodol-
ogy con be used to represent all scales of turbulent flow
using for example a differential k—~ model, or to repre-
sent only the small-scale dynamics using a subgrid-scale
model. This methodology is very robust and can be ap-
plied to a variety of flows as a totally prognostic tool of
analysis, since it requires no apriori known parameters or
any experimental input, which is typically the case with the
currently used turbulence modeling techniques. A compu-
tationally efficient implementation of the RNG methodol-
ogy is obtained if it is combined with spectral or spectral
element discretization methods, which are used today pri-
marily in direct computations of transitional and turbulent
flows. We are currently working in further validating our
RNG/spectral element methodology in unsteady turbulent
flows (e.g. vortex streets) in complex geometries.
5. ACKNOWLEDGEMENTS
This work was supported by AFOSR under Contract F49620-
87-C-0036, by ONR under Contract N00014-82-C-0451, and
by DARPA under contract N00014-86-K-0759. Most of the
computations were performed on the CRAY-X/MP48 at
the Pittsburgh Supercomputing Center.
References
t1] Yakhot V. and Orszag S.A. Renormalization Group
analysis of turbulence.I. Basic theory. J. Sc. Comp.,
1:3, 1986.
t2] Yakhot V. and Orssag S.A. Relation between Kol-
mogorov and Batchelor constants. Phys. Fluids, 30:3,
1987.
[3] Yakhot A., Orszag S.A., and Yakhot V. Renormaliza-
tion Group formulation of large-eddy simulations. J.
Sc. Comp., to appear, 1989.
t4] Piomelli U., Ferziger J., and Moin P. Models for Large
Eddy Simulations of Turbulent channel Bows including
transpiration. Technical Report Report TF-32, Ther-
mosciences Division, Dept. Mech. Engineering, Stan-
ford University, California,94305, 1987.
[5] Karniadakis G.E. Spectral element simulations of lam-
inar and turbulent flows in complex geometries. Appl.
Norm. Math., to appear, 1989.
42
[6] R0nquist E.M. Optimal spectral element methods for
the unsteady three-dimensior~al incompressible Navier-
Stokes equations. PhD thesis, Massachusetts Institute
of Technology, 1988.
t7] Patera A.T. A spectral element method for Fluid Dy-
namics; Laminar flow in a channel expansion. J. Com-
put. Phys., 54:468,1984.
t8] Fischer P., Ho L.W., Karniadakis G.E., R0nquist E.,
and Patera A.T. Recent advances in parallel spectral
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Computers arid Structures, 30:217, 1988.
t9] Prassad A.K. and Koseff J.R. Reynolds number and
end-wall effects on a lid-driven cavity flow. Phys. of
Fluids A, 1:208, 1989.
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of separation and reattachment of a turbulent shear
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port Report MD-37, Dept. Mech. Engineering, Stan-
ford University, California,94305, 1978.
t11] Avva R.K., Kline S.J., and Fersiger J.H. Computation
of turbulent flow over a backward-facing step-zonal
approach. In AIAA-88-0611, 1989.
t12] Abott D.E. and Kline S.J. Experimental investiga-
tion of subsonic turbulent flow over single and double
backward facing steps. J. Basic Engineering, 84, 1962.
t134 Kim J., Main P., and Moser R. Turbulence statistics in
fully developed channel flow at low Reynolds number.
J. Fluid Mech., 177:133,1987.
t14] Kreplin H. and EckelmAn H. Behavior of the three
fluctuating velocity components in the wall region of
a turbulent channel flow. Phys. Fluids, 22:1233, 1979.
t15] Main P. and Kim J. Numerical investigation of tur-
bulent channel flow. J. Fluid Mech., 118:341, 1982.
43
DISCUSSION
by E.P. Rood
For some applications a requirement is
free surface flows near their intersection to
predict with boundary layers or in the
turbulent wake of a ship. What experimental
or other physical information is needed to
develop an adequate turbulence model for such
free surface flows?
Author's Reply
In the renormalization group approach,
full turbulence closure is obtained without a
priori experimental input. This is true for
both free surface and other flows. However,
in developing turbulence models for free
surface flows, it may be best to "tune" the
model using experimental observations of free
surface turbulence. It is hoped that many of
these questions can be answered in the near
future with detailed application of the
renormalization group models.
44
