MULTI-DIMENSIONALFLUCTUATION SPLITTINGSCHEMES FOR THE EULEREQUATIONS ON UNSTRUCTUREDGRIDSbyLisa Marie MesarosA dissertation submitted in partial ful�llmentof the requirements for the degree ofDoctor of Philosophy(Aerospace Engineering and Scienti�c Computing)in The University of Michigan1995Doctoral Committee:Professor Philip L. Roe, ChairpersonAssociate Professor Nikolaos KatopodesAssociate Professor Kenneth G. PowellAssociate Professor Shlomo Ta'asan, Carnegie Mellon UniversityProfessor Bram van Leer

To my one love:James Matthew.

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ACKNOWLEDGEMENTSFirst and foremost I would like to thank my husband, Jamey. It is only with hisunending love, support and understanding that I was able to complete this thesis.I also thank my parents: My father for his pride in my accomplishments and mymother for her faith in my abilities, it is what kept me going when many times Ididn't think I could.I especially thank Professor Roe for being my advisor. While he has been thedriving force behind my research project, most importantly he took the time to guidemy fragile mental state, and for that I am forever grateful. He is able to see a light inme that I can not yet grasp, and it is simply my respect for his opinion that has keptme searching for it. I thank Professor van Leer for his research guidance, especiallythe summer of \preconditioning" at ICASE. I thank Professor Powell whose door wasalways open and who o�ered endless help in the beginning of this research project.I also thank Professor Ta'asan, whose great interest and input into my work hasbeen indispensable. And thanks to Professor Katopodes, who so graciously agreedto serve on my committee on such short notice.I thank my many friends and family around the country who o�ered much neededemotional support throughout my schooling, especially Connie. It is she who has keptme sane throughout my academic endevours and most recently has provided me witha home, albeit temporary. I also thank Tim, whose input into the project helpedpropel it to where it is today, and whose friendship helped keep life in focus. And toiii

Mohit for many deep discussions over very long lunches, I will miss them.Lastly, I thank the Computational Science Graduate Fellowship Program of theO�ce of Scienti�c Computing in the Department of Energy for supporting this work.

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TABLE OF CONTENTSDEDICATION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iiACKNOWLEDGEMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : iiiLIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7LIST OF TABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11CHAPTERI. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : 1II. FLUCTUATION SPLITTING SCHEMES : : : : : : : : : : : 122.1 Fluctuation Splitting for the Euler Equations in One Dimension 122.2 Extension to Two Dimensions? : : : : : : : : : : : : : : : : : 16III. THE TWO DIMENSIONAL EULER EQUATIONS : : : : : 203.1 Simple Wave Solutions : : : : : : : : : : : : : : : : : : : : : : 213.1.1 Six-Wave Models : : : : : : : : : : : : : : : : : : : 253.1.2 Steady/Unsteady Decompositions : : : : : : : : : : 283.2 Hyperbolic/Elliptic Equation Decomposition : : : : : : : : : 333.2.1 \Time-Dependent" System : : : : : : : : : : : : : : 373.2.1.1 Supersonic Regime : : : : : : : : : : : : 383.2.1.2 Subsonic Regime : : : : : : : : : : : : : 413.2.1.3 Sonic Line Continuity : : : : : : : : : : 493.2.2 Summary of Decomposition Method : : : : : : : : : 50IV. UPWINDING SCHEMES FOR ADVECTION OPERATORSIN TWO DIMENSIONS : : : : : : : : : : : : : : : : : : : : : : : 524.1 Linear Schemes : : : : : : : : : : : : : : : : : : : : : : : : : : 524.1.1 Low Di�usion Scheme : : : : : : : : : : : : : : : : : 564.1.2 N Scheme : : : : : : : : : : : : : : : : : : : : : : : 57v

4.2 Nonlinear Schemes via Limiting Functions : : : : : : : : : : : 594.3 Numerical Test Cases : : : : : : : : : : : : : : : : : : : : : : 614.3.1 Accuracy Study : : : : : : : : : : : : : : : : : : : : 614.3.2 Discontinuity Capturing : : : : : : : : : : : : : : : 66V. CELL VERTEX DISCRETIZATIONS FOR ELLIPTIC-BASEDOPERATORS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 725.1 Accuracy : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 765.2 Convergence : : : : : : : : : : : : : : : : : : : : : : : : : : : 78VI. NUMERICAL ALGORITHM : : : : : : : : : : : : : : : : : : : 876.1 Conservation : : : : : : : : : : : : : : : : : : : : : : : : : : : 886.1.1 Wave Modeling Conservation Issues : : : : : : : : : 916.1.2 Hyperbolic/Elliptic Splitting: Conservation Issues : 926.2 Merging of Acoustic Discretization Through the Sonic Line : 956.3 Time-Step Calculation : : : : : : : : : : : : : : : : : : : : : : 966.4 Boundary Conditions : : : : : : : : : : : : : : : : : : : : : : 986.4.1 Solid Walls : : : : : : : : : : : : : : : : : : : : : : : 986.4.2 In ow/Out ow : : : : : : : : : : : : : : : : : : : : 100VII. RESULTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1037.1 Comparative Results : : : : : : : : : : : : : : : : : : : : : : : 1037.1.1 Channel Flow : : : : : : : : : : : : : : : : : : : : : 1047.1.2 Subcritical Flow Over NACA0012 : : : : : : : : : : 1057.2 Hyperbolic/Elliptic Splitting : : : : : : : : : : : : : : : : : : 1107.2.1 Low-Mach-Number Flow Over NACA 0012 : : : : : 1127.2.2 Subsonic Flow Case : : : : : : : : : : : : : : : : : : 1157.2.3 Transonic Flow Over NACA 0012 : : : : : : : : : : 1177.2.4 Supersonic NACA 0012 : : : : : : : : : : : : : : : : 1187.2.5 Low Speed Flow over NACA 0012 at High Angle ofAttack : : : : : : : : : : : : : : : : : : : : : : : : : 120VIII. CONCLUSIONS : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1248.1 Future Work : : : : : : : : : : : : : : : : : : : : : : : : : : : 1298.1.1 Discretization of the elliptic system : : : : : : : : : 1308.1.2 Extension to three dimensions : : : : : : : : : : : : 133APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 136BIBLIOGRAPHY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 141vi

LIST OF FIGURESFigure1.1 Shear ow oblique to grid recognized as shear parallel to grid faceplus a pair of acoustic waves. : : : : : : : : : : : : : : : : : : : : : : 32.1 Linear Data Reconstruction in 1D. : : : : : : : : : : : : : : : : : : 132.2 Fluctuation splitting in two dimensions. : : : : : : : : : : : : : : : : 183.1 Schematic of simple wave displaying orientation and correspondingray speed ~�. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 243.2 Wave orientation diagram of acoustics and shear for discrete six-wavemodels (entropy wave not shown). : : : : : : : : : : : : : : : : : : : 263.3 Characteristics of steady Euler system in supersonic ow. : : : : : : 363.4 Domain of in uence of acoustic waves for preconditioned system nearsonic line. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 483.5 Domain of in uence of acoustic waves for preconditioned system nearsonic line. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 504.1 Example of triangles with one or two in ow sides : : : : : : : : : : 544.2 Splitting of uctuation for LDA scheme : : : : : : : : : : : : : : : : 564.3 Splitting of advection vector for N Scheme : : : : : : : : : : : : : : 574.4 Schematic of normals needed in time-step constraint. : : : : : : : : 584.5 Test case of circular convection of inlet pro�le. : : : : : : : : : : : : 624.6 11 � 11 isotropic grid for circular convection test case. : : : : : : : : 63vii

4.7 Log-log plot of L2 norm of Error vs. grid spacing for circular con-vection test case on isotropic grid. : : : : : : : : : : : : : : : : : : : 634.8 11 � 11 right-running-diagonal grid for circular convection test case. 644.9 11 � 11 30% perturbed right-running-diagonal grid. : : : : : : : : : 654.10 Log-Log plot of error norm vs. grid spacing of LDA and minmodlimited N Scheme for Isotropic (I) and unperturbed Right-running-diagonal grids (R). : : : : : : : : : : : : : : : : : : : : : : : : : : : 664.11 31 � 31 nearly equilateral grid for the discontinuity test cases. : : : 674.12 Contour plots for 26:6� shear ow case (28 contours with umin = 0:98,�u = :04) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 684.13 Cross-cut at x=0 for 26:6� shear ow cases. : : : : : : : : : : : : : : 694.14 Residual history for 26:6� shear ow cases. : : : : : : : : : : : : : : 704.15 Contour plots for Burgers test case (16 contours with umin = �0:8,�u = 0:18.) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 715.1 Stencil for sample Lax-Wendro� scheme. : : : : : : : : : : : : : : : 765.2 Characteristic boundary conditions for Cauchy-Riemann system. : : 815.3 Residual history for model Cauchy-Riemann problem on 21� 21 grid. 825.4 Convergence rates vs. grid size for model Cauchy-Riemann problem. 835.5 Error damping of the Lax-Wendro� scheme. : : : : : : : : : : : : : 856.1 Determination of normal vector at vertex on a solid wall surface. : 996.2 Fluctuation in triangle due to normal uxes through faces. : : : : : 1007.1 10% cosine bump channel consisting of 40 points on the upper andlower surfaces with a total of 495 nodes. : : : : : : : : : : : : : : : 1047.2 Pressure contours (41 contours, Pmin = :55, �P = :005) in co-sine bump channel, M1 = 0:5, (a) Roe Six Wave Model D (b)Steady/Unsteady Splitting and (c) Hyperbolic/Elliptic Decomposi-tion. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 106viii

7.3 Entropy contours (51 contours, smin = :01, �s = :0004) in cosinebump channel, M1 = 0:5, (a) Roe Six Wave Model D (b) SteadyUnsteady Splitting and (c) Hyperbolic Elliptic Decomposition. : : : 1077.4 Residual History of L2 norm of the umomentum for the cosine bumptest cases. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1087.5 Detail of NACA 0012 grid. : : : : : : : : : : : : : : : : : : : : : : : 1087.6 Convergence history of lift coe�cient for subcritical ow over NACA0012,M1 = 0:63, � = 2�. : : : : : : : : : : : : : : : : : : : : : : : : : : : 1097.7 Spurious entropy production along body for subcritical ow overNACA0012, M1 = 0:63, � = 2�. : : : : : : : : : : : : : : : : : : : : 1117.8 Mach Contours for ow over NACA 0012 at M1 = 0:63, � = 2�. : : 1117.9 NACA 0012 at M1 = 0:63, � = 2� showing ow solution in stagna-tion region. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1127.10 Low-speed ow over NACA 0012 at zero angle of attack, M1 = 0:1;Mach contours. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1137.11 Low-speed ow over NACA 0012 at zero angle of attack,M1 = 0:01;Mach contours. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1147.12 Normalized Residual History for M1 = 0:1 and M1 = 0:01 cases. : 1147.13 Cp along airfoil for M1 = 0:1, M1 = 0:01 and the result of a panelmethod. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1157.14 Low-Speed-Flow over NACA 0012 at zero angle of attack,M1 = 0:1computed with a Lax-Wendro� method for the full Euler system:Mach contours. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1167.15 Entropy production along body for M1 = 0:1 case showing H/EDecomposition compared with basic Lax-Wendro� solver. : : : : : : 1167.16 Entropy production along NACA 0012 for subsonic ow case withM1 = 0:5 at 0�. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1177.17 Transonic ow over NACA 0012, M1 = 0:85, � = 1�; Mach contours. 118ix

7.18 Pressure coe�cient along body for ow over transonic NACA 0012,M1 = 0:85, � = 1�. : : : : : : : : : : : : : : : : : : : : : : : : : : : 1197.19 Cl vs. Cd for solutions from GAMM workshop for transonic NACA0012, M1 = 0:85, � = 1�. The dashed circle indicates the region ofacceptably accurate solutions. : : : : : : : : : : : : : : : : : : : : : 1197.20 Supersonic NACA 0012, M1 = 1:2, � = 0� Mach contours. : : : : : 1207.21 Mach number upstream and along body for supersonic ow overNACA 0012, M1 = 1:2, � = 0�. : : : : : : : : : : : : : : : : : : : : 1217.22 NACA 0012, M1 = 0:05, � = 25�; pressure contours and somestreamlines. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1227.23 NACA 0012, M1 = 0:05, � = 25� Mach number along body. : : : : 1227.24 NACA 0012, M1 = 0:05, � = 25�; entropy generation on body. : : : 1237.25 NACA 0012, M1 = 0:05, � = 25�; Cp along body for H/E Splittingcompared with a panel method solution. : : : : : : : : : : : : : : : 123A.1 Sample case of shear crossing a triangle. : : : : : : : : : : : : : : : 138

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LIST OF TABLESTable4.1 Order of accuracy for circular convection of smooth pro�le on isotropicgrid. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 644.2 Order of accuracy for circular convection of smooth pro�le on per-turbed right-running-diagonal grids. : : : : : : : : : : : : : : : : : : 657.1 Lift and drag coe�cients plus maximum entropy production aroundleading edge for the subcritical NACA 0012 test case. : : : : : : : : 110

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CHAPTER IINTRODUCTIONThe most successful numerical schemes developed for computational uid dynam-ics in the past have been based upon physics. While this success is readily seen inone-dimensional ows, there is a de�nite need for improvement in the approximationof solutions in higher dimensions. Not only do we seek a more accurate method ofdiscretizing the equations, but issues of code robustness and rate of convergence arealso important areas that need improvement. The focus of this work is the develop-ment of a two-dimensional scheme for approximating steady solutions of the inviscid uid equations which attempts to make the numerics of the method as harmoniouswith the physics as possible. Throughout the work contained in this thesis, it hasbeen realized that the extension of the schemes successful in one dimension to higherdimensions is not straight-forward, because new physical properties present them-selves as the dimensions are increased. While upwinding in one dimension has metwith great success, and its simple extension to higher dimensions has also provede�ective for some purposes within the aerospace community, we have determinedthat, in two dimensions, upwinding alone is not a su�cient basis upon which tobuild a numerical model. We propose that it is necessary to incorporate, within themethod, a system of elliptic origin, to account for the omni-directional distribution1

2of acoustical information within the subsonic regime. As the road to this conclusionhas not been straight, we will also present some of the other paths taken in searchof a truly multi-dimensional solver, eventually leading to the current method. Aswill be shown in the thesis, improved accuracy over existing methods is realized withthe numerical method presented. However, the issues of robustness and convergencerate are still in need of attention.A very successful class of schemes in one dimension, providing both accuracyand robustness, are Godunov based schemes [9], known as ux-di�erence splitting.However, the current accepted practice to extend these one-dimensional ideas tohigher dimensions is based on one-dimensional physics. It leads to cell-centeredschemes, where interactions between cells take place at the interfaces de�ned by thegrid. At each interface, the states on either side, referred to as the left and rightstates, can either simply be given by the states at the corresponding cell centers or canbe calculated from some higher-order extrapolations. Either way, this de�nes a one-dimensional Riemann problem at each interface, which can be approximately solved,for example, utilizing the Roe method. This method provides an exact solution tothe linearized equations, and decomposes the ux di�erence at the interface into auniquely de�ned set of simple waves, all aligned with the grid face [24]. This is thenutilized to evolve the solution within the cells by sending the disturbance caused byeach individual wave to the cell toward which it is propagating.Although successful codes have been developed utilizing this strategy, clearly thetrue physics of the multi-dimensional system are not modeled correctly, as wavesare only allowed to propagate in directions de�ned by the grid. This causes aninconsistency in the decomposition when the dominant characteristic present in the ow �eld lies oblique to the grid. A classic example of this is the case of a simple

3shear ow oriented oblique to the grid, as shown in Figure 1.1. Because the Riemann= +shear acoustic pair~qLshear ~qR

Figure 1.1: Shear ow oblique to grid recognized as shear parallel to grid face plusa pair of acoustic waves.problem is solved in the direction normal to the grid, the oblique shear will be\recognized" as a shear aligned with the grid, in order to account for the jump inthe velocity component aligned with the grid face, plus a pair of acoustic waves, toaccount for the jump in the velocity component normal to the grid face. The strengthof these acoustic waves will depend upon the angle between the physical shear waveand the grid face, allowing for rather strong spurious acoustics for large angles. Thesespurious acoustic waves bring along unnecessary numerical dissipation, causing theoblique shear layer to di�use over many cells. It is this dependence of the solutionon the structure of the grid that demands a more physically correct algorithm forapproximating multi-dimensional ows.Over the past decade a substantial e�ort has been put into the development of atruly multi-dimensional method for approximating solutions to the Euler equations,

4eliminating the dependence of the accuracy on the grid structure. Unlike in onedimension, where the decomposition into simple waves is unique, in two dimensionsthere are in�nitely many unsteady planar waves present, as there are an in�nitenumber of directions in which the waves can propagate, see Roe [25]. The �rst useof these ideas in a numerical code was in the framework of what are referred toas Rotated Riemann Solvers. In this approach, uxes are still obtained by solvingRiemann problems at grid interfaces, but now in directions not dependent on the gridnormals, but rather determined by the physical nature of the local ow [30, 8]. Whilethese methods were successful in improving accuracy, substantial losses of robustnessand convergence speed prohibited the realization of a universal scheme surpassingexisting methods.It thus became clear that a radically new approach had to be sought in orderto model the multi-dimensional physics adequately. Because the Riemann methodsonly provide two states at each interface, the true two-dimensional character of the ow can not be realized. For example, we do not know if a measured one-dimensionalgradient is balanced by a gradient in some other direction. We need information fromat least three points and thus the Riemann methods have to be abandoned.We consider the family of schemes, known as uctuation-splitting schemes. Inone dimension, Roe showed that these schemes provide an alternative method of im-plementing the ux-di�erence scheme [24]. In this cell-vertex approach, uctuationsare de�ned as the integral of the time derivative over each cell. In one dimension,these uctuations, as in the ux-di�erence method, can be decomposed into a uniqueset of linear waves each governed by an advection operator. The decomposition canbe thought of as \explaining" what the time derivative is due to. Each wave thenupdates the solution at the endpoints of the cell, updating more strongly the point

5toward which it moves.The advantage of the uctuation-splitting approach is that the natural extensionof the intervals in one dimension leads to triangles in two dimensions and tetrahedrain three. In two dimensions, this cell-vertex method operates on triangular grids withthe data stored at the vertices. As in one dimension, uctuations are determined oneach mesh element, de�ned as the integral of the now two-dimensional time derivativeover each triangle. Each cell uctuation then needs to be distributed back to thevertices of the given triangle, providing the changes to the nodal states, used to evolvethe solution in time. This thesis is focused on �nding a method of distributing thecell uctuations to the vertices of the local simplex, which simulates the physics ofthe equations as closely as possible.There are several important ingredients in this family of schemes: one is a con-servative linearization of the Euler equations utilized in calculating the uctuationwithin the local simplex. The known Roe parameter vector in one dimension pro-vides an appropriate linearization, and is shown to have a natural extension to higherdimensions [4]. With the resulting linearized system, a decomposition is performed;in one dimension this led to a set of simple waves, but in two dimensions otherdecompositions are possible and seem more natural. Finally, utilizing the result-ing subproblems, which in one dimension are advection operators, the uctuationis distributed to the vertices of the local simplex. The very compact nature of thedistribution process creates the potential for e�cient coding, and will be adhered toin everything that follows.This extension of the uctuation-splitting schemes to higher dimensions has beencarried out in a strong collaboration between the University of Michigan and theVon K�arm�an Institute for Fluid Dynamics in Brussels. The �rst success was realized

6for scalar advection operators, in the development of upwind biased discretizationson unstructured triangular grids. Original work by Roe, Struijs, et. al. [26, 33]provides nearly-second-order compact schemes, with the ability to capture disconti-nuities sharply, without overshoots even when these lie oblique to the grid. Recently,multi-dimensional upwinding methods developed by Sidilkover, originally based onstructured �nite-volume grids but extended to unstructured cell-vertex grids, havebeen uni�ed with the uctuation-splitting schemes [32]. The formulation of Sidilkoverutilizes symmetric limiting functions, for which any existing one-dimensional limit-ing method can be substituted, providing an elegant framework which encompassesa wide range of schemes. It is within this methodology that the work in this thesisis presented.The success of the scalar advection schemes appeared to con�rm that the naturalextension from one dimension would continue to be the decomposition into advec-tion operators. From the beginning of the present work, up until the past year, thebreaking of the two-dimensional cell uctuations into advection problems has dom-inated the research path. In all ow domains, subsonic as well as supersonic, thedecomposition of the Euler system was based on the idea that all local ow gradi-ents and uctuations present within a mesh simplex could be modeled as a �nite setof unsteady planar waves. Thus, the disturbances present within the mesh simplexwere discretely modeled as linear waves governed by advection operators, based onthe original wave model approach of Roe [25]. Various assumptions have been usedto determine the �nite set of waves. The assumptions are required to select thepropagation directions, out of in�nitely many possibilities, providing a unique set ofwaves for given local ow conditions.While the linearization of the system, as well as the upwind discretizations for

7scalar advective operators, appear to be adequate, solutions obtained with this wavemodeling approach showed a clear lack of accuracy; its cause appeared to lie inthe decomposition of the Euler system into a set of planar wave equations. Severalwave models within this framework were developed and tested, all of which producedresults with too much numerical dissipation [34]. The basic di�culty was that evensteady ow was often interpreted by the model as resulting from cancelation betweena number of unsteady waves.In order to avoid this dissipation, Roe proposed that the scheme should be ableto recognize existing steady patterns in the �eld. It is easily determined that insolutions of the steady system there exist patterns, given by shear and entropy layers,plus potential- ow solutions in subsonic regimes, or acoustic waves de�ned alongthe Mach lines in supersonic ows. Following this reasoning, a combination of bothsteady and unsteady wave patterns were used to model the spatial gradients, with thetime-dependent planar waves accounting for the cell uctuations [27]. In numericalexperiments second-order-accurate results, with low numerical error, were obtained,but convergence was hindered by the ambiguity of wave orientation for cells withsmall residuals [20]. It is also noted that this decomposition only proved stable insubsonic regimes; the development of a supersonic counterpart is a major obstacleyet to be overcome. While the Steady/Unsteady Splitting was not the answer to themulti-dimensionalmodeling problem, it did show the value of a necessary condition tobe imposed on any scheme for the sake of accuracy. This rather obvious condition,not satis�ed by the original wave models, says that, in the instance of vanishingcell uctuations, the wave strengths of all unsteady waves must vanish [18]. Thisguarantees that the scheme will recognize and maintain a given steady solution.It appeared that a major problem contained within both versions of the wave-

8modeling schemes discussed was in the use of unsteady planar waves whose orienta-tion was sometimes ill-de�ned. Therefore, in the present work, the decomposition ofthe Euler system into advective operators is abandoned following the lack of successof the methods discussed.Abandoning the planar wave models calls for a new approach in determining theappropriate decomposition of the Euler system into subproblems. Contrary to theoriginal approach of recognizing simple waves, and therefore scalar subproblems, itattempts to reduce the steady system to a set of subproblems that are decoupled fromeach other as completely as possible. These are similar to the canonical forms �rstproposed by Ta'asan [35], where the subsonic steady Euler Equations are reduced intohyperbolic and elliptic parts. The biggest di�erence between the work of Ta'asanand the method discussed here is that the canonical forms are di�erent. In thiswork we obtain the unique complete decoupling of the linearized system. Also, theextension of this method to supersonic regimes is presented, while not yet addressedby Ta'asan.Because most work in truly multi-dimensional schemes to date has been in thedevelopment of methods for obtaining steady-state solutions, we return to the steadyequations for the determination of the correct decomposition, while simply utilizingthe time dependence as a method of marching to the �nal solution. It is well knownthat the two-dimensional steady Euler equations in supersonic ow are equivalentlyrepresented by four characteristic equations. While these characteristic equations aredecoupled in the linear system, non-linear coupling is present through the coe�cientsof the derivatives. This is similar to the plane-wave decomposition methods becausefour clearly de�ned propagation directions exist. In subsonic ows, however, only twoof these characteristics remain, namely for advection of enthalpy and entropy along

9streamlines. The acoustic characteristics from the supersonic regime become coupledin the subsonic regime, representing the spreading of subsonic acoustical informationwithout preferential directions. Therefore, the steady Euler equations are now rep-resented by two advective equations, providing the hyperbolic part, plus a systemof two coupled equations whose characteristics are imaginary, thus representing theellipticity of the steady acoustical �eld.This decomposition of the steady system exhibits the di�erent mechanisms forpropagating information; that is one-directional convection and omni-directionalwave propagation. In order to solve for the steady state, a \pseudo-time" is intro-duced. Since we are not concerned with time accuracy, the unsteady Euler systemis modi�ed via preconditioning techniques similar to those developed by Turkel andvan Leer et. al. [39, 41]. It is desired to preserve the known decoupled characteristicform of the steady equations even in the unsteady system. It was discovered thatthe original preconditioner of Van Leer already provided this property, allowing theisolation of the distinct information propagation mechanisms in the unsteady system.This decomposition results in unique advective operators for which (as discussedfor the wave modeling schemes) second-order, compact upwind schemes providingnon-oscillatory sharp discontinuities already exist, (see Roe, Struijs, Sidilkover et. al.[26, 33, 32]). However, in the subsonic regimes, the coupled acoustic system callsfor the development of accurate discretizations. This development can be carriedout for an alternate model problem, speci�cally the Cauchy-Riemann equations aug-mented with time terms. For this system, where there is no preferential directionof information propagation, central discretizations can be utilized, converged usinga cell-vertex Lax-Wendro� scheme similar to those developed by Ni, Hall, Mortonet. al. [17, 10, 13].

10It is possible to incorporate both the advective and omni-directional discretiza-tions, including the nonlinear limiting required to avoid shock oscillations, within acompact stencil that permits very e�ective parallelization [37]. Moreover, the decom-position, or generalization of the earlier Riemann problem, needs to be made onlyonce per cell in this approach, rather than once per interface.The fundamental concept put forth in this work is the separation of the ellipticand hyperbolic components present within the system. Utilizing the characteristicform of the steady system to predict supersonic ows accurately is not a new con-cept, but the present work has implemented the ideas in a conservative way. It isfound that the subsonic decomposition into hyperbolic and elliptic parts leads toaccurate solutions, characterized by very low drag for subcritical cases and very lowspurious entropy production. Nevertheless, issues concerning robustness and, mostimportantly, e�ciency still exist.There are numerous details that arise within these splitting schemes which arelikely to be important due to the high ambition of increasing e�ciency and robust-ness as well as accuracy, over existing methods. In this thesis we only present theunderlying framework with current results. We do not claim that all or any of thedetails are yet perfect, and it is noted that quite di�erent approaches may still betaken [19]. We believe that the success of the splitting scheme will depend on �ndingthe proper method of solving the elliptic system that produces textbook convergencerates, whether with single-grid or multi-grid relaxation. This goal is not yet realizedwith the current use of a Lax-Wendro� method for solving the Cauchy-Riemannsystem [16].The focus of this thesis is the development and testing of various methods ofdecomposing the Euler system, incorporated in a cell-vertex uctuation-splitting

11scheme. The issues of robustness and convergence performance are touched uponbut not fully addressed. It is clear, however, that an advance has been made towardthe eventual scheme that will surpass the current standard of industry for comput-ing multi-dimensional ows, with �nally a prevailing feeling that the answer is justaround the corner.

CHAPTER IIFLUCTUATION SPLITTING SCHEMES2.1 Fluctuation Splitting for the Euler Equations in One Di-mensionWe start by reviewing the concept of uctuation splitting as proposed by Roein 1981, in developing an upwind solver for the Euler Equations in one dimension[24, 23]. We analyze any set of hyperbolic partial di�erential equationsUt + Fx = 0; (2.1)in quasi-linear form: Ut +AUx = 0; (2.2)where A = @F@Uis the Jacobian of the ux function F. Given a solution at any time level where dataare stored at nodal points, a piecewise continuous linear representation of the datacan be constructed based on the idea of linear �nite elements in space, (see Figure2.1). From this representation, piecewise constant spatial gradients can be calculatedin each interval.In the same notation as that �rst used by Roe [23], we then de�ne the frameworkfor uctuation splitting schemes: 12

13u xxi xi+1xi�1 xi+2ui�1 ui ui+1 ui+2Figure 2.1: Linear Data Reconstruction in 1D.� Fluctuation: the time residual over a mesh element; in 1D this gives uctuationson each interval. We denote the uctuation by �, given by:�i� 12 = � Z xixi�1 Ut dx (2.3)= Z AUx dx (2.4)= A(Ui;Ui�1)[Ui �Ui�1]: (2.5)To create a conservative scheme a conservative linearization is necessary toevaluate A(Ui;Ui�1) such thatA(Ui;Ui�1)[Ui �Ui�1] = Fi � Fi�1: (2.6)This is known for the Euler equations as the Roe linearization; it is achievedthrough a parameter vector Z, such that if A is evaluated at the algebraic av-erage of the parameter vector, the linearization is conservative. This algebraicaverage is calculated from the assumption of piecewise linear variation in Zover the interval, and thus Zi�1=2 = 12 (Zi�1 + Zi) :The parameter vector is given by, Z = fp�; p�u; p�Hg, where H = �1 p� +12u2 is the total enthalpy.

14� Signal: the action performed on the data to bring it to equilibrium. Thisrepresents the distribution or mapping of the cell uctuations back to the nodes.The distribution schemes discussed within this thesis are local and thereforethe uctuation within each cell is mapped only to the vertices which de�nethat cell. In 1D this de�nes a mapping of an interval uctuation to the twonodes which de�ne its endpoints, i.e.�i� 12 = �+i� 12 +��i� 12 : (2.7)where �+ denotes the portion of the uctuation sent in the positive x directionto xi and �� the portion sent to xi�1.� Residual: the measure of the distance of the solution from equilibrium. Theresiduals are the accumulated signals at each of the nodes after looping overall cells and distributing the uctuations. In one dimension each node receivesa signal from the two intervals of which it is a vertex, and the residual �Ui isgiven by their sum: �Ui = �+i� 12 +��i+ 12 : (2.8)The time evolution of the solution is then given by the following update:Ui(t+�t) = Ui(t)��Ui �t�x i; (2.9)where �xi is the area weighted to node i, given by the sum of half the lengthof the two intervals bordering it, �xi = 12(xi+1 � xi�1):The main issue then within these types of distribution schemes is how to map theinterval uctuations to the nodes to create an e�cient method of obtaining accuratesteady state solutions. In 1D, the most successful method is the upwind Roe scheme,

15where the Euler system is broken into a unique set of linear simple waves, of whicheach can be independently upwinded.By diagonalizing the time dependent system we can uniquely determine the setof waves. We start with the quasi linear system, where we can break A into R�L,where R and L are the matrices containing the right and left eigenvectors of A and� the corresponding eigenvalues: Ut +AUx = 0; (2.10)Ut +R�LUx = 0: (2.11)Multiplying through by L, LUt +�LUx = 0; (2.12)gives a set of nonlinearly coupled scalar advection equations for the characteristicvariables, W, de�ned by @W = L@U,Wt +�Wx = 0: (2.13)Now that the system has been broken into scalar advection problems, we can easilyupwind each. If we denote � = diag(�k), the sign of �k determines the directioneach wave, k, is moving. We therefore split � into� = �+ +�� (2.14)where�+ contains the positive values, (�k � 0), representing those waves traveling inthe positive x direction, and �� contains the negative values, (�k < 0) representingthe waves moving in the negative x direction. From this we can �nd the splitting ofthe ux di�erence by determining the split Jacobian matrix A, given by:A+ = R�+L A� = R��L; (2.15)

16where A+ +A� = A; (2.16)and �+ = A+�U �� = A��U: (2.17)It was shown by Roe [23] that the uctuation splitting scheme described is equiv-alent to a �nite-di�erence scheme utilizing a ux function based on the above wavedecomposition. Therefore, in one-dimension uctuation splitting o�ers nothing new.Fortunately, the extension of this idea to higher dimensions does. The standardextension of the ux-based scheme to higher dimensions is to continue to solve onedimensional Riemann problems in given directions, the most successful being in di-rections determined by the grid structure. This of course does not allow for thetruly multi-dimensional avor of the ow to be correctly modeled. Although variousmethods of �nding more realistic directions in which to solve the Riemann problemwere tried [8, 30], all still resorted to one-dimensional physics and none met withincreased success. It is thus the topic of this work to develop, at least in two dimen-sions, a scheme which more accurately solves the inviscid gas-dynamic equations byincorporating a more physically correct model for two-dimensional ow. The frame-work within which these schemes operate is the extension of the uctuation-splittingidea to higher dimensions.2.2 Extension to Two Dimensions?The natural extension of the assumption of piecewise linear data to two dimen-sions gives rise to triangular elements with data points at the vertices.Using the same terminology, cell uctuations, �, are de�ned as the integral overthe triangular elements, � , which construct the domain, similar to one-dimensional

17splitting schemes where uctuations are de�ned on intervals. Integrating a two-dimensional partial di�erential equation:Ut + Fx +Gy = 0; (2.18)over a triangle � , �� = � Z Z Ut dA = Z Z (Fx +Gy) dA; (2.19)and applying Gauss' Theorem, the evaluation of the uctuation reduces to a pathintegral, �� = I (Fdy �Gdx); (2.20)which is easily computed assuming linear variation of data over the cells. A detaileddiscussion of this computation concerning conservation is given in Chapter VI.Given the cell uctuation, a distribution or mapping back to the vertices isneeded. In order to maintain a compact stencil, the distribution step must onlyutilize information within the local simplex, and distribute signals only to the ver-tices which de�ne the cell. This allows the numerical scheme to be implementedas a loop over mesh simplices, thus making the algorithm very e�cient for parallelapplications as shown by Tomaich [37]. The basis of the splitting schemes is thedistribution of the uctuation to the three vertices of the local triangle, (see Figure2.2). Each vertex receives a portion of the residual, denoted by �i� ; i = 1; 2; 3, suchthat: �� = �1� +�2� +�3� : (2.21)Each of these signals then make a contribution to the nodal residuals. Afterlooping over all triangles, and mapping cell uctuations to nodes, each node has anaccumulated residual consisting of signals gathered from all triangles for which it is

181 2

3 ���3� �2��1�Figure 2.2: Fluctuation splitting in two dimensions.a vertex. These residuals are then used to evolve the solution in time. Consideringthe residual contribution from one triangle, the following conservative evolution intime is used: S1U1 S1U1 ��t�1� + T:F:O:T (2.22)S2U1 S2U1 ��t�2� + T:F:O:T (2.23)S3U1 S3U1 ��t�3� + T:F:O:T (2.24)where Si is the area weight of vertex i, equal to one third of the area of those trianglessurrounding Vi. And T:F:O:T denotes Terms From Other Triangles, as each vertexmay be receiving signals from other triangles surrounding it.The main focus is again on the ideal way to distribute the cell uctuations inorder to obtain accurate solutions with sharp discontinuities. In the one-dimensionaldecomposition of the Euler equations, the system is diagonalizable and thus resultsin a set of scalar advection operators. However, this is not directly extensible tohigher dimensions because the Jacobian matrices A and B are not simultaneouslydiagonalizable, and thus the system can not be reduced in the same manner. The

19issue then becomes what is the appropriate extension. How do we correctly decom-pose the two dimensional system into recognizable subproblems, and then determinethe proper discretizations? This is the focus of the remainder of this thesis.

CHAPTER IIITHE TWO DIMENSIONAL EULEREQUATIONSThe Euler equations in two dimensions are given by:@U@t + @F@x + @G@y = 0 (3.1)where the conservative state vector U and the ux functions F and G are de�nedas: U = 8>>>>>>>>>>><>>>>>>>>>>>: ��u�v�E 9>>>>>>>>>>>=>>>>>>>>>>>; ; F = 8>>>>>>>>>>><>>>>>>>>>>>: �u�u2 + p�uv�uH 9>>>>>>>>>>>=>>>>>>>>>>>; ; G = 8>>>>>>>>>>><>>>>>>>>>>>: �v�uv�v2 + p�vH 9>>>>>>>>>>>=>>>>>>>>>>>; : (3.2)In this system, � is the density, u and v are the velocity components in the x and ydirection, respectively, and p is the static pressure. The speci�c energy and enthalpy,denoted by E and H, respectively, are given by:E = 1 � 1 p� + 12(u2 + v2); (3.3)H = � 1 p� + 12(u2 + v2): (3.4)In one dimension, as discussed in the previous chapter, there is a unique decom-position of the system into scalar subproblems, leading to independent advection20

21operators, but this is not true for the two-dimensional system. We now discuss ex-tensions of these 1D decomposition ideas to two-dimensional ows. Again, the goal isto reduce the system to recognizable subproblems for which accurate discretizationsare available.Unfortunately this issue is non-trivial and the path has been long and tedious.We �rst present the various methods which have unsuccessfully been employed overthe past �ve years. The framework for most of the work in multidimensional methodsto date is based on the discrete wave models �rst presented by Roe in 1986 [25]. Theoriginal idea was to break the system of equations into an equivalent set of scalaradvection operators representing the wave behavior inherent in uid ow. This camefrom considering the simple wave solutions of the Euler system.3.1 Simple Wave SolutionsFirst, for simplicity of the analysis, we will transform the Euler system intoprimitive variable form: Vt +AVx +BVy = 0; (3.5)where, V = 8>>>>>>>>>>><>>>>>>>>>>>: �uvp 9>>>>>>>>>>>=>>>>>>>>>>>; ; (3.6)and A = @V@U @F@V @U@V ; B = @V@U @G@V @U@Vare the ux Jacobians transformed into primitive-variable form.The basis of the \wave-modeling" schemes is that the local ow gradients are

22modeled as the result of a set of simple waves. These simple waves are of the form:V(~x; t) = V(�) = V((~x� ~�t) � ~m) = V(~x � ~m� �mt); (3.7)where ~m is a unit vector in the direction of the gradient imposed by the wave. Wede�ne the characteristic velocity (or ray velocity), representing the physical prop-agation of information, as ~�, with the frontal speed (projection of ray velocity ingradient direction), �m, given by �m = ~� � ~m: (3.8)In assuming local linear variation in V, we recover plane simple waves which aregiven by: V(~x; t) = V(�) = W (�)r+V0 = �(~x � ~m� �mt)r+V0; (3.9)representing a traveling wave of constant strength �, orientated such that its normalis de�ned by ~m = fcos �; sin �g. The projection of this wave onto the primitivevariables, V, is given by r.By de�nition, the simple wave supports a gradient �eld:~rV = @W@� ~mr = �~mr; (3.10)with a time derivative: Vt = �@W@� �mr = ���mr: (3.11)Substituting Equations (3.10) and (3.11) in the Euler system, Equation (3.5),gives the following eigenvalue problem,�(A cos � +B sin � � �mI)r = 0: (3.12)The solution of this problem has the frontal speeds �m as the eigenvalues, with thecorresponding eigenvectors providing r; r represents the projection of the waves ontothe primitive variables.

23The solution of the eigenvalue problem yields four waves given by a pair of acous-tics: �a�m = ~q � ~ma � a , shear: �sm = ~q � ~ms and entropy: �em = ~q � ~me with thecorresponding eigenvectors:ra� = 8>>>>>>>>>>><>>>>>>>>>>>: ��a cos �a�a sin �a�a2 9>>>>>>>>>>>=>>>>>>>>>>>; ; rs = 8>>>>>>>>>>><>>>>>>>>>>>: 0�a sin �sa cos �s0 9>>>>>>>>>>>=>>>>>>>>>>>; ; re = 8>>>>>>>>>>><>>>>>>>>>>>: �000 9>>>>>>>>>>>=>>>>>>>>>>>; ; (3.13)where a is the local speed of sound: a2 = p� : (3.14)To determine the advection operator for the waves, we can equivalently writeEquation (3.12) as: @W@t I+ �(A cos � +B sin �)! r = 0: (3.15)This can be reduced to an advection equation for W by utilizing the following rela-tionship �k (A cos � +B sin �) = �k�kmrk = @W@� �kmrk; (3.16)where �km and rk are a corresponding eigenvalue/eigenvector pair of the eigenvalueproblem given in Equation (3.12). Incorporating the characteristic speed ~� by uti-lizing Equation (3.8), we recover an advection operator for W : @W@t + @W@� (~� � ~m)! r = 0; (3.17) @W@t + ~� � (@W@� ~m)! r = 0; (3.18) @W@t + ~� � ~rW! r = 0: (3.19)

24The ray speeds ~�, shown pictorially in Figure 3.1 for an acoustic front, are given inthe Euler system by: ~�a� = ~q � a~m; (3.20)~�s = ~�e = ~q; (3.21)representing the advection of entropy and shearing waves along streamlines, withacoustic disturbances propagating radially with the speed of sound, relative to thelocal ow velocity.x=dt

y=dt ~ma~m~q~�Figure 3.1: Schematic of simple wave displaying orientation and corresponding rayspeed ~�.Any disturbance in the ow can be considered as due to the superposition of eachof the four wave types k over all possible orientations:~rV = 2�X�=0Xk �k(�)rk(�) ~m; (3.22)where the local uctuation imposed by these waves is given by:Vt = � 2�X�=0Xk �k(�)(~� � ~mk)rk: (3.23)Although in�nitely many waves can exist, the basis of the discrete wave models is thechoice of a �nite number of simple waves to model local ow behavior. Essentially,

25if a valid set of waves is determined, each of these waves is governed by an advec-tion operator. These advection operators can be independently discretized with anupwind biased stencil utilizing schemes developed for scalar advection operators aspresented in Chapter IV. Unfortunately the problem lies in choosing the �nite setof waves.3.1.1 Six-Wave ModelsThe original models proposed by Roe in 1986 [25] were based on the representationof local ow gradients by a discrete set of K unsteady simple waves, thusVx = KXk=1�k cos �krk; Vy = KXk=1�k sin �krk: (3.24)For two-dimensional ow the spatial gradients provided eight degrees of freedom inchoosing the waves: two degrees of freedom for each wave, its strength �k and itsorientation de�ned by an angle �k. Roe proposed that any ow could be modeledby two pairs of mutually orthogonal acoustics (4 strengths plus 1 direction), a shear(1 strength and 1 direction) and an entropy wave (1 strength and 1 direction). Thisresults in 9 unknowns, providing a one-parameter family of wave-models, expressedhere in terms of the shear angle, �s.The direction and strength of the entropy wave are:�e = tan�1 �y � py�a2�x � px�a2 ; �e = vuut �x � px�a2!2 + �y � py�a2!2; (3.25)the strength of the shear wave: �s = vx � uya ; (3.26)and the four acoustics are given by:�a = 12 tan�1 vx + uy � a�s cos(2�s)ux � vy + a�s sin(2�s) ; (3.27)

26R = s�vx + uya � �s cos (2�s)�2 + �ux + vya + �s sin(2�s)�2; (3.28)�a1 + �a2 = ux + vya +R; �a1 � �a2 = px cos �a + py sin �a�a2 ; (3.29)�a3 + �a4 = ux + vya �R; �a3 � �a4 = py cos �a � px sin �a�a2 : (3.30)An example displaying the notation used in identifying the six discrete waves isgiven in Figure 3.2. The two orthogonal acoustic pairs, de�ned by one angle �a,travel with the speed of sound in the direction normal to their orientation relative tothe local ow speed. The shear wave, also depicted in the �gure, is orientated at agiven angle �s, and travels at the local ow speed. The entropy wave, not displayed,also travels at the local ow speed with its orientation aligned with the local entropygradient.shear~q x=dt

y=dt �a acoustic 1acoustic 3acoustic 2 acoustic 4Figure 3.2: Wave orientation diagram of acoustics and shear for discrete six-wavemodels (entropy wave not shown).Numerous models were developed and tested within this family, di�ering within

27their choice of the shear angle, �s. Some examples include:� Wave Model B proposed by Roe [25], where the shear wave propagates per-pendicular to the ow direction. In this model, the shear does not provide acontribution to the cell uctuation, i.e., �s � ~ms = 0.� Wave Model C, of De Palma et. al. [3], with the shear aligned along the pressuregradient. This model has problems in recognizing isolated shear layers throughwhich pressure is constant.� Wave Model D of Roe [28] is the most physical model. It is based on thestrain-rate axes and couples the shear and acoustic wavefronts in the followingform, �s = �a + �4 sgn(vx � uy): (3.31)This model produces acoustic waves aligned with the principal strain-rate ten-sor.In numerical experiments the best results were obtained with Model D, althoughall wave models in this family had excessive numerical dissipation. This led to re-sults similar to a conventional �rst-order solver. It was found by Paill�ere et. al.[18] (generalizing analysis of the supersonic case by Rudgyard [29]) that LinearityPreservation, (LP) within the wave-model is required to avoid unnecessary dissipa-tion. This property requires that the numerical scheme can recognize any cell inequilibrium, i.e., a cell having zero residual. In the context of the wave modelingschemes this requires that in the case of a cell in equilibrium, all wave strengthsmust be zero. The six-wave models do not possess this property. If spatial gradientsexist in an equilibrium cell, the model represents these gradients as a set of unsteadywaves, although they cancel each other in the total uctuation. However, since each

28wave is propagated independently, the resulting change to the vertices of the cell willdestroy the original equilibrium state. Thus, the local steady solution is smeared.Except under special constraints, there can be no more unsteady waves in the modelthan there are cell residuals; in two dimensions this limit is four.Other models falling within this framework were also developed by Rudgyard[29] and Parpia and Michalek [21], neither meeting with universal success. Rudgyardformulated a supersonic splitting, which placed pairs of acoustic wavefronts de�nedalong the Mach lines. This de�nes that one acoustic wave in each pair represents thesteady characteristic and thus makes no contribution to the cell residual. Althoughthe model was not formulated in a linearity preserving framework it did give betterresults than any of the other wave models employed to date. It also presented somemotivation for the necessity to represent steady patterns present within the ow�eldas found by Roe. Therefore, in order to reduce the numerical error in mainly smoothsubsonic ows, Roe proposed to model the local ow as a combination of both steadypatterns and unsteady waves[27], as detailed in the following section.3.1.2 Steady/Unsteady DecompositionsThis splitting was developed to allow discrete decompositions to recognize andkeep steady solutions in subsonic regimes, in hopes of reducing the numerical error.It should be noted here that there is currently no supersonic counterpart for which astable numerical code is achievable. Contrary to the six-wave model schemes, wherethe steady solution is determined by the cancelation of unsteady waves, it was hopedthat by recognizing steady patterns present in the ow, at convergence the unsteadywaves would be small in amplitude, and the solution would be represented by acombination of the steady patterns.

29First, to determine the patterns present in the two-dimensional Euler system, westudy the steady equations in primitive-variable form,AVx +BVy = 0: (3.32)Substituting linear simple waves of the form: V = �(y � �sx)rs, results in an eigen-value problem for A�1B, with the resulting eigenvectors rs representing the steadypatterns given by: P = fVx;Vyg = f��srs; rsg (3.33)where �s are the corresponding eigenvalues. In supersonic regimes this providesfour simple wave patterns. However, in subsonic regimes two of the eigenvalues arecomplex, representing the elliptic nature of the acoustic front, with the patterns,which are not in simple wave form, resulting from taking the real and imaginaryparts. These four patterns P are given by two potential- ow solutions:P1pot = 8>>>>>>>>>>><>>>>>>>>>>>: �v ��u�2uv=M2 (u2 � v2)=M2(u2 � v2)=M2 2uv=M2�a2v ��a2u 9>>>>>>>>>>>=>>>>>>>>>>>; (3.34)P2pot = 8>>>>>>>>>>><>>>>>>>>>>>: �u �v�(u2 + �2v2)=M2 uv(�2� 1)=M2uv(�2� 1)=M2 �(v2 + �2u2)=M2�a2u �a2v 9>>>>>>>>>>>=>>>>>>>>>>>; (3.35)

30plus steady shear and entropy solutions:Pshear = 8>>>>>>>>>>><>>>>>>>>>>>: 0 0�uv u2�v2 uv0 0 9>>>>>>>>>>>=>>>>>>>>>>>; Pent = 8>>>>>>>>>>><>>>>>>>>>>>: v �u0 00 00 0 9>>>>>>>>>>>=>>>>>>>>>>>; (3.36)where M = p(u2+v2)a is the Mach number and �2 =M2 � 1.The discrete modeling of the local ow is now considered to be a combination ofthese steady patterns with a set of unsteady simple waves, such as those found inthe previous section, i.e.,~rV =Xk �k(�)rk(�) ~m+ Xsteady �steadyP: (3.37)We again have eight degrees of freedom in choosing a set of unsteady and steadywaves to represent the local spatial gradients. Roe proposed to utilize three steadypatterns: both potential- ow patterns plus the shear. This gives three unknownsrepresenting their strengths. For the other �ve degrees-of-freedom we choose a pairof unsteady acoustics, providing three unknowns: an orientation angle plus theirstrengths, and an unsteady entropy wave, giving two unknowns: its orientation andstrength. The resulting simple expressions for the strengths and orientations of theunsteady waves needed to evolve the solution are:�a = tan�1 vtut ; (3.38)�a+ �~�a+ � ~ma� = 12 �qu2t + v2t � pt=�a� ; (3.39)�a� �~�a� � ~ma� = 12 �qu2t + v2t + pt=�a� ; (3.40)where ~�a� = ~q � a~ma. The entropy wave strength and orientation are, as in the six

31wave models, �e = tan�1 �y � py�a2�x � px�a2 ; �e =j pt�a2 � �t j : (3.41)By de�nition, the steady patterns do not alter the solution in time. Only theunsteady waves contribute to the uctuation and because only three such waves areutilized in this model, the model is LP satisfying. This is easily seen, since, in thecase of vanishing cell uctuation, all of the wave amplitudes �k�km also vanish.In numerical experiments solutions with very low numerical di�usion were ob-tained. However, problems with robustness and convergence were encountered innearly all test cases performed. Because we attempt to minimize the unsteady can-celation of waves, the cell residuals become small as the steady solution is approached.The wave directions, which depend on these residuals, thus become ill de�ned. Thishindered and usually prevented convergence in all test cases performed.The original concept of the steady/unsteady splitting allows for alternate setsof waves to model the local ow than those originally proposed by Roe. Thereforeother models within this context were experimented with including a model basedon minimum-path-lengths similar to that developed by Rumsey for rotated Riemannsolvers [30].In the minimum-path-length model, the two steady potential- ow solutions plus asteady shear wave are utilized, as in the original Roe splitting. However, this modeldi�ers in the choice of the three unsteady waves. For the minimum-path-lengthmodel, the combination of unsteady waves is either given by:� a pair of acoustics� one entropy waveor

32� one acoustic� one shear wave orientated perpendicular to the acoustic� one entropy wave.The choice between the two possible sets of waves is given by the set for which thepath-length in (ut; vt; pt)-space is minimized. The resulting mathematical constraintsare:� If (pt)2 � [�a (ut cos �a + vt sin �a)]2for �a = tan�1 vtut ;then the �rst option of a pair of acoustics plus an entropy wave provides theshortest path-length.� If (pt)2 � [�a (ut cos �a + vt sin �a)]2 ;again for �a = tan�1 vtut ;the second option consisting of one acoustic, one shear perpendicular to theacoustic and an entropy wave provides the minimum path-length.This alternate splitting did produce smooth solutions with low numerical dissipation,just as the original splitting. Unfortunately, it did not show improvement over theoriginal model of Roe for robustness. For all test cases analyzed, this alternate modelresulted in the same inability to converge to a steady solution.

33It is also noted, that other LP models have been developed by the VKI, mainlywithin the context of the characteristic decompositions; the most recent is the PseudoMach Angle Model Method (PMA) [18]. In this method the supersonic solver isbased on the steady characteristics, while in the subsonic regime four acoustic wavesare employed to model the omni-directional acoustical propagation of information.Therefore the method still attempts to reduce the system to scalar advection op-erators. While this produced the best results to date, it utilizes many waves andtherefore we seek an alternate decomposition which truly takes into account theomni-directional behavior of acoustical waves in subsonic regimes.3.2 Hyperbolic/Elliptic Equation DecompositionIt appears that the use of time-dependent waves to represent steady solutions, oreven the errors in steady solutions, is not an appropriate method. In order to developa better discretization method for solving steady-state problems we �rst study thesteady Euler system.Considering the steady system in primitive variable form:AVx +BVy = 0; (3.42)we desire to reduce this system to a set of simpli�ed problems for which we knowgood discretization techniques. We �nd that for an invertible matrix A:Vx +A�1BVy = 0 (3.43)a decomposition into advection operators again reduces to an eigenvalue problem forA�1B, where we can equivalently represent A�1B by L�1�L:Vx + L�1�LVy = 0: (3.44)

34If we multiply through by the left eigenvector matrix L:LVx +�LVy = 0 (3.45)the system then reduces to four decoupled equations for the characteristic variables@W = L@U: Wx +�Wy = 0: (3.46)These four \advective" equations, equally represented as~� � ~rW = 0; (3.47)are given by:� Two streamline characteristics representing constant entropy and enthalpyalong streamlines,~�s=e = ~q; @We = @p� a2@�@Ws = @p+ �q@q (3.48)where q is the local ow velocity, given by:q = pu2 + v2 @q = u@u+ v@vq : (3.49)� Acoustic characteristics which are real in supersonic ow, but imaginary insubsonic regimes, given by:~�� = (u� � v; v� � u) @W� = @p� �q2� @� (3.50)where � = pM2 � 1 and � is the local ow direction given by:tan � = vu @� = u@v � v@uq2 : (3.51)

35This is the characteristic form of the system. The streamline characteristics arereal, yielding hyperbolic equations governed by advective operators independent ofthe ow regime. However, the acoustic characteristics clearly di�er between super-sonic and subsonic ows, as � = pM2 � 1 becomes imaginary for M < 1. In super-sonic ows the acoustic portion is hyperbolic and also given by advection operators:in subsonic regimes the acoustics are imaginary, yielding an elliptic \sub-system."We thus discuss the two regions independently.� Supersonic FlowIn supersonic ows, the Euler system is hyperbolic in space, and thus is equiv-alently represented by four decoupled advection operators: the two streamlinecharacteristics, plus the acoustic characteristics. The acoustic characteristicsare de�ned along the Mach lines, shown pictorially in Figure 3.3. The acous-tic characteristic variables represent the relationship between the pressure �eldgradients and the turning of the ow.This yields four uniquely de�ned directions on which upwind biased stencils canbe based, representing the physical propagation of information in space. Sincethe equations are completely decoupled, in the linear system, we have no sourceterms and the reduced equation which we need an accurate discretization foris simply the scalar advection operator ~� � ~ru = 0.� Subsonic FlowIn subsonic regimes, the equations are of mixed hyperbolic/elliptic type. Thestreamline characteristic equations, equivalent to those in supersonic ow, arehyperbolic, and upwind discretizations are used. However, the acoustic equa-tions are now complex, physically representing the isotropic spreading of infor-

36~q ~���� ~�s;e~�+

Figure 3.3: Characteristics of steady Euler system in supersonic ow.mation in space. By requiring both the real and imaginary parts of the acousticequations to be zero, the following coupled system is recovered:2664 ��2� 00 �q2 37758>><>>: p� 9>>=>>;s + 2664 0 �q21 0 37758>><>>: p� 9>>=>>;n = 0: (3.52)For simplicity we use streamline coordinates, denoted by @s and @n, repre-senting derivatives along and normal to the streamline, respectively, and �� =p1 �M2. By transforming variables with @u = �q2@� and @v = ��@p, andtransforming into a stretched coordinate system given by @� = ��@s and @� =@n, we recover the Cauchy-Riemann equations:2664 1 00 �1 37758>><>>: uv 9>>=>>;� + 2664 0 11 0 37758>><>>: uv 9>>=>>;� = 0: (3.53)This system, unlike the hyperbolic operators, does not possess a preferen-tial direction along which information is sent, but rather represents an omni-directional behavior. The elliptic nature of the system is clearly seen in the

37Cauchy-Riemann equations, where the solution �eld is made of harmonic func-tions u and v, i.e. r2u = r2v = 0. Central discretizations are used, deviatingfrom the past ideas where upwind techniques were utilized for all portions ofthe ow. Therefore, the subsonic equations have been reduced to two types ofsimpli�ed problems: advection operators which we discretize with upwind bi-ased stencils as in the supersonic regime, and a 2�2 Cauchy-Riemann system,for which symmetric stencils are used.3.2.1 \Time-Dependent" SystemNow that we have successfully decomposed the steady system into recognizablesubproblems, we need to incorporate time-dependent terms into the system in orderto converge the solution by marching in \pseudo-time." We propose to do thiswith preconditioning techniques similar to those developed by Van Leer/Lee/Roe[41] and Turkel [39]. These preconditioners were originally developed as convergenceacceleration techniques, where the time-dependent Euler equations are modi�ed inthe following form: Ut +P (AUx +BUy) = 0: (3.54)This alters the transient system but leads to an equivalent steady state, for positive-de�nite P. Although we still utilize the preconditioner for producing a \pseudo"time-dependent system with near-optimized condition numbers, now, more impor-tantly, we demand that the preconditioner shall preserve, even in the time-dependentsystem, the unique complete decoupling found in the steady equations.For simplicity in discussion we transform the Euler equations into \symmetrizing-variable" form in streamline coordinates, given by:@Usym@t +A@Usym@s +B@Usym@n = 0

38@Usym = (@p�a ; @q; q@�; @p� a2@�)T :where again q is the local ow speed and � is the ow direction, with @s and @nrepresenting derivatives along and normal to the streamline direction, respectively.We note here that for the remainder of this chapter we will denote Usym by U forbrevity.Because the information mechanisms vary between subcritical and supercritical ow we again analyze each region independently and then ensure their analyticalcontinuity through the sonic line.3.2.1.1 Supersonic RegimeWe know that the steady supersonic Euler Equations are convective by nature,however the time-dependent Euler Equations recouple the characteristic equations,even in the linear system. What we attempt to do is �nd an altered time-dependentsystem which is also represented by four scalar convective equations. We also wishto optimize the condition number of the equations, requiring all unsteady waves topropagate at the same speed. To do this we consider the quasi-linear symmetrizingvariable form: Ut +P (AUs +BUn) = 0 (3.55)where P is some positive-de�nite matrix that will diagonalize the system while equal-izing speeds. Performing the following steps:Ut +P (AUs +BUn) = 0;P�1Ut +AUs +BUn = 0;A�1P�1Ut +Us +A�1BUn = 0;A�1P�1Ut +Us + L�1�LUn = 0;

39LA�1P�1Ut + LUs +�LUn = 0;LA�1P�1L�1LUt + LUs +�LUn = 0;LA�1P�1L�1Wt +Ws +�Wn = 0;converts the system to characteristic form. This again de�nes the characteristicvariables W such that @W = L@U, where L contains the left eigenvectors of A�1B,giving: @W = 8>>>>>>>>>>><>>>>>>>>>>>: @p� a2@�1�q@p+ @q��q@p+ q@���q@p� q@� 9>>>>>>>>>>>=>>>>>>>>>>>; ; (3.56)the same as in the steady system. And � is the diagonal matrix of the correspondingeigenvalues, given by: � = 2666666666664 0 0 0 00 0 0 00 0 1� 00 0 0 � 1� 3777777777775 ; (3.57)which are now given in the streamline coordinate system. To diagonalize the system,we require that: D = LA�1P�1L�1 (3.58)be a diagonal matrix, yielding four independent advective equations for the charac-teristic variables, W : DWt +Ws +�Wn = 0: (3.59)The other constraint placed upon the preconditioner is that it perfectly conditionsthe system, i.e. it makes all waves travel at the same speed. This is achieved by re-

40quiring each of the diagonal elements in the matrixD to normalize the correspondingcharacteristic speed to some chosen rate, �c. Thus the elements of D are,For: D = diag(di) and � = diag(�i), di = p1+�2i�c :If we choose to advect all waves at the local ow velocity, giving �c = q then wede�ne D = 2666666666664 1q 0 0 00 1q 0 00 0 1�a 00 0 0 1�a 3777777777775 : (3.60)We can now determine the unique preconditioner P given by:P = L�1D�1LA�1 = 2666666666664 M� � 1� 0 0� 1� 1 + 1M� 0 00 0 �M 00 0 0 1 3777777777775 : (3.61)This is equivalent to the supersonic preconditioner originally derived by Van Leer/Lee/Roe[41] for convergence acceleration.The altered Euler system can now be represented by four scalar advective equa-tions all with a characteristic speed equal to the local ow speed:(@t + q@s) �@p� a2@�� = 0 (3.62a)(@t + q@s) (@p+ �q@q) = 0 (3.62b) @t + �qM @s + qM @n! @p+ �q2� @�! = 0 (3.62c) @t + �qM @s � qM @n! @p� �q2� @�! = 0: (3.62d)These advection equations clearly satisfy the decoupled steady-state operators thatwe previously derived, simply adding the appropriate time terms providing a mecha-

41nism for marching to convergence. Each of the advection equations is thus indepen-dently updated by a forward Euler time integration scheme for the scalar advectionequation: ut + ~� � ~ru = 0; (3.63)providing a preferred direction on which to base an upwind bias. The details coveringconvergence and discretization properties of the now reduced system are shown inthe following chapter.3.2.1.2 Subsonic RegimeFor subsonic ows, we attempt to utilize the same preconditioning technique tomost importantly preserve the decomposition into scalar subproblems found for thesteady-state equations. We also make the additional constraint of optimizing thecondition number of the system, although perfect conditioning is not possible in thesubsonic regime. In this case, the steady system is of mixed hyperbolic/elliptic type,and therefore we hope to extract two advection operators of the form:ut + ~� � ~ru = 0: (3.64)For the remaining equations, we wish to extract a coupled acoustic system of twoequations, which, under the same variable and coordinate transformations as given inthe steady elliptic system, will reduce to the Cauchy-Riemann equations augmentedby time terms in the following way:8>><>>: uv 9>>=>>;t + 2664 1 00 �1 37758>><>>: uv 9>>=>>;x + 2664 0 11 0 37758>><>>: uv 9>>=>>;y = 0: (3.65)This time-dependent system is hyperbolic, allowing for a forward Euler integrationmethod consistent with that used for the advection operators. However, it does result

42in an elliptic steady state. From here on, we refer to this hyperbolic sub-system as the\elliptic-based" system because it is based on the original elliptic acoustical systemin the subsonic steady Euler equations.We again start with the time-dependent Euler equations in symmetrizing variableform, modi�ed with a preconditioning matrix P. However, instead of diagonalizingthe system, we desire the equations to be of the following form:Wt +AwWx +BwWy = 0; Aw = 2666666666664 x 0 0 00 x 0 00 0 x x0 0 x x 3777777777775 ; Bw = 2666666666664 x 0 0 00 x 0 00 0 x x0 0 x x 3777777777775 :This provides the �rst two characteristic/advective equations plus the isolated 2� 2acoustic system.We �rst assume the resulting characteristic variables will be the same as thosefound for the steady system, i.e.@W = L0@U = 8>>>>>>>>>>><>>>>>>>>>>>: @p� a2@�@p+ �q@q1�a@pq@� 9>>>>>>>>>>>=>>>>>>>>>>>; ; (3.66)and the matrices Aw = L0PA �L0��1 (3.67)and Bw = L0PB �L0��1 : (3.68)Thus the preconditioner is given by:P = �L0��1AwL0A�1 (3.69)

43and P = �L0��1BwL0B�1: (3.70)Satisfying both equations for P, while requiring the proposed block form of Aw andBw de�ned previously, we �nd that P must be of the following form:P = 2666666666664 �Mp1;2 p1;2 �Mp2;3 0p1;2 p2;2 p2;3 0�Mp3;2 p3;2 p3;3 00 0 0 p4;4 3777777777775 : (3.71)It appears that this subsonic preconditioner is not unique, and therefore we mustmake some assumptions. The �rst simpli�cation comes from assuming the samesparseness of the preconditioning matrix as that found for the supersonic regime.This is advantageous when considering that the two preconditioners must merge atthe sonic point. However, it is noted here that other terms could be present while stillallowing for sonic line continuity as long as those terms contain a factor of (1 �M2).This constraint requires p3;2 = p2;3 = 0, and results in a symmetric preconditioningmatrix: P = 2666666666664 �Mp1;2 p1;2 0 0p1;2 p2;2 0 00 0 p3;3 00 0 0 p4;4 3777777777775 : (3.72)We now wish to optimize the condition number of the system, and �nd that thesubsonic preconditioner developed by Van Leer/Lee/Roe [41] �ts the desired form

44given in Equation (3.72) given as:P = 2666666666664 M2�� �M�� 0 0�M�� 1�� + 1 0 00 0 �� 00 0 0 1 3777777777775 : (3.73)The resulting decoupled system produced with this preconditioner is given by twostreamline advection equations for entropy and enthalpy, consistent with the super-sonic case: (@t + q@s) �@p� a2@�� = 0; (3.74)(@t + q@s) (@p+ �q@q) = 0: (3.75)Again both quantities are convected with the local ow speed q. The now coupledacoustic system is given by:8>><>>: ��@p�qq@� 9>>=>>;t + 2664 �q�� 00 q�� 37758>><>>: ��@p�qq@� 9>>=>>;s + 2664 0 qq 0 37758>><>>: ��@p�qq@� 9>>=>>;n = 0: (3.76)This system, as desired, reduces to the time-dependent Cauchy-Riemann systemgiven in Equation (3.53). The time-dependent system is hyperbolic producing anelliptical domain of in uence, centered about the point of origin, with a stream-wise range of q�� and a cross-stream range equal to the local ow speed q. Thisis considered to be the optimal conditioning of the subsonic Euler system, with acondition number of 1=��.Unfortunately there are some issues recently discovered which pertain to this pre-conditioned system of equations. When this system was utilized in a numerical code,stability problems in stagnation regions hindered convergence, usually preventingwell-converged solutions from being obtained. It was discovered by Darmofal and

45Schmid [2] that a degeneracy in the system eigenvectors allows for transient energygrowth which is unbounded in the limitM ! 0. Another problem with the precon-ditioner is the sensitivity to the ow angle in the same stagnation point limit [42].While removal of the eigenvector degeneracy is nontrivial, the ow-angle sensitivityis easily removed by requiring that in the preconditioning matrix the p2;2 elementapproaches the p3;3 element in the M ! 0 limit. This is easily seen by rotating thepreconditioner into Cartesian coordinates by the following transformation,R(�) = 2666666666664 1 0 0 00 cos � sin � 00 � sin � cos � 00 0 0 1 3777777777775 ; (3.77)which is a function of the local ow direction. The preconditioner in Cartesiancoordinates is then given by: Pc = R�1(�)PR(�): (3.78)Substituting in the general form of the preconditioner given in Equation 3.72, we�ndPc = 2666666666664 �Mp1;2 p1;2 cos � p1;2 sin � 0p1;2 cos � p2;2 cos2 � + p3;3 sin2 � cos � sin �(p2;2 � p3;3) 0p1;2 sin � cos � sin �(p2;2 � p3;3) p2;2 sin2 � + p3;3 cos2 � 00 0 0 p4;4 3777777777775 : (3.79)From this matrix it is easily seen that only if element p2;2 approaches p3;3 in thelimit of M ! 0, will the two central diagonal terms given by p2;2 cos2 � + p3;3 sin2 �approach a constant; if not, the remaining functions of the ow angle can have largevariations within a stagnation region.

46We again attempt to optimize the condition number, under the same sparsenessconsideration plus this added ow-angle sensitivity constraint [42]. The followingfamily of preconditioners is recovered:P = 2666666666664 �M2�� � �M�� 0 0� �M�� ��� + � 0 00 0 �� 00 0 0 � 3777777777775where � = fcn(M) such that � = 12 in the M ! 0 limit satisfying the ow anglesensitivity constraint. Also, � = 1 in the transonic limit gives the smooth transition ofthe preconditioning matrices between the two regimes. The slightly altered decoupledsystem of equations is now: (@t + �q@s) (@p� a2@�) = 0 (3.80a)(@t + �q@s) (@p+ �q@q) = 0 (3.80b)8>><>>: ��@p�qq@� 9>>=>>;t + 2664 ��q�� 00 q�� 37758>><>>: ��@p�qq@� 9>>=>>;s + 2664 0 �qq 0 37758>><>>: ��@p�qq@� 9>>=>>;n = 0: (3.80c)The characteristic variables remain the same, however the wave-speeds have beenaltered.First we determine a valid function for �(M) with the appropriate limits, wechoose the following:� Incompressible Region, M � 13 � = 12.� Transition Region, 13 �M � 23 � = 12 + 272 (M � 13)2 � 27(M � 13)3.� Transonic Region, M � 23 � = 1.

47This provides for a cubic function within the transition region giving a smooth vari-ation between the incompressible and transonic limits. Many other functions wouldprobably su�ce but were not dealt with in this work.The advective equations now have characteristic speeds given by �q. The acousticdomain of in uence is shown in Figure 3.4. While it remains elliptical in shape, asthe original van Leer preconditioner, the center has moved to 12��(1 � �)q, which isaway from the origin for � 6= 1. The stream-wise range consists of a forward wavemoving at �q and a backward wave traveling at ��q. The cross-stream waves travelat a rate of �p�q normal to the ow. This produces a ray speed given as� = 0@s14�2�(1� �)2 + �1A q: (3.81)Clearly the backward acoustic is the slowest wave, giving �min = ���q with thefastest wave given by the maximum of the forward acoustic and the ray speed forthe transverse waves given in Equation (3.81). For M � :5 the forward acoustic isthe fastest giving a condition number of two, which is twice that of the unmodi�edpreconditioner due to the added stagnation limit constraint. Therefore, for M � :5the transverse acoustics are the fastest giving a condition number of�max�min = q14�2�(1� �)2 + ����q : (3.82)The transonic limit of the condition number, 1� is equivalent to that of the originalVan Leer preconditioner because the modi�ed preconditioner reverts to its originalform for M � 23 .Although this modi�cation has not overcome the problem of degenerating eigen-vectors, also determined by Darmofal and Schmid [2], it has alleviated the stagnationpoint stability issue for the discretizations discussed in this work. Essentially, non-conservative formulations of the method appear stable for decreasing grid sizes, even

48x=t

y=t���qp�q �max �q12�(1� �)q�min

Figure 3.4: Domain of in uence of acoustic waves for preconditioned system nearsonic line.in low speed ows. However, conservative formulations do appear to su�er from theunstable transient growth due to the degeneracy, hindering convergence. More aboutthis will be discussed in the Numerical Algorithm and Results chapters, VI and VIIrespectively.In employing this preconditioning technique the elliptic-based system is isolatedfrom the hyperbolic components even in the transient system. Because the acousticalelliptic-based system no longer has a preferential direction of propagation but ratherrepresents the isotropic spreading of information, damping schemes are appropriateto converge the discrete approximation. This of course eliminates the need to chooseadvection directions, necessary in the past methods, from the in�nitely many thatexist.There is another very useful bene�t in utilizing this form of preconditioning,

49similar to work of Van Leer/Lee/Roe and Turkel: the preservation of accuracy in thecompressible solver even in the incompressible limit [43, 40]. Standard computationalmethods used to approximate the compressible equations generally su�er from aloss of accuracy as the freestream Mach number becomes small, i.e., in the limitof incompressible ow. This is because the numerical scheme does not approachan acceptable method for approximating incompressible ow, as do the di�erentialequations governing the system. Fortunately, with the use of the preconditioningmatrices shown here, the numerical approximation does remain valid in low-speed ows, approaching the correct incompressible limit. This allows for accurate solutionsto be obtained with the compressible solver, even in nearly incompressible ows. Themain advantage of this is in situations of low-speed ow over airfoils at high angles ofattack, where isolated compressible regions are generated near the leading edge, or inpropulsion systems, where ow conditions may range from stagnation in a reservoirto supersonic out ow.3.2.1.3 Sonic Line ContinuityIt is clear that the convection of entropy and enthalpy along streamlines is con-tinuous between the subsonic and supersonic regimes. The continuity of the acousticbehavior is seen by examining the domain of in uence of the acoustic waves in thetransonic limit, shown pictorially in Figure 3.5. In approaching the limit from thesubcritical side the domain of dependence of the acoustic disturbances is an ellipsewhose maximum stream-wise extent is q��, while the cross-stream extent is the owvelocity, q. It is clear that as M ! 1, �� ! 0, and the ellipse attens out intoa line of length q from the origin. The domain of in uence for the characteristicsde�ning the acoustic disturbances in the supersonic regime, shown in Figure 3.5 b).

50are the Mach lines with length equal to the local ow speed. As the sonic point isapproached the Mach lines open up eventually becoming normal to the streamline,giving the same limit as the subcritical case.��x=t x=t

y=ty=tqq��

q a(b)Supersonic Acoustic Wavefront(a)Subsonic Acoustic WavefrontFigure 3.5: Domain of in uence of acoustic waves for preconditioned system nearsonic line.3.2.2 Summary of Decomposition MethodThe decomposition presented here is the unique decoupling of the linear Eulersystem, giving four independent advective operators in supersonic ows, and two ad-vective operators plus an isolated elliptic-based system in subsonic regimes. Throughpreconditioning techniques, this decoupling was preserved even in the \pseudo" time-dependent equations, providing an iterative strategy for solving the decoupled sys-tem. However, while the linear system is decoupled, for non-linear problems couplingbetween the independent systems will occur via the coe�cients.

51In comparison, the canonical form advocated by Ta'asan [35]:2666666666664 q @@s 0 0 00 q @@s 0 00 0 (a2 � q2) @@s a2q @@n1q @@n �1( �1)�q @@n � @@n q @@s 37777777777758>>>>>>>>>>><>>>>>>>>>>>: HSq� 9>>>>>>>>>>>=>>>>>>>>>>>; = 0; (3.83)with @S = @p� a2@�; (3.84)and @H = � 1(@p� � p@��2 ) + q@q; (3.85)results in direct coupling between the \elliptic" and hyperbolic advective operatorseven in the linearized system. With regard to convergence rates and accuracy it isnot clear if any advantage is gained by limiting the coupling to the coe�cients alone.Currently di�erent discretization techniques based on staggered grids are utilized byTa'asan, making comparisons between the canonical forms alone impossible.

CHAPTER IVUPWINDING SCHEMES FOR ADVECTIONOPERATORS IN TWO DIMENSIONSIn the previous chapter, three methods of reducing the Euler equations into sub-problems were presented: simple-wave models, steady/unsteady splitting and hyper-bolic/elliptic decomposition. In the \wave models", the equations were representedby simple waves governed by advection operators, where the Hyperbolic/Ellipticsplitting resulted in hyperbolic advection operators for enthalpy, entropy and super-sonic acoustic disturbances. In the last decade, signi�cant progress has been madein the development of truly multidimensional upwind schemes for advection oper-ators in two and three dimensions. These upwinding schemes are formulated in a uctuation-splitting context, operating on compact stencils, and producing accuratemonotone solutions. Presented in this section will be the formulations for variousschemes developed in two dimensions, including their recent uni�cation with �nitevolume formulations of Sidilkover [32].4.1 Linear SchemesIn analyzing a two dimensional advection equation of the formut + ~� � ~ru = 0; (4.1)52

53the uctuation in a given triangle, � ,�� = Z Z (~� � ~ru) dA; (4.2)is calculated from the basic assumption of piecewise linear data, u, over the element.This signal then must be distributed among the three vertices, V i� , where we denotethe signal sent to each vertex i in a given triangle � by �i� for i = 1:::3 and the sumof the signals equals the uctuation calculated within the element, i.e.3Xi=1�i� = �� : (4.3)A conservative evolution of the solution is then given by:S1un+11 = S1un1 ��t�1� + TFOT (4.4)S2un+12 = S2un2 ��t�2� + TFOT (4.5)S3un+13 = S3un3 ��t�3� + TFOT; (4.6)where TFOT refers to terms from other triangles, because each node can receive asignal from each triangle of which it is a vertex. The area weight Si assigned to eachnode is its median dual area given as one third of the area of the triangles of whichit is a vertex, Si = X8�;i2V� 13S� : (4.7)In developing upwind schemes, the velocity of advection, ~�, de�nes the directionof ow, or direction in which information is propagated. In any given triangle, thelocal ow direction ~� results in two types of triangles as shown in Figure 4.1.� Type I refers to elements with only one in ow side.� Type II triangles have two in ow sides and therefore the cell uctuation mustbe split between the two opposite vertices.

54 ~�~� 132 31 2(a)TypeI (b)TypeIIFigure 4.1: Example of triangles with one or two in ow sidesAll schemes are equivalent in the type I case with the entire uctuation sent tothe one downstream vertex. The various upwind schemes used throughout this workdi�er in how the uctuation within the type II triangle is distributed between thetwo downstream vertices.Before presenting the various distribution methods, some desirable properties ofthe resulting update schemes are presented here to clarify notation.� LP Property (Linearity Preservation)A schemewhich is LP satisfying is globally able to obtain, for any triangulation,an exact linear solution to the advection problem for which the solution is linearin space. This constraint locally reduces to the need for vanishing signals sentto the cell vertices in the case of vanishing cell uctuation, i.e.�� ! 0) �i� ! 0 (8i): (4.8)For regular Cartesian grids broken into triangles using equivalent diagonals alllinear schemes which satisfy LP are at least second order accurate [5]. For

55arbitrary grids, on which accuracy is di�cult to analyze, numerical tests pre-sented later have shown that schemes satisfying LP are more accurate thanthose which do not, emphasizing the important correlation between propertyLP and accuracy.� Positivity PropertyFor a linear scheme, where the solution at a vertex i is given as a function ofthose points surrounding it, with constant coe�cients, i.e.ui = NXk=1 ci;kuk; (4.9)the update scheme is said to be positive if all of the coe�cients are non-negative,i:e: ck � 0; k = 1:::N . This prohibits extrema from developing in the interior ofthe solution, guaranteeing stability and thus convergence of the scheme. Thisconstraint enforces positivity at each vertex, called global positivity from hereon, but actually a stronger constraint of local positivity is desired in order tomaintain the compactness of the scheme. By ensuring that the update sent toeach vertex from each triangle has positive coe�cients, it is guaranteed thatthe global update at a vertex, i.e. the total update to that vertex from alltriangles of which it is a member, will satisfy positivity constraints. This localenforcement allows the algorithm to process each mesh simplex individually,without the need for information about updates to the local vertices from othertriangles of which they are members.The �rst schemes discussed are linear distribution schemes for which the coe�-cients ci;k in Equation (4.9) are constants, i.e. do not depend upon the data.

564.1.1 Low Di�usion SchemeA linear splitting scheme which satis�es property LP but is not positive is theLow Di�usion Scheme (LDA). The cell uctuation is split into two parts, sent to thedownstream vertices, that are proportional to the two areas cut from the triangle bythe advection vector, as shown in Figure 4.2. The area A2 weights the signal to node2, with A3 weighting the signal given to node 3. The resulting split signals are�2 = A2A��� ; �3 = A3A��� : (4.10)This scheme clearly satis�es property LP, given that both �2 and �3 go to zero asthe cell uctuation vanishes. Unfortunately, local positivity constraints are violatedwithin each two-target triangle and in fact even the total updates at each vertex cannot be made positive. This is consistent with a proof given by Deconinck, Struijset. al. [5] showing that a linear scheme can not be both linearity preserving andlocally positive, which is rooted in Godunov's theorem.31 2A2A3 ~�Figure 4.2: Splitting of uctuation for LDA scheme

574.1.2 N SchemeAnother linear splitting of the uctuation, known as the N Scheme (because ofits Narrow stencil and Narrow discontinuity capturing), results from splitting theadvection vector into two components parallel to the sides opposite the downstreamvertices as shown in Figure 4.3. We set~� = ~�2 + ~�3; (4.11)where ~�2 lies parallel to the side opposite vertex 3 and ~�3 lies parallel to the sideopposite vertex 2. 31 2~� ~�2~�3Figure 4.3: Splitting of advection vector for N SchemeThe signals sent to the two vertices are the new uctuations that are calculatedfrom the now split advection problem:�2 = Z Z ~�2 � ru dA (4.12a)�3 = Z Z ~�3 � ru dA (4.12b)where �2 + �3 = �� : (4.13)

58Unlike LDA, this scheme will satisfy local positivity constraints on each triangle,thus ensuring global positivity. The necessary time-step constraint within a giventriangle is: �t � min 2S2~� � ~n2 ; 2S3~� � ~n3! : (4.14)The normals, ~n2 and ~n3, are the inward normals of the sides opposite vertex 2 and3 respectively, scaled to the length of the side, as shown in Figure 4.4.31 2~n3~n2Figure 4.4: Schematic of normals needed in time-step constraint.However, this distribution does not satisfy property LP. The only condition placedupon the signals in the case of vanishing cell uctuation is that they must be equaland opposite, i.e. �� = 0) �2 = ��3; (4.15)but the signals are not necessarily equal to zero.In order to enforce positivity while maintaining accuracy, by Godunov's theoremwe must utilize nonlinear distribution methods. It was �rst proposed by Sidilkover[32] that by employing non-linear limiting functions in the N Scheme formulation,

59the accuracy could be increased while maintaining the ability to preserve monotonic-ity. These limiting schemes, although originally developed for a cell-centered �nite-volume scheme [31], can be applied in a uctuation-splitting context on randomtriangular grids, providing an elegant way of formulating the upwind distributionmethod.4.2 Nonlinear Schemes via Limiting FunctionsIn this approach, the signals calculated from the N Scheme are modi�ed by sub-tracting from one signal and adding to the other a non-linear function of the twosignals. The resulting new signals are of the form:f�2glim = �2 �(�2;��3) (4.16a)f�3glim = �3 +(�2;��3) : (4.16b)The sum of the new signals is still the cell uctuation and therefore the conservationproperty of the distribution scheme is preserved. The desired conditions placed onthe function (a; b) are:� Linearity Preservation (LP): producing vanishing signals in the case of vanish-ing cell uctuation: (a; a) = a: (4.17)� Symmetry: in two dimensions neither edge should be favored over the other,and therefore the resulting signals should be independent of which edge ischosen �rst, giving the condition:(a; b) = (b; a): (4.18)

60Some well-known limiters which meet the above constraints are MinMod, Harmonicand Superbee.MinMod : (a; b) = sgn(a) max [0;min (j a j; sgn(a) b)]Harmonic : (a; b) = 12 (1 + sgn(ab)) 2aba+ b (4.19)Superbee : (a; b) = 12 (1 + sgn(ab)) b max �min�2ab ; 1� ;min�ab ; 2�� :All of the resulting limiting schemes are linearity preserving and therefore willproduce nearly second-order schemes. The other desired property for an upwindscheme, positivity, can also be preserved. However, only the minmod limiter meetslocal positivity constraints, which are preserved for the same time-step restrictionas that given for the unlimited N Scheme. While local positivity is not guaranteedon all triangulations for the other harmonic limiter and the compressive superbeelimiter, it is shown by Sidilkover and Roe that local positivity can be achieved forsu�ciently regular grids [32]. Speci�cally, if a type II triangle has type I triangles asneighbors across each of its in ow sides, then a bound can be placed on the degreeof compression, [jj=min(a; b)], allowed to the limiter. This bound usually exceedsone. Also, the timestep constraint used for stabilizing the N scheme, coupled withthe compressive limiters, is the same as that given for the unlimited N Scheme.It is easily seen that the N Scheme, limited with minmod, is equivalent to thePositive Streamwise Invariant (PSI) Scheme developed by Struijs et. al. [33], againunifying the �nite-volume work of Sidilkover with the original uctuation-splittingmethods. We have chosen to present the schemes in the limiter format because ofthe elegant formulation, providing numerous distribution schemes within one simpleframework.

614.3 Numerical Test CasesIn order to compare the upwinding schemes discussed previously including:� Linear LDA Scheme,� Linear N Scheme,� N Scheme with various limiting functions: minmod, harmonic and superbee,three types of test cases are studied. First, the advection of a smooth pro�le is usedto determine the accuracy of the schemes. Next, the ability of the various methodsto capture discontinuities, including both shear layers and shocks, is discussed.4.3.1 Accuracy StudyThe �rst test case consists of a circular convection problem where a given smoothinlet pro�le is convected about the origin with ~� = fy;�xg. In Figure 4.5 the domainis shown with the imposed boundary conditions given by:� The in ow pro�le of uinlet = e2x sin2(�x) on y = 0.� u = 0 on x = �1.The other two boundaries, which have out ow, need no extra condition and their so-lution is determined solely by the interior scheme. In order to determine numericallythe order of accuracy of the various upwinding schemes, the error of the solution iscomputed on several di�erent grids. For this study the error of the scheme is deter-mined by the L2 norm de�ned by the integral of the error over the computationaldomain: jjerrorjj2 = sZ Z (u� uexact)2 dA: (4.20)

62This integration is numerically computed by assuming piecewise-constant data overthe median dual mesh. We choose piecewise-constant data because the integral normreduces to an area-weighted point norm, providing equivalent error values to thosefound for linear elements but at a lesser cost.u = e2x sin(�x)

outflowoutflowu = 0 0

1�1 0Figure 4.5: Test case of circular convection of inlet pro�le.The solutions were computed on isotropic grids which consist of a structuredsquare grid broken into triangles along alternating left and right diagonals, e.g. seethe 11 � 11 grid shown in Figure 4.6. Test cases were performed on N � N gridsfor N = 11; 15; 21; 31; 41; 61. The log(error) plotted vs. the log(h), where h denotesthe grid spacing, is shown in Figure 4.7. To determine the order of accuracy of theschemes, m, de�ned by error = Chm, a least-squares �t of a straight line is madethrough the data points for the four largest grids. The calculated values are givenin Table 4.1.

63�1:00 �0:60 �0:200:000:400:80

xy Grid Plot.

Figure 4.6: 11� 11 isotropic grid for circular convection test case.�1:80 �1:40 �1:00 �0:60�4:00�3:20�2:40�1:60

log(h)log(error) LDAN SchemeMinmodHarmonicSuperbee

Figure 4.7: Log-log plot of L2 norm of Error vs. grid spacing for circular convectiontest case on isotropic grid.

64Scheme Order of accuracyLow Di�usion 1.98Nscheme 0.91Nscheme (minmod) 1.71Nscheme (harmonic) 1.97Nscheme (superbee) 1.88Table 4.1: Order of accuracy for circular convection of smooth pro�le on isotropicgrid.�1:00 �0:60 �0:200:000:400:80

xy Grid Plot.

Figure 4.8: 11 � 11 right-running-diagonal grid for circular convection test case.The importance of property LP is clear from the low accuracy demonstrated bythe unlimited N Scheme. When the limiting is included the schemes gain nearly anorder of accuracy.To see the sensitivity of the accuracy of the schemes to the grid, we now performthis same analysis on a grid consisting of quadrilaterals broken into triangles withright-running diagonals, (see the 11�11 grid shown in Figure 4.8). Also, to determinethe sensitivity of the accuracy for irregular grids, the regular grid is perturbed by

65�1:00 �0:60 �0:200:000:400:80

xy Grid Plot.

Figure 4.9: 11 � 11 30% perturbed right-running-diagonal grid.Distribution Scheme Order of accuracyunperturbed 10% 20% 30%Low Di�usion 2.14 2.02 1.85 1.75Nscheme (minmod) 1.67 1.61 1.48 1.57Nscheme (harmonic) 1.85 1.84 1.76 1.71Table 4.2: Order of accuracy for circular convection of smooth pro�le on perturbedright-running-diagonal grids.allowing each interior point to move a given fraction of the regular grid spacing ina random direction. Results from grids perturbed by 0%, 10%, 20%, and 30%, aregiven in Table 4.2. The 11 � 11, 30% perturbed grid is shown in Figure 4.9. Thesolutions using LDA and the limited N Scheme with both the minmod and harmoniclimiters, are computed for grids of size N = 11; 21; 31; 41:The unperturbed grid shows order-of-accuracy values very close to those givenfor the isotropic grid. For the LDA and N Scheme limited with minmod, the errornorms for grids constructed with right-running diagonals are compared with the

66�1:80 �1:40 �1:00 �0:60�4:40�3:60�2:80�2:00

log(h)log(error) LDA(R)LDA(I) MinMod(R)MinMod(I)

Figure 4.10: Log-Log plot of error norm vs. grid spacing of LDA and minmod lim-ited N Scheme for Isotropic (I) and unperturbed Right-running-diagonalgrids (R).error norms from the isotropic grids and are shown in Figure 4.10. Clearly the right-running-diagonal grids provide more accurate solutions which is understandable asless dissipation is introduced for diagonals aligned with the direction of advection.Also, it is noted from Table 4.2 that the order of accuracy of the schemes doesnot degrade much with growing perturbation, still providing nearly second-ordersolutions even on fairly random grids.4.3.2 Discontinuity CapturingAnother important feature of the upwind methods is their ability to sharplycapture discontinuities. Two types of discontinuities exist: shear layers, where thecharacteristics lie along the discontinuous surface; and shocks, where the character-istics come into the discontinuity from both sides. Both types are studied here to

67compare the various schemes.The �rst test case, a shear case, consists of a discontinuity advected at 26:6�. Theleft state is given by u = 1:0, the right state by u = 2:0. The case was analyzed ona nearly equilateral 31� 31 grid, shown in Figure 4.11, produced with an advancingfront Delaunay grid generation method developed by J.-D. M�uller [15].�1:00 �0:60 �0:200:000:400:80

xy

Figure 4.11: 31 � 31 nearly equilateral grid for the discontinuity test cases.The solutions obtained with the various upwind schemes are shown in Figure 4.12,with a cross-cut at x = 0 shown in Figure 4.13. It is clear that the harmonic andcompressive superbee limiters are very e�ective in hindering the inherent numericalspreading of a contact surface, producing solutions superior to the LDA scheme aswell. In the test case shown, most of the grid does meet the regularity requirementfor positivity found by Sidilkover [32]. Due to this, the overshoots for both limitersare very small. Unfortunately these overshoots can not be predicted and therefore intrue applications enforcement of positivity will be necessary. However, consideringthe increase in sharpness of the shear layer, the bene�t of utilizing a weaker of even

68�1:00 �0:60 �0:200:000:400:80

xy LDA�1:00 �0:60 �0:200:000:400:80

xy N Scheme with minmod limiter

�1:00 �0:60 �0:200:000:400:80 N Scheme with harmonic limiter�1:00 �0:60 �0:200:000:400:80 N Scheme with superbee limiter

Figure 4.12: Contour plots for 26:6� shear ow case (28 contours with umin = 0:98,�u = :04)

69compressive limiter may outweigh the added cost of ensuring grid regularity withinthe vicinity of the discontinuity.0:00 0:40 0:800:901:301:702:10

yu LDAminmodharmonicsuperbee

Figure 4.13: Cross-cut at x=0 for 26:6� shear ow cases.The convergence histories of the shear ow cases are shown in Figure 4.14. Thetest cases were initialized with u = 0 everywhere except on the two in ow boundarieswhere the discontinuity is de�ned. All of the methods converge quickly. The NScheme limited with minmod is the quickest, taking less than 100 iterations to reachmachine zero. The slowest scheme is the superbee limiter, for which the residualhangs up at a slightly higher level than for the other limiters. The probable causeof this is that superbee, unlike the harmonic limiter, is not a smooth limiter. Thisis known to be a source of limit cycles and general convergence slow-down.The last test case consists of a shock ow. This is based on a scalar nonlinear

700: 80: 160: 240:�10:0�6:0�2:02:0

iterationsL2norm Line Contours LDAminmodharmonicsuperbeeFigure 4.14: Residual history for 26:6� shear ow cases.advection equation related to Burgers' Equation given byut + uux + uy = 0; (4.21)which supports steady discontinuities in u. The speci�c case analyzed is referredto as the \Tree Burgers Case" which is de�ned by the following in ow boundaryconditions:� u = 1:5 on x = �1� u = �:5 on x = 0� u = 1:5 � 2(x+ 1) on y = 0:The boundary y = 1 is an out ow boundary where the solution is again determinedby the interior scheme. Results of the LDA scheme and the N Scheme limited with

71minmod, harmonic and superbee, performed on the same grid as in the shear owcase, are shown in Figure 4.15. For this type of discontinuity, all of the schemesappear to perform equally well, capturing the shock over 2-3 cells without spreading.It therefore appears for these discontinuities, similar to those determined by acousticsin the Euler system, where the characteristics run into the shock hindering spreading,the locally positive minmod scheme is su�cient for obtaining quality solutions.�1:00 �0:60 �0:200:000:400:80

xy LDA�1:00 �0:60 �0:200:000:400:80

xy N Scheme with minmod limiter

�1:00 �0:60 �0:200:000:400:80xy N Scheme with harmonic limiter

�1:00 �0:60 �0:200:000:400:80xy N Scheme with superbee limiter

Figure 4.15: Contour plots for Burgers test case (16 contours with umin = �0:8,�u = 0:18.)

CHAPTER VCELL VERTEX DISCRETIZATIONS FORELLIPTIC-BASED OPERATORSIn Chapter III, the steady subsonic Euler Equations were reduced to hyperbolicand elliptic components. The hyperbolic portion of the system is governed by advec-tion operators, whose discretization and time marching strategies were discussed inthe previous chapter; the elliptic portion of the system was shown to be equivalentto the Cauchy-Riemann Equations. With the use of preconditioning matrices, thisreduced system was augmented with time terms producing the following equivalentsystem 8>><>>: uv 9>>=>>;t + 2664 1 00 �1 37758>><>>: uv 9>>=>>;x + 2664 0 11 0 37758>><>>: uv 9>>=>>;y = 0: (5.1)This elliptic-based system represents the distribution of information via the subsonicacoustical properties of the ow. An appropriate discretization technique for thissystem is needed in order to build a truly two-dimensional inviscid ow solver.Because this system is elliptic in the steady state, central discretizations are ap-propriate. There exists no preferential direction upon which to base a bias in theweighting functions. In the current work, a forward Euler integration scheme is cho-sen, for simplicity in combining the marching to steady state of both the hyperbolicand elliptic-based components. A viable choice then for approximating the system72

73is an unstructured cell-vertex implementation of a Lax-Wendro� scheme in the samestyle as that proposed by Ni in 1982 [17] and later developed by Hall [10] and Mortonet. al.[13]. Employing these schemes on triangular grids poses problems di�erent tothose encountered by the above authors on quadrilateral grids. Rather than pos-ing, as they did, the under-determined problem of driving cell uctuations to zero,giving rise to various spurious mode patterns, we only enforce uctuation balanceat vertices. In order to predict the accuracy and convergence rates for the acousticEuler system we �rst study this family of Lax-Wendro� schemes applied to the linearCauchy-Riemann model system.The standard Lax-Wendro� scheme has the equivalent equation:Ut +AUx +BUy = �2 A @@x +B @@y! (AUx +BUy) ; (5.2)where � = �t produces the time-accurate version, and we note that the right-handside, which represents the arti�cial dissipation, should vanish in the steady state.Assuming linearity and expanding the dissipation term, we getUt +AUx +BUy = �2 �A2Uxx + (AB+BA)Uxy +B2Uyy� : (5.3)For the Cauchy Riemann system,AB = �BA;and A2 = B2 = I, resulting in:ut + ux + vy = �2�uvt � vx + uy = �2�v:The dissipation terms are therefore the Laplacian operators acting on u and v.

74The method is easily implemented in a uctuation splitting format thus main-taining the favorable compactness of the scheme. Given the uctuation in a cell,�� = Z Z (AUx +BUy)dxdy; (5.4)it is distributed to the three vertices of the triangle. We de�ne the proportion orsignal received by each vertex by matrix weights, Ai, and thus,��i = Ai�� ; i = 1:::3; (5.5)is the signal sent to node i. There are two terms,�Ut = (AUx +BUy) + ��2 A @@x +B @@y! (AUx +BUy)central term + dissipation termin the equivalent equation. For the �rst term a central distribution sends equal pro-portions to the three vertices, each receiving one third of the total residual. Thedissipation or second-order term in the Lax-Wendro� scheme is discretized in the uctuation-splitting framework based on a Galerkin Finite-Element method[38], re-sulting in total weights given by:Ai = I3 + �2S� (A;B) � 12~ni; (5.6)where ~ni is the inward scaled normal to the side opposite the vertex i. Conservationof the method is ensured since, 3Xi=1 ~ni = ~0;therefore 3Xi=1Ai = I:An analysis of a one-dimensional Lax-Wendro� scheme to approximate the fulltransonic Euler equations was performed by Morton/Rudgyard/Shaw [14]. Although

75this one-dimensional analysis displayed the desirability of optimizing the convectiveproperties of the scheme to eliminate errors through boundaries, the extension of thisanalysis to higher dimensions is not so straightforward. Here we shall discuss issuesraised by a two-dimensional analysis of the model system, adopting the notation ofMorton et. al.When time-accuracy is not a concern, there exist two distinct time terms in theLax-Wendro� scheme; a nodal time-step, �t, which drives the overall evolution ofthe scheme; and a cell time scale, � , which determines the amount of dissipation.These time scales are given by:�t = �nh� � = �ch�;with �n and �c representing the nodal and cell CFL numbers respectively. The locallength scale is denoted by h, with � representing the maximum local wave speed.For Cauchy-Riemann augmented as in Equation (5.1) the local wave speed is unityin all directions.For simplicity we analyze a uniformly spaced, structured, rectangular grid, brokeninto triangular elements along right-running-diagonals. A section of the grid, shownin Figure 5.1, provides a stencil of 6 neighbors about all interior points.Applying the Lax-Wendro� method to the model system, an update for point U0is given by: �U0 = 6Xi=0MiUi; (5.7)where Mi is the matrix weight placed on the state at node i. For the stencil shownin Figure 5.1 the resulting weighting matricesM are given by:M0 = 2664 �2�n�c 00 �2�n�c 3775 (5.8a)

7604 5 63 21

h hFigure 5.1: Stencil for sample Lax-Wendro� scheme.M1 = 2664 �n(12�c � 13) �n6�n6 �n(12�c + 13) 3775 M4 = 2664 �n(12�c + 13) ��n6��n6 �n(12�c � 13) 3775(5.8b)M2 = 2664 ��n6 ��n6��n6 �n6 3775 M5 = 2664 �n6 �n6�n6 ��n6 3775 (5.8c)M3 = 2664 �n(12�c + 13) ��n6��n6 �n(12�c � 13) 3775 M6 = 2664 �n(12�c � 13) �n6�n6 �n(12�c + 13) 3775 :(5.8d)The terms proportional to �c contribute a simple �ve-point smoothing, the othershave an advective e�ect. Utilizing this stencil, we can study issues concerning bothaccuracy and convergence in order to predict viable choices for the two parameters,�n and �c, existing in the update scheme.5.1 AccuracyPerforming a Taylor series expansion about U0 shows second-order accuracy ismaintained in the steady state for all choices of parameters. It is interesting to note

77that, if we analyze the discretization in the limit of in�nite dissipation, i.e., �c )1,the Cauchy-Riemann system is given by:ut = !�uvt = !�v;where ! is proportional to the product �n�c. This product remains �nite, under thestability constraint of �n�c < 12, derived in the following section. This system, nowparabolic in time, shows the limit of the Lax-Wendro� discretization as a paraboliciteration. This system is easily discretized with a Galerkin FEM for second-orderspatial accuracy and is compared in convergence properties with the standard Lax-Wendro� implementation, �c = O(1), in the next section. However, because theanalysis was performed on a regular grid, it was found that when the method wasutilized on irregular grids, accuracy was lost with increasing dissipation, i.e., increas-ing �c. It is unclear to the author how to implement the method to preserve theaccuracy on irregular grids for large damping coe�cients. Therefore, we are limitedto choosing �c = O(1) for irregular grids, where it appears second-order accuracy ispreserved even when utilized for Euler problems, (see Results in Chapter VII).Other issues of importance for Lax-Wendro� methods include the use of extranon-linear damping mechanisms to ensure damping in all directions as well as thepreservation of monotonicity constraints. Because we apply this method only to theelliptic-based subsystem, which has no real characteristics in the steady state, i.e.,there exists no real direction such that(A @@x +B @@y) = 0;damping of errors will occur in all directions. This reduces the need for additionalarti�cial damping necessary when Lax-Wendro� is used to approximate steady sys-

78tems with a hyperbolic component. However, because the elliptic-based system isutilized on the subcritical side of discontinuities, we must have enough dissipationpresent to prevent post-shock oscillations.In one dimension, the Lax-Wendro� method is monotone for �c � 1, and thuswe search for an equivalent criteria in two dimensions. The standard criterion formonotonicity of systems is given by the condition that all coe�cient matrices, M,have positive eigenvalues. In analyzing the coe�cient matrices given in Equation(5.8a)-(5.8d), the corner point weights, M2 and M5 have eigenvalues,�M2;M5 = � �n6p2 (5.9)which are clearly negative for all parameters excluding the limit of �c ! 1. Inthis limit the nodal CFL must be zero, due to the stability constraint shown in thefollowing section. This gives zero eigenvalues for the matrix weights at the cornerpoints and also produces positive eigenvalues for all of the other matrix weightsprovided that �n�c � 1=2. Therefore, the only extension of the one-dimensionalmonotonicity constraint to higher dimensions, is the limit of a parabolic iterationof the second-order system. Therefore, if other values for the Courant numbers arechosen, it does appear that additional arti�cial dissipation would be necessary. Itis needed to provide su�cient damping for complete elimination of the oscillationsabout discontinuities resulting from the non-positive property of the scheme. Thisis not addressed in this work.5.2 ConvergenceContinuing with the same sample grid shown in Figure 5.1, we now address theissues of stability and convergence rate, including error elimination via convectionand damping. To determine the domain of parameters, �n and �c, for stability, and

79also to predict damping capabilities of the method, a Von Neumann stability analysisis performed, assuming periodic boundary conditions. The ampli�cation matrix,G,where Un+1 = GUn; (5.10)describes the evolution of the discrete approximation from time level n to time leveln + 1. The spectral radius of the ampli�cation matrix, for the grid considered, isdetermined to be:�2 (G) = [1 � �n�c (2� cos kxh� cos ky h)]2+�2n9 [sin kxh � 2 sin kyh � sin(kxh + kyh)]2+�2n9 [sin kyh� 2 sin kxh � sin(kxh + kyh)]2 :Requiring the spectral radius to be less than one over the entire frequency domainde�nes the stability region. First analyzing the spectral radius at corner points, i.e.(0; 0); (0; �), (�; �) and (�; 0) frequencies, we determine the following bound:�n�c � 1=2: (5.11)Because the lowest frequencies, (0; 0); give �(G) = 1 for all parameters, we alsorequire that the spectral radius be a decreasing function with increasing frequencynear this point to ensure that the stability bound is not exceeded. This provides theother bound of �n � �c: (5.12)In substituting these limits into the spectral radius calculation it is found that allother frequencies remain stable, proving the above bounds are su�cient for ensuringstability.

80If we �rst assume error elimination via damping properties alone, we �nd for agiven �c, the smallest ampli�cation rate, �(G) = 1 � 12�2h2(1 � 1=(2�2c )) + O(h3),is obtained for �n = 1=(2�c). This is found by equalizing the (low,low) and the(high,high) frequencies for a �nite grid of size h = 1=M , i.e. requiring�� �M ; �M � = � �(M � 1)M ; �(M � 1)M ! :The optimization of this over possible values of �c gives the limit �c !1, with �n !0 but �c�n = 1=2. This is in fact equivalent to a Jacobi iteration scheme with a �nite-element discretization of the remaining Laplacians, predicting convergence in O(M2)iterations for grids consisting of M �M vertices. Therefore, this predicts O(M2)iterations till convergence for the Lax-Wendro� scheme, with optimal performancefor the limiting Laplacian discretizations, if we consider damping alone.However, numerical tests con�rmed the importance of advecting the errors. Testcases were performed on the structured standard grid described previously. For theLax-Wendro� scheme, �c = �n = 1p2 was chosen to maximize advection of errors, asit provides the maximum advection rate within the stability bounds. Also, charac-teristic boundary conditions were easily imposed, as the grid boundaries were alignedwith the axes, thus reducing wave re ections. These conditions are found by reducingthe system to one dimension in the directions normal to the boundaries:x direction : ut + ux = 0vt � vx = 0 )) prescribe u on x = 0prescribe v on x = 1y direction:ut + vy = 0vt + uy = 0 ) (u+ v)t + (u+ v)y = 0(u� v)t � (u� v)y = 0 )) prescribe u+ v on y = 0prescribe u� v on y = 1as shown in Figure 5.2

81u u+ vu� v v x

yFigure 5.2: Characteristic boundary conditions for Cauchy-Riemann system.Two iterative methods are compared with the Lax-Wendro� scheme for theparabolic system: Point Jacobi and Gauss-Seidel. By utilizing a Gauss-Seidel relax-ation the work per iteration is greater than it is for the Point Jacobi or Lax-Wendro�methods. This is because the Gauss-Seidel relaxation sweeps across the grid updat-ing the solution at vertices in the process, and utilizing the new updated values tocompute the solutions at the remaining mesh points. Therefore, all triangles aboutthe current vertex must be visited to evaluate the new state at that vertex. To com-pute the new solution at all mesh points, each of the triangles in the domain will thusbe visited three times, to compute the solution at each of its three vertices. Also,because of the way the Gauss-Seidel relaxation must be implemented, the desirableproperty of the compactness of the algorithm, present for both the Point-Jacobi andLax-Wendro� methods, is lost. No longer can the numerical scheme be implementedas a single loop over triangles, where each triangle is analyzed independently makingthe algorithm very e�cient for parallel implementation. These issues would need tobe addressed when evaluating the true advantage of the speed-up associated withthis relaxation.For the test cases, both the Point Jacobi and Gauss-Seidel methods are applied

82with their corresponding optimal relaxation parameters [11],!PJopt = 1:0!GSopt = 2�1 � �M � ;for an M �M grid.The example case analyzed has an exact solution given by:u = 4x3 � 12xy2;v = 4y3 � 12yx2: (5.13)The initial condition consists of random numbers in the range (�16; 8), which in-cludes initial maxima of twice the solution maxima. The boundary conditions forthe Lax-Wendro� scheme are the characteristic conditions described previously. Forthe parabolic systems, both u and v are given on all boundaries.For a 21 � 21 grid the residual histories are given in Figure 5.3. Clearly the0: 800: 1600: 2400:�14:0�8:0�2:04:0

iterationsjj ut jj2 Lax Wendro�Point JacobiGauss Seidel

Figure 5.3: Residual history for model Cauchy-Riemann problem on 21� 21 grid.advection of errors dominates the convergence, as the Lax-Wendro� scheme converges

83much quicker than the Jacobi relaxation which was predicted to be superior in theanalysis of damping rates only. The fastest convergence is achieved by the optimizedGauss-Seidel relaxation, however, it is repeated that this destroys the compactnessof the scheme.Next, a study to predict convergence rate as a function of the grid size is per-formed on grids of size M = 21; 41; 81; 161; 321. The ln(M) vs. ln(#iterations) isshown in Figure 5.4. As predicted by theory the Jacobi scheme converges in O(M2)iterations, while the optimized Gauss-Seidel converges in O(M) iterations. The Lax-3:00 4:00 5:00 6:004:08:0

12:0ln(M)

ln(iterations) slope 1slope 2Lax Wendro�Point JacobiGauss SeidelFigure 5.4: Convergence rates vs. grid size for model Cauchy-Riemann problem.Wendro� method also shows a convergence of O(M) iterations, emphasizing theimportance of the ability to advect errors.For single-grid calculations, as opposed to multi-grid, the advantage of the stan-dard Lax-Wendro� technique is that fast convergence due to advection is realized,without the necessity of optimizing the relaxation parameter for the Gauss-Seidelmethod, which is not trivial for arbitrary grids. With non-optimal coe�cients, themethod can quickly deteriorate to O(M2) behavior. However, the extension to Euler

84for arbitrary grids also poses the non-trivial problem of developing non-re ectingboundary conditions. It is thus concluded that a compromise between damping andadvection is needed, given the current status of boundary conditions. Also, the ex-ibility of the Lax-Wendro� scheme makes it appealing, not to mention the ease ofcombining it with the hyperbolic iterative schemes for the advective terms existingin the full system.Another viable option for improving convergence performance is multi-grid accel-eration, where damping rates in the high frequency range, i.e., kxh � �=2 S kyh ��=2 govern the e�ciency. For a given �c, optimizing over this domain in the limit ofdecreasing mesh size, we �nd the lowest ampli�cation rate,�(G) = ����� 1 + 9�2c15�2c � 1 ����� for �n = 6�c15�2c � 1 :Again the minimization over all �c, giving �(G) ! :6, yields a Jacobi iterationmethod for the remaining parabolic equations, with �c ! 1, but �n�c ! :4. Thisagain leads to the loss of convection capabilities that proved very useful providedthat boundary conditions could be e�ectively imposed.Finally, we choose parameters for the Lax-Wendro� scheme that compromisebetween damping and convection properties, due to the eventual problem of bound-ary re ections in full Euler calculations. Assuming a �nite damping coe�cient, e.g.�c = 1, high frequency optimization gives �n = :43 and �(G) = :71. For this casethe ampli�cation rates over the entire frequency spectrum are shown in Figure 5.5.For multi-grid considerations where the block containing the (low,low) frequencies isomitted, maximum damping occurs for equalization of spectral radii for the f�; �gand f�=2; �g frequencies.Although � = :714 is higher than values usually sought for elliptic solvers, we

850.0

0.5

1.0 0.0

0.5

1.0

0.0

0.5

1.0

�LW (G)kxh� kyh�Figure 5.5: Contour plot of the error damping of the Lax-Wendro� scheme over theentire frequency range for the CFL-number combination �c = 1; �n = :43.have now recovered the ability to advect errors as well. In comparison, a Gauss-Seidel relaxation optimized for multi-grid enhancement provides a high frequencyspectral radius of � = :447 [11]. But again, for the Gauss-Seidel method, the lossof compactness needs to be addressed in terms of overall best e�ciency, precludingthe bene�t of enhancing e�ciency via parallel implementation. For a complete studyof multi-grid methods for this model system, as well as the extension to Euler, seeM�uller [16].To summarize, a standard Lax-Wendro� iterative method for converging themodel Cauchy-Riemann system is shown to provide second-order solutions in an ef-�cient manner, by utilizing both damping and advection of errors. Unfortunately, todate no e�cient characteristic boundary conditions exist that limit re ections in fullEuler calculations on arbitrary domains. With the incorporation of the discretizationdiscussed here into the full Euler system, shown in Chapter III, numerical experi-ments show that indeed a compromise between advection and damping provides the

86best convergence times.The other issue of importance addressed here is monotonicity. As implemented,the Lax-Wendro� method does not provide the damping necessary to prohibit os-cillations on the subsonic side of discontinuities. Therefore, additional non-lineardissipation must be incorporated if this discretization is utilized for the subsonicacoustical system contained in the Euler system. This is left to future studies. How-ever, in the test cases shown in the Results (Chapter III), with only weak shockspresent, no noticeable oscillations occur.

CHAPTER VINUMERICAL ALGORITHMThe basic concept of reducing the Euler Equations into subproblems, discussedin Chapter III, de�ned three distinct approaches. The �rst method of reducing thesystem utilized discrete sets of simple waves to model the local ow �eld; the sec-ond used fewer simple waves plus steady solution components; the third decomposedthe equations into elliptic-based and hyperbolic systems determined from the steadyequations. In the preceding chapters, discretizations were proposed for the two dif-ferent types of resulting operators. However, these reduced system were discussed interms of a non-conservative formulation of the equations; some details concerning theimplementation of these ideas in a conservative numerical code need to be addressed.An appropriate linearization is needed in order to maintain conservation for thewave modeling schemes as well as the Hyperbolic/Elliptic splitting. There are otherissues concerning conservation for each type of equation decomposition which are alsodiscussed. Also, for the Hyperbolic/Elliptic splitting, the current practice of mergingthe discretizations of the elliptic-based and hyperbolic acoustic systems between thesupersonic and subsonic regimes is shown. Other important issues within a discretescheme include the boundary conditions and the method of calculating local time-step constraints for marching to steady state; these are also addressed in this chapter.87

886.1 ConservationTo analyze the discretizations of Euler equations in quasi-linear form:Ut +AUx +BUy = 0 (6.1)we need an average state at which the Jacobian matricesA andB are to be evaluated,in order to maintain a conservative ux balance. This is of course analogous to theone-dimensional case, treated in Chapter II, with the standard Roe linearizationgiven by the parameter vector,Z = p�8>>>>>>>>>>><>>>>>>>>>>>: 1uvH 9>>>>>>>>>>>=>>>>>>>>>>>; = 8>>>>>>>>>>><>>>>>>>>>>>: z1z2z3z4 9>>>>>>>>>>>=>>>>>>>>>>>; : (6.2)Considering a one-dimensional transition between two states, U1 and U2, the con-servation property �F = A�U; (6.3)is satis�ed for A = A(Z), where Z = 12 (Z1 + Z2).In multidimensional cases, speci�cally in two dimensions, there is a natural ex-tension of the Roe linearization [4]. For each cell we need the uctuation givenby: � = Z Z @F@x + @G@y ! dA; (6.4)with a conservative linearization providing cell average gradients such that:cFx = �AdUx; dGy = �BdUy: (6.5)If we linearize the two-dimensional system with respect to the parameter vector Z,Z Z (Fx +Gy) dA = Z Z (AZZx +BZZy) dA; (6.6)

89where AZ and BZ are the Jacobians of the ux functions F and G with respect toZ, we �nd that these Jacobians:AZ =MFZ = 2666666666664 z2 z1 0 0 �1 z4 +1 z2 � �1 z3 �1 z10 z3 z2 00 z4 0 z2 3777777777775 (6.7)and BZ =MGZ = 2666666666664 z3 0 z1 00 z3 z2 0 �1 z4 � �1 z2 +1 z3 �1 z10 0 z4 z3 3777777777775 ; (6.8)are linear functions of the parameter vector. If we assume linear variation in Zover mesh elements, de�ning constant gradients cZx and cZy over the cell, the averagegradients of the ux functions, cFx and dGy are easily calculated:cFx = Z Z AZdA cZx = AZ cZx = AZ(Z) cZx (6.9)and dGy = Z Z BZdA cZy = BZ cZy = BZ(Z) cZy: (6.10)Therefore, the average state at which AZ and BZ are calculated for conservation isthe average state of Z in a cell, given by the assumption of linear variation as:Z = Z Z ZdA = 13(Z1 + Z2 + Z3); (6.11)where Zi denotes the parameter vector evaluated at node i. From this average stateof Z we can also obtain the consistent average state of the primitive variables in each

90cell: V = 8>>>>>>>>>>><>>>>>>>>>>>: ���u�v�p 9>>>>>>>>>>>=>>>>>>>>>>>; = 8>>>>>>>>>>><>>>>>>>>>>>: �z12�z2= �z1�z3= �z1 �1 � �z1 �z4 � 12( �z22 + �z32)� 9>>>>>>>>>>>=>>>>>>>>>>>; : (6.12)The corresponding average spatial gradients of the primitive variables are thus cal-culated by: cVx =MVZ bZx cVy =MVZ bZy; (6.13)where MVZ is the transformation matrix from the parameter vector to primitivevariables, MVZ = @V@Z = @V@Z (�Z) (6.14)which is also calculated at the Roe average state. This gives the following for thegradients of the primitive variables:d~rV = 8>>>>>>>>>>><>>>>>>>>>>>: d~r�d~rud~rvd~rp 9>>>>>>>>>>>=>>>>>>>>>>>; = 8>>>>>>>>>>><>>>>>>>>>>>: 2p��~rz1���u~rz1 + ~rz2� =p�����v~rz1 + ~rz3� =p�� �1 p�� � �H~rz1 � �u~rz2 � �v~rz3 + ~rz4� 9>>>>>>>>>>>=>>>>>>>>>>>; : (6.15)This linearization seems to be the simplest way of converting conservative residu-als to the forms needed by the decomposition, and it guarantees conservation howeverlarge the local gradients are. However, it is to be expected that any linearizationin the presence of large gradients will have some inconsistency, as this one does. Ifwe consider an isolated shear ow where pressure and density are constant, in mostinstances there will be a numerical pressure gradient calculated from the above for-mulation due to the assumption of linear variation in the velocity and total enthalpy

91�elds, rather than the pressure �eld, (see Appendix A for example). Other alter-natives such as assuming linear variation in the primitive state vector as opposedto the Roe parameter vector have been considered, but pose problems in the cal-culation of the Jacobian matrices, as they are not linear functions of the primitivevector. Therefore, despite the discrepancy of spurious pressure gradients, the Roelinearization does provide the average state that yields a conservative formulation.This linearization is necessary for all of the schemes discussed in this work. How-ever, other issues concerning conservation which di�er between the Wave ModelSchemes and the Hyperbolic/Elliptic Decomposition now are also addressed.6.1.1 Wave Modeling Conservation IssuesIn the discrete wave models given in Chapter III, the local primitive spatialgradients were modeled as a set of planar waves:dVx =Xk �k cos �krk; dVy =Xk �k sin �krk; (6.16)with the corresponding time derivative given by:Vt = �Xk �k�kmrk: (6.17)Because the scheme does not produce vanishing cell uctuations, but rather providesa method of converging accumulated signals at nodes, the distribution within eachcell must be of the conservative uctuation �U = �UtS� . The linearization providesthe appropriate primitive spatial gradients and average state from which the wavestrengths, orientations and corresponding eigenvectors are determined. A simpletransformation of variables from the primitive to conservative state will then producethe appropriate time gradients of the conserved quantities:Ut =MUVVt = �MUVXk �k�kmrk = �Xk �k�km MUVrk; (6.18)

92where MUV is the Jacobian of the conservative state with respect to the primitivestate, MUV = @U@V : (6.19)Essentially what we recover is the transformation of the eigenvectors rk into theirconservative counterparts rUk =MUVrk; (6.20)and the conservative time change for each wave,Ukt = ��k�kmrUk: (6.21)This provides a uctuation, �kU, for each wave,�kU = ��k�kmrUk S� (6.22)which is distributed among the vertices of the triangle via the multidimensionalupwinding schemes discussed in Chapter IV.6.1.2 Hyperbolic/Elliptic Splitting: Conservation IssuesThe method of decoupling the preconditioned Euler system into hyperbolic andelliptic-based subsystems resulted in a set of \characteristic equations." Althoughthe analysis and underlying framework of the scheme is based on this characteristicform, conservation is easily enforced in the �nal numerical algorithm.Given the preconditioned characteristic form,Wt + (PA)WWx + (PB)WWy = 0; (6.23)we decompose the system into elliptic-based and hyperbolic components by splitting(PA)W and (PB)W into their diagonal and block diagonal components representing

93the advective and elliptic-based operators respectively, i.e.(PA)W = (PA)HW + (PA)EW; (PB)W = (PB)HW + (PB)EW: (6.24)Appropriate discretizations for the di�erent operators are then utilized to distributethe preconditioned uctuation, fP�gW = �WtS� . These distribution matrices,AEiand AHi , representing the proportion of the elliptic and hyperbolic uctuations sentto each of the vertices, determined from the corresponding discretization method,i.e. fP�giW = AEi �(PA)EWWx + (PB)EWWy�+ AHi �(PA)HWWx + (PB)HWWy� ; (6.25)can be combined into one distribution matrix Ai representing the proportion of thetotal uctuation sent to each vertex such that:fP�giW = Ai fP�gW ; Xi Ai = I: (6.26)The preconditioned uctuation, fP�g�W in a triangle � is thus given by:fP�g�W = �Xi A�iWtS� (6.27)= Xi A�i ((PA)WWx + (PB)WWy)S� : (6.28)In looping over all cells and accumulating signals at nodes, an explicit integrationscheme evolves the solution in time byWi(t+�t) =Wi(t)� �tSi X� A�i fP�g�W : (6.29)It is clear that the above formulation will produce incorrect jump relations whencapturing discontinuities, because converged nodal solutions will balance � (PW)rather than �U. As before, this discrepancy is a consequence of the fact that, when

94steady state is reached, cell uctuations do not converge to zero, but rather a balanceat vertices is obtained.Therefore, to establish conservation, the non-preconditioned uctuation in con-served variables needs to be distributed in each cell and balanced at vertices for aconservative steady state. Again considering a single triangle, the preconditionedcharacteristic uctuations are transformed into conservation form via:�U = �UtS� =MUWP�1 fP�gW ; (6.30)where MUW is the Jacobian of the conserved state U w.r.t. the characteristic stateW, i.e. MUW = @U@W : (6.31)After accumulating the change to the conservative state at vertices, the precondi-tioner can be utilized in its original context to speed up convergence. The evolutionof the conservative state vector U is given by:Ui(t+�t) = Ui(t)� �tSi PiX� MUWP�1� A�i fP�g�W ;where Pi is the preconditioner evaluated at the old nodal state Ui(t).Both conservative and non-conservative formulations of the presented decoupledmethod have been implemented and tested in numerical experiments. While thenon-conservative formulation is very robust in obtaining steady state solutions, theconservative formulation su�ers from stability issues within the stagnation region. Apaper by Darmofal and Schmid [2] gives a detailed look at stability issues arising fromthe preconditioned system, based on eigenvector degeneracy in the limit of M ! 0.This degeneracy appears to manifest itself more readily in the conservative scheme.

956.2 Merging of Acoustic Discretization Through the SonicLineIn Chapter III, the smooth transition through the sonic line, of the analyticalacoustic system in the Hyperbolic/Elliptic decomposition, was shown. We now dis-cuss the steps taken to ensure that the discrete representations of the acoustics inthe subsonic and supersonic regimes, also smoothly merge in the sonic limit. In thecurrent state of the work, with a Lax-Wendro� solver utilized in subsonic ows andthe limited N Scheme utilized for the characteristics in supersonic ow, a slightlyad-hoc �x is needed to merge the two discretizations. Some speci�cs of this mergingare discussed, although it is noted that many other alternatives are possible.The method utilized for the results shown in this work consists of choosing aregion about the sonic line, which we denote as the sonic region, where both thesupersonic and subsonic methods are employed simultaneously. We de�ne the sonicregion by: qjM2 � 1j = � � 0:1:The discretization methods proposed in this thesis are upwind methods alongthe steady characteristic directions in the supersonic region and a Lax-Wendro�method for the subsonic system. However, when we enter the sonic region, from thesupersonic side, i.e. M > 1 such that � = 0:1, the use of the characteristic directionsfor upwinding is phased out while the Lax-Wendro� method is phased in. Eventually,when the other limit of the sonic-region is reached, i.e. M < 1 such that again� = 0:1, the use of the characteristic directions has been completely phased out andthe Lax-Wendro� method is utilized for the full system. A cubic blending function, , is de�ned between the two endpoints of the sonic region. This function scalesthe proportion of the acoustic residual discretized with the Lax-Wendro� method,

96and therefore goes from a value of zero at the supersonic limit of the sonic regionto one at the subsonic limit. Also, a zero derivative in , with respect to � at bothendpoints is enforced. Therefore, the proportion of the acoustic residual discretizedby the upwind scheme is equal to (1 � ).An important issue in the sonic region is that both the supersonic and subsonicdiscretizations have vanishing dissipation in the streamline direction. This can allowfor such non-physical phenomena as entropy-violating expansion shocks. Thus, thestreamwise dissipation is frozen. For the advection operators, this is done by freezingthe characteristic directions as those de�ned at the sonic region limit. Therefore, thedirection of the characteristics, relative to the streamline, remains attan � = � 1�fr ; �fr = 0:1:This prevents them from becoming completely perpendicular to the streamline. Forthe system solver, the streamwise dissipation terms are simply frozen at their valuesgiven at the sonic region limit on the subsonic side. This prevents the elliptical do-main of in uence from completely attening out, but rather maintains its streamwiserange given for � = 0:1.6.3 Time-Step CalculationThe evolution of the solution to steady state, given in the previous section, isachieved by marching forward in time with a standard local forward-Euler timeintegration. Therefore, at each vertex i in the domain, a local maximum time-step isneeded. Because the system we are discretizing is hyperbolic in time, the maximumtime-step per iteration is limited by the time it takes the fastest wave to traverseone computational cell. We thus de�ne the nodal time-step, �ti as the minimum

97cell time-step �t� over all cells, � , for which i is a vertex, i.e.�ti = CFL min�;(i2�)�t� ; (6.32)where the CFL number is the global Courant number or scaling of the time-step forstability.In each cell, thus, we need the time it takes the fastest local wave, with speed�max, to traverse the cell length h, �t� = h�max ; (6.33)where h is de�ned as the length of the shortest side of the mesh simplex. In the wave-model methods, this maximum wave speed is given by an acoustic wave moving inthe direction of ow, �max =j q j + a: (6.34)For the Hyperbolic/Elliptic Decomposition, where the time-dependent system is pre-conditioned, the speeds at which the waves travel are altered in order to optimizethe condition number of the system and thus maximize convergence performance. Inthis work, all wave speeds are scaled to the local ow velocity q and therefore�max =j q j : (6.35)In stagnation regions, where the ow speed goes to zero, a limit must be placedto bound the time-step away from in�nity. We utilize a fraction of the freestreamvelocity q1 in order to obtain proper scaling for all non-dimensional forms of thesystem, providing a modi�ed maximum speed�max = max �j q j; 10�3q1� : (6.36)This bound on the wave speed proved su�cient for the cases tested and presentedin this work. It is noted that the non-conservative formulation of the scheme must

98be utilized in ows with stagnation points in order to prevent instability during theinitial transients.6.4 Boundary ConditionsThe last area of importance in the actual implementation of the uctuation split-ting methods discussed in this work is that of boundary conditions. Because wemust solve problems on �nite domains, boundary conditions are necessary in orderto de�ne information which is sent from the outside world to the �nite domain beingmodeled. The two basic categories of boundaries discussed and utilized are solidwalls and in ow/out ow boundaries.6.4.1 Solid WallsWe utilize a slip boundary condition along solid surfaces which constrains the ow to be tangent to the wall. This is easily enforced in a strong way for cell vertexschemes since the nodes lie on the solid surface. It simply reduces to forcing owtangency on the wall at every iteration, by initializing the ow �eld to this stateand not allowing the interior scheme to make changes to the normal component ofvelocity, i.e., assuming the direction of ow is known at boundary vertices.To determine the normal to a curved surface at each vertex we used the averageof the scaled normals to the two faces of which that vertex is a member, see Figure6.1. These normals are scaled to the length of their corresponding face. Therefore,the direction of tangency at vertex i, ~ti is given as~ti = ~xi+1 � ~xi�1; (6.37)with ~ni de�ned as perpendicular to ~ti.Another alternative is to use a weak boundary condition to drive the �nal solution

99i� 1 ~ni i+ 1iFigure 6.1: Determination of normal vector at vertex on a solid wall surface.to tangency along solid surfaces, by enforcing no ow through faces which lie on asolid surface. This is done by modifying the uctuation in cells with a face on a wall.Essentially the total uctuation can be interpreted as a sum of uxes normal to thethree faces of the triangular cell, see Figure 6.2, such that� = F 1n + F 2n + F 3n : (6.38)These normal uxes are calculated under the same assumption as interior cells, i.e.,linear variation of the Roe parameter vector. To enforce the boundary condition,the normal ux through the edge on the wall, F 3n , is found by knowing that thereis no ow through the edge and therefore the only ux through the face is dueto the pressure force at the wall. This alters the uctuation in the cell, whichis in turn distributed to the vertices via the desired scheme, i.e., Wave Model orHyperbolic/Elliptic Decomposition, in the same manner as for all interior cells.In numerical experiments the weak condition appeared to su�er from a spuriousmode where velocity vectors would not approach tangency, but rather an oscillatorymode where the velocity averaged over each face had zero normal component. Mostlikely, a more sophisticated approach, which not only modi�es the cell uctuationbut also the distribution scheme to account for the solid wall, is needed. In the

1001 2

3F 3nF 2n F 1n�

Figure 6.2: Fluctuation in triangle due to normal uxes through faces.test cases displayed in this thesis, the strong condition was utilized because of itsrobustness, and e�ect on convergence and accuracy. It is noted that other solidboundary conditions have been developed, such as a vanishing-ghost-cell approach[18]. However, in a comparison with the strong condition this approach proved toprovide equivalent solutions with a slight slow-down in convergence rate.6.4.2 In ow/Out owAt in ow and out ow boundaries, characteristic conditions are applied dependingon whether the local ow conditions are supersonic or subsonic. The numericalboundary conditions are determined by those physical quantities given outside ofthe computational domain, which must be speci�ed at the boundary. The fourindependent scenarios are discussed.� Supersonic In ow

101All four characteristics run into the domain from the exterior and therefore allvariables are speci�ed.� Supersonic Out owNo incoming characteristics from the exterior domain and therefore no bound-ary conditions are speci�ed. The interior scheme alone determines the exitconditions.� Subsonic In owIn this case, the two hyperbolic characteristics are incoming and therefore theentropy and enthalpy at the inlet must be speci�ed. The elliptic or acousticportion of the ow represents one incoming and one outgoing characteristic, asdiscussed for the model Cauchy-Riemann system in Chapter V. As in the modelsystem, assuming a one-dimensional analysis along the incoming streamline, wesee that the direction of the ow, �, is carried in from the exterior and thereforemust be speci�ed at the in ow boundary, while the pressure is determined fromthe interior scheme.For internal ow problems, speci�cally the channel ow test case analyzed in theResults (Chapter VII), specifying the ow direction to be constant across theinlet cut does not physically model true conditions. Although this is standardpractice it violates the true physics if it is assumed that a straight channel ofin�nite length is upstream of the inlet cut.For the NACA 0012 cases also shown in the Results, the ow direction givenby the boundary condition is found from a vortex correction applied at the far�eld.

102� Subsonic Out owFor subsonic out ow conditions the two hyperbolic characteristics come fromthe interior domain and therefore the interior scheme will determine the exitconditions for enthalpy and entropy. The acoustic or elliptic system will againhave one incoming and one outgoing characteristic. However, now the exitpressure is determined from the incoming wave and therefore must be speci�edby a boundary condition and the direction of ow through the exit plane isdetermined by the interior scheme.

CHAPTER VIIRESULTSThe uctuation-splitting schemes based on the various methods of reducing theEuler equations presented in Chapter III are now tested in a variety of ow regimes.In all cases, the hyperbolic advective operators for both the simple waves utilizedin the wave models, as well as the hyperbolic characteristic equations found in theHyperbolic/Elliptic Splitting Method, are discretized using the N Scheme coupledwith the minmod limiter. This produces nearly second-order, monotone, solutionsas shown for scalar advection operators in Chapter IV. The elliptic-based subsys-tem contained within the H/E Splitting is solved using the Lax-Wendro� scheme,as discussed in Chapter V, utilizing a cell CFL of 1:2. All cases are initialized withfreestream conditions; in the case of the H/E Splitting for ow �elds containing astagnation region, the ow �eld is initially updated using a non-conservative formu-lation of the method, in view of the stability issues discussed in Chapter VI.7.1 Comparative ResultsTwo test cases are analyzed in order to compare the three main methods ofreducing the Euler system of equations to recognizable subproblems: Roe's WaveModel D, the Steady/Unsteady Splitting and the Hyperbolic/Elliptic Decomposition103

104method. All cases analyzed were run on unstructured triangular grids generated witha frontal Delaunay method developed by J.-D. M�uller [15].7.1.1 Channel FlowThe �rst comparison of methods is performed on subsonic ow through a channelwith 10% cosine shaped bumps on both the upper and lower walls, the grid consistingof 495 total nodes is shown in Figure 7.1. The in ow Mach number is given asM1 = 0:5; where the theoretical result is a subsonic, isentropic, symmetric solutionabout the bump. The boundary conditions are: the upper and lower surfaces aresolid walls, and characteristic conditions are applied at the subsonic inlet and outlet.0:00 1:33 2:67 4:000:000:330:671:00

Figure 7.1: 10% cosine bump channel consisting of 40 points on the upper and lowersurfaces with a total of 495 nodes.The pressure contours for the three splitting methods are shown in Figure 7.2,with the corresponding entropy contours shown in Figure 7.3. Clearly the six-wavemodel produces the most dissipative result. The asymmetry in pressure about thebump is the most pronounced and corresponds to the most entropy production.The Steady/Unsteady Splitting is less dissipative as seen in the pressure �eld andthe decrease in entropy production along the walls, however, the pro�les are veryrough due to the problem with this method that wave directions depend upon small,oscillating residuals. The best result, seen by the near symmetry in the pressure �eld

105as well as the least entropy production is for the Hyperbolic/Elliptic DecompositionMethod. This decomposition method produces a very smooth solution �eld evenon this coarse grid, as opposed to the rough solutions produced by the other twomethods. It is also the only method to converge completely, as seen in the residualhistory plot shown in Figure 7.4. The six-wave model converges about one and a halforders before entering some type of limit cycle, although at this point the solution�eld has stabilized. The Steady/Unsteady Splitting has the most trouble in obtaininga solution, as the residual barely drops one order in magnitude. It does not appearto enter into a limit cycle; the solution �eld continues to change, and no consistentpattern is ever observed.7.1.2 Subcritical Flow Over NACA0012The standard subcritical test case of ow over a NACA 0012 given by a freestreamMach number, M1 = 0:63, at a 2� angle of attack is performed to again comparethe three methods from the previous test case. The grid consists of 131 nodes on thebody, with a far �eld located at 30 chords, giving a total of 2133 nodes. The gridis shown in Figure 7.5. This is a fairly coarse grid, relative to the standard gridsused in workshops which usually contain at least twice as many points on the body.The boundary conditions imposed are a solid wall tangency condition on the surfaceof the airfoil; with the freestream values plus a vortex correction imposed at the far�eld boundary.The convergence history of the calculated lift coe�cient is shown in Figure 7.6.Again, the problem with the Steady/Unsteady Splitting is clearly seen, as the liftcoe�cient does not converge to a given value, but enters some random oscillatorymode. For the six-wave model, the lift coe�cient converges to a constant value,

1060:00 1:33 2:67 4:000:000:330:671:00 pmin 0.6337pmax 0.73640:00 1:33 2:67 4:000:000:330:671:00 pmin 0.558pmax 0.7370:00 1:33 2:67 4:000:000:330:671:00 pmin 0.552pmax 0.731

Figure 7.2: Pressure contours (41 contours, Pmin = :55, �P = :005) in cosine bumpchannel, M1 = 0:5, (a) Roe Six Wave Model D (b) Steady/UnsteadySplitting and (c) Hyperbolic/Elliptic Decomposition.

1070:00 1:33 2:67 4:000:000:330:671:00 smin 0.0smax 0.008510:00 1:33 2:67 4:000:000:330:671:00 smin -0.00784smax 0.004600:00 1:33 2:67 4:000:000:330:671:00 smin -0.00021smax 0.00298

Figure 7.3: Entropy contours (51 contours, smin = :01, �s = :0004) in cosine bumpchannel, M1 = 0:5, (a) Roe Six Wave Model D (b) Steady UnsteadySplitting and (c) Hyperbolic Elliptic Decomposition.

1080: 400: 800: 1200:�6:0�4:0�2:00:0

iterationsjj (�u)t jj2 6 waveSteady/UnsteadyHyperbolic/Elliptic

Figure 7.4: Residual History of L2 norm of the u momentum for the cosine bumptest cases.�0:70 0:10 0:90 1:70�1:20�0:400:401:20

x/cy/c

Figure 7.5: Detail of NACA 0012 grid.

1090: 10000: 20000: 30000:0:0000:1400:2800:420

iterationsCl 6waveSteady/UnsteadyHyperbolic/Elliptic

Figure 7.6: Convergence history of lift coe�cient for subcritical ow over NACA0012,M1 = 0:63, � = 2�.albeit much too low. Unlike the channel ow, where the residual was shown to entera limit cycle, in this case the residual still hangs up after only dropping three ordersbut enters an oscillatory pattern with smaller amplitude allowing the solution to con-verge to nearly steady conditions. In Table 7.1 the results of the three decompositionmethods are given, including the lift and drag coe�cients as well as the maximumspurious entropy production, occurring at the leading edge of the airfoil for all cases.Also included in the table are the accepted values from the AGARD [22] test data.Again, the unnecessary dissipation of the six-wave model is seen by the low lift co-e�cient of only approximately half of the accepted value, as well as the high dragcoe�cient 50 times higher than that produced by the Hyperbolic/Elliptic Decompo-sition solution. Also, the maximum entropy production is nearly 40 times that of theH/E Decomposition, as shown in the plot of the entropy production along the body

110Model CL CD �smax6 wave D :156 :019 :031Steady/Unsteady :339� :003 �:0003 � :0005 :0021Hyperbolic/Elliptic :334 :00036 :0008Hyperbolic/Elliptic (261 body nodes) :334 :0000 :0003Accepted Values [22] :335 0:0 0:0Table 7.1: Lift and drag coe�cients plus maximum entropy production around lead-ing edge for the subcritical NACA 0012 test case.for the three methods shown in Figure 7.7. The Steady/Unsteady Splitting, whileshowing a marked improvement over the six-wave model, still produces nearly threetimes the entropy of the H/E Decomposition, and of course fails to reach a convergedsolution. Clearly, the H/E Decomposition produces the best results, showing goodagreement with the accepted values. Also included in Table 7.1 are values computedfor the H/E Decomposition on a grid with twice the resolution, i.e. 261 body nodeswith a total of 11; 572. The solution is nearly the same as for the coarse grid, givingthe same lift coe�cient with a decrease in drag (now zero to four decimal places).Also, a decrease in maximum entropy production is seen.The Mach contours for the solution obtained with the H/E Splitting on the coarsegrid (131 body nodes) are given in Figure 7.8, including a blow-up of the stagnationregion in Figure 7.9. Both plots show the smooth solution obtained with this method,and the ability to maintain the quality of the solution even in the stagnation region,where the ow is changing rapidly.7.2 Hyperbolic/Elliptic SplittingBecause of the superiority of the Hyperbolic/Elliptic Decomposition Method overthe other methods discussed in this work, various other ows over a NACA 0012 were

1110:00 0:33 0:67 1:00�0:00100:01000:02100:0320

x/centropy 6waveSteady/UnsteadyHyperbolic/Elliptic

Figure 7.7: Spurious entropy production along body for subcritical ow overNACA0012, M1 = 0:63, � = 2�.�0:70 0:10 0:90 1:70�1:20�0:400:401:20 Fmin 0.005Fmax 0.984

Figure 7.8: Mach Contours for ow over NACA 0012 at M1 = 0:63, � = 2�.

112�0:060 �0:020 0:020 0:060�0:060�0:0200:0200:060 Fmin 0.005Fmax 0.984

Figure 7.9: NACA 0012 at M1 = 0:63, � = 2� showing ow solution in stagnationregion.computed with the H/E Decomposition in order to demonstrate the versatility androbustness of the method. Several solutions, including subcritical ow, low speed ow, transonic and supersonic cases, are presented on the same grid shown in theprevious section, Figure 7.5.7.2.1 Low-Mach-Number Flow Over NACA 0012In order to demonstrate the accuracy of the scheme even in the incompressiblelimit, two low-speed NACA 0012 airfoil cases at zero angle of attack are analyzed.Mach contour plots for a freestream Mach number of .1 and .01 are shown in Fig-ures 7.10 and 7.11 respectively. The solutions appear self-similar, by a scaling factorgiven by the freestream Mach number, as predicted by incompressible theory. Acomparison of the pressure coe�cient along the body for both cases with that givenby a panel method code [12] is shown in Figure 7.13. Clearly the solutions produced

113with the compressible code are nearly equivalent to the solution in the incompress-ible limit. The numerical dissipation is very low giving coe�cients of drag for bothcases at 0.0005, with maximum entropy generation on the order of 10�5. The con-vergence histories (Figure 7.12), as well as the solutions, are identical, con�rmingthe theoretical analysis of this form of preconditioning.�0:70 0:10 0:90 1:70�1:20�0:400:401:20 Fmin 0.0001Fmax 0.1188

Figure 7.10: Low-speed ow over NACA 0012 at zero angle of attack, M1 = 0:1;Mach contours.For comparison, a solution of the M1 = 0:1 ow over the airfoil computed witha Lax-Wendro� method used to solve the full Euler system is presented: a Machcontour plot is shown in Figure 7.14. This computation is performed without theuse of preconditioning matrices, which are known to preserve the accuracy of thenumerical scheme even in low speed ows [40]. It can be seen from the contour plotthat the solution computed with the full Lax-Wendro� method is degraded for thisnearly incompressible ow �eld, with excessive numerical dissipation generated about

114�0:70 0:10 0:90 1:70�1:20�0:400:401:20 Fmin 0.00000Fmax 0.01187

Figure 7.11: Low-speed ow over NACA 0012 at zero angle of attack, M1 = 0:01;Mach contours.0: 5000: 10000: 15000:�8:0�6:0�4:0�2:00:0

iterationsk�tk2 M=0.01M=0.1

Figure 7.12: Normalized Residual History for M1 = 0:1 and M1 = 0:01 cases.

115:00 :33 :67 1:00:90:30

�:30�:90x=c

Cp H/E M = :01H/E M = :1panelFigure 7.13: Cp along airfoil for M1 = 0:1, M1 = 0:01 and the result of a panelmethod.the body. Also shown is a plot of the spurious entropy production along the airfoil,Figure 7.15, found in the computations from both the H/E Decomposition and thefull Lax-Wendro� method. The Lax-Wendro� solution results in a drag coe�cientof 0:069 compared with 0:0005 for the H/E splitting, as well as entropy productionat the leading edge of nearly 35 times the magnitude of that produced by the H/Emethod.7.2.2 Subsonic Flow CaseThe spurious entropy generation along a NACA 0012 for a subcritical case witha freestream Mach number of M1 = 0:5 at no angle of attack is shown in Fig-ure 7.16, for both the H/E Decomposition and for a standard Lax-Wendro� solver.Again, the entropy near the leading edge of the airfoil is greater for the standardLax-Wendro� scheme, by approximately a factor of 20. The decrease in entropy

116�0:70 0:10 0:90 1:70�1:20�0:400:401:20 Fmin 0.0021Fmax 0.1381

Figure 7.14: Low-Speed-Flow over NACA 0012 at zero angle of attack, M1 = 0:1computed with a Lax-Wendro� method for the full Euler system: Machcontours.0:00 0:40 0:80 1:20�0:000100:000300:000700:00110

x/centropy HELW

Figure 7.15: Entropy production along body for M1 = 0:1 case showing H/E De-composition compared with basic Lax-Wendro� solver.

117produced by separating the system and only utilizing the Lax-Wendro� scheme forthe elliptic-based subproblem, is not as pronounced as in the low-speed airfoil casediscussed previously, although still signi�cant. The better results produced by theH/E splitting method for the low-speed case are most likely due to the increase inaccuracy in these ow regimes from the use of preconditioning matrices, as foundin their original development for convergence acceleration. This test case, however,demonstrates that the increase in accuracy found in the low-speed case was not dueto preconditioning alone, but to the splitting of the system as well. If it had been,we would expect to �nd little gain in the present test.0:00000 0:00100 0:00200 0:00300�0:00200:00200:00600:0100

x/centropy HELW

Figure 7.16: Entropy production along NACA 0012 for subsonic ow case withM1 =0:5 at 0�.7.2.3 Transonic Flow Over NACA 0012A standard transonic case ofM1 = 0:85 at a 1� angle of attack was also computed,Mach contours are shown in Figure 7.17, and, pressure coe�cient along the body is

118shown in Figure 7.18. The lift and drag coe�cients computed as 0:384 and 0:059,are plotted versus solutions found in a GAMM workshop [7], shown in Figure 7.19.The dotted circle represents what were considered good solutions from the workshop.Considering the coarseness of our grid, with approximately half the number of bodynodes, our solution falls very near the spectrum of accepted good solutions. In thesolution plots, no noticeable oscillations occur on the subsonic sides of the shocks,even though monotonicity is not guaranteed for the Lax-Wendro� method at thechosen Courant numbers.�0:70 0:10 0:90 1:70�1:20�0:400:401:20 Fmin 0.019Fmax 1.448

Figure 7.17: Transonic ow over NACA 0012, M1 = 0:85, � = 1�; Mach contours.7.2.4 Supersonic NACA 0012A standard supersonic case of M1 = 1:2 at a 0� angle of attack was also com-puted. Mach contours are shown in Figure 7.20 and the Mach number upstream ofshock and along the body are shown in Figure 7.21. Again, the solution displays the

1190:00 0:33 0:67 1:001:200:40�0:40�1:20

x/cCp

Figure 7.18: Pressure coe�cient along body for ow over transonic NACA 0012,M1 = 0:85, � = 1�.0:0500 0:0520 0:0540 0:0560 0:0580 0:06000:3000:3200:3400:3600:3800:400

CdCl H/E SplitGAMM WkshpFigure 7.19: Cl vs. Cd for solutions from GAMM workshop for transonic NACA0012, M1 = 0:85, � = 1�. The dashed circle indicates the region ofacceptably accurate solutions.

120ability of the method to capture discontinuities sharply, considering the coarsenessof the grid. The stagnation region is clearly de�ned, with an expansion occurringalong the surface of the airfoil. The sonic expansion is smoothly produced with thenumerical scheme, demonstrating the smoothness of the merging of the subsonic andsupersonic discretizations of the acoustical system.�1:00 0:00 1:00 2:00�1:50�0:500:501:50 Fmin 0.000Fmax 1.523

Figure 7.20: Supersonic NACA 0012, M1 = 1:2, � = 0� Mach contours.7.2.5 Low Speed Flow over NACA 0012 at High Angle of AttackLow speed ow over a NACA 0012 with M1 = 0:05 at a 25� angle of attack iscomputed to demonstrate the method for a case in which an isolated compressibleregion (Mmax = 0:29) is contained in a nearly incompressible ow �eld. The pres-sure contours (Figure 7.22) display the isolated compressible region in the expansionaround the leading edge. The computation gives a Cd = 0:038 and Cl = 2:8 (forcomparison 2�� = 2:74; and a panel method code produced Cl = 2:87:) The Mach

121�1:00 �0:20 0:60 1:000:000:601:201:80

x/cMach BodyCut at Y=0

Figure 7.21: Mach number upstream and along body for supersonic ow over NACA0012, M1 = 1:2, � = 0�.number and entropy generation along the body (Figures 7.23 and 7.24) show theability of the method to capture the expansion with low numerical error consideringthe coarseness of the grid, giving a maximum entropy growth on the order of 10�4.Also shown is a comparison of the pressure coe�cient along the airfoil for the H/Esplitting with a panel method solution [12] in Figure 7.25. This shows the ability ofthe compressible code to accurately predict a mostly incompressible ow.

122

Figure 7.22: NACA 0012, M1 = 0:05, � = 25�; pressure contours and some stream-lines.

0:00 0:33 0:67 1:000:0000:1000:2000:300x/c

MachFigure 7.23: NACA 0012, M1 = 0:05, � = 25� Mach number along body.

1230:00 0:33 0:67 1:00�0:000100:000130:000370:00060

x/centropy

Figure 7.24: NACA 0012, M1 = 0:05, � = 25�; entropy generation on body.0:00 0:33 0:67 1:002:0�10:0�22:0�34:0

x=cCp H/E splitpanel method

Figure 7.25: NACA 0012, M1 = 0:05, � = 25�; Cp along body for H/E Splittingcompared with a panel method solution.

CHAPTER VIIICONCLUSIONSWe have compared three di�erent approaches in the development of a truly multi-dimensional algorithm for the Euler equations. The wave-modeling ideas provided areduction of the system to a set of planar waves governed by scalar advection oper-ators. This ideology dominated the research e�ort for many years, mainly becauseit was a direct extension of the successful splitting method in one dimension. Whilenumerous models were developed and tested, none met with universal success. Con-trary to this philosophy, a radically new approach in the Hyperbolic/Elliptic splittingwas recently advocated, following original ideas of Ta'asan [35]. We presented a de-composition of the linearized steady system into hyperbolic and elliptic operators.With the use of preconditioning matrices, this decoupling of the steady system waspreserved even in the iterative scheme. In supersonic regimes, the decompositionresulted in a set of four decoupled purely hyperbolic or advective operators, repre-senting the well-known characteristics of the steady equations. This provides fourclearly de�ned directions along which upwind-biased discretizations can be made,without the need for choosing sets of waves as in the wave model schemes. In sub-sonic ows, two advective operators remain, plus an elliptic-based coupled system,which is isolated from the hyperbolic operators in the linear form. Again, the hyper-124

125bolic operators provide distinct directions to bias the discretizations along. However,the elliptic-based system represents isotropic spreading of information and thereforewe propose central discretizations. Most importantly, the need to choose sets ofwaves to model the local ow, of the in�nitely many that exist, has been eliminated.The decoupled system presented in this work provides an alternative to the formproposed by Ta'asan [35]. He proposes a canonical form with direct coupling betweenthe elliptic and hyperbolic operators even in the linearized system. Our system, whenapplied to non-linear problems, will only have coupling due to the solution-dependentcoe�cients. It is unclear yet whether this indirect coupling, as opposed to the directcoupling of Ta'asan, poses any advantage regarding accuracy or convergence rates. Itseems that in supersonic regimes our complete decoupling should provide the optimalscheme, as it essentially reduces to pure characteristic theory applied in a conservativeway. It is this physical implication of the complete decoupling in supersonic owsthat leads us to continue to advocate our formulation, with the natural extensioninto the subsonic regime preserving the decoupling.In order to solve the system based on the decomposed forms, we �rst analyzed theresulting reduced operators, both hyperbolic and elliptic. For the advective opera-tors, scalar analysis shown in Chapter IV provided a nearly second-order, monotone,compact spatial discretization based on the N Scheme limited using minmod. This isthe chosen method of upwinding for the Euler results shown in this work. While testcases for weaker or even compressive limiters such as harmonic or superbee showed aslight increase in spatial accuracy, their main advantage is in their ability to minimizethe spreading of shear layers. Unfortunately, a price must be paid for the improveddiscontinuity capturing in that local positivity constraints can be met, only on su�-ciently regular grids. Therefore, if a compressive limiter is desired for the capturing of

126contact surfaces, the grid regularity conditions must be met within the vicinity of thediscontinuity. It is not yet clear whether the added bene�t of sharper discontinuitieswill outweigh this added cost.While the discretization of the hyperbolic operators seems clear the optimalscheme for the elliptic-based system has not yet been found. For this system, wherethere is no preferential direction of propagation, we desire a central discretization,such that errors are e�ectively damped in all directions. By adding hyperbolic timeterms to the spatially elliptic system, as is done in this work, producing the Cauchy-Riemann equations augmented with time terms shown in Chapter V, this coupledsystem is easily solved using the same basic framework as for the advective opera-tors. Because of the exibility and compactness of utilizing a Lax-Wendro� methodfor time marching, it is employed for the Euler results shown in this work. Theanalysis of the model system showed this does provide a second-order scheme, how-ever, contrary to one-dimensional analysis, in two dimensions there is no choice of�nite coe�cients that guarantees oscillation-free solutions, and therefore non-linearlimiting will eventually need to be incorporated if this method is to be utilized forproblems with stronger discontinuities.Besides accuracy, another area of importance concerning the elliptic-based systemis the rate at which solutions converge. The scheme allows for the expulsion of errorsvia damping and advection through boundaries. The two parameters, nodal and cellCourant numbers, can be adjusted to favor one over the other. Although the analysisof the model Cauchy-Riemann system provides no de�nitive answers in choosing thevariable coe�cients, with current problems of re ections at boundaries it seems clearthat a compromise between damping and advection is needed.Following the success of the chosen discretizations for the model problems repre-

127senting the reduced systems of the various Euler decompositions, these were utilizedin the full Euler code to evaluate the accuracy of the decompositions. The originalwave-modeling ideas, consisting of six waves representing all local ow disturbances,in space as well as time, had too much dissipation, producing results similar to �rst-order dimension-split solvers. This loss of accuracy, attributed to the presence ofunsteady waves even in the absence of time residuals, was alleviated by allowingsteady patterns to account for part of the spatial gradients present within a lo-cal simplex. The Steady/Unsteady splitting did greatly improve the accuracy of themethod, providing solutions with very low entropy generation along bodies. However,probably due to the erratic dependence of wave orientations on small cell residuals,convergence proved impossible to achieve in all test cases analyzed. As shown in theresults, for ows over a NACA 0012 airfoil, even the pressure coe�cient failed tostabilize.While little success was realized with the wave modeling approach, the Hy-perbolic/Elliptic decoupling produced accurate solutions with numerical dissipationlower than that obtained with the wave models. Clearly the elimination of advectionoperators to govern acoustic wave propagation in subsonic regimes where the steadysystem is elliptic, is the advantage over previous multi-dimensional models.In Chapter VII, results from the Hyperbolic/Elliptic decomposition, for a range of ow conditions about a NACA 0012 airfoil, were shown. For the standard AGARDsubcritical case, the computed solution provided lift and drag coe�cients in goodagreement with the accepted values from a potential code. Also shown are theresults of a transonic ow case from a GAMM Workshop [7]. Although our solutionwas computed on a grid with half, and in some case one fourth, the re�nement ofthose contained in the workshop, results close to the range of accepted values were

128achieved.In order to demonstrate the ability of the compressible code to remain accurateeven in low-speed ow cases, several computations were presented. Each providedsolutions with very low spurious entropy production and resulted in small drag coef-�cients. These solutions compared very well with those given by an incompressiblepanel method, showing that the correct incompressible limit is found. Also, oneof the low-speed solutions of the preconditioned Hyperbolic/Elliptic decompositionwas compared with that produced by a standard Lax-Wendro� scheme used for thefull Euler equations. For this case, with M1 = 0:1, the excessive dissipation thatmanifests itself within standard compressible codes in low speed cases, is seen in theLax-Wendro� solution. The decomposition method produced a drag coe�cient lessthan one percent of that given by the Lax-Wendro� solution.While some of the improvement in accuracy of the decomposition method overa Lax-Wendro� solver is due to preconditioning, some is also due to the splittingof the equations into hyperbolic and elliptic-based components. To display this,a compressible ow over a NACA 0012 airfoil was calculated with both methods.Again the decomposition method resulted in a more accurate solution, where thefull Lax-Wendro� method produced twenty times more spurious entropy along theairfoil.While the accuracy of our method for subcritical ows appears to compare wellwith results shown by Ta'asan, the convergence rates utilizing a multi-grid methodimplemented by M�uller [16] are well below the multi-grid performance achieved byTa'asan [36]. In our decomposition, complete decoupling of the linear system, evenin the iterative scheme, would allow the individual operators, both elliptic and hy-perbolic, to converge independently, providing convergence rates predicted in the

129analysis of the model systems. However, for conservative Euler calculations couplingdoes occur via the coe�cients, and studies by M�uller [16] show that convergence isa�ected by this, reducing performance. This subject still requires further study. It isnoted that completely di�erent discretization methods are utilized by Ta'asan versusour method. Before a clear judgment can be made between the di�erent canoni-cal forms, each must be implemented utilizing the same discretization and iterativetechniques.To summarize, although there remain issues of major exploration within themethod presented in this work, preliminary results show great promise, encouraginga belief that this path of breaking the system of equations into canonical forms is thekey to multi-dimensional ow simulation. Accurate solutions were presented for avariety of ow situations, including nearly incompressible ow �elds, displaying theversatility of the method. This is a desirable quality especially when considering verylow-speed airfoil problems at high angles of attack. The issue of the loss of robustnessof the conservative implementation, most likely a consequence of the degeneracydescribed by Darmofal and Schmid [2], needs to be addressed in continuing work.8.1 Future WorkIt seems that the main problem in the existing method is in solving the ellipticsub-system found in the steady equations. The current method of converting thisacoustic system into a hyperbolic \pseudo" time-dependent system and employinga Lax-Wendro� method, does not appear to be the optimal choice. We discuss analternate path in the discretization and solving of this system that could be followedin future work. Also, the extension of the Hyperbolic/Elliptic decomposition to threedimensions is discussed.

1308.1.1 Discretization of the elliptic systemWhile great strides were taken to isolate the subsonic acoustical system from thehyperbolic characteristics, the iteration on the elliptic-based system, in this thesis,essentially resorts to what we know best: a hyperbolic time system. While we ad-vocate the use of non-biased discretizations, we utilize a Lax-Wendro� method forconverging the system creating symmetric stencils but with biased weights. Physi-cally the elliptic system represents the isotropic distribution of information and thusshould require symmetric weights providing for symmetric solutions. Also, with thedi�culty in developing non-re ective boundary conditions for real ow problems, thedamping properties for this system are most likely the key in optimizing convergence.The damping rates for the Lax-Wendro� method, even when incorporating multi-grid techniques by optimizing over high frequencies, predict convergence rates worsethan those achieved for the Cauchy-Riemann system by Brandt [1]. This suggestsan alternate approach to the elliptic-based system is necessary to achieve accept-able convergence rates as well as more accurate (or symmetric) solutions. Somepreliminary ideas on where the e�ort should focus are discussed here.We return to the model elliptic system given by the Cauchy-Riemann equations:2664 @x @y@y �@x 37758>><>>: uv 9>>=>>; = LU = 0: (8.1)In reapplying the operator L, L (LU) = 0; (8.2)we �nd that this 2 � 2 elliptic system decouples into2664 @xx + @yy 00 @xx + @yy 37758>><>>: uv 9>>=>>; = 0: (8.3)

131If we augment this system with time terms,Ut +r2U = 0; (8.4)we now have a parabolic system in time, as opposed to the original hyperbolic timesystem. This follows work of Brandt [1] in which optimal convergence is achieved forsolving Cauchy-Riemann in a parabolic \time-marching" sense.These equations can be discretized by a standard Galerkin Finite-ElementMethod,providing the desired symmetric stencils, as expected for an elliptic system. Eithera Jacobi or Gauss-Seidel relaxation, can be used to solve the system, and when cou-pled with a standard multi-grid scheme should produce the fastest solutions. Becausethe Gauss-Seidel method violates the local compactness of the algorithm, a detailedanalysis into the overall cost versus a compact Jacobi iteration will need to be per-formed. In this analysis, both the added work per iteration and the loss of parallelenhancement for the Gauss-Seidel relaxation, needs to be taken into account.Unlike utilizing a hyperbolic time marching method, where one numerical bound-ary condition is needed per boundary, for the parabolic system both equations needboundary conditions on each boundary. However, the information needed at eachboundary does not change, in that one condition is given by a Dirichlet condition forthe known variable and the other is a Neumann condition coming from the �rst-orderequations on the boundary. For example, if u is given on a boundary aligned withthe x axis, the corresponding Neumann condition for v, in this case the derivative ofv in the x direction, is found from the �rst order system:� vx + uy = 0 ) vx = uy: (8.5)So the physical coupling of the variables, u and v, remains, but is only enforced inthe numerical scheme by the boundary conditions.

132In the Euler equations, it is the pressure �eld and the local ow angle, �, whichare contained in the elliptic system. By converting the system to its second-order,or parabolic, form, the natural boundary conditions following those expressed forthe model system are surprisingly similar to familiar conditions utilized in existingnumerical codes. For a solid wall, the tangency condition places a Dirichlet condi-tion on � as the ow direction is de�ned. From the �rst-order system we obtain aNeumann condition for pressure given by:@p@n = ��q2@�@s = �q2R (8.6)where R is the radius of curvature of the body.Unfortunately the implementation of these ideas into a full Euler code is not triv-ial. The simultaneous solving of this parabolic system with the hyperbolic marchingschemes used for the advective operators is not clear, especially how to do so in a con-servative way. Essentially, the preconditioning matrix which preserves these desiredforms in the unsteady equations will now contain di�erential operators, convert-ing the �rst-order acoustic system into second-order parabolic equations, providinga complete decoupling of all characteristic variables (in the linearized sense). Ofcourse, nonlinear coupling between systems occurs through the coe�cients, as wellas coupling at boundaries between the parabolic operators. It is highly probablethat this coupling will reduce the convergence rates predicted by the linear modelproblems. However, it may produce a much more e�cient method, and slightly moreaccurate solutions due to the symmetry in the weights for the acoustic system, thanthat presented in this work.The other area of concern in the existing method is the robustness of the algorithmin stagnation regions. Due to the degeneracy of the acoustic eigenvectors of the

133preconditioned system in this limit, unbounded spurious transient growth can occur.However, in the proposed system, where the acoustic �eld is now converged viaparabolic operators, this unbounded transient growth should no longer be an issue.8.1.2 Extension to three dimensionsThe extension of the Hyperbolic/Elliptic decomposition to three dimensions isnot straightforward. First of all, complete decoupling of the linearized system, as inthe two-dimensional system, is not possible. We �nd that in the three-dimensionalsystem, the two streamline characteristic equations for entropy and enthalpy stilldecouple, and for these two decoupled advective operators, the upwind schemes pre-sented for two-dimensional scalar advection in this work have a natural extensionto three dimensions, as shown by Deconinck, Struijs et. al. [6]. However, the otherthree equations are not further reducible when considering �rst-order operators.Unfortunately, the analysis of the three remaining coupled equations is not easilyextendible from the two-dimensional methods. This system of equations, in stream-line coordinates, is given by:�2ps + �q2 (�n1 + �n2) = 0�q2�s + pn1 = 0�q2�s + pn2 = 0;where s is de�ned as the local streamline direction, with n1 and n2 perpendicularto s, forming a right-handed Cartesian coordinate system. We denote the local ow angles by � and �, both necessary in de�ning the direction of ow in threedimensions. Again � = pM2 � 1 provides the distinction between supersonic andsubsonic regimes.In supersonic ows, of the three real characteristics of the coupled system, one is

134de�ned in the ow direction. The other two de�ne the Mach cone generated aboutthe streamline. An ordinary di�erential equation along the streamline can only bewritten if di�erential operators are used. This results in the advection of helicity(the component of vorticity in the direction of ow) , where = ~q � ~! = q @�@n1 � @�@n2!along the streamline. However, the other equations representing the Mach coneremain coupled. The resulting advection of helicity and the system representing theMach cone are not direct extensions of model systems found in the two-dimensionalanalysis. Therefore, the proper discretizations are not straightforward.In subsonic regimes, the system of three equations now contains one real eigen-value plus two complex ones. If we allow the use of �rst-order di�erential operators,the system can be written as second-order, or Laplacian, operators in the followingform: ��2pss + pn1n1 + pn2n2 = 0q ���2�ss + �n1n1 + �n2n2� = � n2q ���2�ss + �n1n1 + �n2n2� = n1:Unlike in two dimensions, two of the resulting stream-wise biased Laplacian equationshave source terms which are given by gradients in helicity. As in the supersonic case,this is not a direct extension of what was recovered in the two-dimensional analysis.Again, research into the development of proper discretization techniques for thissystem must be performed.Although reductions of the full system are thus attainable in three dimensions,the discretizations of the resulting sub-systems are not known. In this thesis, it was

135shown that the jump from a successful model based on physics in one dimension, totwo dimensions, was not trivial. It is the belief of the author that the extension ofthe successful two-dimensional ideas presented in this work, to three dimensions, isjust as hard. However, the improvement over existing methods should prove to beeven greater in the three dimensional case, and be well worth the e�ort.

APPENDICES

136

137APPENDIX AInconsistency due to LinearizationThe spatial gradients of the primitive state, determined from the parameter vectorZ = 8>>>>>>>>>>><>>>>>>>>>>>: p�p�up�vp�H 9>>>>>>>>>>>=>>>>>>>>>>>; ; (A.1)are given by:d~rV = 8>>>>>>>>>>><>>>>>>>>>>>: d~r�d~rud~rvd~rp 9>>>>>>>>>>>=>>>>>>>>>>>; = 8>>>>>>>>>>><>>>>>>>>>>>: 2p��~rz1���u~rz1 + ~rz2� =p�����v~rz1 + ~rz3� =p�� �1 p�� � �H~rz1 � �u~rz2 � �v~rz3 + ~rz4� 9>>>>>>>>>>>=>>>>>>>>>>>; ; (A.2)where V = 8>>>>>>>>>>><>>>>>>>>>>>: ���u�v�p 9>>>>>>>>>>>=>>>>>>>>>>>; = 8>>>>>>>>>>><>>>>>>>>>>>: �z12�z2= �z1�z3= �z1 �1 � �z1 �z4 � 12( �z22 + �z32)� 9>>>>>>>>>>>=>>>>>>>>>>>; ; (A.3)and �H = �z4= �z1: (A.4)

138In the assumption of linear variation of the parameter vector, the above formula-tions provide gradients and average values for the primitive state. The inconsistencyin this linearization is seen when analyzing a cell which crosses a shear layer throughwhich density and pressure are constant. The following example demonstrates thisproblem.In Figure A.1 a shear layer crossing a triangle is shown. The shear layer is de�ned12

~q13h

hshear~q3 ~q2Figure A.1: Sample case of shear crossing a triangle.with velocities given by, ~q1 = �~q2 = �~q3 = (u; 0); (A.5)with � equal to some constant representing the strength of the shear. The densityand pressure are constant throughout and therefore,�1 = �2 = �3 = �� and p1 = p2 = p3 = p�: (A.6)

139The stagnation enthalpy, given by H = �1 p� + 12q2 is thusH1 = � 1 p��� + 12u2; H2 = H3 = � 1 p��� + 12�2u2: (A.7)This de�nes the parameter vector at each vertex, given by:Z1 = 8>>>>>>>>>>><>>>>>>>>>>>: p��p��u0p�� � �1 p��� + 12u2� 9>>>>>>>>>>>=>>>>>>>>>>>; ; Z2 = Z3 = 8>>>>>>>>>>><>>>>>>>>>>>: p��p���u0p�� � �1 p��� + 12�2u2� 9>>>>>>>>>>>=>>>>>>>>>>>; ; (A.8)and thus,Z = 8>>>>>>>>>>><>>>>>>>>>>>: p��(1+2�)3 p��u0p�� � �1 p��� + 1+2�26 u2� 9>>>>>>>>>>>=>>>>>>>>>>>; and bZy = 8>>>>>>>>>>><>>>>>>>>>>>: 01��h p��u01��22h p��u2 9>>>>>>>>>>>=>>>>>>>>>>>; : (A.9)The gradients in the x direction are zero. Substituting Equation A.9 into EquationsA.2 and A.3 gives the average primitive state and gradients for the cell as:V = 8>>>>>>>>>>><>>>>>>>>>>>: ��(1+2�)3 u0p� + �1 (1��)29 ��u2 9>>>>>>>>>>>=>>>>>>>>>>>; and cVy = 8>>>>>>>>>>><>>>>>>>>>>>: 01��h u0 �1 (1��)26h �u2 9>>>>>>>>>>>=>>>>>>>>>>>; : (A.10)Clearly a spurious pressure gradient exists which is proportional to (�u)2, where �uis the jump in the velocity across the shear. It is noted that the average pressure statefound in the cell is not the constant pressure given at the vertices, but is altered by afactor due to the linearization also proportional to (�u)2. Existence of the spuriouspressure gradient, not balanced by any gradient of v, implies that the decomposition

140will predict a pair of acoustic waves traveling vertically. These give an additionalnumerical dissipation that prevent the code from preserving this shear ow as anexact solution. It is likely that inconsistencies of this order are inevitable features ofany discontinuity-capturing technique.

BIBLIOGRAPHY

141

142BIBLIOGRAPHY[1] A. Brandt, \Multigrid techniques, 1984 guide," in 14th VKI Lecutre Series onComputational Fluid Dynamics, Lecture Series 1984-04, Von K�arm�an Institutefor Fluid Dynamics, 1984.[2] D. Darmofal and P. Schmid, \The importance of eigenvectors for local precon-ditioning of the Euler equations," AIAA Paper 95-1655, June 1995.[3] P. de Palma, \Investigation of Roe's 2D wave decomposition models for the Eulerequations," Tech. Rep. 1990-08, Von K�arm�an Institute for Fluid Dynamics, June1990. Supervisors: H. Deconinck, R. Struijs.[4] H. Deconinck, P. Roe, and R. Struijs, \A multidimensional generalization ofRoe's ux di�erence splitter for the Euler equations," Journal of Computersand Fluids, vol. 22, no. 2, pp. 215{222, 1993.[5] H. Deconinck, R. Struijs, and G. Bourgois, \High resolution shock capturingcell vertex advection schemes on unstructured grids," in Computational FluidDynamics, Von K�arm�an Institute for Fluid Dynamics,Lecture Series 1994-05,1994.[6] H. Deconinck, R. Struijs, G. Bourgois, H. Paill�ere, and P. Roe, \Multi-dimensional upwind methods for unstructured grids," in Unstructured GridMethods for Advection Dominated Flows, AGARD Report 787, 1992.[7] A. Dervieux, B. van Leer, J. Periaux, and A. Rizzi, \Numerical simulation ofcompressible Euler ows," Notes on Numerical Fluid Mechanics, vol. 26, 1989.Vieweg, Braunschweig.[8] D.W.Levy, K.G.Powell, and B.vanLeer, \An implementation of a grid-independent upwind scheme for the Euler equations," AIAA Paper 89-1931-CP,1989.[9] S. K. Godunov, \A di�erence scheme for numerical computation of discontinuoussolutions of hydrodynamic equations,"Math. Sbornik, vol. 47, pp. 271{306, 1959.[10] M. Hall, \Cell-vertex multigrid schemes for solution of the Euler equations,"in Numerical Methods for Fluid Dynamics II (K. Morton and M. Baines, eds.),(Clarendon, Oxford), pp. 303{365, 1986.

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145[38] G. T. Tomaich and P. L. Roe, \Compact schemes for advection-di�usion prob-lems on unstructured grids," in Twenty-Third Annual Numerical Modeling andSimulation Conference, vol. 5, 1992.[39] E. Turkel, \Preconditioned methods for solving the incompressible and low speedcompressible equations," Journal of Computational Physics, vol. 72, 1987.[40] E. Turkel, Fiterman, and B. van Leer, \Preconditioning and the limit to theincompressible ow equations," Tech. Rep. 93-42, ICASE, 1993.[41] B. van Leer, W. Lee, and P. Roe, \Characteristic time-stepping or local precon-ditioning of the Euler equations," AIAA Paper 91-1552, 1991.[42] B. van Leer, E. Turkel, C. H. Tai, and L. Mesaros, \Local preconditioning in astagnation point," AIAA Paper 95-1654, June 1995.[43] G. Volpe, \Performance of compressible ow codes at low mach numbers," AIAAJournal, vol. 31, pp. 49{56, 1993.

ABSTRACTMULTI-DIMENSIONAL FLUCTUATION SPLITTING SCHEMES FOR THEEULER EQUATIONS ON UNSTRUCTURED GRIDSbyLisa Marie MesarosChairperson: Philip L. RoeSchemes for approximating steady solutions to the two dimensional inviscid gas-dynamic equations on unstructured triangular grids are presented. This family ofcell-vertex schemes, known as uctuation-splitting or distribution methods, incorpo-rates solutions to multi-dimensional subproblems as building blocks, and is developedin hopes of providing more accurate solutions than standard methods, which solvemulti-dimensional problems utilizing one-dimensional physics.Three di�erent methods of decomposing the Euler equations into recognizablesubproblems are presented. The �rst decomposition utilizes simple planar waves tomodel the local ow, where each wave is governed by an advection operator. Thesecond decomposition utilizes fewer planar waves, while utilizing steady solutioncomponents. The last decomposition is based on the recognition of di�erent mech-anisms of information propagation present in the steady system of equations. It is

1shown that the steady Euler equations can be decoupled into hyperbolic and ellipticcomponents. In supersonic regimes, this yields four advective operators representingthe steady characteristics. In subsonic regimes, besides the two hyperbolic compo-nents, an elliptic system of two coupled equations representing the acoustical �eld isrecovered. Preconditioning matrices are then employed to preserve this decouplingeven in the transient system.Next, methods are presented for discretizing the resulting reduced operators, bothhyperbolic and elliptic, with analysis and results for model systems. For advectiveoperators, a nearly second-order, compact, monotone, upwind distribution schemefor scalar problems is presented. For the elliptic-based system, without a directionof propagation, central discretizations advanced with a Lax-Wendro� scheme areutilized. Following the success of the discretizations in solving the model problems,they were utilized for full Euler calculations, with results presented. While the wave-model schemes did not meet with universal success, results for the hyperbolic/ellipticdecoupling method display accurate solutions for a variety of ows including nearlyincompressible cases.