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AIAA 92-0325 A Nearly-Monot one Genuinely Multidimensional Scheme for the Euler Equations I.H. Parpia and D.J. Michalek The University of Texas at Arlington Arlington, TX

30th Aerospace Sciences Meeting & Exhibit

January 6-9,1992 / Reno, NV For permission to copy or republish, contact lhe American Institute of Aeronautics and Astronautics 370 LEnfant Promenade, S.W., Washington. D.C. 20024

v A NEARLY-MONOTONE GENUINELY MULTIDIMENSIONAL SCHEME FOR THE EULER EQUATIONS

Ijaz H. Parpiat and

Donna J . Michalek' The University of Texas at Arlington

Arlington Texas 76019

Abstract In this paper we describe the development of an al- gorithm in which a third data point is introduced into

We report on recent progress made in the devel- the local reconstruction. The method, which is mod- opment of a genuinely multidimensional finite-volume eled after Roe's reconstruction of gradient data,5 is ap- method for the Euler equations. In this method, wave plied directly to the data on a triangle. We use the re- information is derived from a reconstruction procedure construction primarily to derive wave orientations; the applied to the data on a triangle. The algorithm is wave strengths are still obtained from two neighboring therefore best suited to application on an unstructured states. grid. We also present a flux formula which provides nearly-monotone solutions. The new method is tested In the next' sectioti, we describe the discretization on three simple two-dimensional flows, and the results procedure, followed hy the wave model and the flux for- show that dominant-wave resolution is high, and that ' ' mula. We then present three test cases to demonstrate strong-wave transitions are nearly 'monotone.

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Introduction Discretization . ,. v :c'

d on a family' ' We shall restrict' all of bur descriptions to the two- of genuinely multidimensional finitevolume algorithms dimensional case. The reconstruction of gradient in- in which wave strength and orientation information is formation on the mesh fit$ naturally into the standard derived from two neighboring states.'-* T h q e methods unstrueiured mesh discretization, in which the cell in are designed specifically to 'recognize' single Rankine- two spack dimensions is a triangle. Hugoniot wave transitions, and where the data fails t o indicate the presence of a single wave, a plausible set of waves, derived from various rules of thumb is used to reconstruct the jump between the states. None-of a].6 is Used t o generate the grids we show below.

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The Delaunay node generation and triangulation procedure for general boundaries developed by Hase et

the several variations on this idea has proved to be suf- ficiently robust to replace the classical dimensionally- The 'Ow solver described in this paper is node- split (grid-aligned wave) approach. " . centered, with computationalcells built from the triangle-

centroid mesh dual shown in Figure 1 Each edge or face /I..,I

Two-state methods rely by construction completely of a polygonal cell is shared by adjacent triangles in the on only two neighboring data points, and this is p&r- patch surrounding a node, and the wave model together haps why these methods perform so poorly. I t might with the flux formula described helow are used to write simply be unreasdnable to suppose that two states c k i y a nu'merical approximation to the flux on eveiy half- proyide rea!istic account o f the local wave content';f'' face (the portion of the face contained in a triangle).

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. . theflowfield:_,;. $I .,> .. ~. ' .. ~ , i . ~

' ~ ! ...',,: ,. I, t t Assistant Professor of Atmspsce Engineering. ' - , . . . . , . . , . , , . . )Graduate Research,Assistant. .,,..

Copyright 01992 American h?ritute of Aero+mticg,md Ast nautics, Inc. All iighte reserved.

'4 1

mesh dual linearized equtaions are

and

2" = ( i ) , Figure 1: Finite volume cell

,. . p, = ir,n"*a . ,. ....

Wave Model .. ., & . ,*h (3) ' p h ~ = e : , i * n = c . ,,en.

There are three familieb of planar wave solutions of , , x i ,

the twoAimknsiona1 Euter 'kquitions: 1 , . acoustic, shear, H~~~ the overbar denotes an arithmetic average, + = and entropy waves. It is our intention to use these waves G, and the RankineHugoniot averaged speed ofsound as building blocks in an algorithm for the description is ,j = m, and evolution of local variations in the flow properties.

Of the infinitely many combinations of waves which For computational simplicity, we define a local lin- might be used in 'the reconstruction of the data on a

eariaation of the governing equations, and then recon- triangle, out guide in the choice of a particular set is struct the data using superposition. One such linearim- guided primarily by the complexity of the algebra which tion procedure is Roe averaging? which has proved to arises in the specification of the wave parameters. The be remarkably effective in Ck.9iCal grid-aligned meth- data provide eight derivatives, to which are matched ods, and is also central to the recently developed multi- the variations described by the chosen wave pattern. dimensional fluctuation distribution alogorithm ofStru- one rrJght therefoSe suppose that the minimum i j s et aLS We use a different IinWriZation here, fOk%'- ber of waves is four (each Wave has two free parameters, ing the Rankine-Bwniot averaging techniqueg which strength and orientation), but no choice of four waves is described in an earlier paper? can dwr ibe general data. A study of the eigenvectors

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(2) shows thisminimum number is in fact five, consist- equations are Of ing of two acoustics, two shears, and an entropy wave,

and this is the choice we make in the algorithm de- Elementary Of the

the form - scribed below. ~ ( z , t ) = o(n)e(n)(z-Xt),

:.!: .i.\. wh.ere 'V is the vector of primitive variables (p , u , p ) . . .. (d.ensity, velocity and pressure). Equation (1) defines a wave of strength a, with wavefront normal n. e is the eigenvector which describes the transition ac ros the wave in state space, and X is the wavespeed. The eigen- vectors 2 and wavespeeds X of the Rankine-Hugoniot

Noting that only eight wave parameters follow from the data, we specify that the acoustics waves have a common normal naC, and that the two shear waves are mutually perpendicular, with normals nlhr and dh2. The orientation of the entropy wave is defined by its normal ne", The system of eight equations from which the unknown wave parameters are found is

2

c

A

Figure 2: Wavefront passing through a triangular cell . . * . fragment ~.

flux jump across the wave, and Q is the area-averaged value of the vector of conserved'variables (mass, mo- mentum, and energy per unit area). The result (20) is used below in the development of the numerical flux function.

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Figure 3 depicts two triangular fragments of neigh- boring cells I and r. For algorithmic simplicity, we wish to account for the entire effect of the waves which arise between the states I and r through the numerical flux a t the common face. This would obviate the need to explicitly calculate the effect of these waves on the r e maining faces of the triangulaf segments. The mannei in which this can be accomplished is to set the flux at the common face, normal to the face, to . . - . / , : , . . , . , . ,~ . .

. .

where si is an estimate of the length of the wave seg- ment contained in the left triangle, and S, is the face length. We note that i t is equally valid to write

F = F, - AFWG, (22) SI

and i t is perhaps most reasonable to use the average of (21) and (22). Finally, for simplicity, we assume that ratio of the length of the wave segment to the cell face length is' zer6 or unity, depending on the direction of

Figure 3: Neighboring triangular cell fragments

motion of the wave. This yields the flux formula

6

F = - Fj + F, - x s g n ( X k ) A F r 2 [ k=1

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for a system of five waves. W .. ... . , .

It is interesting to'note that the only difference in the flux formula used in References 1-4,

and equation (23) is the dot product term.

Test Cases

The new method has been tested on a variety of simple flow problems, including several which we have used in earlier papers. The array of test cases presented below consists of two'supersonic flow problems and a subsonic ~ ehannel problem. The first two cases show the wave tesolukCon performance of tke'method in low and higlisupersonic flows. The last case is included be- cause the omnidirectional nature of the propagation of acoustic information in subsonic flow appears to present a severe challenge to the solution methodology. The wave model is designed to identify dominant waves in the data,'and'an'added level of sophistication may be necessary to reconstruct data in which no strongly pre- ferred acoustic directions exist.

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L d u I 1.w E 3 98 D 1.8,

t .5 0.5

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0.0 -1

0.5

Figure 4: Channel geometry and grid 0.0

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L M P, F 2.89 E 2.77 D @.e4 Figure 6: Mach number contours . .

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Figure 5: Pressure contours

Simple forward time integration with local time s t e p ping is used for every case shown below.

Supersonic Channel Flow

This is the problem of Levy et, al.'O The geome- try of the channel, and the 1074 node grid we used is shown in Figure 4. The ramp angle on thelower wall is 15'. Pressure and Mach number contours for an inflow Mach number of 2 are shown in Figures 5 and 6. The flow properties are monotonic, and the resolution of the shockwaves is generally good (2-3 triangles). TLiS ' is confirmed in Figure 7, which is a close-up view of the mesh overlaid onto the Mach number contours in a re- gion near the wedge shock. The density residual history, shown in Figure 8, indicates a two order-of-magnitude convergence in about 1000 steps.

. . . .,._I/ . ," l ~ . " . > ._.,. . . . '.., , . Figure . . 7: Mach nu,mber contours .. *_ .~ i . near . . the weage.shock

5

.5.Q

0 250 500 750 1000 1250 tlme step

Figure 8: Density residual history for the supersonic channel flow problem

Figure 9: Ramp geometry and grid Double Ramp

The monotonicity of the algorithmis tested far more severely in this flowfield, which is a Mach 4 flow over a 20-35' double ramp. The uniform 1052 node grid we used is shown in Figure 9. Pressure and Mach num- ber contours are plotted in Figures 10 and 11. The Mach number contours in Figure I1 clearly indicate the presence of the slip surface. These contour plots also show nearly-monotone variations in the flow properties. The density residual history in Figure 12 shows a two order-of-magnitude reduction in 500 timesteps.

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Subsonic Channel Flow

The last test case we show here is a fully subsonic flow in the channel shown in Figure 13. The bump on the lower wall is a 10% circular arc, and there are 29 nodes on the bump. The stretched grid contains a total of 1108 nodes. Pressure and Mach number contours for an inflow Mach number of 0.6 are shown in Figures 14 and 15. The more serious convergence problem alluded to'earlier is apparent from the 'density residual history plot in Figure 16.

1.00 r

0.76 -

050 -

0.25 -

r

Figure 10: Pressure contours

6

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0.50

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0.25

0.00 0 .m 0.25 ' 0.50 0.75

2 5

"" .2.5 0.0 2.5

..11.. ~ . , Figure.13: ChanLel geometry and grid

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., . 1: .I .mi*:. ; . . > * ; : ? . : I ;..'.c?:t.: :,,. . . ,.

ach number contours

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Figure 12: Density residual history for the double-ramp problem

Figure 15: Mach number contours

L". F. F 1.11

f 0.82

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.2.5 I \ kW# lb~lO(max(res(1)))

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loplO(rms(res(t ))) -

Concluding Remarkr

In this paper we present the development and ap- plication of a new genuinely multidimensional finite- volume algorithm for unstructured grids. Numerical examples show that the method represents a substan- tial improvement over earlier algorithms of this type. Wave resolution is high, and strong wave transitions are nearly monotone. Although convergence to steady- state is also improved, more effort is needed to find ways t o further enhance the convergence behaviour of the algorithm.

,LJ 2. C.L Rumsey, B. van Leer, and P.L. Roe, “A grid-independent approximate Riemann solver with ap- plications to the Euler and Navier-Stokes equations,” AIAA Paper 91-0299, 29th Aerospace Sciences Meet- ing, Reno, NV 1991.

3. C.L. Rumsey, B. van Leer, and P.L. Roe, “Effect of a multidimensional flux function on the monotonic- ity of Euler and Navier-Stokes Computations,” AIAA Paper 91-1530-CP, AIAA 10th Computational Fluid Dynamics Conference, Honolulu, HI, June 1991.

4. I. Parpia, “A planar oblique wave model for the Euler equations,” AIAA Paper 91-1545-CP, AIAA 10th Computational Fluid Dynamics Conference, Honolulu, HI, June 1991.

5. P.L. Roe, “Discrete models for the numerical analysis of time-dependent multidimensional gasdynam- ics,” Journal of Cornpufational Physics, 63, 1986, p. 458.

6. J.E. Base, D.A. Anderson, and I. Parpia, “A De- launay triangulation method and Euler solver for bod- ies in relative motion,” AIAA Paper Sl-159O.CP, AIAA 10th Computational Fluid Dynamics Conference, Hon- olulu, HI, June 1991. u

7. P.L. Roe, “Approximate Riemann solvers, pa- rameter vectors, and difference schemes,” Journal of Computafional Physics, 43, 1981, p. 357.

8. R. Struijs, B. Deconinck, P. de Palms, P. Roe, and K.G. Powell, “Progress on Multidimensional Up- wind Euler Solvers for Unstructured Grids,” AIAA Pa- per 91-1550-CP, AIAA 10th Computational Fluid Dy- namics ‘~onfe~enc~~’Honolu1u. HI, June 1991. . .

Acknowledgement ,:y . , , . . , , . . I..

9. C.P. Kentzer, “Quasi-linear form of the Rankine- This research was supported in part by NASA Lang- Hugoniot jump conditions,” AIAA Journal, 24,4,1986,

lev Research Center under research nrant NAG-1-1207, mi r. ””.. with James D. Keller, Technical Officer.

10. D.W. Levy, K.G. Powell, and B. van Leer,“An implementation of a grid-independent upwind scheme for the Euler equations,” AIAA Paper 89-1991, 9th Computational Fluid Dynamics Conference, Buffalo, NY, 1989.

References

1. LH. Parpia, and D.J. Michalek, “A shock cap Luring method for multidimensional flow,” AIAA Poper 90-9016, 8th Applied Aerodynamics Conference, Port- land, OR, 1990.