NASA Technical Memorandum 83481

NASA-TM-83481 19840003763

7

Embedding Methods for the' Steady Euler Equations

Shih-Hung ChangCleveland State UniversityCleveland, Ohio

and

Gary M. JohnsonLewis Research CenterCleveland, Ohio

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EMBEDDINGMETHODSFOR THE STEADY EULER EQUATIONS

Shah-HungChang*ClevelandState UniversityCleveland,Ohio 44115

and

Gary M. 3ohnsonNationalAeronauticsand Space Administration

Lewis ResearchCenterCleveland,Ohio 44135

SUMMARY

A recentapproachto the numericalsolutionof the steady Euler equationsIs to embed the flrst-orderEuler system in a second-ordersystem and then torecapturethe original solutionby imposingadditionalboundaryconditions.Initialdevelopmentof this approachand computationalexperimentationwlth Ithave been based on heuristicphysicalreasoning. This has led to the con-structlonof a relaxationprocedurefor the solutionof two-dlmenslonalsteady

o flow problems. In the presentreport the theoreticalJustificationfor theembeddingapproach is addressed. It is proven that, with the appropriatechoice of embeddingoperatorand additionalboundaryconditions,the solutionto the embedded system is exactlythe one to the originalEuler equations.Hence, solvingthe embeddedversionof the Euler equationswill not produceextraneoussolutions.

INTRODUCTION

In the developmentof numericalsolutionproceduresfor the steady Eulerequations,the common approachis to replacethe steadyequationsby their un-steady counterpartsand then to seek a temporally-asymptotlcsteady solution,either in real tlme ([1], [2]) or in pseudo time ([3] - [5]). Due to the dlf-flcultlesassociatedwith the numericalsolutionof a direct finite differencerepresentationof the steady Euler equations,relativelyfew departuresfromthis approachare to be found in the literature. Steger and Lomax [6] devel-oped an Iteratlveprocedurefor solvinga nonconservatlonform of the steadyEuler equationsfor subcrltlcalflowwith small shear. Deslderland Lomax [7]investigatedprecondltlon_ngprocedureson the matrix systemarisingfrom thefinitedifferencingof the Euler equations. Bruneau,Chattot,LamlnleandGulu-Roux[8] have used a variationalapproachto transformthe Euler equa-tions into an equivalentsecond-ordersystem. Preliminarynumericalresultshave been presentedfor two-dlmenslonalinternalflows. Recently,Jespersen[9] has made significantprogresstoward adaptingmultlgrldtechniquesto thesolutionof the Euler equationsand has presentedresultsfor transonicflowsover airfoils.

*Summer FacultyFellowat Lewis ResearchCenter,1982-1983(work partiallyfunded by NASA Grant NAG3-339).

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Johnson [10] - [12] proposed a surrogate-equation technique, in which thefirst order steady Euler equations are embeddedIn a second order system ofequations. The solutton of the original Euler equations is then recaptured byimposing appropriate additional boundary conditions on the embeddedsystem.The advantages of such an approach are that the difficulties of solving thedirect difference representation of the steady Euler equations can be bypassedand the resulting second-order embeddedsystem can be solved by a variety ofwell-proven numerical procedures. Inttial development of this approach andcomputational experimentation with it have been based on heuristic physicalreasoning. This has led to the construction of a relaxation procedure for thesolutton of two-dimensional steady flow problems. All the numerical resultsin [10] : [12] suggest that this is a viable and potentially more economicalapproach than the alternative of solving the unsteady equations of motion.

In this report the theoretical Justification for the embedding approach isaddressed. It is proven that, with the appropriate choice of embedding opera-tor and addltlonal boundary conditions, the solutton to the embeddedsystem isexactly the one to the original Euler equations. Hence, solving the embeddedversion of the Euler equations will not produce extraneous solutions. Thefollowing section contains the proof for the two-dimensional Euler equationsand shows that for the Cauchy-Rtemannequations a similar result follows im-mediately. Generalizations to three dimensions and to other systems of par-tlal differential equations are mentioned subsequently.

EMBEDDINGPROCEDURE

The steadyEuler equationscan be written in vector form as

fx + gy = 0 (1)

where x and y are Cartesiancoordinates,

FPU2 l and g = F_v ]pu +P/f

ipuv / Ipv2+pIL(E+ p)uj L(E+ p)vJ

Here p, p, u, v, and E denote respectively the density, static pressure,velocity components in the x and y directions, and the total energy per unitvolume. Furthermore

E = p [e + I/2 (u2 + v2)]

where the specific internalenergy e Is relatedto the pressureand densitybythe gas law

p = (y - l)pe

with y denotingthe ratio of specificheats.

Let

w=r,:lLo j

By Euler's theorem on homogeneousfunctions, f and g can be expressed (see,for example, [13], [14]) as f = Aw and g = Bw, where A and B are the Jacobianmatrtces

af and B _ agA _ aw aw

We have

0 -l 0 0

3-y u2 l-y v22 + 2 (y - 3)u (y- l)v 1 - y

A = - uv -v -u 0

yEu. (I - y,u,u2 * v2, - + (,3u2 + v2,_ (y - l)uv -yu

P P

and

0 0 -l 0

uv -v -u 0

-Y 3)v lB = 3-y v2 1 2- 2 . 2 u (y - l)u (y - - y

yvE* (I - y)V(U2 . v2) (y - 1)uv _ yE + v _Ij______(3v2 + u2) -yv

P p

Now, Eq. (1) can be written as

a 8(Aw) + _-_(Bw) = 0

or

[° °°I_-_x(A) + _-_y( ) w = 0 (2)

Let L denote the differentialoperator

B a

L = _-_(A) + _-_(B) (3)

Then the Euler equations become

Lw= 0 (4)

Now, let L* be the formal adJolnt operator to L defined by

L* IAT a BT _-_I: - + (s)

where AT and BT are the transposesfo A and B respectively. We may thenconsiderthe Euler equations(4) as embeddedin the second-ordersystem

L* Lw = 0 (6)

Let D be a boundedclosed regionwith a plecewisesmooth boundary. Forsimplicityof argument,assume that Eq. (6) is defined in a domain containingD.

aD

We now show that with an additionalconditionon the boundary,BD, of D, solu-tlons of Eq. (6) are also solutionsof Eq. (4).

Theorem Let L and L* be definedas in (3) and (5) respectively. If w is asolutionof

L* Lw = 0 In D

and satisfiesthe additionalrequirement

Lw = 0 on BD,

then It Is also a solutionof

Lw = 0 in D.

Proof Let (.,.)denote the Euclideaninner productin four-dlmenslonalspace. It can be shown that (see AppendixA for details) for any w,

a a(Lw, Lw) - (w, L*Lw) = _-_(Aw, Lw) + _-_(Bw, lw) (7)

Integratingover D and using Green'stheorem,we obtain

D_f((Lw, Lw) - (w, L*Lw))dxdy

= a (Aw, Lw) + _-_(Bw, Lw) dxdy (B)

=f((Aw, Lw)dy - (Bw, Lw)dx)aD

Here the llne integralin the last expressionof Eq. (8) is evaluatedIn thecounterclockwisedirectionover the closed contour aD. Now, if w satisfiesthe hypothesesof the theorem,i.e. L*Lw = 0 in D and Lw = 0 on aD, thenEq. (B) reducesto

D_I(Lw, Lw)dxdy=0

This impliesthat

(Lw, Lw} = 0

in D and hence

Lw = 0

in D.

Q.E.D.

Now, considerthe specialcase of the Cauchy-Riemannequations

ux + Vy = 0 (9)

vx - Uy = 0 (lO)

Let

'=[:],and rewriteEqs. (9) and 110) in vector form

fx + gy = 0 (ll)

If we choose

w_f

then we havei

aw

and

;]owEq. (11) can then be expressed as

a a_-_ lAw) + _-_ (Bw) = 0

or

Hence, If we again use L to denote the dlfferentlal operator

a aL = _-_(A) + _-_(B) (12)

the Cauchy-Riemannequationscan also be writtenas

Lw = 0 (13)

Let

= - (14)

Then Eq. (13) can be consideredas embedded in

L*Lw = 0 (15)

Note that a few simplematrix multiplicationswill reduce Eq. (15) to

a2 a2-- w + -- w = 0 (16)ax2 By2

which demonstratessimply the well-knownfact that Eqs. (g) and (lO) are em-bedded In the second-ordersystem (16).

Now, let D be the same region as definedpreviously. Then the Introduc-tlon of the differentialoperatorsL and L* for the Cauchy-Riemannequationssuggeststhe followingimmediateconsequenceof the previoustheorem.

Corollary If w is a solutionof Eq. (16) in D and if, on the boundaryof D,It satisfiesEqs. (9) and (lO), then it Is also a solutionof Eqs. (9) and(lO) in D.

6

Thus if one wishes to obtain the unique solutton to a boundary value prob-lem of the Cauchy-Rtemannequations (9) and (10), one can also solve Eq. (16)together with the original boundary conditions and the additional requirementthat Eqs. (9) and (lO) be satisfied on the boundary.*

• GENERALIZATIONS

Generalizationof the resultdiscussedabove to the three-dlmenslonalsteady Euler equationsis straightforward. Supposethe equationsof motionare expressedas

fx . gy + hz = 0

This can then be written as

a (A) + _-_(B ) + _-_(C q = 0

and hence the operator L can be introduced

a a aL = _-_(A ) + _-_ (B) + _-_ (C )

The further details are analogous to the case of two dimensions.

The embedding concept can be used on any partial differential equationsexpressible in the form

Lq = 0 or Lq = f

However, we shall not pursue this idea further here.

In Deslderl and Lomax[7], preconditioning matrices are investigated. Inour Eq. (6), L* may be considered as a preconditioning operator. Hence, theembedding method is a preconditioning procedure for the continuous model,whtle Desidert and Lomax's approach is one for the corresponding discretemodel.

Based on the mathematicalformulationpresentedhere, two-dlmenslonalsteady Euler solversare currentlybeing developed. A detaileddescriptionofthis work, includingnumericalresults,will be presentedin a forthcomingreport.

CONCLUSIONS

It has been proven that, for the numericalsolutionof the two-dlmenslonalsteady Euler equations,one can solve a second-orderembedded systemtogether

*The authorsunderstandthat Dr. T. N. Phillipsof ICASE - NASA LangleyRe-search Center has recentlyobtaineda similarresult for the non-homogeneousCauchy-Riemannequations.

with appropriate additional boundary conditions. Thls provides a theoreticalJustification for the recent computational experimentation with the surrogate-equation technique.

The proof presented here Is extendible to three dimensions and the embed-ding technique is applicable to a wider class of partial differential equa-tions than the Euler equations of motion considered here.

REFERENCES

I. Magnus,R.; and Yoshihara,H.: InvlscidTransonicFlow Over Airfoils.AIAA J., vol. 8, no. 12, Dec. 1970, pp. 2157-2162.

2. Warming, R.F.; and Beam, R.M.: Upwind Second-OrderDifferenceSchemesandApplicationsin AerodynamicFlows. AIAA J., vol. 14, no. 9, Sep. 1976,pp. 1241-1249.

3. Cou_ton,M.; McDonald,P.W.; and Smolderen,J.J.: The DampingSurfaceTechniquefor Time-DependentSolutionsto Fluid Dynamic Problems. TN-IO9,von Karman Institutefor Fluid Dynamics (Belgium),Mar. 1975.

4. Essers,J.A.: MethodesNouvellespour le Calcul Numeriqued'EcoulementsStatlonnairesdeFluides ParfaitsCompressibles. Th_se de DoctoratenSciencesAppliqudes,Unlversltdde Liege (Belgium),1977.

5. Veuillot,J.P.; and Vivland,H.: Pseudo-UnsteadyMethod for the Computa-tion of TransonicPotentialFlows. AIAA 3., vol. l?, no. 7, July 1979,pp. 691-692.

6. Steger,J.L.; and Lomax,H.: Calculationof InvlscldShear Flow Using aRelaxationMethod for the Euler Equations. AerodynamicAnalysesRequiringAdvanced Computers,NASA SP-347, Pt. 2, Mar. 1975, pp. BII-838.

7. Deslderl,3.A.; and Lomax,H.: A PreconditioningProcedurefor theFinite-DlfferenceComputationof Steady Flows. AIAA Paper Bl-lO06,3uneIgBl.

8. Bruneau,C.H.; et al.: Finite ElementLeast SquareMethod for SolvingFullSteady Euler Equationsin a Plane Nozzle. Proceedingsof the EighthInternationalConferenceon NumericalMethodsin Fluid Dynamics,Krause,E., ed., LectureNotes in Physics,vol. l?O, Spr_nger-Verlag,1982,pp. 161-166.

9. Jespersen,D.C.: A MultigridMethod for the Euler Equations. AIAA Paper83-0124,Jan. 1983.

lO. Johnson G.M : A NumericalMethod for the IterativeSolutionof Inv.iscid' " \ #

Flow Problems. These de Ooctorat en Sciences Appliquees, Untversit_ Llbrede Bruxelles (Belgium), 1979.

11. Johnson,6.M.: Surrogate-EquatlonTechniquefor Simulationof SteadyInviscidFlow. NASA TP-1866,Sep. 1981.

12. Johnson,G.M:: RelaxationSolutionof the Full Euler Equations. Proceed-%ngs of the Eighth InternationalConferenceon NumericalMethods in FluidOynamics,Krause,E., ed. LectureNotes in Physics,vol. 170, Springer-Verlag,1982, pp. 273-279.

13. Beam, R.M.; and Warming, R.F.: An Implicit Finite-DifferenceAlgorithmfor Hyperbolic Systems in Conservation-Law Form. 3. Comput. Phys.,vol. 22, no. 9, Sep. 1976, pp. 87-110.

14. James, G. and James, R.C., eds.: Mathematics Dictionary. Third ed.Van Nostrand-Reinhold, 196B.

APPENDIXA

Derivationof Eq.(7) For any dlfferentlablevector-valuedfunctionsU and V,we have

<_.v><_ovo=- ;;>._<o.v>and

°<u,v><_,v>:_<o,_>+_Hence, we have

8

<_w.,w>=<_x(Aw_+_ (_w_.,,>

(a a= _;(Aw),Lw>.<_;(Bw),Lw>

=-<_w._(,w_>+_°<Aw.,w>-<_w._(,w_>,_ <_w.,w>

<W, AT a a=- _ (,w_>._ <_w._w>-<w.__ (_w_>._°<_w.,_>

<W, (AT 8 BT _) a 8 <Bw, Lw>= - _;. ,w>._;<Aw._w>,;;

=<w,_,,w>.°;;<_w,,w>.°_;<_w,_w>This is Eq. (7).

lO

1. Report No. 2. Government Accession No. 3. Reclplent'e Catalog No.

NASATM-83481

4. Title and Subtitle 5. Report Date

Embedding Methods for the Steady Euler Equations July 19836. Performing Organization Code

505-31-07. Author(s) 8. Performing Organization Report No.

Shih-Hung Chang and Gary M. Johnson E-180310. Work Unit No.

9. Performing Organization Name and Address

National Aeronautics and Space Admini stration 11. Contract or Grant No.Lewis Research Center

C1eve1and, 0hi o 44135 :13.Type of Report and Period Covered

12. Sponsoring Agency Name and Address

Technical MemorandumNational Aeronautics and Space Administration 14SponsoringAgency CodeWashington, D.C. 20546

15. Supplementa_ Notes

Shih-Hung Chang, Cleveland State University, Cleveland, Ohio and SummerFacultyFellow at Lewis Research Center, 1982-1983 (work partially funded by NASAGrantNAG3-339); Gary M. Johnson, NASALewis Research Center.

16. Abstract

A recent approach to the numerical solution of the steady Euler equations is toembed the first-order Euler system in a second-order system and then to recapturethe original solution;by imposing additional boundary conditions. Initial develop-ment of this approach and computational experimentation with it have been basedon heuristic physical reasoning. This has led to the construction of a relaxationprocedure for the solution of two-dimensional steady flow problems. In the presentreport the theoretical justification for the embedding approach is addressed. Itis proven that, with the appropriate choice of embedding operator and additionalboundary conditions, the solution to the embedded system is exactly the one to theoriginal Euler equations. Hence, solving the embedded version of the Euler equa-tions will not produce extraneous solutions.

17. Key Words (Suggested by Author(s)) 18. Distribution Statement

Numerical analysis; Computerized Unclassified- unlimitedsimulations; Steady flow; Compressible STARCategory 64flow; Euler equations of motion

19. Security Classlf. (of this report) 20. Security Classif. (of this page) 21. No. of pages 22. Price"

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