The main objective of the NASA sponsored Aerothermal Modeling Program, Phase 2--Element A, is to develop an improved numerical scheme for predicting combustor flow fields. This effort consists of the following three technical tasks. Task 1 involves the selection and evaluation of various candidate numerical techniques. Task 2 involves an in-depth evaluation of the selected numerical schemes. Task 3 involves the convection-diffusion scheme and the direct solver that will be incorporated in the NASA 3-D elliptic code (COM3S)

Karki, K. C.

Mongia, H. C.

Patankar, Suhas V.

Runchal, A. K.

English

NASA Technical Reports Server

N89- 1 89 0.-
AEROTHERMAL MODELING PROGRAM--PHASE II*
ELEMENT A: IMPROVED NUMERICAL METHODS FOR TURBULENT VISCOUS RECIRCULATING FLOWS
K.C. Kacki, H.C. Mongia, S.V. Patankar_, and A.K. Runchal$
Allison Gas Turbine Division
General Motors Corp.
The main objective of the NASA-sponsored Aerothecmal Modeling Program, Phase II--
Element A, is to develop an improved numerical scheme for predicting combustor flow
fields. This effort consists of the following three technical tasks. Task 1 has
been completed and Task 2 is in its final stage.
Task 1--NUMERICAL METHOD SELECTION
Task i involved the selection and evaluation of various candidate numerical tech-
niques. The criteria for evaluation included accuracy, stability, boundedness,
and computational efficiency. These schemes were used to solve a number of simple
test problems. On the basis of these preliminary results, the following three
schemes were chosen for detailed evaluation:
(i) flux-spline techniques
(ii) CONDIF
(iii) bounded flux-spline
To make the solution algorithm more efficient, it was decided to evaluate the per-
formance of a fully coupled approach in which the continuity and momentum equations
are solved directly, rather than sequentially as in SIMPLE or SIMPLER (ref. I).
Task 2--TECHNIQUE EVALUATION
Task 2, currently in progress, involves an in-depth evaluation of the selected
numerical schemes. The numerical accuracy and computational efficiency were judged
using the test cases that have either analytical solutions, fine-grid numerical
solutions, or experimental results. The following three classes of test problems
were investigated:
(i) convection-diffusion (scalar transport)
(ii) laminar flows
(iii) turbulent flows
The results for each of these groups are summarized in the following paragraphs.
*Work done under NASA Contract NAS3-24350.
_Onivecslty of Minnesota.
#Analytical and Computational Research, Inc.
105
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Convection-Diffusion
The test problems for convection-diffusion included (a) the transport of a step
change in a scalar in a uniform velocity field, and (b) recirculating flow in a
cavity with prescribed source for temperature. For both these problems, the
higher-order schemes (flux-spline and CONDIF) give more accurate results than the
hybrid scheme over the entire range of Peclet numbers (see Figures 1 through 4).
For the transport of the step change in the scalar, the flux-spline schemes exhibit
undershoots and overshoots.
Laminar Flows
Two selected laminar flow test cases are: (a) driven cavity and (b) flow over a
backward facing step.
The driven cavity problem was solved usinK a 22 x 22 uniform grid and the results
were compared with a 82 x 82 hybrid solution. The higher-order schemes give much
better accuracy and show substantial advantage over the hybrid scheme (see Figures
5 and 6). Computations for the flow over a backward facing step were made at two
Reynolds numbers (Re = i00 and 715) and the results were compared with the experi-
mental data (ref. 2).
At Re = I00, there is negligible false diffusion and the results of all schemes
compare well with experiments. At Re = 715, the higher-order schemes predict a
longer reattachment length, compared to the hybrid scheme, indicating smaller
numerical diffusion. There is, however, disagreement between the numerical and
experimental results. These deviations are probably due to the presence of three-
dimensional effects in the experiments (ref. 2).
Turbulent Flows
The selected schemes were used to compute the Stanford Conference test case 0421
(flow over a backward facing step) (ref. 3). The computed reattachment lengths
from various schemes are listed in Table I. In these computations, plug flow was
assumed at the inlet. It is seen that the flux-spline scheme approaches a grid-
independent solution with fewer grid points than the hybrid scheme. The improve-
ment shown by the flux-spline technique, however, is not as large as in laminar
flow cases. A similar trend in the results was noticed when the experimentally
measured velocity profile was specified at the inlet.
Performance of the Coupled Solution Approach
The efficiency of a numerical technique based on the primitive variables depends
to a great extent on the manner in which the velocity-pressure coupling is treated.
The iterative methods (e.g., SIMPLE, SIMPLER) derive an equation for pressure and
solve the momentum and pressure equations in a sequential manner. The convergence
of such an approach is found to be slow. An alternative to this sequential
approach is the direct solution of the whole set of continuity and momentum equa-
tions (ref. 4 and 5). This study evaluates the performance of a direct or coupled
approach in conjunction with a flux-spline scheme for convection-diffusion.
106
In the present approach, the discretized continuity and momentum equations are
treated as simultaneous equations and solved using the Yale Sparse Matrix Algorithm
(YSMP) (ref. 6). The nonlinearities in the equations are handled using the suc-
cessive substitution technique. In turbulent flow computations, the solution of
the flow equations (using YSMP) is followed by the solution of the equations for
the turbulence quantities (k and c). The equations for k and c are solved
sequentially in a decoupled manner using a line-by-line tridiagonal matrix algor-
ithm (TDMA). The sequence of calculations is as follows: (i) the continuity and
momentum equations are solved using YSMP, (2) with the given velocity field, the k
and c equations are solved to provide a new viscosity field for the momentum
equations. This procedure is repeated until convergence.
Preliminary Results
The details of the test problems selected for the evaluation of the coupled solu-
tion approach are given in Table II. The number of iterations required for con-
vergence and the execution times for the coupled solver are compared with those
for the SIMPLER approach in Table III. The results indicate that the direct solver
gives a speed-up factor of about three for laminar flows and five for turbulent
flows. Further evaluation on a finer grid is under progress.
TASK 3
The convection-diffusion scheme with superior performance in Task 2 and the direct
solver will be incorporated in the NASA 3-D elliptic code (COM3D). A test case
will be run to assess the accuracy and computational efficiency of the selected
scheme/algorithm for three-dimensional situations.
REFERENCES
I. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere, 1980.
2. Armaly, B. F., Durst, F., Pereira, J. C., and Schonung, B., "Experimental and
Theoretical Investigation of Backward Facing Step Flow," Journal of Fluid
Mechanics, Vol 127, 1983, pp 473-496.
3. Kline, S. J., Cantwell, B. J., and Lilley, G. M. (eds), The 1980-81AFOSR-HTTM
Stanford Conference on Complex Turbulent Flows, Stanford University.
4. Vanka, S. P., "Block Implicit Calculation of Steady Turbulent Recicculating
Flows," International Journal of Heat and Mass Transfer, Vol 28, 1985, pp
2093-2103.
5. Braaten, M., Development and Evaluation of Iterative and Direct Methods for
the Solution of the Equations Governing Recirculating Flows, Ph.D Thesis,
University of Minnesota, May 1985.
6. Eisenstat, M. C., Gursky, M. C., Schultz, M. H., and Sherman, A. H., "Yale
Sparse Matrix Package, II, The Nonsymmetric Codes," Research Report No. 114,
Yale University, Department of Computer Science, 1977.
107
Table I.
Calculated reattachment lengths [XRlh] (pluE flow at the step).
Grid Hybrid CONDIF Flux-spline
32 x 32 4.4 4.2 4.6
40 x 40 5.0 4.5 5.3
57 x 57 5.2 --- 5.3
Case No.
1
2
3
4
Table II.
Laminar flow test cases.
Flow Reynolds number
Driven cavity
Driven cavity
Sudden expansion
Sudden expansion
400
i000
400
4OO
Turbulent flow test cases
Backward facing step 5.6 x 105
Grid
(uniform)
22 x 22
22 x 22
22 x 12
22 x 22
22 x 22
Case No.
Table III.
No. of iterations required and execution times.
No. of iterations
SIMPLER Direct
Execution times
(seconds_
SIMPLER Direct
62 17
84 30
106 47
122 48
800 39
18 6
24 8
16 5
35 I0
408 79
IBM 3084
108
!
I
t_
O
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t.
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i=.
60 -
40 -
20 -
0
T=O
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Y
1 1=0
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im
0.1 1.0
T.O
1 x
Hybr_
COND[F__ _Y
_. F1i-spllne 1
I0 100
Grid Peclet number Pe
t TE86-1290
Figure i. Comparison of the average error in the temperature calculated using
different schemes. Recirculating flow with prescribed source for temperature.
109
ii
Z
L
O
Z
1.0'
.9
.8
.7
.G
.5
.4
.3
.2
.1 I
-1
- Pe-lO _ Fexact solution
/ _ _ _lFlux-spllne 1
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i
0 1
y (x-O)
A
(u
f,,,,i
L
0
Z
.9
.8
.7
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.5
.4
.3
.2
.1
Pe - 100 /--'----"'----exact
solution
- 1
I
0 1
y (x.O)
TE8_-1291
FiEure 2. Comparison of the temperature profiles calculated usin E different
schemes, with the exact solution. Recirculatin8 flow with prescribed
temperature source and velocity.
II0
1" =: 1 ---'="
T = 0--_. Y
A/
TE85-4217
Figure 3. Transport of a step change in temperature in a uniform flow field.
I
I
L,J
,2
0
L.
Iv
Iv
L
22
20
18
16
14
12
10
6
6
4
2
0
Hybrid
F1ux-spllne 3
F1ux-spllne 1
OND[F
0 4 8 12 16 20 24 28 32. 36 40 44 48
Flow angle, O--deg
T£86-1292
Figure 4. Comparison of the average error in the temperature calculated
from the different schemes.
III
1.0
0.8
O
O
0.6
y (x - 0.5)
0.4
0.2
0 - hybrid
(2} - spliue
- COSD_
-- exact
I.O0
TE86-3511
Figure 5. Normalized U-velocity at x = 0.5.
112
O. 50
0.25
¥ 0.00
0 = hybrid
[3 - spllne
m CONDIF
-- exsct
0.2 0.4 0.6 0.8 l.O
x (y = 0.5)
TE86-3512
F£gure 6. Normalized V-veloc£ty at y = 0.5.
113



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