The construction of a reliable, shock capturing finite difference method to solve the Euler equations for inviscid, supersonic flow past fighter and missile type configurations is highly desirable. The numerical method must have a firm theoretical foundation and must be robust and efficient. It should be able to treat subsonic pockets in a predominantly supersonic flow. The method must also be easily applicable to the complex topologies of the aerodynamic configuration under consideration. The ongoing approach to this task is described and for steady supersonic flows is presented. This scheme is the basic numerical method. Results of work obtained during previous years are presented

Osher, S.

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Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19850002605 2020-03-20T21:17:42+00:00Z
r,	 1
ANNUAL STATUS REPORT
Sub-nitted to
NASA LANGLEY RESEARCH CENTER
Hampton, Virginia 23665
GRANT NUMBER:	 NAG1 -270
T--TLE:	 White Paper Entitled: SHOCK CAPTURING FINITE
DIPPERENCE ALGORITHMS FOR SUPERSONIC FLOW PAST
FIGHTER AND MISSILE TYPE CONFIGURATIONS
PRINCIPAL
INVESTIGATOR:
	 Stanley J. Osher
Department of Mathematics
UCLA
GRANTEE
INSTITUTION:	 The Regents of tr ,a University of California
Los Angeles, CA 9CO24
PERIOD COVERED
BY THIS REPORT:	 3/1/84 - 2/28/85
DISTRIBUTION:
	 3 copies:
	 Dr. David Rudy
Chief, Computational Methods Branch (MS 409)
NASA Langley Research Center
Hampton, Virginia 23665
2 copies:
	 NASA Scientific and Technical
Information Facility
P.O. Box 8757
Baltimore/Washington International Airport
Maryland 21240
ce
l
N85-109 13(uASA-CR-174051) SdOCK CAPTUhING FINITE
UiFFxkENCE ALGQt1TtibS FOR SUPELSJNIC FLOW
PAST FIGHILR AND IlISSiVE 'ITiE CONFIGURATICNS
Annual Status Report, 1 Md" 1984 - 28 Feb.
1965 (Cdlitornia Univ.) 7 F HC A02/ fIF A01
unc;la S
G3/02 01208
SEMI ANIMAL- STATUS REPORT
I4LSA-XIG1-270
1. Introduction
It is highly desirable to construct a reliable, shock capturing
finite difference method to solve the Euler equations for iuviscid,
supersonic flow past fighter and missile type of configurations. The
numerical method must have a firm theoretical foundation and must
be robust and efficient. It should be able to treat subsonic poczets
in a. predominantly supersonic flow. The method must also be easily
applicable to the complex topologies of the aerodynamic configuration
under consideration. In the rest of this report, we briefly describe
our ongoing approach to this task, We first present in Section
2, a background sketch on a scheme developed by Oster and his co-
workers. This scheme and analogous extensions to steady supersonic
flows is the basic numerical method. Some results of the first year's
effort are presented in Section 3. Results obtained during the
second year's effort are presented in Section 4. Results obtained
during the third year's effort are presented in Section 5. Results
obtained during this year's effort are presented in Section 6. A list of
references is complied in Section 7. A list of papers written during this
period is attached.
2. The Osher Scheme: A background
Engquist and Osher [1,2,3] developed circa 1980 a monotone, upwind
scheme for scalar conservation laws and applied it to the small
disturbance equation of transonic flow. This scheme is now incorporated
in produezion computer programs at both NASA's Ames Research Center
and Langley Research Center and has been found to be much more robust
than the conventional Mu_rman-Cole algorithm. The Engquist-Osher
algorithm was also more reliable in the expansion shocks were excluded
s.:
and stead flows were computed more quickly.y	 pu	 q k]y. Oster then developed
h	 a generalization of the scalar a.'Agorithm for hyperbolic systems of
conservation laws [41 which was further studied by Osher and Solomon
L7)
2The Osher scheme for systems is an upwind shock "capturing"
algorithm applied to the unsteady Eisler equations. It resembles
several others, such as Godunov's [6, 11 ], Roe's [8] and Steger and
Warmings's [9]. It can be applied to essentially all hyperbolic
systems of conservation laws arising in physics, but becomes
relatively s-hple for Euler's equations in general geometries using
body fitted coordinates. The latter extension resulted from a
collaboration between Osher and Chakravarthy [10,11]. The Osher
scheme is based on a Riemann solver, as in Godunov's but compression
waves are used to approximate shocks. This leads to a smoother and
simpler algorithm. The numerical flux functions are written in
closed form and include various switches which make them upwind.
Osher's algorithm reaches steady shock solutions exactly (for constant
states on either side of the shock) on the grid with a one or two
point monotone transition (the two point transition property carries
over for problems with nonzero gradients on either side of the shock).
The Osher algorithm for the unsteady Euler equation is currently
first order accurate and explicit in time. An extension to second
order accuracy and implicit temporal differencing is possible. While
the Osher methodology has so far been applied extensively to the
unsteady Euler equations, the scheme was a,..tually developed for
general systems of conservation laws. Thus an algorithm may also be
developed for steady supersonic flows where the marching direction
would be a spatial coordinate. These extensions are now incorporated
into the proposed task outline.
3. Results of the First Year's Effort
So far we have extended the basic algorithm to be second order
accurate. This involves using and modifying flux limiting techniques
first developed by Van Leer [12], cf. also Harten [13]. The technique
fits very well into our basic algorithm because of its inherent use
of a nonlinear field by field decomposition.
3The resulting algorithm is now at least second order accurate and
variation diminishing, hence no overshoot is possible. It still possesses
steady, di3crete shock profiles having a one or two point transition.
Moreover, the entropy condition, which ensures that only physically correct
limit solutions occur, is still valid. The full scheme involves, at most,
a five point discretization.
Explicit calculations were implemented through Richtmyer's version of
the Lax-Wendroff time discretization. The bouAidary treatment involves
using the first order scheme one point from the boundary, and using a
natural boundary condition Riemann problem solver at the physical and far
field boundaries.
4. Some Tesults Obtained During the Second Year's Effort
During this period, the details of the first order Osher algorithm
were worl.-=%! out for hyperbolic space marching for Euler's equations
in body-fitted coordinates. Further work was done to make it second
order accurate and "variation diminishing." Numerical comparisons with
Roe's algorithm were performed for this problem.
Implicit time marching algorithms were devised for the first and
second order unsteady algorithms. Using the upwind nature of the space
differences, it was found that extremely fast convergence to steady state
was possible. This was done using techniques borrowed from elliptic
equations to invert the implicit scheme for very large time steps. No
approximate factorization was used.
Algorithms for the unsteady problem based on triangulation of the
space region were devised. The control volume approximations use the
basic first order Osher algorithm together with multidimensional,
coordinate free, flux limiters. A very flexible and powerful algorithm
is being developed. Implicit calculations, using the algorithm withour
approximate factorization, can lead to rapid convergence to steady state
in very general geometries.
Finally, schemes based on Riemann solvers are known to have an
"entropy glitch," i.e. an O(Ax) expansion shock at sonic expansions.
For the Osher scheme, we have removed this by adding a simple multiplier
to our usual viscosity in the appropriate field. Besides cosmetics,
this also helps the robustness of implicit calculations.
5. Some Results Obtained L_ l ring the Third Year's Effort
During this period we devised MUSCL type high resolution algorithms
[171 	These are similar ( 5 point,
to those we devised earlier [18],
TVD, entropy condition satisfying)
but are somewhat more flexible, and
simpler to implement. The original MUSCL idea is due to van Leer [15].
We also constructed high resolution supersonic space marching
algorithms for Euler's equations.
We began to extend our techniques to laminar compressible Navier-
Stokes flow, avoiding spurious boundary layrers using results of [16],
1191.
The efficient implementation of these algorithms is proceeding
quite smoothly.
6. Some Results Obtained During the Present D--riod
During this period we made our high resolution algorithms even
more accuratF. In [20], we gave a systematic procedure for constructing
semi-discrete approximations to scalar conservation laws. Except for
isolated critical points, these schemes will have 2m - 1 order accuracy,
2m order dissipation, and a bandwidth using 2m + 1 points, for m any
integer between two and eight. They are in conservation form and TVD.
In [21] and [22] we included the simple 5 point, thirde order accurate,
TVD method into our Euler and Navier-Stokes codes, and much improvement
was found. Shocks were very crisply resolved, and phenomena such as
boundary larger separation were computed accurately.
There are improvements in time discretization to be obtained. It
would also be good to remove the degeneracy to first order accurac, which
is a problem in the efficient use of the higher than third order methods.
Worm is proceeding in this direction. The goal of a very efficient
compressible Navier-Stokes code is within reach.
57. References
1. Enquist, B. and Osher, S., "Statle and Entropy Satisfying
Approximations for Transonic Flow Calculations," Mathematics of
Computation, Vol. 34, 1980, pp. 45-75.
2. Engquist, B. and Osher., 'One Sided Difference Schemes and
Transonic Flow." Proceedings of National Academy of Sciences, USA,
Vol. 77, 1980, Pp. 3071-3074.
3. Engquist, B. and Osher, S., `One sided Difference Equations for
Nonlinear Conservation Laws," Mathematics of Computation, Vol. 36,
1981, pp. 321-352.
4. Osher, S., "Numerical Solution of Si1gular Perturbation Problems
and Hyperbolic Systems of Conservation Laws," North Holland
Mathematical Studies #+7, 1981, pp. 179-205.
5. Osher, S. and Solomon, F., "Upwind Schemes for Hyperbolic Systems
of Conservation Laws," Mathematics of Computation, 1982, Vol.
38, PP . 339-374.
6. Godunov, S.K., "A Finite-Difference Method for the Numerical
Computation of Discontinuous Solutions of the Equations of Fluid
Dynamics," Mat. Sb., Vol. 47, 1959, PP• 271-290.
7. van Leer, B., "Upwind Differencing for Hyperbolic Systems of
Conservation Laws," ICASE Internal Report Document #12, 1980.
8. Roe, P.L., "Approximate Riemann Solvers, Parameter Vectors, snd
Difference Schemes," Journal of Computational Physics, 'co appear.
9. Steger, J.L., and Warming, R.F., "Flux Vector Splitting of the
Inviscid Gasdynamic Equations with Applications to Finite
Difference Methods," NASA Technical Memorandum 78605, 1979.
10. Osher, S. and Chakravarthy, S.R., "Upwind Schemes and Boundary
Conditions with Applications to Euler Equations in General
Geometries," Submitted to J. Comp. Phys.
L'. Chakravarthy, S.R. and Usher, S., "Numerical Experiments with
the Osher Upwind Scheme for the Euler Equations," AIAA-82-0975,
St. Louis, Mc. (1982).
12. van ?peer, B., "Towards the Ultimate Conservative Difference
Scheme, II. Monotonicity and Conservation Combined in a Second-Order
Scheme," J. Comp. Phys., 14, (1974), 361-37^.
13. Harten, A., "High Resolution Schemes for Hyperbolic Conservation
Laws," J. Comp. Phys., to appear.
6References Cont.
14. Harten, A. "The Artificial Compression Method for Computation
of Shocks and Contact Discontinuties. I. Single Conservation
Laws," Comm. Aare Appl. Math., 30, (1977), 611-638.
15. •ran Leer, B. "Towards the Ultimate Conservative Difference
Scheme, V. A Second-order Sequel to Godunov`s Method," J.
Comp. Phys., Vol. 32, (1979), PP . 101-136.
16. Michelson, D. "Initial-Bo-.indary Value Problems for Incomplete
Singular Perturbations of Hyperbolic Systems," to appear,
Proceedings of Large Scale Computations in raid Mechanics.
17. Osher, S., "Convergence of Generalized MUSCL Schemes," SIAM. J.
Num. Anal., (submitted).
18. Osher, S., and Chakravarthy, S., "High Resolution Schemes and
the S)atropy Condition," SIAM J. Num. Anal., (to appear).
19. Gustafsson, B. and Sundstrom, A., "Incompletely Parabolic
Problems in Fluid Dynamics," SIAM J. Appl. Math., Vol. 35,
(1979), PP .
 343-357.
20. S. Osher and S.R. Chakravarthy, "Very High Order .Accurate TVD
Schemes .," I CASE Report x/84-44 (1984), Sutm i- ted to SINUM.
21. S. Chakravarthy, and S. Osher, "Computinf with High Resolution
Upwind Schemes for Hyperbolic Equations, ` to appear in Proceedings
of A.M.S. - SIAM Summer Seminar, La Jolla, CA (198,3)
22. S.R. Chakravarthy , K.Y. Szema, S. Osher, and J. Gorski, "A New
Class of High Accuracy Total Variation I-iminishing Schemes for the
Navier-Stokes Equations", in preparation.
8. Papers Written During the Present Period
[1) Sukumar Chakravarthy and Stanley Osher, "Computing with high
Resolution Upwind Schemes for Hyperbolic Equations," to appear
in Proceedings of AMS - SIAM Summer Seminar, La Jolla, CA (1983)
[2] Stanley Osher and Sukumar Chakravarthy, "Very High Order Accurate
TVD Schemes," ICASE Report X84-4!L (1984), Submitted to SINUM.
[3] S.R. Chakravarthy, K.Y. Szema, S. Osher, and J. Gorski., "A New Class
of High Accuracy Total Variation Diminshing Schemes for the Navier-
Stokes Equations", in preparation.


Shock capturing finite difference algorithms for supersonic flow past fighter and missile type configurations

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