An existing two-dimensional, compressible flow, Navier-Stokes computer code, containing a full Reynolds stress turbulence model, was adapted for use as a test bed for assessing and improving turbulence models based on turbulence simulation experiments. To date, the results of using the code in comparison with simulated channel flow and over an oscillating flat plate have shown that the turbulence model used in the code needs improvement for these flows. It is also shown that direct simulation of turbulent flows over a range of Reynolds numbers are needed to guide subsequent improvement of turbulence models

Minh, H. HA

Rubesin, M. W.

Vandromme, D.

Viegas, J. R.

English

NASA Technical Reports Server

Center for Turbulence Research
Proceedings of the Summer Program 1987
N 88 - 23 1 0 5 't
185
Test Code for the Assessment and
Improvement of Reynolds Stress Models
By M. W. RUBESIN 1, J. R. VIEGAS 1,
D. VANDROMME 2 and H. HA MINH 3
An existing two-dimensional, compressible flow, Navier-Stokes computer code, con-
taining a full Reynolds stress turbulence model, has been adapted for use as a test
bed for assessing and improving turbulence models based on turbulence simulation
experiments. To date, the results of using the code in comparison with simulated
channel flow and over an oscillating flat plate have shown that the turbulence model
used in the code needs improvement for these flows. They also show that direct
simulation of turbulent flows over a range of Reynolds numbers are needed to guide
subsequent improvement of turbulence models.
1. Introduction
Various turbulence models are suggested in the literature to close the different
terms in the Reynolds stress budget, each with its advocates and its critics. To
properly assess these models, direct simulation of turbulent flows can be used to
compute the terms in the Reynolds stress budget and directly compare the model
expressions with the terms (see Mans0ur et al., 1987). Direct comparison is not
enough, however, because perfect agreement can rarely be achieved, and for a com-
plete evaluation, the models should be used in an actual computation of the flow. A
code previously developed by the authors (see. Vandromme eta/., 1983) has been
modified to simulate the channel flow of Kim et al. (1987) and the oscillating flat
plate of Spalart & Baldwin (1987). The code is based on a bidiagonal predictor-
corrector time marching algorithm (MacCormack, 1981) that has been modified to
solve the nonconservative equations that result from the introduction of Reynolds
stress turbulence models. This algorithm has been used to calculate several complex
compressible flows, including shock wave-boundary layer interactions ( HaMinh et
a/., 1985, and Viegas & Rubesin, 1983). Because the code is quite robust and can
accept turbulence modeling changes conveniently, it was decided to adapt it to the
low speed conditions of the existing turbulence simulations rather than to write a
new code. For the selected cases of flow in a channel (Kim et al., 1987) or over an
oscillating infinitely long flat plate (Spalart & Baldwin, 1987), meshes having the
1 NASA Ames Research Center
2 Facult4 des Sciences de Rouen
3 Institut de M$canique des Fluides de Toulouse
https://ntrs.nasa.gov/search.jsp?R=19880013721 2020-03-20T06:11:01+00:00Z
186 M. W. Rubesin, J. R. Viegas, D. Vandromme and H. Ha Minh.
dimensions of 5 (axial) and 50 (transverse) were employed. The small number of
mesh points in the axial direction are adequate for these problems since there are no
variations of dependent variables, except for pressure, in this direction. The code
uses two mesh points at each of the upstream and downstream boundaries to define
the conditions there. Computations with this mesh arrangement for the channel
flow took about 1 minute of Cray cpu time for about 1000 iterations and about
10000 iterations to reach steady state. The code, when used to yield time-accurate
oscillating plate solutions, took about an hour of cpu time to reach a steady periodic
state.
2. Results and Discussion
To test the code's ability to handle low speed flows, both the channel and the
oscillating plate were first run with laminar flow. The numerical results in both
cases agreed quite well with the corresponding analytical solutions (Schlichting,
1960). When the same flow cases were run with the turbulence model developed
by Vandromme et M. (1983), it was found that the numerical results did not agree
with the statistical output from the simulations.
2.1 Channel Flow
The channel flow computations at Re=2800 gave results that differed considerably
from the statistical quantities corresponding to the simulation of Kim eta/., (1987).
For example, Fig. la shows a comparison of the distribution of the turbulent shear
stress, normalized by the wall shear as a function of the distance from the surface
in wall parameters. The solid symbols are the output from the simulation, whereas,
the open symbols represent the computed results from the modeled equations. The
difference between the results indicate that the current model in the code needs
considerable improvement. When similar computations were performed for channel
flows with higher Reynolds numbers, namely, Re=13000 and 50000, the stresses
shown in Figs. lb and lc resulted. Here the solid symbols are the same as on
Fig. la and are used as reference for the levels of the computed results, the open
symbols. These figures indicate that the current model in the code shows a large
sensitivity to the Reynolds number of the channel flow. If the LES calculations of
Moin and Kim (1982) are examined, it is noted that the maximum Reynolds shear
stress at Re=13800 found there is about 0.86. This is larger than the corresponding
value of 0.7 in the complete simulation at Re=2800, and is in the same direction of
increased normalized shear stress with higher channel Reynolds number exhibited
by the turbulence model shown in Figs. la and lb. There exist questions regarding
the near wall treatments of the subgrid model in the LES, and this suggests that
the process of improving the turbulence model to handle these Reynolds number
effects could benefit from additional accurate channel flow simulations at Reynolds
numbers higher than 2800.
Another aspect of the process of using the simulated results to improve statistical
turbulence models becomes evident in Fig. 2. Here is shown the rate of dissipation
Assessment of Reynolds stress models 187
1.0
.8
^.6
V.4-
.2-
4
Re- 2800
..............: ... e e 0 • •
• o o
o
• o
ooO • CHANNEL SIMULATIONo MODELING RESULI_
| i i "1
1.0
.8
A .S
V.4
.2
0.
oO_° ,-" "o',',
o o •
o •
o
o •
o •
o •
o •
o
_ (b)
Re - 13,000
%'°°'so•• 80 o°o • • •
1,0 "
.8
/%.6
V.4
.2"
o ,9 o '_ o o
o
o
° Re = 50,000
o
o
° •••oeoo°•••••• • o0• • •
00• • • •
(P
,9
_' (C)
y+
FIGURE 1. Dimensionless Reynolds shear stress in the near wall region of the
channel, a) Re = 2800. b) Re = 13000. c) Re = 50000.
in the equation for the normal stress, Y-_, expressed in dimensionless form. Again,
the open and closed symbols represents the results of the model calculations and
the simulations, respectively. The figure shows considerable difference between the
two computations, and at first glance suggests difficulties with the particular model
used for the turbulence dissipation rate. It was found (Mansour, Kim and Moin,
188 M. W. Rubesin, J. R. Viegas, D. Vandromme and H. Ha Minh.
iii t f' • CHANNEL SIMULATION
o MODELING RESULTS
o 12o
y+
FIGURE 2. Distribution of dimensionless dissipation rate in _-_ equation.
1987), however, that the model used in the computations evaluated with the mo-
ments found from the simulations, rather than from the modeled computations,
gave results quite close to those of the simulations. Thus the error of the modeled
computations shown on this figure result from the errors of the Reynolds stresses
used in the evaluation of the dissipation rather than the functional form of the dis-
sipation model. This is an example of the highly interactive behavior of elements
of turbulence models and indicates that model improvements must be made simul-
taneously in all elements of the model so that it improves in the aggregate. The
wealth of information available in the simulations should facilitate this process in
future studies.
2.2 Oscillating Flat Plate
Representative results of the comparison of the computation of the flow over an
oscillating flat plate with the direct simulation of Spalart & Baldwin (1987) are
shown in Fig. 3. Here are shown the time-dependent skin friction over a cycle of
time for the model computations as well as the simulation. Although the computed
results are periodic and show similar phase relationships as do the simulation, there
are significant differences between the results which, again, indicate the weaknesses
of the current turbulence model.
3. Future Directions
The authors plan to continue to use direct turbulence simulations to improve the
Reynolds stress model used in the code. The directions this will take depends a
great deal on the availability of simulations for a range of Reynolds numbers. The
range in Reynolds number is needed to minimize the number of assumptions used
Assessment of Reynolds stress models 189
1.0
.5
o
:D
-.5
-1.0
OSCILLATING FREE STREAM ..f'_l
_'_ _ VELOCITY / t'
-'_, '_ Uo = AMPLITUDE /.._.
;. _-_ SKIN FRICTION ,,_.:_T_- -,,_
: ". _ T\ -SIMULATION _.."_P"/
,"- ,_ _, i , , , ._'-"_, I, ,
' "k%'\ ' ' ' I , I
...-.""/'/
- / "VN /'/ -
SKIN FRICTION _ '_ _",,---._ SKIN FRICTION
LAMINAR FLOW "_._r _'" /-MODELING CODE"
I I _--_J I I
1 2 3 4 5 6
_, radians
2.5
.5
%
-1.5 _"
-3.5
FIGURE 3. Skin friction as function of phase angle ¢ for an oscillating flat plate.
Comparison of model results with simulation data.
regarding the functional form of of the various coefficients in the turbulence model.
A test for the universality of these coefficients will be agreement with simulation
data from a variety of flow fields.
REFERENCES
HAIVIINH, H., RUBESIN, M.W., VANDROMME, D., AND VIEGAS, J. R., 1985
On the Use of Second Order Closure Modeling for the Prediction of Turbulent
Boundary Layer/Shock Wave Interactions: Physical and Numerical Aspects,
International Symposium on Computational Fluid Dynamics, Tokyo, Japan,
Sept. 9-12.
KIM, J., MOIN, P., AND MOSER, R., 1987 Turbulence Statistics in Fully De-
veloped Channel Flow at Low Reynolds Number . J. Fluid Mech. 177, 133 -
166.
MACCORMACK, R. W., 1981 A Numerical Method for Solving the Equations of
Compressible Viscous Flows, AIAA Paper 81-0110.
MANSOUR, N.N., KIM, J. AND MOIN, P., 1987 Reynolds-stress and Dissipation
Rate Budgets in a Turbulent Channel Flow, to appear in J. Fluid Mech.
MOIN, P. AND KIM, J., 1982 Numerical Investigation of Turbulent Channel
Flow. J. Fluid Mech. 118, 341 -377.
SPALART, P. R., AND BALDWIN, B. S. 1987 Direct Simulation of a Turbulent
Oscillating Boundary Layer, NASA TM 89460, May.
SCHLICHTING, H., 1960 Boundary Layer Theory, Mc-Graw Hill, New York.
VANDROMME, D., HAMINH, H., VIEGAS, J. R., RUBESIN, M. W., AND KOLL-
M. W. Rubesin, J. R. Viegas, D. Vandromme and H. Ha Minh.
MANN, W., 1983 Second Order Closure for the Calculation of Compressible
Wall Bounded Flows with an Implicit Navier-Stokes Solver, Fourth Symposium
on Turbulent Shear Flows, Karlsruhe, GFR, Sept. 12-14.
VIEGAS, J. R., AND RUBESlN, M. W. 1983 Wall-Function Boundary Conditions
in the Solution of the Navier-Stokes Equations for Complex Compressible Flows,
AIAA 83-1694.


Test code for the assessment and improvement of Reynolds stress models

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