Streamline patterns calculated from transonic flow solutions which were generated using a nonconservative finite difference scheme showed a net streamtube area increase far downstream of the disturbance indicating that the global mass balance was destroyed. Similar calculations using a conservative finite difference scheme did not show this defect. Comparative calculations were made at several free-stream Mach numbers for nonlifting flow over a 10% parabolic arc airfoil. In a transonic internal flow, this nonconservation of mass may be of greater concern than in an unconfined external flow

Newman, P. A.

South, J. C., Jr.

English

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A	 . A
4%SA TECI-#NjCAL	 NASA TM X-72827
MEMORANDUM
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CONSERVATIVE VERSUS NONCONSERVATIVE DIFFERENCING:
TRANSONIC STREAMLINE SHAPE EFFECT'S
Perry A. Newman and Jerry C. South, Jr.
February 1976
(NASA—TH—X-72827) CONSERVATIVE VERSUS	 N76-21154
NONCONSERVATIVE DIFFERENCING: TRANSONIC
STREAMLINE SHAPE EFFECTS Interim Report
(1,ASA) 15 p HC $3.50	 CSCL 01A	 UnclasG3/02 21481
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vised, or may be incorporated in another publication.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATIN
LANGLEY RESEARCH CENTER, HAMPTON, VIRGINIA 23665
J
I
1. Report No.
_NA9A IM X-7282
2, Government Accession No, 3. Recipient's Catalog No.
4. Title and Subtitle 5. Report Date
CONSERVATIVE VERSUS NONCONSERVATIVE DIFFERENCING: February 1976
TRANSONIC STREAMLINE SHAPE EFFECTS 6. Performing Organization Code
7. Author(s) 8. Performing Orgaetalion Report No.
Perry A. Newman and Jerry C. South, Jr.
10. Work Unit No.
9. Performing Organization Name and Address
NASA-Langley Research Center 505-06-11-02
Hampton, Virginia
	
23665 11. Contract or Grant No.
13. Type of Report and Period Covered
NASA Technical Memorandum
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, D. C.	 20546 14. Sponsoring Agency Cade
15. Supplementary Notes
Interim technical information release, subject to possible revision and/or later
formal publication.
16. Abstract
Streamline patterns calculated from transonic flow solutions which were generated
using a nonconservative finite difference scheme show a net streamtube area increase
far downstream of the disturbance indicating that the global mass balance has been
destroyed.	 Similar calculations using a conservative finite difference scheme do not
show this defect.	 Comparative calculations have been made at several free-stream Mach
numbers for nonlifting flow over a 10% parabolic arc airfoil.	 In a transonic internal
(or confined) flow, this nonconservation of mass may be of greater concern than in
an unconfined external flow.
I
I
I
17. Key Words (Suggested by Author(s) ) 18. Distribution Statement
Aerodynamics	 •Interference
Transonic Flow	 Conservation form
Compressible Flow Unclassified unlimited
Internal Flow
Wind-Tunnel Walls
19. Securitv Clasdf. (of this report) 20. Security Classif. (of this page) 21. No, of Pages 22. Price'
Unclassified Unclassified 13 $3.25
^For sale by the National Technical Information Service, Springfield, :Virginia 22161
i
f,6
L,___
I I I
CONSERVATIVE VERSUS NONCONSERVATIVE DIFFERENCING:
TRANSONIC STREAMLINE SHAPE EFFECTS
By Perry A. Newman and Jerry C. South, Jr.
Langley Research Center
SUMMARY
Streamline patterns calculated from transonic flow solutions which were
generated using a nonconservative finite difference scheme show a net stream-
tube area increase far downstream of the disturbance indicating that the
global mass balance has been destroyed. Similar calculations using a conserv-
ative finite difference scheme do not show this defect. Comparative calcu-
lations have been made at several freestream Mach numbers for nonlifting flow
over a 10% parabolic arc airfoil. In a transonic internal (or confined) flow,
this nonconservation of mass may be of greater concern than in an unconfined
external flow.
INTRODUCTION
The basic relaxation technique introduced by Murman and Cole (ref. 1) has
been used extensively in computing transonic flow fields. A deficiency was
later recognized in the original finite difference scheme and Murman (ref. 2)
introduced a conservative finite difference scheme, which included a shock-
point operator, in order to correct it. Nevertheless, the original nonconser-
vative scheme continues to be used by most investigators since it seems to
give shock jumps and locations at the surface of the configuration more nearly
like.those observed in experiments.- The purpose of this note is to demon-
strate that these nonconservative shocks, which may extend well into the flow
field, destroy the global mass balance by producing mass at the shock. In a
transonic internal (or confined) flow this lack of mass balance may prove to
be more crucial than is the case for an unconfined external flow.
The present results and observations were prompted by streamtube anom-
al •iesfirst noted in calculations pertaining to wind tunnel flows in both 2
and 3 dimensions. The influence of conservative versus nonconservative finite
difference formulation on the global mass balance is readily observed by com-
aputing streamline shapes in an external 2-D transonic flow. This note presents
1such sample results.
PROBLEM DESCRIPTION
A NASA Langley study of one concept for minimizing wind tunnel interfer-
ence involves contouring the upper and lower (initially solid) walls of a
small 2•-D facility according to numerical results obtained from nonlinear flow
solutions. Figure 1 shows calculated free-air streamline deflections (from
straight lines) at the proposed tunnel-wall locations for 2-D subsonic lifting
flow (M = 0.5, cx = 3") about a 10% parabolic arc airfoil. Arrows along the
abcissa denote the test section and airfoil chord lengths in the streamwise
direction. In this figure the deflections have been taken to be zero at the
front end of the variable wall test sectioi*. The point to be made here is
that a streamtube roughly the dimensions of the tunnel returns to its far-
upstream size far downstream of the model. That is, there is a global mass
balance in the calculation. Figure 2 shows the same kind of results for a
supercritical lifting flow (M = 0.8, a = 3 0 ). In this case, however, the size
of the streamtube has increased from far upstream to far downstream indicating
that mass has been introduced. These results were generated using a program
which employed the Garabedian and Korn (ief. 3) nonconservative finite dif-
ference scheme. In the M = 0.5 case there is no shock wave in the flow whereas
in the M = 0.8 case there is one on the upper surface of the airfoil and it
extends about 3/4 of a chord length into the flow field.
A computer program recently developed by South and Brandt (ref. 4)
contained the Murman (ref. 2) conservative finite difference scheme and was
easily modified to use the Garabedian and Korn (ref. 3) nonconservative finite
difference scheme. This program solves the transonic small disturbance equa-
tion for only symmetric flow but incorporates several iterative solution
techniques. For the results presented here, the equally-spaced computational
grid was analytically stretched so that the physical grid extended to infinity
in both the streamwise and normal directions. Streamline shapes were obtained
along several grid lines by a streamwise integration of the normal component
of the perturbation velocity using a trapezoidal rule in the computational
grid.
Comparisons of conservative and nonconservative results were made with
all aspects of the computation identical except the finite differencing scheme.
*Note that for an unconfined 2-D lifting flow there is an appreciable stream-
tube deflection from upstream infinity to the front; of the test section. One
can infer this from the finito streamline slopes at large JX! in figures 1 and 2.
This phenomena presents an operational problem for those intent on producing
"interference-free" confined flow.
Go-PA'
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iIn all cases the calculations were iterated until the root-mean-square value
of the true residual was of the order of the truncation error. Although the
solution itself was observed to change somewhat for a given case when the
analytical stretching was changed, it is felt that the relative streamline
shape effects presented here indicate that a nonconservative finite difference
scheme destroys the global mass balance in a supercritical flow calculation.
RESULTS
Comparison cases were run for a 10% parabolic arc airfoil at zero inci-
dence for free stream Mach numbers of 0., 0.70, 0.84 and 0.95. These represent
incompressible, subcritical, mild supercritical, and strong supercritical flow
conditions respectively. The computational grid was 128 streamwise by 33 in
the normal direction (the physical half-space, airfoil mean plane to infinity).
Streamline deflections were computed along several grid lines for all cases;
not all of them are shown in the figures.
The incompressible (M = 0) and subcritical (M = 0.70) results for pressure
distributions and streamline deflections were identical for conservative and
nonconservative finite differencing. In all cases the computed streamtubes
returned to their proper size. Since the conservative and nonconservative
schemes differ only at points where the flow is supersonic, the results were
expected to agree.
Results for the mild supercritical case (M = 0.84) are shown in figures
3 and 4. Figure 3 shows streamwise pressure coefficients along the symmetry
line y = 0. Conservative results are given by the solid curves while noncon-
servative results are given by dashed curves. As others have shown in the
past, one effect of the conservative scheme is to locate the shock wave further
downstream on the airfoil surface. Figure 4 compares computed streamline
deflections along three grid lines in.the flow field. As can be seen from the
figure, the displacements at downstream infinity indicate that the conservative
streamtubes return to their upstream infinity value whereas the nonconservative
ones do not. For this case, the shock wave extends about 1/2 a chord length
into the flow.
Similar results for the strong supercritical case (M = 0.95) are shown in
figures 5 and 6. In figure 5, which shows streamwise pressure coefficients
along the symmetry line y = 0, it can be seen that the flows are no longer
similar. In the nonconservative case there is a normal shock at the airfoil
trailing edge whereas conservative differencing gives a weak oblique shock
wave at the trailing edge followed by a normal shock about 1/2 chord length
behind the airfoil. The streamline deflections for this case are shown in
figure 6. Here it is very evident that the nonconservative streamline
deflections do not become zero far downstream of the airfoil. On the other
hand, however, the conservative streamtubes are seen to return to their proper
size far downstream.
GINS 
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The extent to which the conservative and nonconservative flow fields
differ in this strong supercritical case can be seen by comparing figures 7
and 8. Here Mach number contours in the supersonic bubble are shown for the
nonconservative and conservative flows respectively. The multiple shock
"fishtail" pattern shown in figure 8 is very similar to one computed by Murman
(ref. 2) when he introduced his conservative scheme. The triangular region of
nearly-constant, supersonic velocity between the weak oblique shock coming off
the trailing edge and the shock wave further downstream is.clearly evident.
This downstream shock is normal at the symmetry line, becomes a strong oblique
shock somewhat further out in the flow, and appears to again become normal
after merging with the weak oblique shock coming off the airfoil trailing edge.
The Mach number pattern shown in figure 7 was computed using the nonconserva-
tive scheme and has a normal shock coming off the trailing edge. The super-
sonic bubble is smaller than that for the conservative scheme and it appears
to be terminated almost everywhere by a normal (or strong oblique) shock.
CONCLUDING REMARKS
Use of a nonconservative finite difference scheme in transonic flow cal-
culations destroys the global mass balance. The mass created by the noncon-
servative operator at the shock produces a non-physical swelling of the
inviscid streamtubes which persists far downstream. Perhaps the fortuitous
agreement between the nonconservative and experimental results comes about
because this streamtube swelling effect simulates a viscous wake or thickened
boundary layer downstream of a shockwave. In any case, it is felt that the
conservative finite difference scheme should be used in applications where
streamtube effects are important, such as internal or confined flows.
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REFERENCES
1. Murman, E. M.; and Cole, J. D.: Calculation of Plane Steady Transonic
Flows. AIAA J., vol. 9, no. 1, Jan. 1971, pp. 114-121.
I
	
2. Murman, E. M.: Analysis of Embedded Shock Waves Calculated by Relaxation
Methods. Proceedings of AIAA Computational Fluid Dynamics Conference,
Palm Springs, Calif., July 1973, p p . 27-40.
3. Garabedian, P. R.; and Korn, D. G.: Analysis of Transonic Airfoils.
Comm. Pure Appl. Math., vol XXIV, 1971, pp. 841-851.
4. South, J. C.; and Brandt, A.: Application of a Multi-Level Grid
Method to Transonic Flow Calculations. Presented at the Project
SQUID Workshop on Transonic Flow Problems in Turbomachinery, U. S.
Naval Postgraduate School, Monterey, Calif., Feb. 11-12, 1976.
(Conference proceedings to be published).
0


Conservative versus nonconservative differencing: Transonic streamline shape effects

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