A RESIDUAL DISTRIBUTIONAPPROACH TO THE EULEREQUATIONS THAT PRESERVESPOTENTIAL FLOWbyMani Rad

A dissertation submitted in partial ful�llmentof the requirements for the degree ofDoctor of Philosophy(Aerospace Engineering and Scienti�c Computing)in The University of Michigan2001Doctoral Committee:Professor Philip L. Roe, ChairProfessor Nikolaos KatopodesProfessor Kenneth PowellProfessor Bram van Leer

This dissertation is dedicated to my parents and sister.

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ACKNOWLEDGEMENTSGraduate study has been a journey well worth taking, complete with its ownpitfalls, lessons and enjoyment. As such, I would like to acknowledge the people whocontributed to make the whole experience positive.My parents and sister provided the conviction that may have sometimes beenmissing, continuously pointing me in the right direction. Their wisdom helped mecope with many aspects of graduate life and I am grateful for that. Thanks to thepeople around me, never did I feel I was living the journey alone. I am particularlythinking of all my friends at FXB who were accomplices in the good times.I consider myself very fortunate for having Professor Phil Roe as advisor, heguided my work with a never-ending supply of ideas. In addition to providing tech-nical knowledge, he more importantly impressed on me his thought process. Hisother students will probably con�rm that the right answer to a certain question issometimes another question 1.I would like to thank Professor Bram van Leer and Professor Ken Powell fortheir insight and feedback in this research as well as the excellent courses in CFDand aerodynamics. My appreciation is also extended to Professor Nik Katopodes foragreeing to be a member of my doctoral committee and his continued interest in thesubject of my dissertation.1or in Professor Roe's case, half a dozen questions.iii

TABLE OF CONTENTSDEDICATION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iiACKNOWLEDGEMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : iiiLIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viiCHAPTERI. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Areas of Focus . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Low-Mach Number Flows . . . . . . . . . . . . . . . 21.1.2 Treatment of Multidimensional Problems . . . . . . 41.2 Our Manifesto . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 The Main Pillars . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 The Hyperbolic/Elliptic Decomposition . . . . . . . 71.3.2 The Fluctuation-Splitting Paradigm . . . . . . . . . 81.4 Wish List for the Ideal Euler Solver . . . . . . . . . . . . . . 101.4.1 Physically Correct Solution in a Large Mach Num-ber Range . . . . . . . . . . . . . . . . . . . . . . . 111.4.2 Multidimensional Treatment . . . . . . . . . . . . . 121.4.3 Shock-Capturing . . . . . . . . . . . . . . . . . . . . 131.4.4 Convergence . . . . . . . . . . . . . . . . . . . . . . 141.4.5 Preserving Potential Flow . . . . . . . . . . . . . . . 14II. FORMS OF THE EULER EQUATIONS . . . . . . . . . . . . 172.1 The Choice of Variables . . . . . . . . . . . . . . . . . . . . . 192.1.1 The Conservative Variables . . . . . . . . . . . . . . 192.1.2 The Primitive Variables . . . . . . . . . . . . . . . . 212.1.3 The Natural Variables . . . . . . . . . . . . . . . . . 232.1.4 The Characteristic Variables . . . . . . . . . . . . . 242.1.5 The Symmetrizing Variables . . . . . . . . . . . . . 252.2 Steady-State Euler in Diagonal Form . . . . . . . . . . . . . . 272.3 System Decomposition, a By-product of Local Preconditioning 292.4 Potential Flow, a Building Block for Euler . . . . . . . . . . . 32iv

III. FLUCTUATION-SPLITTING SCHEMES . . . . . . . . . . . 363.1 Description of Fluctuation-Splitting . . . . . . . . . . . . . . 373.1.1 Fluctuation Calculation . . . . . . . . . . . . . . . . 373.1.2 Fluctuation Distribution . . . . . . . . . . . . . . . 393.1.3 Desired Properties . . . . . . . . . . . . . . . . . . . 413.1.4 Geometrical Interpretation . . . . . . . . . . . . . . 443.2 Upwind Schemes . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.1 Low Di�usion A . . . . . . . . . . . . . . . . . . . . 463.2.2 N Scheme and its Limited Versions . . . . . . . . . 473.3 Other Familiar Schemes . . . . . . . . . . . . . . . . . . . . . 503.3.1 Lax-Wendro� . . . . . . . . . . . . . . . . . . . . . 503.3.2 Streamwise-Upwind Petrov Galerkin . . . . . . . . . 513.3.3 Least-Squares . . . . . . . . . . . . . . . . . . . . . 523.4 Matrix Fluctuation-Splitting . . . . . . . . . . . . . . . . . . 573.5 A Third-Order Approach . . . . . . . . . . . . . . . . . . . . 61IV. SEARCHING FORA CAUCHY-RIEMANNDISCRETIZA-TION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1 A Look at Lax-Wendro� . . . . . . . . . . . . . . . . . . . . . 684.1.1 The Update Procedure . . . . . . . . . . . . . . . . 684.1.2 Some Drawbacks . . . . . . . . . . . . . . . . . . . . 704.2 The Least-Squares Choice . . . . . . . . . . . . . . . . . . . . 734.2.1 The Update Procedure . . . . . . . . . . . . . . . . 734.2.2 General Properties . . . . . . . . . . . . . . . . . . 754.3 A Generalized Framework for Least-Squares . . . . . . . . . . 784.4 Examples of Application . . . . . . . . . . . . . . . . . . . . . 804.4.1 Incompressible Potential Flow . . . . . . . . . . . . 804.4.2 Compressible Potential Flow . . . . . . . . . . . . . 84V. ASSEMBLING AN EULER SOLVER . . . . . . . . . . . . . . 875.1 Conservative Linearization of the Euler equations . . . . . . . 875.2 The Hyperbolic Residuals . . . . . . . . . . . . . . . . . . . . 895.3 The Elliptic Residual . . . . . . . . . . . . . . . . . . . . . . 925.4 Forcing the Euler Solver to Preserve Potential Flow . . . . . . 975.4.1 A Constrained Minimization . . . . . . . . . . . . . 975.4.2 Treatment at Discontinuities . . . . . . . . . . . . . 105VI. IMPLEMENTATION AND RESULTS . . . . . . . . . . . . . . 1096.1 Pseudo-Time Iteration . . . . . . . . . . . . . . . . . . . . . . 109v

6.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 1136.2.1 Solid Walls . . . . . . . . . . . . . . . . . . . . . . . 1146.2.2 In ow/Out ow . . . . . . . . . . . . . . . . . . . . 1186.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 1206.3.1 Ringleb Flow . . . . . . . . . . . . . . . . . . . . . . 1216.3.2 Cylinders . . . . . . . . . . . . . . . . . . . . . . . . 1256.3.3 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . 1406.3.4 Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . 141VII. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.1 Contributions over Previous Fluctuation-Splitting Work . . . 1517.2 Summary of our Approach . . . . . . . . . . . . . . . . . . . 1537.3 Highlight of Computational Results . . . . . . . . . . . . . . 1547.4 Practical Issues for the Future . . . . . . . . . . . . . . . . . 1557.4.1 Multigrid Acceleration . . . . . . . . . . . . . . . . 1567.4.2 Extension to Three-Dimensions . . . . . . . . . . . 1577.4.3 Some Comments on Navier-Stokes . . . . . . . . . . 159BIBLIOGRAPHY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 162

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LIST OF FIGURES1.1 This ow chart shows each element that entered in the design of theEuler solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1 Mach angles ��, streamline, and natural coordinate system (s; n). 253.1 Samples of structured and irregular triangular meshes. . . . . . . . 383.2 Patch of cells around node i showing the median dual cell area. . . 403.3 One and two-target scenarios. . . . . . . . . . . . . . . . . . . . . . 413.4 The fraction of the residual produced in region FCEi contributes tothe update of node i. This fraction has a geometrical interpretationsince it is the ratio of area BAD to BAi . . . . . . . . . . . . . . . 453.5 Geometrical interpretation of the LDA scheme. . . . . . . . . . . . 463.6 Geometrical interpretation of the N and PSI schemes. . . . . . . . 493.7 Geometrical interpretation of Lax-Wendro� (left) and SUPG (right).For Lax-Wendro�: since pointD is in the interior of the triangle, �T;iwill be non-zero. For SUPG: this version of SUPG is truly upwindsince point D is on the out ow edge. . . . . . . . . . . . . . . . . . 513.8 Matrix-distribution LW-PSI scheme applied to Euler. Potential owaround the cylinder should be perfectly fore-aft and top-down sym-metric, but shows an arti�cial wake. This numerical solution pro-vides empirical evidence that, even in the irrotational limit, theacoustic part of the problem generates entropy. . . . . . . . . . . . 603.9 Same numerical experiment, this time conducted with a second-orderupwind �nite-volume method. The symmetry of the ow is barelyrecognizable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.10 The Hermite cubic element. . . . . . . . . . . . . . . . . . . . . . . 623.11 Sample of fundamental Hermite elements, from top to bottom: H1,H12 and H123. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.1 Stencils used in the least-squares, Lax-Wendro� �nite di�erence dis-cretization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2 Superimposed fore and aft pressure pro�les for a cylinder ow inthe incompressible limit. Lax-Wendro� (left) shows poor symmetrycompared to least-squares (right). . . . . . . . . . . . . . . . . . . 71vii

4.3 Left: Correct velocity magnitude pro�le for a point vortex problemin the radial direction. Right: Solution produced by least-squares orcentral-di�erence schemes with tangency conditions . . . . . . . . . 765.1 Projection of the natural basis onto the (z2; z3) plane. . . . . . . . 1025.2 Projection of the natural basis onto the (z1; z4) plane. . . . . . . . 1025.3 Projection onto a three dimensional space. . . . . . . . . . . . . . . 1045.4 Cells where the constrained minimization will be neglected. . . . . 1065.5 The velocity vectors evaluated at each node translate the triangle.and either produce a contracted cell (left) or an in ated one (right).This test accompanied with 5.4.2 is a robust way of di�erentiatingbetween entropy producing shocks and isentropic expansions. . . . 1076.1 Boundary treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.2 NACA0012 at M1 = 0:85; � = 0�. . . . . . . . . . . . . . . . . . . 1176.3 Streamlines in Ringleb ow. The limit line is de�ned by the locus ofpoints where the streamlines form cusps. . . . . . . . . . . . . . . . 1236.4 Zoom on the sonic line and supersonic region. . . . . . . . . . . . . 1236.5 Accuracy of second-order Euler solver . . . . . . . . . . . . . . . . 1246.6 Pressure isolines (left), ow direction contours (right). . . . . . . . 1266.7 First and second Riemann invariants. . . . . . . . . . . . . . . . . . 1266.8 Ringleb streamlines (left) and grid used for this set of Euler calcu-lations (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.9 Sample of triangular grids used for cylinder ow calculations. Cylin-der is of unit radius, far-�eld boundary taken at 10 radii. Thestructured mesh (left) forms the basis for polar coordinates and isstretched to capture the high gradients close to the body. The un-structured mesh (right) is generated by a frontal Delaunay methodas described in [50]. . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.10 Incompressible cylinder ow, Mach isolines. Potential ow shown insolid lines, Euler ow shown in dashed. . . . . . . . . . . . . . . . 1286.11 Incompressible cylinder ow, ow direction isolines. Potential owshown in solid lines, Euler ow shown in dashed. . . . . . . . . . . . 1286.12 Coe�cient of pressure distribution on cylinder surface. . . . . . . . 1296.13 Log-log plot of L2 and L1 error norms versus characteristic cell sizeshows third-order accuracy. . . . . . . . . . . . . . . . . . . . . . . 1316.14 Numerical entropy generation on cylinder surface. Note that theleast-squares-PSI scheme produces entropy levels several orders ofmagnitude lower than Lax-Wendro�-PSI (version found in earlierEuler solvers based on hyperbolic-elliptic decomposition). The abovecalculations are done with double precision. . . . . . . . . . . . . . 131viii

6.15 Near-critical cylinder ow, Mach isolines. Potential ow (solid line)and Euler (dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.16 Mach number distribution on the cylinder surface. Potential ow(upper curve) and Euler (lower curve). . . . . . . . . . . . . . . . . 1336.17 Numerical entropy production on cylinder surface (left). Close-up ofthe leading edge, showing a non-dimensionalized pressure of exactly1.0 at the stagnation point (right). . . . . . . . . . . . . . . . . . . 1346.18 Super-critical ow around cylinder, M1 = 0:60. . . . . . . . . . . . 1356.19 Pressure distribution (left) and entropy levels (right) along cut 'AA'.Note the jump in pressure due to lack of 'smoothing' at the sonic line,it could be considered minor given the magnitude of the correspond-ing entropy change of 6:7 � 10�4. Note the absence of post-shockoscillation even though least-squares in not monotone. This seemsto indicate that least-squares has satisfactory damping properties inall directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.20 Cartoon of non-uniform enthalpy in ow problem. Due to the linearin ow velocity pro�le, the vorticity is constant everywhere in the ow-�eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.21 Mach contours around cylinder and some streamlines showing circu-lation zones around the trailing-edge stagnation point. . . . . . . . 1386.22 Streamlines superimposed with contour lines of enthalpy (left). Sinceenthalpy is advected along streamlines, the two should coincide. Thefore-aft symmetry of the ow is well captured as indicated by thecirculation zones. Order of accuracy of the Euler solver on thisproblem shows third-order behavior. . . . . . . . . . . . . . . . . . 1396.23 Grid used for ow calculations around ellipse. . . . . . . . . . . . . 1406.24 Streamlines and some Mach contours, clearly indicating no lift onthe ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.25 Mach number pro�le over the ellipse body shows good symmetry.The trailing edge suction peak is almost captured with same intensityas the leading-edge one. . . . . . . . . . . . . . . . . . . . . . . . . 1426.26 Sample grids used for NACA 0012 ow calculations. Airfoil has unitchord and far-�eld boundary is usually taken at around 10 chordlengths away from the airfoil. . . . . . . . . . . . . . . . . . . . . . 1436.27 NACA 0012 calculations in the incompressible regime. Mach contourplots are superimposed for comparison, showing that all solutions areself-similar in the incompressible limit. . . . . . . . . . . . . . . . . 1446.28 NACA 0012 calculations in the incompressible regime. Pressure co-e�ecient (left) and numerical entropy generation over body (right). 1446.29 Comparison of convergence history between M1 = 0:1, M1 = 0:01,M1 = 0:001 and M1 = 0:0001 shows that number of iterations isnot inversely proportional to Mach number . . . . . . . . . . . . . 145ix

6.30 NACA 0012 calculations in the supercritical regime. Mach contoursfor M1 = 0:70; � = 1:65� . . . . . . . . . . . . . . . . . . . . . . . 1476.31 Pressure coe�cient distribution . . . . . . . . . . . . . . . . . . . . 1476.32 Distribution of entropy along the body surface (left) and close-up ofthe leading edge stagnation point . . . . . . . . . . . . . . . . . . . 1486.33 Transonic ow around NACA 0012 at M1 = 0:85 with � = 1:0�angle of incidence. The plot of Mach contours emphasizes the upperand lower surface shock formation. . . . . . . . . . . . . . . . . . . 1496.34 Coe�cient of pressure distribution on upper and lower bodies of theairfoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.35 High-lift subsonic ow around NACA 0012: M1 = 0:05; � = 25�.Mach contours and some streamlines. . . . . . . . . . . . . . . . . 1506.36 Pressure coe�cient (left) compares numerical solution obtained againstpanel method. Mach number distribution (right) shows strong suc-tion peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

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CHAPTER IINTRODUCTIONIf one looks at the evolution of industrial and commercial software used to simu-late compressible uid ow, it seems at �rst that progress in Euler solvers during thenineteen-nineties has considerably tamed compared to the vigorous growth it enjoyedprior to that. The core of all internal and external ow solvers used by designers stillrelies on CFD technology developed roughly two decades ago. The research com-munity has been able to produce innovations that incrementally improve accuracyand computational time, yet an ever reduced portion of this work is �nding its wayto mainstream applications. The state of the art compressible ow techniques arenot yet drastically more e�cient than their older counterparts, therefore providinglittle justi�cation to make the move from one to the other. Nonetheless, users arewell aware of the limitations of their computer programs and shortfalls are oftenmade up through corrections, tuning or even physical experiments. In some extremecases, Euler simulations become an art of grinding performance out of a code byarti�cially adapting it to situations it wasn't designed for. Much of this justi�es thedevelopment of more versatile compressible ow solvers.

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21.1 Areas of FocusIn the last several years researchers have turned their interest to low-Mach number ows and truly multi-dimensional Euler solvers. These are areas of interest which, itis believed, may have considerable impact on industrial applications aimed at solvingcompressible ows. In this section, we brie y review some of the numerical methodsthat are currently used to treat those particular problems.1.1.1 Low-Mach Number FlowsFlow problems exist that simultaneously contain locally low-Mach numbers re-gions, whereas signi�cant compressibility e�ects occur in other parts of the ow.One example speci�c to aerodynamics is maximum lift ow over a wing, such as ahigh-cambered or multi-element airfoil. Another case would be in engines where thenear-stagnant ow inside the combustion chamber contrasts with the compressible ow due to heat transfer e�ect around turbine blades. In the past, ow solvers weredesigned for either purely compressible or incompressible problems and the regionin between was of secondary importance. Nowadays, physical problems are diverseenough that Euler codes must behave well in the whole range of Mach numbers.To achieve that versatility, the CFD community is divided over two approaches.One extends existing numerical algorithms designed for compressible ows, and theother adapts current incompressible solvers. In the latter, pressure is held as theprimary variable and the e�ect of acoustics on the ow �eld is treated via a Tay-lor expansion of the pressure in terms of Mach number ([39], [51]). Recent workfound in [4] represents well the current status of incompressible solvers which havebeen extended to work in compressible regimes. That method is a pressure-baseddiscretization on a staggered grid in general boundary-�tted coordinates. Dimen-

3sionless variables are well-behaved and remain �nite for all Mach numbers, even inhighly compressible regions. The time integration is implicit and combines with apressure correction method. The drawback of such a scheme is that it poorly adaptsto unstructured grids around complex geometries. More importantly, the excessivedissipation and lack of monotonicity property makes it a poor candidate for shockcapturing. For these reasons, the group which has had the upper hand is the onethat advocates handling the problem from the compressible side. Extending existingcompressible ow produces better results and turns out to be more consistent overa broader range of ow conditions.When looking for the converged solution of the steady Euler equations, a com-mon strategy has been to time-march the corresponding unsteady equations. Theoriginal problem which is mixed hyperbolic-elliptic then becomes purely hyperbolic,with all acoustic and uid information being advected along di�erent characteristics.The di�culty in computing low-Mach number ows with compressible solvers is at-tributed to the large disparity of the acoustic wave speeds (u + a; u � a) and the uid speed u. As a remedy, the acoustic waves could be damped arti�cially, thusimproving convergence to a steady-state.Modifying the strength of acoustic waves by changing the eigenvalues of the Eulersystem has been the main idea behind preconditioning techniques ([14], [15], [84], [41],[79], [11]). When a preconditioning operator is applied to the time-derivative, thetransient properties change, but the steady-state solution remains una�ected. This istrue only at the level of the partial di�erential equation (p.d.e). Preconditioning thediscrete equations not only changes the convergence rate but can also improve theaccuracy of the steady state solution at M ! 0. Indeed, the preconditioner modi�esthe numerical ux functions such that the arti�cial viscosity terms balance with the

4inviscid ux terms. Furthermore, some researchers have suggested similar schemeswhere local preconditioning changes the numerical ux functions in a time-consistentmanner ([86], [71]). However, 'preconditioned-type' strategies for unsteady problemsstill seem to su�er from lack of robustness.One problem associated with local preconditioning is loss of stability at stag-nation points. However that issue has been addressed and [14] shows that it isrelated to the non-normality of the eigenvector structure at very small Mach num-bers. Overall, preconditioning the discrete Euler equations has been quite successfuland is considered the standard technique for treating ows with regions of di�eringcompressibility.1.1.2 Treatment of Multidimensional ProblemsAt the heart of present day upwind schemes for computing compressible ows isthe solution of the one-dimensional (1D) Riemann problem or some simpli�cation ofit. Its solution describes the evolution of the ow which results from bringing intocontact two uids at constant but di�erent states. The �rst aspect of a conservative�nite-volume method is to discretize the computational domain in a number of con-stant states and write the conservation law as a cell average for each control-volume.At each cell interface, the uxes are evaluated using the data found on left and rightstates. The ux di�erences are then split into waves and wave speeds along direc-tions normal to cell-faces. Finally cell averages are modi�ed to either side by thecontributions of waves entering each element.With the 1D Riemann solvers being extremely successful, the question rises howtruly multidimensional physics could be brought into this picture. Early work byGodunov [28] attempted to generalize the process to higher dimensions but it was

5quickly realized that truly 2D or 3D Riemann problems were too complicated to betractable. Instead, the idea of applying the Riemann problem dimension by dimen-sion emerged and the resulting family of schemes has been the key to multidimen-sional compressible solvers. Common approaches include applying the 1D operatorsmultiplicatively (operator splitting) or additively (all directions at the same) to allthe linearly independent space directions. These directions are usually the (x; y)coordinate directions or the cell normals of the �nite volumes. An original versionof such methods is discussed in [83] and derivatives include [70]. Variations of thesame family include transverse Riemann solvers where, in addition to the dimensionby dimension application of 1D operators, linear waves are used to model the cross ow in transverse directions [13].Another group of multidimensional schemes are based on the method of transportdescribed in [22] and [23]. In a �rst step, the non-linear system is decomposedinto a �nite set of linear multidimensional advection equations. The decompositionis done at the level of the partial di�erential equation, but unlike the hyperbolic-elliptic decomposition discussed in this work, it is not unique. In a second step, theresulting linear scalar advection equations are solved. Since this step is independentof the decomposition, any solver may be used, and the one which surfaces mostoften is multidimensional upwind �nite-volume (again involving 1D Riemann solver).The potential of the method of transport lies in its exibility in adapting to anysystem of conservation laws. However, one is left with the task of choosing from amultitude of possible decompositions the one that most closely mimics the physics.A more important drawback is that the process can only be �rst-order accurate dueto the linearization. Higher-order calculations come at the expense of costly Taylorexpansions only possible in smooth regions.

6No matter how you look at the techniques mentioned above, they essentially repli-cate multidimensional problems with 1D physics. This creates excessive dissipationwhen discontinuities are not aligned with the mesh. The weakness is most visible onunstructured grids where the edge normals are directed randomly. Many attemptsof solving Riemann problems in rotated directions have produced somewhat disap-pointing results so it should be concluded that a multidimensional upwinding theoryhas to built without considering the 1D Riemann problems at all.1.2 Our ManifestoIn an e�ort to depart from Riemann problems, the work presented in this disser-tation suggests a radical approach to the design of compressible ow solvers. Ourtarget is an Euler solver which performs to a high degree of accuracy for multidi-mensional problems, and over a large range of Mach numbers starting with the verylow-end. If we were to state a thesis which will guide us through the design of sucha scheme, this is how it would be formulated:Starting with the steady-state Euler equations, we seek a stationary solution.With the problem being of mixed hyperbolic-elliptic nature, no preconditioning isnecessary. Indeed, with no such thing as an 'acoustic wave' at low-Mach numbers,speed disparity between acoustic and convective waves is meaningless. If we hada way to completely decompose hyperbolic parts from elliptic ones, each portioncould be attacked separately using its own specialized scheme. In fact, a uniquehyperbolic-elliptic decomposition does exist at the p.d.e level and is used as a steppingstone to a full decoupling at the residual level. Consequently, it becomes possible totightly control the numerical update procedure such that acoustic variables do nota�ect advective ones and vice versa. In the case of uniform entropy and enthalpy

7in ows and for shock-free �elds, the solver should recover potential ow. This isparticularly advantageous to incompressible ows where numerical errors usuallystem from excessive entropy generation and large dissipation.To preserve multidimensionality, advective parts of the Euler system are treatedwith upwind uctuation-splitting while the elliptic part is discretized using least-squares which conveniently blends into the same family of schemes. Least-squaresis optimal for di�usion problems because it propagates disturbances without anydirectional bias. Each piece of the split Euler residual has its exclusive discretizationto insure that the correct physical phenomenon is captured. In turn, we expect thisprocedure to produce solutions independent of grid orientation or irregularity.From the points above, it is clear that despite a substantial change in our modusoperandi, the paradigms are quite conventional. Some of the elements we use are al-ready well-understood, namely hyperbolic-elliptic decomposition of the Euler systemand multi-dimensional upwinding. Others constitute the impetus of this research, asthe full decoupling of residuals or the discretization of the subsonic acoustic subsys-tem. Still a few remain untouched and could be the subject of further investigation.1.3 The Main Pillars1.3.1 The Hyperbolic/Elliptic DecompositionFor the pseudo-unsteady Euler equations, a unique hyperbolic-elliptic decomposi-tion is possible through local preconditioning. In [47] and [56], the van Leer-Lee-Roepreconditioner is rederived from the perspective of that decomposition. Althoughnot originally expected, this is just another nice side-product of preconditioning. Inthe steady case, however, the Euler equations may easily be diagonalized. In bothcases, the decomposed equations are similarly formulated in terms of the natural

8variables. Refer to chapter II for further details.In the supersonic regime, Euler fully decouples into four advection equationscarrying information for entropy, total enthalpy (S; h) and two acoustic variables.Interestingly enough, the idea of upwinding (S; h) along streamlines is totally ab-sent in conservative �nite-volume methods for solving the steady system. It is avery attractive feature that (S; h) advection are ordinary di�erential equations, andcan be integrated by marching along the streamline starting at the inlet boundary(commonly known as the method of characteristics).In the subsonic case, only (S; h) are advected and the acoustic part becomes a2� 2 elliptic subsystem of Cauchy-Riemann type. This prohibits a strict decouplingas we would have desired. To help better control the properties that we want toinstill in our solver, we must take the decomposition ideas further to the residuallevel. If we truly extract the elliptic part of the residual, then it should be insensitiveto variations in (S; h). On the ip side, there should be zero contribution to (S; h)from the elliptic residual. This is something we enforce by constraining the updateprocedure. Chapter V discusses these issues.1.3.2 The Fluctuation-Splitting ParadigmA major di�erence between uctuation-splitting and ux-di�erence splitting or ux-vector splitting schemes is in the handling of multidimensional problems. Theselatter methods begin by making a decomposition of the spatial gradients, typically ofthe di�erences between the adjacent computational cells or nodes. In the former class,any elaborate analysis is deferred until the spatial gradients in di�erent directionshave been combined to form a residual. Thus, a gradient that is precisely cancelledby a gradient in some other direction need not be analyzed. These residuals are then

9used to update the vertices forming the cells from which they derive. A given nodemay be updated by any or all the cells that it belongs to. In this dissertation, wequalify such techniques as distributive.Nowadays the understanding of scalar upwind uctuation-splitting is mature, andcombined with non-linear limiting, both monotonicity and accuracy properties areguaranteed ([65], [73]). Chapter III gives an overview of these methods. Although uctuation-splitting for scalar advection is well-understood, the key remaining issueis its extension to hyperbolic systems which cannot be diagonalized (such as unsteadyEuler).To answer that void, there have been two principal directions of research. Re-cent e�orts include a matrix generalization of the �rst-order upwind schemes tonon-commuting hyperbolic systems. The matrix version reduces to scalar schemesapplied to each of the decoupled equations, if the system is diagonalizible. Lim-ited second-order schemes have also been developed but they are not satisfactory interms of convergence and robustness ([59], [81], [8]). The other channel of research isbased on the hyperbolic-elliptic decomposition of the Euler equations described previ-ously. The advection equations are handled with standard scalar upwind uctuation-splitting and the acoustic part with a matrix distribution form of Lax-Wendro� ([47],[56]). In both approaches, coupling between acoustic and advective parts cannot beavoided in subsonic ow. Due to this interaction, it has been reported that strictmonotonicity is lost even though monotonic upwind advection is used. Also, in theirrotational limit, solutions are corrupted by the presence of non-uniform entropy�elds, a problem present in compressible �nite-volume schemes at an even greaterextent. Overall, excellent results are obtained by using any of the two methods butthere still is room for improvement.

10The work presented here is very closely related to the second path describedabove. Having fully decoupled the advective and acoustic residuals, multidimensionalupwinding remains the method of choice for (S; h) convection. Meanwhile, a newapproach is taken for distributing the elliptic residual, the least-squares minimization.Its advantage is that it can be cast in uctuation-splitting form which permits a verymodular design of computer programs. Remember that the acoustic part of subsonicEuler is of Cauchy-Riemman type so least-squares needs to behave very well onthat type of problem. When applied to the linear Cauchy-Riemann problem on auniform grid, least-squares attempts to solve two heat equations, one for each velocitycomponent. This means that least-squares treats the Cauchy-Riemann equations astwo decoupled Laplacians which at �rst seems advantageous (see chapter IV). Whenthe entire Euler solver is applied to subcritical ows for which potential solutionsshould be obtained, the enthalpy is constant to machine zero, and the entropy isconstant to an extremely high degree of accuracy (see results in chapter VI).1.4 Wish List for the Ideal Euler SolverThe ideas presented in the previous sections provided the broad brushes of thepainting. In this section, we will brie y look at some of the other elements whichshould be incorporated in our Euler solver. The presentation is in the form of anabridged list of properties desired from a solid Euler code. Nestled in this discussion,there are areas where primary material was available from previous research, as wellas areas that we chose not to touch. The pictorial complement, shown in �gure 1.1,contains all the elements important to our solver, as well as references to chapterswhich contain that material. Boxes drawn with bold frames are areas where webelieve to have made a contribution. One message we try to convey through that

11illustration is that the ultimate Euler scheme should be very modular. The contentsof some boxes might be allowed to change without a�ecting the rest of the design.1.4.1 Physically Correct Solution in a Large Mach Number RangeAs mentioned earlier, many of the problems we want to solve demand a largerange of applicability. The supersonic and intermediate subsonic regimes pose lit-tle problem because our discretizations and numerical techniques are well adaptedto those. Instead, one should focus on avoiding defects in regions considered 'di�-cult', such as the incompressible regime (particularly stagnation points) and sonictransitions.The Incompressible LimitLoss of accuracy near stagnation points is usually associated with spurious en-tropy generation. Put roughly, the typical Godunov type method causes the passageof transient compression and rarefaction waves in the neighborhood of stagnationpoints. These arti�cial waves cause excessive dissipation and corrupt the solution inthat region.If applied appropriately, preconditioning is successful in reducing the numericale�ects that corrupt incompressible regions. Nonetheless, Euler preconditioners nor-mally lose robustness around stagnation points, and as explained in [14], this relatesto the degeneracy of eigenvectors. Modi�cations to the preconditioning matrix havebeen suggested to avoid instabilities.All of this indicates that stagnation solutions are di�cult to compute withoutsome special treatment. Our treatment of the stagnation point is automatic becauseit involves the solution to a Cauchy-Riemann system, the equations which governirrotational incompressible ow. Having split away the acoustic residual, we show

12in chapter IV that least-squares minimization produces a satisfactory discretizationof Cauchy-Riemann. Computational results show that pressure non-dimensionalizedby its stagnation value smoothly reaches unity at the stagnation point. The fact thatour Euler solver does not need special handling at stagnation points is indicated in�gure 1.1, by crossing out the corresponding box.The Sonic LimitThe sonic line is usually perceived as a singularity because it is the meeting pointof physical phenomena of very di�erent nature. As such, most schemes perform somespecial treatment in the sonic region, maybe an entropy �x, to prevent non-physicalsolutions such as expansion shocks. In our case, the di�culty is re ected by an ellipticscheme meeting an advective one. Since they distribute signals very di�erently, oneisotropically and the other in an upwind manner, one would expect a mismatch atthe sonic line. However, experimental results show that even without any specialtreatment, a practically smooth transition takes place. The distribution of pressureand entropy across the sonic transition show jumps that may be considered minor, ifmeasured by the magnitude of entropy increase. One area that was not touched uponis taming the di�erence between the two schemes at the sonic line, maybe throughsome smoothing or blending of the two sides. There is no question however, that inthe ultimate perfect scheme, some sort of sonic line treatment will eventually haveto be implemented.1.4.2 Multidimensional TreatmentAs discussed earlier, we hope that combining decomposition and uctuation-splitting ideas will allow a scheme that re ects genuinely multidimensional physics.The key is to decompose the Euler residuals into their simplest parts, both in su-

13personic and subsonic cases. As shown in �gure 1.1, specialized schemes may thenbe applied to each portion of the residual, in a way to correctly capture the propa-gation of information. For strongly discontinuous ows, it means multidimensionalupwinding and for incompressible ow, least-squares. This should lead to very smallsensitivity to grid topology and orientation.1.4.3 Shock-CapturingFor a correct capturing of shock positions and strengths, we must assure that ourscheme is conservative. At the level of the p.d.e, conservation is expressed when theEuler equations are written in terms of the conservative variables which exactly sat-isfy jump relations across discontinuities. Once we move to a discrete form, however,the necessity of a conservative linearization for the calculation of cell residuals �xesthe choice of variables to the parameter vector, p�(1; ~q; h)t. Even though the naturalvariables (S; h; p; �) are of particular importance to this work since they decomposethe Euler equations, we must hesitate to use them in calculations. The advantage ofour formulation is that once a residual is available in a set of variables, we can simplymove to another set of variables through the transformation matrix. For purposesof computational e�ciency, we will only hold a single set of variables, and it will bethe parameter vector.Concerning monotonicity, it should strictly be enforced on the supersonic sideas long as the multidimensional upwind schemes we apply are positive. This holdstrue independent of our subsonic acoustic treatment, given that the residuals oneither side are decoupled. On the subsonic side, computational results indicate nooscillations in post-shock regions. At �rst, this is a surprising observation given thefact that least-squares is not monotone, but it seems to possess enough damping to

14prevent undershoots or overshoots.1.4.4 ConvergenceThe time-dependent Euler equations exhibit sti�ness that is strongly dependenton the Mach number. The sti�ness of the system is measured by the ratio of thelargest to smallest characteristic speed, and for very small Mach numbers this ratioincreases without bounds. This slows down the convergence rate of any marchingmethod. Preconditioning answers this by modifying the transient properties of thetime-dependent equations, reducing the disparity in characteristic speeds.We have rid ourselves of the sti�ness problem inherent to the unsteady equationsby solving the steady ones. However, we introduce another problem, that of quicklyconverging a Cauchy-Riemann type problem to steady-state using an elliptic solver.Without the luxury of advecting errors out of the domain, we are left with no otherchoice than error reduction through damping. This causes the poor convergence rateof least-squares for a wide range of Mach numbers, not just at the low end. Wehave been able to somewhat accelerate that very slow process by implementing aquasi-Newton iterative method. However, the elliptic problem needs further con-vergence acceleration e�ort, which could maybe materialize in the form of multigridtechniques. That sort of work was not undertaken in this research. Since we areusing two separate schemes for the elliptic and hyperbolic part, another issue is howto simultaneously optimize convergence on both.1.4.5 Preserving Potential FlowIf we accomplish complete decoupling at the update level between the ellipticand hyperbolic parts, the acoustic part of the residual should only contribute tochanges in pressure and ow angle and not to entropy and enthalpy. Conversely the

15advective residuals should change only the entropy and enthalpy. Here, we constrainthe least-squares minimization such that the update is along a descent directionwhich leaves (S; h) una�ected. In this way, our Euler code is able to compute withvery high accuracy potential ows for which (S; h) should be exactly constant. Evenon di�cult test cases, such as those suggested by Pulliam in [61], spurious entropygeneration is low enough to get correct aerodynamic coe�cients.Euler solvers which use potential ow as a building block are nowhere to be foundin the compressible ow literature, except in the very recent work by Hafez ([32],[33]). His formulation is not equivalent to the Euler equations per se but is based ona combination of Cauchy-Riemann and Crocco's relation. For isentropic, irrotational ows, only the Cauchy-Riemann equations are solved for the velocity componentsand all thermodynamic variables are derived from Bernoulli's law. Indeed, this isa fair approximation in the far-�eld where the ow is smooth or shock waves areweak. Otherwise, an entropy correction is needed for density and obtained via thetangential momentum equation.

16Conservative formulationof the Euler equations

Local preconditioning (VLLR)

Hyperbolic / Elliptic decomposition (using natural variables)

Conservative linearization usingparameter vector

Euler residual in each element

Acoustic residual(elliptic part)

E

Entropy, enthalpyresiduals

hS

Entropy, enthalpy andacoustic residuals,

hS W+ W-

M<1M>1

Multidimensional upwinddistribution: positive,LP, conservative. For both system and scalaradvection problems

Multidimensionalisotropic distribution:LP, conservative.For both system andscalar elliptic problems

Convergence acceleration: - preconditioning - more efficient iterative method

Preserving potential flow: constraining the elliptic update onto surface of constant S and h

Stagnation point treatment,through preconditioning orother

Sonic line treatment: - blending - entropy fix

Update

pieces already available,mature understanding

pieces proper to this thesis

pieces left untouched butthat should eventually beadded

Ch. II

Ch. V

Ch. II

Ch. V

Ch. V Ch. IV, VCh. V

Ch. III Ch. III Ch. III, IV

Ch. V

Ch. VI

Multidimensional upwinddistribution: positive,LP, conservative. For both system and scalaradvection problems

Figure 1.1: This ow chart shows each element that entered in the design of theEuler solver.

CHAPTER IIFORMS OF THE EULER EQUATIONSThere exist various ways to express the Euler equations, each associated with adi�erent set of unknowns, and each revealing a distinct character for the set of partialdi�erential equations. Despite the multitude of formulations, the most common andprobably most fundamental expression is the conservative form because it is in theconservative variables that the Rankine-Hugoniot relations are upheld. Hence, theusual approach to design compressible ow solvers with shock capturing capabilityis to discretize the conservative form and solve it in a consistent manner. Then onehopes that the numerical solution obtained is within a reasonably small truncationerror of the other discretized Euler forms.We know of many features of Euler solutions which are exposed more clearlyin formulations other that the conservative one. These alternative formulations arefrequently used to check solution quality and accuracy but seldom designed into thesolver itself. Capitalizing on the advantages provided by all set of variables in asingle scheme will probably prove to be elusive, but we set forward to incorporate asmany bene�ts as possible.For instance, we make use of the natural variables (S; h; p; �) because they enablea full diagonalization of the Euler equations in the supersonic regime and a partial17

18decomposition in subsonic. With advection equations extracted for both entropy andenthalpy, one may explicitly enforce their constancy along streamlines (constancy ofentropy only along shockfree paths). This is done by two-dimensional characteristicupwinding, or uctuation splitting schemes, as described in chapter III. Another setof variables which is important in this work is the parameter vector p�(1; vecq; h).For accurate capturing of ow discontinuities, we need to compute and distributeconservative residuals. The parameter vector allows a linearization of the Eulerequations which is both algebraically attractive and conservative.Much of the research conducted in [47], [56] and related papers [46], [57] hadthe goal to reduce the Euler system to recognizable subproblems for which accuratediscretizations have proven to succeed. Up to 1994, the framework for most multidi-mensional Euler methods was based on discrete wave models �rst presented in [64]and also discussed in [55]. The idea presented in these various papers is to break thesystem of equations into equivalent set of scalar advection operators representing thewave behavior characteristic of gas ow. With better understanding of local precon-ditioning came its application to Euler decomposition into acoustic and advectivesubproblems. Soon wave modeling soon became obsolete. With hindsight, it is ap-parent that representing the elliptic part of Euler with a discrete set of wave modelswould not be completely successful, because even in the steady-state it continues toemphasize the transient behavior.For the expressions in this chapter and in all the following ones� � is density� u and v are velocity components in the x and y direction� p is static pressure

19� a = q p� is acoustic speed� q = pu2 + v2 is the ow speed magnitude� � is the ow angle� = 75 is the speci�c heat ratio for idealized air� S is the entropy� e = 1 �1 p� + q22 the total speci�c energy� h = �1 p� + q22 the total speci�c enthalpy� Non-dimensionalized quantities (�)? are evaluated with respect to stagnationvalues (�)o. For example, p? = ppo , �? = ��o , q? = qao . From this point on, for thepurpose of conciseness, all superscripts (�)? will be dropped.2.1 The Choice of Variables2.1.1 The Conservative VariablesFor computing compressible ow, the most fundamental choice is probably theconservative form since a weak solution to it captures shocks that satisfy the Rankine-Hugoniot relations. @u@t + @f@x + @g@y = 0The conservative state vector u and the associated ux vectors f and g are de�nedby u = 2666666666664��u�v�e3777777777775 f = 2666666666664

�u�u2 + p�uv�uh3777777777775 g = 2666666666664

�v�uv�v2 + p�vh3777777777775 (2.1)

20In chapter III, we discuss residual-distribution schemes in greater detail but here wejust want to comment on the importance of the variables held at cell vertices onthe residual calculation. Determining the residual based on the conservative form iscomputationally expensive because it would involve a direct evaluation of Jacobianintegrals. The Jacobian matrices �A and �B are nonlinear functions of the conservativevariables. �T = Z ZT @f@x + @g@y! dS= ST �A@u@x + �B@u@y !Given that the integrals �A = 1ST Z ZTAdS and �B = 1ST Z ZTBdS are quite di�cultto compute, we follow a simpler procedure presented in [63] which introduces theparameter vector. z = 2666666666664z1z2z3z43777777777775 =

2666666666664p�p�up�vp�h

3777777777775The computational convenience in using the parameter vector stems from the factthat all components of the conservative variable u and ux vectors f , g are bilinearin it. It follows that C = @f@z , D = @g@z , and @u@z shown below are linear in z.C = 2666666666664

z2 z1 0 0 �1 z4 +1 z2 �1� z3 �1 z10 z3 z2 00 z4 0 z23777777777775 D = 2666666666664

z3 0 z1 00 z3 z2 0 �1 z4 �1� z2 +1 z3 �1 z10 0 z4 z33777777777775

21@u@z = 2666666666664

2z1 0 0 0z2 z1 0 0z3 0 z1 0z4 �1 z2 �1 z3 z1 3777777777775The cell residual can then easily be computed as�T = Z ZT @f@x + @g@y!dS= Z ZT @f@z @z@x + @g@z @z@y!dS= ST �C@z@x + �D@z@y!Since z is piecewise linear on each cell, @z@x and @z@y are trivial. The integrals �C and �Dare calculated exactly by evaluating the matrices at the center of the cell �z = 3Xi=1zi.2.1.2 The Primitive VariablesWe tend to think in terms of primitive variables v = (�; u; v; p)t, maybe becausethey are the most tangible ones, but certainly because they can be controlled inan experimental setup. Sophisticated numerical methods for the Euler equationsseldom make use of the primitive variables since they do not lend themselves to veryaccurate discretizations, particularly for compressible problems. Instead, they arewidely used in boundary conditions. Generally, we have a good understanding ofthe behavior of v at solid wall boundaries and in the far-�eld of a computationaldomain. In addition, a vast amount of experimental data has been taken in termsof pressure, velocity and density, therefore making the primitive variables a primecandidate for enforcing boundary conditions.A very familar form of the Euler equations in terms of v is@v@t +A@v@x +B@v@y = 0 (2.2)

22where we use the notationv = 2666666666664

�uvp3777777777775 A = 2666666666664

u � 0 00 u 0 1�0 0 u 00 �a2 0 u3777777777775 B = 2666666666664

v 0 � 00 v 0 00 0 v 1�0 0 �a2 v3777777777775 (2.3)

Yet another form of the Euler equations in terms of primitive variables would beexpressed in a natural coordinate system. Natural coordinates (s; n) are a convenientway of writing ow equations because they rid the system of directional componentsand emphasize ow angle and velocity magnitude. The streamline direction s andthe normal to streamline n are obtained from the cartesian coordinates through arotational transformation.2664 @@s@@n 3775 = 2664 cos � sin �� sin � cos � 3775 2664 @@x@@y 3775 = 2664 uq vq�vq uq 3775 2664 @@x@@y 3775 (2.4)Transforming equation 2.2 into natural coordinates,@v@t +An@v@s +Bn@v@n = 0where An and Bn are the Jacobian matrices in natural coordinates . Note thatthe velocity magnitude q dominates the diagonal of An, appropriately weightinggradients in the streamline direction. The Euler system may then be expressed asv = 2666666666664

�uvp3777777777775 An = 2666666666664

q � 0 00 q 0 1�0 0 q 00 �a2 0 q3777777777775 Bn = 2666666666664

0 0 �q 00 0 0 00 0 0 1�q0 0 �a2q 03777777777775 (2.5)

232.1.3 The Natural VariablesThe Euler equations are seldom explicitly expressed in terms of natural variablesbut they are very important to the present approach. As explained in sections 2.2and 2.3, it is in these variables that the Euler equations are maximally decoupled.In the subsonic regime, the Euler equations are split into a 2�2 elliptic subsystemand two scalar advection equations, one for entropy and the other for enthalpy. Notethat the elliptic part represents the acoustic aspect of the ow and is completelyindependent from the rest since it only involves pressure and ow angle (see equation2.7). Since hyperbolic problems exhibit wave propagation and elliptic problems showsmoothing, separate discretizations must be used to capture each behavior. Ourdecomposition enables the essentially di�erent hyperbolic and elliptic behavior to becomputed independently with the minimum of crosstalk. In the following expressions,�? = p1�M2. @x@t + ~A@x@s + ~B@x@n = 0 (2.6)In equation 2.6, the vectors and matrices are de�ned as@x = 2666666666664

@S@h@p@�3777777777775 ~A = 2666666666664

q 0 0 00 q 0 00 0 �?2 00 0 0 �q23777777777775 ~B = 2666666666664

0 0 0 00 0 0 00 0 0 ��q20 0 1 03777777777775 (2.7)

where @S = @p� a2@� and @h = �q@q + @p.In the supersonic case, the acoustic part transforms into separate advection equa-tions. Then the Euler equations are completely diagonalized and we can write them(section 2.1.4) in terms of characteristic variables w. Since x and w only involve(S; h; p; �) we will often refer to both as natural variables throughout the rest of this

24thesis. One decomposes Euler into its simplest homogeneous part for subsonic ow,the other does it for supersonic ow.2.1.4 The Characteristic VariablesIn the supersonic regime, the steady Euler equations may fully be expressed incharacteristic form and the transformation simply requires a diagonalization of thesystem. The characteristic version is simple and well known, a very close version ofit is given in [34] @w@s +�@w@n = 0 (2.8)where we use the notation@w = 2666666666664

@S@h�q2� @� + @p��q2� @� + @p3777777777775 � = 2666666666664

0 0 0 00 0 0 00 0 1� 00 0 0 � 1�3777777777775 (2.9)

In equation 2.9, the acoustic Riemann invariants are advected along Mach linesde�ned by tan(�) = 1� . The Mach lines are separated by an angle �� from the localstreamline, as shown in �gure 2.1.The physics dictate a discretization of the entropy, enthalpy and acoustic equa-tions by a space operator upwinded along streamlines and Mach lines. This is verysimilar to the widely used method of characteristics. The point of deviation from themethod of characteristics is to achieve this in a conservative formulation, such thatshocks and contacts can be handled without any special treatment. The hallmark ofmultidimensional upwinding schemes described in 3.2 is that unlike state of the art�nite-volume approaches, they do not rely on locally one-dimensional physics to per-

25

x

yn

s

q

Figure 2.1: Mach angles ��, streamline, and natural coordinate system (s; n).form conservative upwinding. Therefore, it is now possible to directly use streamlinesand Mach lines to advect the quantities of interest.Since the system is fully diagonalized forM > 1, it lends itself perfectly for prac-tical problems where the ow is guaranteed to be purely supersonic and conditionsat in ow or out ow are known. An important application is the design of supersonicnozzles where the outlet Mach number is enforced but the inlet conditions and inter-nal ow boundaries are iteratively solved for until the desired outlet Mach numberis matched.2.1.5 The Symmetrizing VariablesIn [27], it is shown that hyperbolic systems of conservation laws are symmetriz-able given the necessary condition that there exist an entropy function which itselfsatis�es an additional conservation law. The hyperbolic conservation law in onespace dimension is of the form ut + fx = 0

26In quasi-linear form, this is ut +A(u)ux = 0where the Jacobian matrix A is de�ned as A = @f@u . If s and f s are the entropyfunction and entropy ux, the question is whether we can writest + f sx = 0The condition set forth in [27] is that the following relation must hold.suA = f suFor any system strictly larger or equal than two dimensions, the above equality is welldetermined only if the Jacobian matrix is symmetric Aij = Aji. The entropy functionis particularly important since it di�erentiates between solutions that are physicallymeaningful and those that are not. An additional bene�t of symmetrizability is thatit provides a su�cient condition for the stability of the system.We will now proceed to express the Euler equations, a hyperbolic system ofconservation laws, in terms of symmetrizing variables. Apparent in this form, morethan any other, is the complete directional symmetry of Euler. As a rule of thumb,when developing new schemes or update formulas for solving the Euler equations,directional symmetry should be a �rst check.@q@t + �A@q@s + �B@q@n = 0 (2.10)where @q = 2666666666664@p�a@qq@�@S

3777777777775 �A = 2666666666664q a 0 0a q 0 00 0 q 00 0 0 q

3777777777775 �B = 26666666666640 0 a 00 0 0 0a 0 0 00 0 0 0

3777777777775

27The symmetrizing variables for the Euler equations are not explicitly used in ourapproach but they are a good starting point for preconditioning the system (moreon this is 2.3).2.2 Steady-State Euler in Diagonal FormThe point of view that is followed in this work is that many simple problems areeasier to solve that one di�cult problem. From there stems the motivation to reduceEuler into simple subsystems so that we eventually solve equations with which weare very familiar. This leads to the diagonalization already shown in 2.8.@f@x + @g@y = 0 , An@vn@s +Bn@vn@n = 0, @vn@s +An�1Bn@vn@n = 0, @vn@s + L�1�L@vn@n = 0, @w@s +�@w@n = 0L is the matrix of left eigenvectors shown below, � = diag h0; 0; 1� ;� 1� i is the ma-trix of eigenvalues with only non-zero diagonal terms, and @w is the characteristicvariables given in 2.9. L = 26666666666640 0 ��q2� 10 0 �q2� 1�a2 0 0 10 �q 0 1

3777777777775For supersonic ow, when � = pM2 � 1 is real, the system is completely diagonalizedand will be written as four independent scalar advection equations, two for entropyand enthalpy and another two for the acoustic variables. This formulation lends itselfperfectly to multidimensional upwinding methods. One such family of schemes is

28 uctuation-splitting (see description in chapter III) which is based on the calculationof a residual on each cell and its distribution to nodes in an upwind manner. Theorder of accuracy, monotonicity and conservation properties of such schemes dependson the choice of distribution.When the Mach number crosses M = 1 and the ow becomes subsonic, two ofthe eigenvalues become complex: � = �i�? = �ip1�M2. This signi�es that partof the problem is now elliptic so the Euler equations will decompose di�erently in thesubsonic case. The entropy and enthalpy advection equations remaining identical,the acoustic part becomes2664 1 �q2�i�?1 � �q2�i�? 3775 2664 @p@� 3775s + 2664� 1i�? ��q2�? 2� 1�i�? �q2�? 2 3775 2664 @p@� 3775n = 0 (2.11)To simplify the system further, take the real part of either equation to be zero.Alternatively, one could also take the imaginary part to be zero, both will producethe same outcome which is the following 2� 2 elliptic subsystem.2664 �?2 00 �q2 3775 2664 @p@� 3775s + 2664 0 ��q21 0 3775 2664 @p@� 3775n = 0 (2.12)A form similar to equation 2.12 also exists for M > 1, easily obtained using thelast two equations of 2.8. The only di�erence from its subsonic counterpart is that�? should everywhere be substituted by �. An eigenvalue analysis shows that forMach number greater than 1, the coupled acoustic subsystem goes from elliptic tohyperbolic. Even though equation 2.12 mathematically behaves di�erent on eitherside of M = 1, a scheme could be designed which provides a single discretization,one that covers all ranges of Mach number. The least-squares scheme does exactlythat, producing solutions which match seamlessly at M = 1 (see IV). However, it is

29not optimal to treat hyperbolic equations the same way as elliptic ones: upwindingmust be used to take advantage of their advective character.The subsonic acoustic equation 2.12 can be cast into a Cauchy-Riemann typesystem by replacing @u = �q2@� and @v = �?@p, and transforming into a stretchedcoordinate system @� = �?@s and @� = @n.2664 1 00 �1 3775 2664 @u@v 3775� + 2664 0 11 0 3775 2664 @u@v 3775� = 0 (2.13)In past work ([47]), the acoustic system was discretized using Lax-Wendro� (sec-tion 4.1), essentially a central scheme with additional arti�cial viscosity terms. Thistreatment relies on the non-physical assumption that there are preferred directions ofpropagation and is contrary to the correct nature of elliptic equations. In this work,our main issue has been the design of a scheme which mimics the omni-directionalpropagation of information. This e�ort has lead to the least-squares scheme (section4.2) which solves equation 2.12 in a physically correct way. Actually, in chapter IVwe see that when solving Cauchy-Riemann on a uniform grid, Lax-Wendro� updatesare equivalent to least-squares updates but with the addition of advective terms.2.3 System Decomposition, a By-product of Local Precon-ditioningUnlike the steady case where the Euler equations are mixed elliptic and hyper-bolic, the time-dependent version is purely hyperbolic and the decomposition is notas straightforward anymore. That problem was laid to rest in [47] and [56] withthe use of local preconditioning. The advantages of preconditioning in relation toconvergence acceleration and incompressible ows were brie y discussed in chapterI. As its understanding developed for those purposes, it was discovered in parallel

30that it also uniquely splits the unsteady Euler equations into simple homogeneousparts. Coincidentally, the preconditioner of van Leer-Lee-Roe which gives an optimalcondition number [40] also produces the desired decomposition. Because of di�erentmathematical behaviors, the matrix that decomposes supersonic Euler will not bethe same as for the subsonic case. Starting with the governing partial di�erentialequations in symmetrizing variables (equation 2.10), modify the spatial operators asfollows @q@t +P �A@q@s + �B@q@n! = 0 (2.14)As the transient approaches the steady-state solution, time derivatives go to zero,ensuring that the system reduces to the original stationary equations. Therefore,modi�cations brought to the system a�ect the transient solution only. Using thenotation D = L�A�1P�1L�1 and L�1�L = �A�1�B, the system is transformed toD@w@t + @w@s +�@w@n = 0 (2.15)The diagonal matrix � and the vector @w are identical to the ones given in equation2.9.In supersonic regimes, matrix P is constrained such that the Euler system willbe diagonalized and all waves travel at the same speed. These conditions uniquelydetermine the preconditioner to be exactly the one derived by van Leer, Lee, Roe forconvergence acceleration. The resulting diagonal matrix D which weights the timederivatives follows immediately.P = 2666666666664

M� � 1� 0 0� 1� 1 + 1M� 0 00 0 �M 00 0 0 13777777777775 D = 2666666666664

1� 0 0 00 1� 0 00 0 1�a 00 0 0 1�a3777777777775 (2.16)

31Since each advection equation is independent and requires upwinding with forwardtime integration, scalar uctuation-splitting would be an appropriate choice.For the subsonic regime, a similar approach is taken although we are now lookingfor only two advection equations and an elliptic subsystem. The subsonic precondi-tioner is constrained accordingly and in [47] it is derived to beP = 2666666666664

�M2�? � �M�? 0 0� �M�? � + ��? 0 00 0 �? 00 0 0 �3777777777775 (2.17)

A quantity � is introduced in the preconditioner to �x problems arising at the stag-nation points. Numerical experiments showed that the preconditioned system ofequations exhibited instability in those regions and therefore did not converge. Theproblem is associated to two sources. One is discussed in [14] and is related to thedegeneracy of the system eigenvectors in the limit of M ! 0. The other is thatwithout � = fcn(M), the preconditioner is highly sensitive to the variations of the ow angle in the neighborhood of the stagnation point. Taking these issues intoconsideration, the time-dependent subsonic Euler equations preconditioned by 2.17decompose to (@t + �q@s)S = 0(@t + �q@s)h = 02664 �?@p�qq@� 3775t + 2664��q�? 00 q�?3775 2664 �?@p�qq@� 3775s + 2664 0 �qq 0 3775 2664 �?@p�qq@� 3775n = 0It is evident that unsteady Euler decomposes in terms of the natural variables 2.7,much the same way as in the steady case. One di�erence is that the subsonic acoustic

32equations have time derivatives augmenting the Cauchy-Riemann part, making themhyperbolic by nature. In the past, this system was treated using a pseudo-timemarching scheme such as Lax-Wendro�. The correct approach to solving the subsonicacoustic part of Euler is to apply an elliptic solver, one that is customized to discretizeCauchy-Riemann in its original form. With the advent of the least-squares scheme,this is now possible and further discussed in chapter IV.Another important aspect of the decomposition in advective and acoustic partsis that the latter fully describes irrotational ow. Therefore, we must ensure thatour Euler solver accurately recovers potential ow in the case of zero vorticity �elds.2.4 Potential Flow, a Building Block for EulerSuccessful methods for steady compressible potential ow equations are well es-tablished and documented in the literature. The extensive array of techniques thathave been proposed usually validate each other and produce results to a high de-gree of accuracy. One familiar method is the �nite-di�erence discretization of thesmall-perturbation equation 2.18 of which a good account is given in [31]h1�M12 � ( + 1)M12�xi�xx + �yy = 0 (2.18)The equation above is the non-conservative form, � is the velocity potential andM1 is the in ow Mach number. There also exist popular techniques based on theweak formulation which forms the basis for �nite-element [16] and �nite-volume [9]discretizations. For any smooth functionW , the weak form of the potential equationin conservation form is obtained after multiplication by W and integration over thecomputational domain .Z �~r� � ~rWd� I� ��nWd� = 0 (2.19)

33In equations 2.19, � marks the boundary of domain and n is any direction normalto it.We suggest yet another approach to the solution of potential ow equations butour aim is not as much novelty as creating a framework to test our ideas for the moredi�cult Euler equations. The basis of our procedure is that potential ow shouldsimply be a special case of the Euler equations which occurs when the ow hasuniform enthalpy and entropy everywhere in the domain. With these assumptions,compressible potential ow is clearly a subset of Euler ow, directly governed by theacoustic part only. q@S@s = 0q@h@s = 02664(1�M2) 00 �q237752664@p@�3775s + 2664 0 ��q21 0 3775 2664@p@�3775n = 0 (2.20)The same argument should be made in the reverse direction, in other words, an Eulercode must be able to preserve a potential ow solution in the absence of vorticity.Much of this research has been steered in that direction since, with the exception of[32], Euler solvers that rigorously reduce to potential ow solvers are non-existentin research and industrial codes. Chapter V describes how an Euler discretization isforced to produce potential ow results in the irrotational limit.It is possible to further reduce equations 2.20 using the basic potential assump-tions. Irrotationality means isentropic and isenthalpic ow along streamlines, andresults in the following relations.h = ho , e + p� + 12q2 = eo + po�o, � 1 p� + 12q2 = � 1 po�o

34, 2 � 1p+ �q2 = 2 � 1po ��o, �q2 = 2 � 1po ppo! 1 � 2 � 1p, �q2 = 2 � 1 �p1� 1 o p 1 � p� (2.21)where we used the the assumption of constant enthalpy along streamlines ppo = ���� .The same condition also yields � 1 p� + 12q2 = � 1 po�o , a2 � 1 + 12q2 = ao2 � 1, q2a2 = 2 � 1 ao2a2 � 2 � 1, M2 = 2 � 1 ao2a2 � 1!, 1�M2 = + 1 � 1 � 2 � 1 pop ��, 1�M2 = + 1 � 1 � 2 � 1 ppo! 1� 1 (2.22)Substituting 2.21, 2.22 in 2.20, we derive a particular form of the compressible po-tential ow equations which only have static pressure and ow angle as unknowns.2664 +1 �1 � 2 �1p 1� 00 2 �1 �p 1 � p�3775 2664@p@�3775s + 2664 0 � 2 �1 �p 1 � p�1 0 3775 2664@p@�3775n = 0Equation 2.8 already shows how the steady Euler equations diagonalize in thesupersonic regime. The �rst two scalar advection equations that form the acousticpart will fully determine the supersonic potential ow equations. Hence, potential ow for M > 1, can be written in characteristic form.@w�@n � �@w�@s = 0 (2.23)where @w� = � � �q2@p + @�. With the further assumption of constant entropy andenthalpy along streamlines, the Riemann invariants w� can also be expressed purely

35in terms of the ow angle � and the Prandtl-Meyer function �(M). Indeed, forsupersonic potential ow w� = � � �(M). In this notation, equation 2.23 becomesdw� = d (� � �(M)) = 0 along dsdn = ��which will be used for supersonic regions only. In subsonic regime, we revert backto the Cauchy-Riemann type system. Even though the equations are quite di�erent,they only depend on two independent variables, the velocity components.

CHAPTER IIIFLUCTUATION-SPLITTING SCHEMESFluctuation-splitting (FS), alternatively called residual-distribution, is a cell ver-tex discretization technique which originally aimed at a truly multidimensional treat-ment of scalar advection equations and hyperbolic systems. These methods are ma-ture for steady-state solutions and their time-accurate version is still a topic of activeresearch. As such, all the transient solutions produced by the schemes described herehave no meaning, only the �nal solution is useful. The term ' uctuation-splitting'is usually associated to multidimensional upwind techniques. In this thesis, we willloosely think of FS in a more general sense, one where the distribution does not nec-essarily need to be upwind. This broader de�nition allows us to include distributivemethods for parabolic problems which are considerably more di�usive.We look at some upwind FS schemes (N, LDA, PSI) used for scalar advectionequations. These have low dissipation and depending on the properties satis�ed,they permit sharp, oscillation-free shock capturing. Some more familiar schemes(Lax-Wendro�, SUPG) may be cast in the FS framework when the residual is splitcentrally instead of a purely upwind fashion. The application of FS schemes tohyperbolic systems follows from the direct extension of the scalar case. Much ef-fort has recently been dedicated to develop system FS (or matrix FS) schemes for36

37hyperbolic systems of conservation laws and successes include [81] and [8] for theEuler equations and [59] for the shallow water equations. As it is probably clear bynow, we believe that such a bulk treatment of the cell uctuation is inappropriatebecause it fails to make a distinction between the elliptic and hyperbolic portions.A more subtle approach is to split the cell uctuation along those lines and treateach sub- uctuation separately. In this presentation, we do not emphasize upwindmatrix distribution schemes because the only hyperbolic problems we encounter arescalar advection equations. However, when it comes to solvers for the 2� 2 ellipticsubsystem, we formulate them (Lax-Wendro�, least-squares) as system FS schemes.3.1 Description of Fluctuation-SplittingWhat all residual distribution approaches share in common with each other is thatthe computational domain is divided into elements in a structured or unstructuredway (�gure 3.1). In our case, the elements will be triangles in 2-D and tetrahedra in3-D. FS schemes have also been developed for arbitrary quadrilaterals in 2-D [80].Unstructured meshes around complex geometries are usually preferred because theyare easier to generate than body-�tted grids and use simpler boundary conditionsthan cartesian meshes. Whatever loss of accuracy is induced by the geometry mustthen be compensated by higher �delity solvers.3.1.1 Fluctuation CalculationConsider the scalar advection equation as our model problem.ut + ~a � ru = 0 (3.1)When solving 3.1, we are actually discretizing the equivalent equation 3.2 with dif-fusion terms on the right-hand-side which vanish in the steady state. Usually, this

38

Structured set of triangles

around node i

i

T

T

T

i

i-1

i+1

Unstructured set of triangles

around node i

i

T

T

T

i

i-1

i+1

Figure 3.1: Samples of structured and irregular triangular meshes.modi�ed equation is used to analyze the particular errors introduced by a certaindiscretization, but here, it will serve a di�erent purpose. Section 3.1.4 uses 3.2 as aninstrument to present interesting geometrical interpretations of some FS schemes.ut + ~a � ru = (~d � r)(~a � r)u (3.2)Iteration toward a steady solution takes place from an initial guess which maybe chosen arbitrarily. A common practice for treating the initial conditions is to setall unknowns equal to either the far�eld in ow conditions or the velocities to 0 andnon-dimensionalized pressures and densities to 1. Quantities of interest are storedonly at the nodes and the solution is approximated by a continuous piecewise linearrepresentation on each triangle. For steady-state problems,u(x; y) = 3Xi=1 uibi(x; y)where bi is the standard linear �nite-element basis function and ui is the vertexvalue. We compute an average residual or uctuation (the two terms will be usedinterchangeably) �T for each cell T , and then change the current solution at each

39node by an amount proportional to �T .�ui = �! �T;i�T (if node i belongs to cell T ) (3.3)where ! is a relaxation factor. The distribution coe�cient �T;i is the fraction of theresidual which is aimed at node i. Intuitively, the notion of a residual is the measureof how 'far' we are from the converged solution. More precisely, �T is de�ned as thecell area integral of @u@t or, equivalently, of the ux divergence.�T = Z ZT ut dS= � Z ZT ~a � ru dS= �ST ~a � ru= � 3Xi=1 kiuiwhere ki = 12 ~a � ~ni and ~ni is the outward vector normal to the edge opposite node iscaled by the length of that edge. It is clear that the residual is a linear combinationof vertex values ui=1;2;3. Using the property P3i=1 ki = 0, it may also be expressed asdi�erent permutations of ui.�T = �k1(u3 � u1)� k2(u3 � u2)= �k1(u2 � u1)� k3(u2 � u3)= �k2(u1 � u2)� k3(u1 � u3)The sign of ki is an indication of ow direction, so ki > 0 means positive in ow forthe side opposite node i and ki < 0 signi�es out ow.3.1.2 Fluctuation DistributionHaving used the vertex values to calculate the uctuation, the �rst step is nowcomplete. The next step is to distribute that uctuation back to the vertices in the

40i

TiT

i-1

Ti+1

Patch of cells around node i.

The shaded area S is the median

dual cell around node ii

Si

A

BF

E

C

Figure 3.2: Patch of cells around node i showing the median dual cell area.form of signals. Even though all FS schemes share the same residual calculation, itis in the distribution of �T that they di�er from each other. The update at meshpoint i may be cast into the following formun+1i = uni + �tSi XT2Ti �T;i �T (3.4)The summation extends over a patch of cells Ti surrounding node i. Si is the areaof the median dual cell around node i. For some choices of �T;i, the scheme may beupwind since no contribution is sent to upstream nodes. Figure 3.3 shows the twopossible con�gurations that can occur. In the one target case (type I), there is onlyone in ow side and the entire uctuation is sent to the unique downstream node. Inthe two target case (type II), the uctuation is split between the two downstreamnodes.For other choices of the distribution coe�cients, we can recover familiar schemesthat have served as workhorses in the �nite-di�erence and �nite-element communi-

411

3

22

3

1a

a

Type I

Upwind FS schemes

Type II

Figure 3.3: One and two-target scenarios.ties. In particular, we will see in a later section how Lax-Wendro� and SUPG may becast in the language of uctuation-splitting. The distribution coe�cients then con-tain a central term and a dissipation term; therefore all three nodes are adjusted nomatter what the direction of the characteristic. The parallel between �nite-elementand upwind FS schemes is natural since they share the same compact stencils andthe same linear data representation.The residual can be distributed to the nodes of a triangle in many ways but toinsure satisfactory shock-capturing properties and accuracy, the splitting must meetsome conditions. The properties that are designed into FS schemes are described inthe next section.3.1.3 Desired PropertiesContinuityContinuity requires that the distribution coe�cients �T;i be continuous for changesin advection speed ~a as well as for changes in the solution. This property comes into

42play only in non-linear FS schemes where the distribution coe�cients are limited andthe limiting functions contain logical operations.Linearity PreservationAlso known as the residual property in the �nite-element context, linearity preser-vation is related to the accuracy of the scheme. It requires that the scheme preservethe exact steady state solution when this is a linear function of the space coordinatesx and y for an arbitrary triangulation of the domain. A necessary and su�cient con-dition is that the distribution coe�cients vanish in the limit of �T ! 0. Numericalexperiments reveal that linearity preserving FS schemes are just under second-orderaccurate on irregular triangular grids such as in �gure 3.1. On uniform triangulargrids illustrated in the same �gure, the schemes having this property show at leastsecond-order accuracy ([18]).PositivityPositivity means that every new value un+1i may be written as a convex combi-nation of values at the previous time step. This guarantees a maximum principlewhich prohibits the occurrence of new extrema and ensures stability of the explicitscheme. Equivalently, positivity can be expressed asun+1i = NXk=1ci;kunkwhere N is the number of nodes surrounding vertex i and the coe�cients ci;k are re-quired to be non-negative. This condition e�ectively states that if un+1i is a weightedaverage of the node values around it, then the scheme observes a maximum principle.The above de�nition is called global positivity, it ensures that un+1i is bounded bythe minimum and maximum values of unk in its stencil. There also exists a stronger

43constraint called local positivity which looks at a particular cell instead of all thetriangles of which a node is member. Local positivity ensures that the update sentto each vertex from each mesh simplex has positive coe�cients. The local updatesall add up to produce a global update which satis�es monotonicity requirements.When we mention positivity in the rest of this thesis, we usually mean it in theweaker sense. The stronger condition guarantees oscillation-free results in additionto maintaining compactness.ConservationWhen solving hyperbolic problems, solution quality is often measured by theaccuracy of wave locations and strengths. For homogeneous problems, a necessarycriterion to be observed must then be conservation. Consider the sum of adjustmentsbrought to all nodes, weighted by the median dual cell area Si. Take NV to be thenumber of nodes in the domain, NT the total number of triangles, and T 2 Ti thetriangles surrounding node i.NVXi=1 Si �ui = NVXi=1 �tXT2Ti�T;i �T= �t NTXT=1 3Xj=1�T;i �T (3.5)When the sum of the distribution coe�cients add up to unity, P3i=1 �T;i = 1, wehave a conservation property of type I. The right-hand side of equation 3.5 may besimpli�ed by expressing cell uctuations in terms of contour integrals. It is clear from3.6 that when uctuations are summed over all triangles, contributions from commoninterior edges cancel. We then say that the uctuations �T telescope, meaning thatthat only e�ects from the domain boundary � are present.NTXT=1�T = NTXT=1 IT u ~a � d~n = I� u ~a � d~n (3.6)

44Combining 3.5 and 3.6, the net change to all nodes depends only on boundary terms.The same result can be obtained for the non-linear Euler equations given that theyare conservatively linearized as in section 5.1.NVXi=1 Si �ui = �t I� u ~a � d~nWhen the sum of the distribution coe�cients add up to zero, P3i=1 �T;i = 0, wehave a conservation property of type II. In this case, no interior triangle changes theinterior nodes, and all e�ects are due to the boundary.NVXi=1 Si �ui = 03.1.4 Geometrical InterpretationIn this section, the modi�ed equation 3.2 will be used to investigate the geomet-rical interpretation of FS schemes (see [68]). Consider a direct FS type discretizationof that equation on the patch of triangles shown in 3.2. We chose triangle ABi in�gure 3.4 as our representative cell and calculate the portion of the residual whichwill contribute to node i.Z ZFCEi ut dS = � Z ZFCEi~aru dS + Z ZFCEi(~d � r)(~a � r)u dS= � I u~a � d~n+ I (~a � ru)(~d � d~n)= 13 �ABi + 12 �ABiSABi ~d � ~ni (3.7)In equation 3.7, we converted the surface integral to a contour integral using thedivergence theorem. The �rst term is simply a third of the total cell uctuation,since area FCEi is a third of cell ABi. The second term follows from the fact that~FE = 12 ~AB and also that terms from the integration along Fi and Ei cancel out

45A

B

i

E

F

C

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nA

nB

ni

A

B

i

C

a

d

D

Figure 3.4: The fraction of the residual produced in region FCEi contributes to theupdate of node i. This fraction has a geometrical interpretation since itis the ratio of area BAD to BAiwith neighboring cells. Therefore the fraction of residual from cell ABi that goes tonode i is �ABi;i = 13 + ~d � ~ni2SABiThe geometrical interpretation of this result follows and all notation refers to �gure3.4 . For any point D within the triangle that is de�ned by ~CD = ~d, �i is the ratio ofareas BAD to BAi. Actually the set (�A; �B; �i) corresponds to the homogeneouscoordinates of point D with respect to the triangle BAi. So we also have �A =BDi=BAi and �B = AiD=BAi. Some special choices for the position of point Dand therefore of vector ~d reproduce familiar schemes.For all upwind schemes, the choice of ~d depends on whether we have a one ortwo-target situation. In the case of two in ow sides (type II triangle), point D willlie on the third edge and could be one of the two vertices forming that edge. In theother scenario (type I triangle), point D will coincide with the out ow vertex. For uctuation-splitting schemes that are not upwind, the vector ~d = ~CD will point to

46A

B

i

aD

C

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d

Figure 3.5: Geometrical interpretation of the LDA scheme.somewhere in the interior of the triangle. In a geometrical sense, what distinguishesvarious schemes apart from each other is the location of point D. In the next couplesections we study upwind methods (LDA) as well as schemes that come from acentral-di�erencing (Lax-Wendro�) or �nite-element (Petrov-Galerkin) heritage. Wesee how they di�er from each other in their particular choice of vector ~CD.3.2 Upwind Schemes3.2.1 Low Di�usion AWhenever there are two in ow faces, point D is placed on the out ow edgewhere the local characteristic crosses it. Otherwise it is placed on the out ow node.Following the same geometrical reasoning as in the previous section and using �gure3.5 as illustration, the fraction of the uctuation that gets distributed to downwindnodes A and B is 8>><>>: �A = BDiBAi �T�B = DAiBAi �T () 8>><>>: �A = kAki �T�B = kBki �T (3.8)

47Equivalently, the LDA distribution may be expressed in terms of the ki as shownabove. If the vertices of a triangle are labeled i = 1 � � �3, a general expression whichis compactly implemented in computer programs would be�T;i = max(0; ki)P3i=1max(0; ki)LDA is linearity preserving since individual updates �T;i �T vanish when the cellresidual �T does. Indeed, numerical results performed in previous work con�rm thatLDA is second-order accurate on a uniform mesh. The LDA scheme is not monotonesince linear schemes cannot be both linearity preserving and locally positive. This isa corollary to Godunov's theorem.3.2.2 N Scheme and its Limited VersionsThe N scheme is another FS schemes which provides linear distribution coe�-cients. The idea behind the N scheme is that an optimal scheme with Narrow stenciland Narrow discontinuity capturing follows from a natural splitting of the advectionvector along some local coordinates. This results in splitting the advection vector ~ainto two components parallel to the sides opposite the downstream vertices. Usingthe notation of �gure 3.6, we have ~a = ~aiA+~aiB. Accordingly, the signals sent to thetwo vertices are the new uctuations that are calculated from the now split advectionproblem. �T = �A�T + �B�T= Z ZT (~aA � r)u dS + Z ZT (~aB � r)u dS (3.9)In compact form, for a general triangle with vertices labeled i = 1; 2; 3, the distribu-tion coe�cients of the N scheme take the form�T;i = �i�T where �i = � max(0; ki)P3l=1max(0; kl) 3Xj=1min(0; kj)(ui � uj) (3.10)

48The N scheme satis�es local positivity constraints on each patch of triangle arounda node, provided that the time step for marching in the advection direction satis�esthe condition �t � min �SAkA ; SBkB �, where SA, SB are the dual cell areas for nodes Aand B. Hence global positivity is ensured as well. However the scheme is not linearitypreserving: the signals destined to the vertices may become equal and opposite, butthey do not necessarily vanish when the cell residual becomes zero. To gain accuracywhile maintainingmonotonicity, the distribution coe�cients must be made non-linearfunctions of the nodal states.Various non-linear schemes have been suggested, �rst in [65] and more recentlyin [73]. In both these references, the authors discuss the NN scheme and the levelscheme. A more widely used non-linear method is the Positive Streamwise Invariant(PSI) scheme, developed and discussed in [75]. The idea behind these schemes isthat the residual remains the same if we add to the scaled advection velocity �~a acomponent of the advection velocity parallel to the local solution gradient ~ak. So themodi�ed ki will now be de�ned aski = 12 (~a � ru) � rujruj2 + �~a!~niThe residual is unchanged with this modi�cation but the signals sent to the nodesare such that the scheme can now be linearity preserving. As the understandingof these schemes matured, [73] indicated a common ground between limited �nite-volume and non-linear FS schemes. It turns out that each non-linear FS scheme canbe written as a limited version of the N scheme. For example, PSI is equivalent tothe N scheme limited with minmod (see [56]). The minmod limiter may either beexpressed as (r) = max(0;min(r; 1)) or as (r) = 12(1+ sign(r))min(r; 1). Building

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iAFigure 3.6: Geometrical interpretation of the N and PSI schemes.on equation 3.10, the distribution coe�cients for PSI are�T;i = �i�T !The limiting bounds �T;i between 0 and 1, hence enforcing linearity preservation.Note that local positivity properties are preserved with the addition of the minmodlimiter, for the same time-step restrictions as for the N scheme.The geometrical explanation of the N scheme is not as elegant as for LDA becauseequation 3.9 does not point to a clear position for D. Due to the upwind nature ofthe N scheme, point D must lie somewhere on the line de�ned by the out ow edge.Because of the limiting e�ect, point D will be bounded to the out ow segment for thenon-linear versions of the N scheme. However, the geometrical interpretation stopsthere as the direction of the vector ~d = ~CD may vary from cell to cell depending onthe state of the unknown variables at each node.

503.3 Other Familiar SchemesIn the family of residual-distribution schemes, the fully upwind methods havebeen recognized to produce very satisfactory results for hyperbolic problems. Un-fortunately, most uid problems are not purely convective in nature so for thosesituations involving mixed physics, one must return to more symmetric methods.Some of these familiar schemes had already been cast in the residual-distributionframework and used in research codes and applications to a certain degree of suc-cess. The techniques closest to our discussion are probably the ones developed byJameson [37], Hall [30], Morton [49] and Ni [52]. Here we look at the Lax-Wendro�,SUPG and least-squares which are all linearity-preserving but not positive.3.3.1 Lax-Wendro�The distribution coe�cients of the standard Lax-Wendro� scheme follow fromthe identity Si�ui = Z ZSi �t~a � ru� 12�t2(~a � r)(~a � r)u dS (3.11)Note that the above equation follows from 3.2 with the choice ~d = 12~a�t whichproduces the unique time-accurate version of Lax-Wendro�. Integrating over themedian dual cell area leads to a central (equidistribution) term supplemented withdissipation which is proportional to the time step. In this case, there is a single CFLnumber, �c, which is dependent on the characteristic size of triangle edges.�T;i = 13 + 12�t kiSTWhen time-accuracy is not a concern, there exist two distinct CFL numbers, �c whichdetermines the amount of dissipation and �n which drives the overall evolution ofthe scheme. Solutions produced by Lax-Wendro� are dependent on the ratio of �n

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Figure 3.7: Geometrical interpretation of Lax-Wendro� (left) and SUPG (right). ForLax-Wendro�: since point D is in the interior of the triangle, �T;i will benon-zero. For SUPG: this version of SUPG is truly upwind since pointD is on the out ow edge.to �c and this sensitivity accentuates on arbitrary grids. Stability analysis for theseCFL numbers is very di�cult on an unstructured mesh but [69] derived the followingbound on uniform stencils. �n�c � 12No matter if we look at the time-accurate version or not, the typical Lax-Wendro�scheme places point D on the interior of triangle ABi, with ~CD parallel to ~a. There-fore, all three nodes usually receive an update and the scheme will not be purelyupwind.3.3.2 Streamwise-Upwind Petrov GalerkinFor the classical Galerkin �nite-element method it can easily be shown that thedistribution coe�cient is �T;i = 13 , which is not surprising since this method isknown to be a central type discretization. However, a simple equidistribution of the uctuation is unstable.

52One version of SUPG places the integration point D such that ~CD = �~a. Thismight seem similar to Lax-Wendro� but the contrast is that � can be uniquelychosen such that only downwind nodes are updated (see �gure 3.7). We then obtaindistribution coe�cients which are related to those of LDA:�T;i = 13 �1� �T;iLDA� = 13 1� max(0; ki)P3i=1max(0; ki)!Hughes ([35], [36]) suggests yet another version of SUPG that is similar to theprevious but with an additional dissipation term. The extra damping helps curboscillations near discontinuities, therefore permitting better shock capturing. Geo-metrically, point D is chosen such that ~d = �1~a + �2~ak where ~ak is the projection ofthe advection vector ~a onto the direction of the local gradient of the solution. Thetime-like terms �1 and �2 must be appropriately chosen to satisfy stability criteria.Note that for �1 = 12�t and �2 = 0, this scheme reduces precisely to Lax-Wendro�,while with �1 = 0 and �2 = 12�t we have Lax-Wendro� based on the advection speed~ak. The distribution coe�cients for this method can be expressed as�T;i = 13 + �1 kiST + �2 (ki)kSTwhere (ki)k = 12~ak � ~ni.3.3.3 Least-SquaresThe motivation behind the development of the least-squares (LS) scheme wasto obtain an elliptic solver which could easily be integrated with FS type schemes.In order to correctly solve the Cauchy-Riemann equations, least-squares has to dis-tribute the residuals in a symmetric way. One way to achieve this is to drive cellresiduals as close to zero as possible, through minimization. Even though LS targetselliptic equations, it is not restricted to a speci�c type of partial di�erential equation

53(p.d.e.). Let's consider the time-dependent system where F is a �rst-order di�er-ential operator. The p.d.e. is locally linearized in each element using whatever setof variables we chose to store at vertices. We do not require to write the system inany particular form because our aim is to create a method that exploits the di�erentproperties of the di�erent sets of unknowns. Hence, let u be the set of unknowns weselect, ut + F(u) = 0Just as before, de�ne the uctuation by the cell area integral of ut,�T = Z ZT ut dS = Z ZT �F(u) dSA simple analysis inspired by [68] shows that individually minimizing all cell uctuations is an over-determined problem. Denote by NE = NEi + NEb , NV =NVi + NVb and NT respectively the number of edges, vertices and triangles in ourplanar domain, (�)i and (�)b being interior and boundary contributions. Then theyare related by the formula NE = NV +NT � 1 (3.12)For each interior edge there are two associated triangles and for each boundary edgethere is only one so the following relation holds2NEi +NEb = 3NT (3.13)Combining equations 3.12 and 3.13 and rearranging, one obtainsNT = 2NVi +NVb � 2 (3.14)When solving an m � m system, there are mNV unknowns, a number m at eachvertex. If we were to individually drive a prede�ned norm of the uctuation down to

54zero in each cell, there would be NT conditions. The di�erence between the numberof unknowns, mNV , and the number of equations, NT , determines the number ofboundary conditions, NBC , that must be imposed.NBC = mNV �NT= mNVi +mNVb �NT= (m� 2)NVi + (m� 1)NVb + 2 (3.15)In the case of Cauchy-Riemann (m = 2), at every point on the boundary we imposethe two components of velocity. So with 2NVb conditions, the problem will be over-determined for any computational domain with more than two boundary nodes.Individually driving a prede�ned norm of the uctuation down to zero in each cellis not the right approach. Instead, the best we can hope to achieve is to minimize acertain norm of �T over a patch of cells surrounding node i (see �gure 3.1). De�ne�i as the function to be minimized. It is the sum over all cells Ti surrounding nodei of local norms depending on �T .�i = XT2Ti 1ST (�T )tQ (�T ) (3.16)For Cauchy-Riemann, when minimizing �i with respect to velocity components ateach node, we have 2NVi equations and with the addition of 2NVb boundary condi-tions, the problem becomes well-posed.The square matrix, Q, in expression 3.16 is symmetric and positive de�nite. Itplays an important role in LS since it fully determines the functional to be minimized.In the case of a system, Q assigns relative weights to the di�erent equations. Indeed,with the choice Q = DtD, we are essentially performing a LS minimization of D�T ,which is some weighted combination of the residuals. Also, through a scaling factor,

55the matrix Q weights the errors in each triangle relative to the other. For example,if there is any reason to believe that a small triangle should contribute more to thesolution than its area permits, the weight of that triangle is increased. If there is aneed to combine the LS method with another FS scheme, one can make Q come intoplay by either neutralizing or emphasizing LS.We must implement an iterative procedure that �nds ui which minimizes �i. Atthat point, the solution is converged and �i no longer depends on the state of node i.We expect each local minimization to produce a steady-state solution ui which will bepart of the correct global solution 1. This point could be the subject of more rigorousanalysis because solving a number of local problems does not necessarily guaranteea global solution. However, with constraints matching the number of degrees offreedom and a unique global solution, we believe that our problem is well-behaved.To drive the norm in 3.16 to a minimum, we must calculate its gradient withrespect to ui. Assume that Q is independent of the unknown variables u. Further-more, we do not allow grid points to move so cell areas are constant. The vector ofgradients is then @�i@ui = XT2Ti 1ST (�T )tQ @�T@ui (3.17)The vanishing of the above sum would signify that a local equilibrium has beenattained. At each iteration, signals are distributed to vertices such that we movedownward along the gradient. Contributions to a certain node come from all trianglesthat share it as a vertex. The adjustment due to a single triangle T is�ui = �(�T )tQ @�T@ui (3.18)1This is an example of what's called in computer science a greedy algorithm. In each locationor during each phase, a decision is made that appears good, without regard to global or futureconsequences

56Each change takes the state vector ui one step closer to convergence, and at steady-state, a �nal ui is found such that �i over the patch of cells surrounding node ihas been locally minimized. Note that for each triangular element, the net sum ofadjustments going to the three vertices will vanish. This is another expression of theconservation property (type II) seen earlier.3Xi=1 �ui = 0The acoustic part of subsonic Euler is a Cauchy-Riemann type system so it car-ries information in an isotropic manner. On a symmetric stencil, LS minimizationprovides symmetric weights so it represents the correct treatment for elliptic prob-lems. LS is e�ectively a purely di�usive scheme, hence it suits the heat equationut = c(uxx+ uyy) perfectly. Since error reduction takes place through damping only,its greatest advantage is when used in combination with some multigrid acceleration.Multigrid is known to work best with schemes that act as smoothing processes.When applied to advection problems, LS in its standard form produces poor re-sults because of excessive dissipation and oscillations near discontinuities. AlthoughLS is not optimal for solving advection equations, it is interesting to look at itsdistribution coe�cients in that case:�T;i = 12 !ST ki (3.19)where ! can be thought of as a pseudo-time parameter or relaxation factor. Thisshows that LS acts through a purely di�usive process. In fact, �T;i for LS is almostidentical to the the dissipative part of Lax-Wendro� or SUPG.

573.4 Matrix Fluctuation-SplittingThe matrix version of FS schemes follows from a direct extension of the scalarcase. Consider the following hyperbolic systemut + fx + gy = 0 , ut +Aux +Buy = 0 (3.20)Any attempt to perfectly replicate 3.20 in discrete form is vain. Because of theunavoidable di�usion intrinsic to any scheme, the equivalent equation which we wouldbe solving is ut +Aux +Buy = (C@x +D@y) (Aux +Buy)Note that the above equation closely resembles its scalar counterpart, i.e. expression3.2. Since the right-hand side vanishes in the steady-state, we recover the originalpartial di�erential equation 3.20. The dissipation term can be expanded asut +Aux +Buy = (CAuxx + (DA+CB)uxy +DBuyy) (3.21)Ignoring the dissipative right-hand side for now, we concentrate on discretizingthe advective part only using matrix-distribution schemes. A detailed account ofthese methods in the context of Euler equations is given in [81] and [8]. The Eulerresidual is conservatively linearized over each cell T using the parameter vector z(see section 5.1). This permits to write the uctuation �T as a linear combinationof the state vectors at each vertex�T = Z ZT utdS= � Z ZT Aux +BuydS= � 3Xi=112 (A;B)~niui= � 3Xi=1Kiui

58where ~ni is the normal vector to the edge opposite node i scaled by the length ofthat edge. The uctuation must then be distributed along directions of preferentialwave propagation to re ect the hyperbolic nature of the system. The matrix Ki issplit in its positive and negative parts� �RK ��K�LK�i = � �RK ��+K�LK�i + � �RK ���K�LK�i= K+i +K�iIn the above equation, �RK and �LK are the right and left eigenvector matrices ofKi, whereas ���K;i and ���K;i are the corresponding positive and negative eigenvaluematrices. The distribution matrices particular to each system FS mimic the scalarcase, and below we provide a couple of examples. The coe�cient matrix for thesystem version of LDA is given byBT;i = K+i0@Xj K+j1A�1For the system Lax-Wendro�, the splitting of the uctuation is according to centralterms in addition to dissipative terms.BT;i = 13I+ �t2ST (A;B) � ~ni (3.22)If we are indi�erent to time-accuracy, a more general expression would beBT;i = 13I+ !h2 Ki0@Xj jKjj1A�1 (3.23)In current Euler solvers which are based on hyperbolic-elliptic decomposition,common practice is to use a hybrid treatment where the matrix PSI is applied insupersonic regions and either Lax-Wendro� or LDA in subsonic cases. Since Eulercan be fully diagonalized for M > 1, the matrix-distribution scheme identically re-covers scalar PSI for each of the four separate advection equations. In the subsonic

59situation however, the acoustic part does not decouple, so matrix-distribution actsas scalar advection for entropy and enthalpy only. We believe that this hyperbolictreatment of the acoustic system is incorrect because of the elliptic nature of thesubproblem. We end up in a situation where the acoustic information is propagatedalong a �nite number of preferential directions instead of omnidirectionally. Thesebiased directions will inherently depend on all variables, including entropy and en-thalpy. Therefore, it is clear that despite a decoupling of the acoustic and advectiveparts at the PDE level, the acoustic updates are allowed to a�ect entropy and en-thalpy at the discrete level. Figure 3.8 (top) shows results found in some recent workthat applies matrix-distribution schemes to the decomposed Euler equations. Eventhough it simulates an irrotational ow where the entropy �eld should be constant,one can clearly see the wake produced due to spurious entropy generation. That�gure is actually a considerable improvement compared to the results (3.9, bottom)produced by multidimensional �nite-volume techniques where the entropy wake is solarge that the ow symmetry is barely recognizable.The alternative suggested in this thesis is to take the PDE decomposition onestep further to the residual level. Splitting the residuals proves to be a more versatileapproach. It provides control at the numerical level all the while permitting the use ofFS schemes. All advective residuals (resulting from entropy, enthalpy and supersonicacoustic equations) are distributed with some scalar upwind FS. Meanwhile, theresidual originating from the elliptic problem is treated with a more di�usive schemesuch as Lax-Wendro� or LS. Various numerical results may be found in section 6.3.2and should be compared to 3.8.

60

Figure 3.8: Matrix-distribution LW-PSI scheme applied to Euler. Potential owaround the cylinder should be perfectly fore-aft and top-down symmetric,but shows an arti�cial wake. This numerical solution provides empiri-cal evidence that, even in the irrotational limit, the acoustic part of theproblem generates entropy.

Figure 3.9: Same numerical experiment, this time conducted with a second-orderupwind �nite-volume method. The symmetry of the ow is barely rec-ognizable.

613.5 A Third-Order ApproachUp to now, the general emphasis of the FS community was to control a scheme'saccuracy with the distribution method alone, not with the actual uctuation calcu-lation. The material �rst introduced in a recent paper [5] provides a fresh view onthis subject. It exposes the importance of uctuation accuracy on that of the schemeitself. The proposition laid out by the authors is that the accuracy of a FS schemeis determined by the accuracy with which the uctuation is calculated over the cell,given that the distribution coe�cients are bounded. This section describes howto compute the uctuation in order to obtain third-order results, for any system ofequations. It turns out that third-order extension comes at very little additional costcompared to computational expenses of a second-order method. As explained below,the only modi�cation is a correction term to the standard uctuation, so the extraoverhead is minimal. If doubling the mesh re�nement causes the solution accuracyto increase eight-fold, we need no other motivation to implement these changes.In order to obtain more accurate estimates of the uctuation, one must recurto a higher order quadrature of the data. Nodes will not only carry solution valuesbut also gradient information. Variables are represented on each cell not by piece-wise linear elements but by linear combinations of cubic Hermite elements. Thereare ten degrees of freedom in a cubic representation so there exist ten fundamentalHermite cubics with which one can completely specify any data in a cell. Workingwith Hermite elements requires storing a total of nine quantities for each cell if wewere solving a scalar equation, three unknowns at cell vertices and six terms sijcontaining gradient information. Each sij is the projections of the average gradient(ru)i onto edge vectors ~sj which claim i as common vertex. There are many ways

62i

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Figure 3.10: The Hermite cubic element.to calculate (ru)i but a simple way is just to average the gradients of u in all thecells surrounding node i.sij = (ru)i � ~sj; (i; j) 2 f1; 2; 3g2; i 6= j (3.24)Note that sij are not treated as unknowns, they are just computed at each iterationbased on the most recently updated variables. After calculating all sij, the next stepin the process is to reconstruct the variable u(x; y) within each cell.Instead of using a cartesian coordinate system (x; y) to represent a variable, weexpress it in terms of the local triangular coordinates (L1; L2; L3). The completecubic element is shown in �gure 3.10. The quantities L1, L2 and L3 are the areas ofthe respective triangles 23i, 31i and 12i normalized by the total surface of the triangle123. Therefore, we must have L1 + L2 + L3 = 1. The Hermite basis functions canbe expressed in terms L1, L2 and L3 as follows� Hi = Lj2(3 � 2Lj) is the basis function that represents the solution at thevertices of the triangle. It has vanishing derivatives at each vertex, unit value

63at vertex j and zero value at both other vertices. See �gure 3.10, 3.11.� Hij = LiLj2 is the basis function that represents the gradients of the solutionat the vertices of the triangle. It has zero value at all vertices, �k ~nik2ST gradientat vertex j and other gradients zero.� Hijk = LiLjLk is the basis function that represents the average value of thesolution at the centroid of the triangle. It has zero value and gradient at allverticesEach variable is then reconstructed on a cell through a linear combination of theHermite basis functions Hi, Hij and Hijk. Here, we write the expression for a scalarbut the extension to a vector variable is straightforward.u(L1; L2; L3) = u1H1 + u2H2 + u3H3 + kH123�s21H31 � s32H12 � s13H23 + s31H21 + s12H32 + s23H13Turning our attention back to the Euler equations, let's consider how we couldapply the Hermite data representation to obtain higher-order estimates of the Eulerresiduals. To reconstruct the cubic data for all four variables on each cell, the totalnumber of quantities held increases to thirty-six, out of which twelve are vertexunknowns. Since the residual may be expressed in terms of an integral over thecell contour, the only useful part of the reconstruction comes from edges. At �rst,that seems awkward since all the information from the interior of the cell goes towaste. We will use the cubic Hermite representation along the cell edges to calculateappropriate line integrals �12, �23, and �31.�T = Z ZT (Awx +Bwy) dS= I Aw dy +Bw dx

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65= �12 + �23 + �31To compute �ij, introduce a coordinate system local to each edge ijx = xi� + xj(1� �)y = yi� + yj(1� �) 9>>=>>;) dx = �xd�dy = �yd�Suppose that we have available the value of the state vector at the midpoint of eachedge, wm. Since we have constructed a Hermite polynomial wH along each edge,wm is easily obtained by evaluating wH at midpoints:wm = wH �12� = �w� 18(Rj �Ri)where (Rj � Ri) represents the di�erence of gradients evaluated at nodes i and jand then projected on side ij (vector called ~sk). In the case of a scalar problem, itjust reduces to (sjk � sik).Rj �Ri = 2666666666664

(rw1)j � ~sk � (rw1)i � ~sk(rw2)j � ~sk � (rw2)i � ~sk(rw3)j � ~sk � (rw3)i � ~sk(rw4)j � ~sk � (rw4)i � ~sk3777777777775Using Simpson's rule on side ij, a new relationship for the edge integral is derivedusing the convention �() = ()i+()j2 and �() = ()j � ()i�ij = Z ji (Aw�y +Bw�x)d�= 16�yA(wi + 4wm +wj)� 16�xB(wi + 4wm +wj)= (A �w�y �B �w�x)� 23 [A( �w�wm) �y �B( �w�wm) �x]= (A �w�y �B �w�x)� 112A(Rj �Ri) �y + 112B(Rj �Ri) �xThe three edge integrals add up to give a net uctuation inside each cell. Thesums are over the three edges of the triangle and �R are di�erences for the particular

66edge considered.�T = X(A �w�y �B �w�x)| {z }standard second� order uctuation � 112 X (A �R �y +B �R �x)| {z }corrective term (3.25)The second term in expression 3.25 simply acts as a correction to the standardsecond-order residual. It should be apparent that third-order accuracy comes atvery little additional cost because the only extra quantity is �R. What limits usto third-order accuracy is the calculation of average nodal gradients rwi, which weestimate at best to second-order on su�ciently smooth grids. If we had the exactvalue of the nodal gradients, we could actually reach higher accuracy (fourth-orderin numerical experiments)In this section we used the Euler system as an illustration, but remember thatall the relationships shown also hold for scalar advection and Cauchy-Riemann. Inas much as the Euler residual is split in entropy, enthalpy and acoustic parts, athird-order estimate must be performed for each portion separately. Section 5.3uses equation 3.25 and tailors it speci�cally to a Cauchy-Riemann type uctuation.Obtaining higher-order accuracy simply through a better estimate of the cell residualis very convenient because the changes to the computer program are minimal. Nomodi�cations are made to the distribution method so �T;i and BT;i particular to eachscheme remain the same.

CHAPTER IVSEARCHING FOR A CAUCHY-RIEMANNDISCRETIZATIONIn chapter III, we saw how various schemes of upwind, central and purely dif-fusive type all unify under the umbrella of uctuation-splitting. Yet in chapter II,we studied how the Euler equations decompose in hyperbolic and elliptic parts, eachexhibiting inherently di�erent physical phenomena. As a result, it would be advanta-geous to integrate these decomposition ideas in the uctuation-splitting framework,hence obtaining separate discretizations for the two parts but under a single updateprocedure. The hyperbolic portion of the system is governed by advective operators,for which discretization and time marching strategies are mature and satisfactory.In this chapter, we turn our attention to the discretization of the elliptic part andstudy potential candidates. An optimal elliptic discretization might or might notexist but our goal is to identify the elements that make one discretization superiorto another. Since the acoustic part of the subsonic Euler equations is of non-linearCauchy-Riemann type, we pick the linear Cauchy-Riemann equations as our modelproblem and attempt to learn the most from them.

67

684.1 A Look at Lax-Wendro�In past work ([47], [56], [3]), the state of the art was to rely on a central typescheme for the 2 � 2 elliptic subsystem. In that approach, hyperbolic time termswere added to the spatial elliptic system producing the Cauchy-Riemann equationsaugmented with time terms. The equivalent equation that we are actually solving isgiven in 3.21 and rewritten herewt +Awx +Bwy = (CAwxx + (DA+CB)wxy +DBwyy)Just as in the scalar case, the amount of di�usion in Lax-Wendro� is set such that(A;B) = (C;D). Due to the linearity and symmetry of the Cauchy-Riemann system,AB = �BA and A2 = B2 = I, so the source terms take the form of Laplacianoperator. wt +Awx +Bwy = !h2 (wxx +wyy)In implementation, Lax-Wendro� is usually chosen because of exibility and com-pactness, and analysis of the model system shows that it provides second-order re-sults. Equivalently, other central schemes such as SUPG may be applied to obtainsimilar behavior.4.1.1 The Update ProcedureFor the linear Cauchy-Riemann equations, the uctuations will be an array oftwo elements, one residual for the continuity terms and the other for vorticity. Inany triangle with vertices indexed i = 1; 2; 3, the residual vector is given by�T = 2664 3775 = 266664 12ST 3Xi=1ui�yi � vi�xi12ST 3Xi=1ui�xi + vi�yi 377775 (4.1)

69Then, Lax-Wendro� yields the following distribution matrixBi = �n3 2664 1 00 1 3775+ �c4ST 2664nx nyny �nx3775 2664 3775 (4.2)When time-accuracy is not an issue, there are two time steps to be selected. The �rstCourant number, �n, dictates the overall convergence of the solver and �c controlsthe amount of dissipation locally introduced. The selection of these two types of timesteps naturally depends on the grid and ow conditions. Applying the Lax-Wendro�method to the linear Cauchy-Riemann equations on the uniform stencil pictured in�gure 4.1 yields the following update formula for the u component of velocity. Asimilar expression may be found for the other component.u0n+1 = u0n � �n�c(4u0n � u1n � u3n � u4n � u6n) +13�n(�2u1n + 2u3n + 2u4n � 2u6n � u2n + u5n)13�n(v1n � v2n � v3n � v4n + v5n + v6n)The terms proportional to �n�c contribute to a �ve-point smoothing, representingthe dissipative character of Lax-Wendro�. They are identical to the update formulawe would obtain from a forward time central-space discretization of the Laplaceequations. This di�usive part of the update is responsible for reducing the errorsthrough damping. The terms multiplying �n alone follow from the central (13) partand are associated with wave-like e�ects. These terms are responsible for propagatingerrors outside the domain. In general, when designing a discretization technique foran elliptic set of equations such as Cauchy-Riemann, we are confronted with thedilemma of how much propagation and di�usion to incorporate. For that problem,dissipation e�ects mimic the correct physical phenomenon but propagation carrieserrors faster and helps convergence. These issues are discussed further in the next

700

Stencil for sample least-squaresand Lax-Wendroff discretization

2

14

3

5

6

FD

E

C

B

A

h

h

Figure 4.1: Stencils used in the least-squares, Lax-Wendro� �nite di�erence dis-cretization.section.4.1.2 Some DrawbacksThe application of Lax-Wendro� to Cauchy-Riemann is successful to a certainextent but maybe there exists a superior elliptic discretizations. The discussion thatfollows is not restricted to Lax-Wendro� but generalizes to the family of schemesthat have the same central term in addition to some modi�ed di�usion coe�cient.Having tried SUPG for example, we noticed similar behavior since only dissipationterms are di�erent there.As shown in �gure 4.2, Lax-Wendro� does not preserve the symmetry of the ow.This behavior is particularly noticeable for an incompressible cylinder ow governedsolely by the Cauchy-Riemann equations. Due to the fore and aft asymmetry, thedrag coe�cient will be corrupted, unlike similar calculations with least-squares whichproduce a CD of machine zero. The lack of symmetry is introduced from advective

71

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00.97

0.975

0.98

0.985

0.99

0.995

1

x:dowstream distance

p/p

o: st

atic

pre

ssu

re o

ver

sta

gn

atio

n p

ress

ure

aft

fore

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 00.97

0.975

0.98

0.985

0.99

0.995

1

x:dowstream distance

p/p

o: st

atic

pre

ssu

re o

ver

sta

gn

atio

n p

ress

ure

fore & aft

Figure 4.2: Superimposed fore and aft pressure pro�les for a cylinder ow in the in-compressible limit. Lax-Wendro� (left) shows poor symmetry comparedto least-squares (right).e�ects in Lax-Wendro�, for this reason it fails to be an ideal candidate to treatCauchy-Riemann type problems. Empirically, one observes that the quality of thesolution produced by Lax-Wendro� is extremely sensitive to the level of irregularityof the grid. Hence, loss of accuracy due to arbitrary grids is a common symptomof the Lax-Wendro� method. Elliptic phenomena have no preferential direction ofinformation propagation, so we may instead look for a scheme respecting that.Another important concern is the choice of the two time steps and their inherentconsequence on stability. Stability analysis for Lax-Wendro� on an arbitrary gridis intractable. As a result, the choice of time steps is a trial and error exercisewhich consists of tuning the code from one execution to another. A more automaticselection of time steps is desirable but, currently, we can only derive relations in thecase of a uniform mesh such as in 4.1. On that sort of grid, [30] shows that the nodaland cell time steps needs to satisfy �n�c � 12 . Otherwise, for any irregular grid usedin practice, that inequality does not hold and one resorts to manually tuning (�n; �c),balancing the advective and dissipative forces against each other.

72If we were to apply Lax-Wendro� to a hyperbolic problem, we would adjust �nand �c to satisfy certain stability criteria which again dictate a certain compromisebetween di�usion and convection. In that case, increasing �c would hurt the solutionsince it would introduce more dissipation in the wave propagation problem. For theCauchy-Riemann problem, the rules are reversed as di�usion e�ects are welcome.Here, when �n is decreased to zero and �n is increased, we recover the least-squaresscheme and the solution improves considerably. The catch is that one then encountersslow convergence.Convergence of Lax-Wendro� occurs through two mechanisms, error reductionvia damping and convection out of the domain. Error elimination via dampingcannot be relied on solely since it is prohibitively slow. The rate of error dampingmay be considerably improved by multigrid techniques but [47] reports that overallspeedup is not nearly as great as one would expect. Indeed, unless one is solving atrivial problem where grid boundaries are aligned with the axes and characteristicboundary conditions are imposed, the advective part of Lax-Wendro� will re ecterrors o� boundaries instead of carrying them out. For practical Euler calculationson irregular grids, there is a need for e�cient characteristic boundary conditionswhich reduce wave re ections. The subject of wave re ections o� boundaries, thedistinctively jagged convergence histories, and consequences on convergence rates areextensively covered in [45].The above description hopes to provide a qualitative idea of the di�culties en-countered when solving Cauchy-Riemann, as well as to illustrate the properties ofLax-Wendro� and its numerical drawbacks. We now have a motivation for movingto a technique that better captures the physics of elliptic problems.

734.2 The Least-Squares ChoiceAn elliptic-based system represents isotropic spreading of information so we seeka scheme which damps errors equally in all directions. It would be a good propertyif our solver when applied to Cauchy-Riemann on the grid in �gure 4.1 behaved astwo separate Laplace discretizations . The updates for the components of velocity(u; v) would then be completely decoupled. As a result, when computing the acousticpart of the subsonic Euler equations at very low Mach numbers, we would correctlyrecover incompressible potential ow. In fact, the least-squares discretization doespossess exactly that property as shown by its update formula.@ui@t = �@2ui@x2j (4.3)The above heat equation (4.3), when discretized on a uniform grid, in a forward-timecentral-space �nite-di�erence fashion would produce the same updates as our least-squares scheme on the Cauchy-Riemann equations. The equations are completelydecoupled and therefore the updates will be independent.4.2.1 The Update ProcedureThe uctuations corresponding to the continuity and vorticity equations are eval-uated and are given in 4.1. Least-squares yields the following distribution matrixwhich is identical, within a multiplicative relaxation factor, to the dissipation termof the Lax-Wendro� method.Bi = !4ST 2664nx nyny �nx3775 2664 3775 (4.4)where nx and ny are the components of the scaled normal vector opposite side i.With ! denoting some iteration constant, the least-squares updates on the simple

74grid shown in 4.1 may then be written asu0n+1 = u0n � !h2 (4u0n � u0n � u1n � u3n � u4n � u6n)v0n+1 = v0n � !h2 (4v0n � v0n � v1n � v3n � v4n � v6n) (4.5)It is evident that the update formulas produced by least-squares and �nite-di�erencetechniques are identical. This was exactly our original aim, a method that wouldcompletely decouple the updates for u and v from each other and provide meansfor the propagation of information in in�nite set of directions. Another aspect ofsolving two parabolic equations is that convergence is slow because errors are reducedthrough damping only, not propagation. The ip side of that is that the smoothingproperty becomes advantageous in combination with multigrid acceleration.In light of the equivalency of the two schemes, intuition tells that the boundaryconditions that must be imposed for least-squares must follow the interior schemeand also be of decoupled kind. This has to be true if there is going to be anyparallel between least-squares on Cauchy-Riemann and central �nite-di�erence onthe Laplace equations. Indeed, any incompressible potential ow technique thatsolves the Laplace equations does so for the potential and stream functions, not foru and v as expressed in 4.3. A very good reason for this choice is that separateNeumann, Dirichlet or Robin (mixed) conditions exist for the potential and streamfunctions at solid boundaries. Hence decoupled boundary conditions agree withthe decoupled discretizations to which they are applied. However, our approachbreaks this consistency because it combines a least-squares discretization of Cauchy-Riemann with tangency conditions at wall boundaries. The tangency constraintmeans ignoring the velocity component normal to the wall, so it essentially couplesthe u and v components on boundaries. Our original intuition which suggested a

75boundary treatment consistent with the interior scheme turns out to be correct.The next section looks at symptoms which would otherwise appear in the numericalresults.4.2.2 General PropertiesIn a series of numerical experiments which tested the e�ectiveness of least-squareson Cauchy-Riemann, we simulated simple ow around a cylinder in the incompress-ible limit. What stands out the most is that numerical results converge to the exactsolution at less than second-order accuracy. Even more discouraging, for the slightlymore complicated situation of a lifting cylinder, the problem is accentuated and re-sults are less than �rst-order accurate. The inaccuracy is re ected in its failure tocapture suction peaks.Since the problem worsens in lifting cases, one might question least-squares' abil-ity to capture a point vortex problem, the fundamental lifting component of cylinder ow. Figure 4.3 pictures the numerical solution produced for solid body rotationby the least-squares scheme with tangency condition. The vortex strength is chosento be unity to simplify the graph. The correct solution for the velocity magnitudein the radial direction should be q = 1r . Computational results evidently show amixed behavior, a region in the far-�eld where q / r and a region close to the solidwall where q / 1r . Also despite the correct 1r behavior near the wall region, thevelocity magnitude is considerably underestimated there. Actually, both 1r and rare legitimate solutions of the Laplace equation, therefore their superposition willalso be so. On the other hand, the linear Cauchy-Riemann equations only admit assolution q = 1r , which is the only one desired for the point vortex problem. Thisexperiment points out that the Cauchy-Riemann equations are better solved using

76

r0

r1

1

r0

r1

1

r

qProfile of velocity

magnitude versus radial

distance for point vortex

problem. The exact solution

has 1/r behavior.

r0

r1

1

r0

r1

1

r

q Profile of velocity

magnitude versus radial

distance for point vortex

problem. Solution produced

by least-squares is linear

in the far-field

Figure 4.3: Left: Correct velocity magnitude pro�le for a point vortex problem in theradial direction. Right: Solution produced by least-squares or central-di�erence schemes with tangency conditionsa scheme which unlike the standard least-squares technique described in section 4.2,preserves the coupling between u and v.Up to the current stage of investigation, we believe that the (u; v) coupling dis-parity between the interior and boundaries is the main cause of our troubles. Wehave not yet pinpointed a very rigorous solution to the problem. Nonetheless, it isclear that a logical path of study should either involve improving boundary condi-tions or otherwise modifying least-squares so that (u; v) updates are coupled. As faras boundary modi�cations are concerned, quite a bit of e�ort was invested in de-veloping either strong or weak boundary formulations which would split away (u; v)constraints, but none seemed fruitful. Although ad-hoc, one boundary conditionwhich partially succeeds consists of enforcing zero vorticity at the wall only, there-fore ignoring continuity. However, the inaccuracy still remains for lifting cases.Another way to correct the fundamental discrepancy between the tangency con-ditions and the solver used in the interior domain is to recouple the u and v updates.Ironically enough, we originally set forth to create a method that decouples the two

77velocity components, but now all clues seem to indicate that their interdependency isnecessary in the presence of coupled wall conditions. We achieve this by maintainingthe least-squares scheme but minimizing a slightly di�erent norm, one where the areaweight is ignored. The new quantity to be minimized is the norm without the areaweighting. �i = XT2Ti(�T )tQ(�T ) instead of �i = XT2Ti 1ST (�T )tQ(�T ) (4.6)For an arbitrary stencil that has uneven spacings, the least-squares approach onthe non-weighted norm is analogous to solving the following equations for u and anequivalent one for v:@u@t = �@2u@x2j + a@3u@x3j + b@3v@x3j +H:O:C:D:T: (4.7)Therefore, it is clear that by removing the area weight, we are solving an ap-proximation to the Laplace equations that couples the two components of velocity.The additional pieces that provide coupling are higher order cross-di�usion terms(H:O:C:D:T:). For a patch of cells with arbitrary geometry, the coe�cients multiply-ing them may become very complicated but [53] studies some special con�gurationswhere the equivalent equation is slightly less messy.With changes to the norm, the least-squares minimization produces satisfactoryresults for lifting airfoils. The order of accuracy improves considerably, closing inon second order as the standard least-squares scheme should. Multiplying the norm� by the cell area ST is successful but there are in�nitely other ways to recouplethe updates, the simplest one coming to mind is to multiply by higher powers of STor even any power of 1ST . However, near high velocity-gradient regions where cellareas are made small to bene�t accuracy, these alternative choices might introduceinstability or spurious e�ects such as oscillations. Therefore, we stay clear of those

78options and always use the new norm de�ned in expression 4.6 whenever we applyleast-squares to a Cauchy-Riemann type problem.4.3 A Generalized Framework for Least-SquaresAs a follow up to section 3.3.3 which gives a general overview of the least-squarestechnique, in the proceeding discussion the method can be formalized for use withCauchy-Riemann type systems. At the center of this framework is the update matrixwhich fully describes the contribution from one node to another in a given cell.Denote by wi the vector of unknowns at node i. De�neW as a vector comprisedby (wi=1;2;3), it essentially groups together all the variables at the three vertices of acell. For exampleW will be a vector of size six for a 2� 2 system and of size twelvefor the Euler system. W = � w1 w2 w3 �t (4.8)In each cell, the vector of uctuations is a linear combination, arising from themultiplication of (wi=1;2;3) with the coe�cient matrices Ri=1;2;3. The matrices Rirepresent local derivatives that are frozen during the update. To simplify notation,de�ne R as a rectangular matrix that completely determines the residual as RW.It is comprised of the coe�cients matrices Ri=1;2;3 and takes the form of three 4� 4blocks for the Euler equations and three 2� 2 blocks for potential ow.R = � R1 R2 R3 � (4.9)We can now compactly write the residual �T as a linear function of the state vectorsat each vertex, �T = 3Xi=1Riwi = RW (4.10)In the spirit of the least-squares scheme, we de�ne a norm to be minimized and

79according to the earlier discussion, it will not be area-weighted. As a mean tointroduce notation for a later section, the weighting matrixQ is written as Q = DtD.This is simply equivalent to rede�ning the uctuation as D�T . The norm to beminimized is 1 then � = XT2Ti(WtRt)Q(RW)= XT2Ti(WtRt)(DtD)(RW) (4.11)For any gradient-based minimization technique, one needs the local derivative of thenorm with respect to the state vector at each vertex.@�@wi = XT2Ti(RtDtDR)wi (4.12)On the practical side, notice that for a Cauchy-Riemann type problem, at eachiteration and for each cell we must calculate two uctuations and six gradients.For the Euler equations these numbers become four and twelve respectively. Thisindicates that given a certain number of cells in our domain, the computational costof each least-squares iteration is �xed. Since no savings are to be made in oatingpoint operations per iteration, any code optimization must lie in e�cient iterativetechniques. In any case, at each iteration until equilibrium, the contribution to cellvertex i that implements the required relaxation procedure isWn+1 �Wn = �!(RtDtDR)Wn = �!UWn (4.13)where ! is a global relaxation factor speci�c to the iterative method. It is clear thatthe update from cell T to its nodes can be fully represented as a matrixU = RtDtDRwhich is constant within T . The update matrix U is a convenient way of thinking1Given two square matrices A and B, (AB)t = BtAt.

80about uctuation-splitting schemes because it consists of sub-matrices Ui;j whichdescribe the e�ect of node j on node i. To get a visual idea of the update procedurein equation 4.13, consider�W =

2666666666666666666666666664�w1�w2�w3

3777777777777777777777777775= �!

2666666666666666666666666664U1;1 U1;2 U1;3U2;1 U2;2 U2;3U3;1 U3;2 U3;3

3777777777777777777777777775

2666666666666666666666666664w1w2w3

3777777777777777777777777775

n

4.4 Examples of ApplicationNow that we have selected least-squares as our choice of discretization for Cauchy-Riemann type equations and de�ned a framework to express it, we can move onto concrete examples. This section provides a precise recipe for applying the least-squares method to Cauchy-Riemann and its non-linear counterpart, the compressiblepotential system. Notation that may have previously appeared cumbersome or re-dundant will �nd its justi�cation here. The procedure applied to Cauchy-Riemannis very similar in nature to the one for potential ow, the latter discretization simplybeing a generalization of the former.4.4.1 Incompressible Potential FlowThe Cauchy-Riemann equations can written using primitive variables or naturalvariables. For purposes of compactness, (�)s;n and (�)x;y denote partial derivatives

81with respect to (s; n) and (x; y) respectively.Awx +Bwy = 0 , 2664 1 00 �13775 2664 uv 3775x + 2664 0 11 0 3775 2664 uv 3775y = 0 (4.14)~A xs + ~B xn = 0 , 2664 1 00 q2 3775 2664 p� 3775s + 2664 0 �q21 0 37752664 p� 3775n = 0 (4.15)The above formulations are obtained one from the other using the transformation Tde�ned by T = @x@w = 2664 �u �v� vq2 � uq2 3775 (4.16)One of the �rst questions that arises is whether to use natural variables x or primitivevariables w in the Cauchy-Riemann solver. At �rst, it might seem attractive tohold natural variables since pressure and ow angle are completely decoupled fromeach other. This advantage would be very attractive in solving the Euler equationswhere the acoustic part is expressed in these variables only. Another considerationis that the ow angle � at any solid wall is easily imposed since the surface geometryis known. Moreover, it is interesting to hold (p; �) because they decouple at theboundary, therefore the least-squares discretization of 4.15 would use the standardarea weighted norm. Indeed, as discussed in section 4.2.2, the resulting couplingbetween variables should be consistent for both boundary and interior treatments.At second glance however, it seems that a direct discretization of equations 4.15 interms of nodal p and � will produce inaccurate solutions. This is con�rmed throughempirical results which produce ow angle solutions near stagnation points with lessthan �rst-order accuracy. For dimensional consistency, all ow angle gradients aremultiplied by q2. This is detrimental to accuracy because in the neighborhood ofstagnation points, these coe�cients rapidly approach zero whereas those for pres-

82sure gradients are O(1). A similar e�ect takes place in the acoustic part of theEuler equations where the coe�cients weighting � gradients are proportional to M2,therefore vanishing in the incompressible limit. Using ow angle also presents theambiguity that it is a periodic function so all solutions modulo 2� are physicallyidentical. These reasons are self-su�cient for discarding the natural variables whensolving Cauchy-Riemann, and instead only working with (u; v). Incidentally, thatchoice generalizes well since it is akin to the parameter vector p� [1; u; v; h]t whichconservatively linearizes the Euler equations.We now determine the cell uctuations corresponding to equations 4.14. Ourapproach will probably seem roundabout because we start with system 4.15 andrederive 4.14 through a chain of matrix multiplications. However, this detour helpsbetter understand the procedure used for the Euler equations. In the followingexpressions, D assigns di�erent weights to the continuity and vorticity residuals, andfor the simple Cauchy-Riemann system, it is naturally equal to the identity matrix.It might seem strange to introduce that notation here but it is a formalization thatwill �nd its importance for compressible potential and Euler problems.~Axs + ~Bxn = 0 , D ~ATws +D~BTwn = 0, �D~ATu�D~BTv�wx + �D ~ATv +D~BTu�wy = 0The uctuation vector �T associated to cell T is then�T = 12ST �D ~ATu�D~BTv� 3Xi=1�yiwi � 12S �D ~ATv +D~BTu� 3Xi=1�xiwi= 12ST 3Xi=1hD ~AT [u(�y)i � v(�x)i]�D~BT [u(�x)i + v(�y)i]iwi= 12ST 3Xi=1 hD ~ATqni �D~BTqtiiwi= 12ST 3Xi=1 [Nqni � Sqti ]wi

83= 3Xi=1Riwi (4.17)With (t; n) now representing directions tangent and normal to an edge, we use theconcise notation qni = [u(�y)i � v(�x)i]qti = [u(�x)i + v(�y)i]where ((�x)i; (�y)i) is the vector representing the side opposite vertex i. Equation4.17 reiterates that the residual vector may be written as a linear combination of thevertex values wi with derivative matrices Ri. That was the notation introduced insection 4.3 and we now have an explicit expression for Ri.Ri = 12ST [Nqni � Sqti ]Despite the fact that a series of matrices are multiplying each other, the residualand update formulas are su�ciently simple that explicit formulae can be written forthem. N = D ~AT = 2664 �u �v�v u 3775 S = D~BT = 2664 v �u�u �v 3775Ignoring the cell area terms 12ST , we now have all portions Ui;j of the update matrix,each describing the update contribution of node j on i.Ui;j = [Nqni � Sqti ]thNqnj � Sqtj i= hqniqnj + qtiqtj iNtN� hqniqtj � qtiqnj iNtS (4.18)To arrive to equation 4.18, we exploiting some special properties of N and S, moreprecisely, NtN = StS and StN = �NtS.Remember that the variables being adjusted are (u; v), the ones stored at cellvertices. In spite of that, it helps to identify those portions of the update which

84change the thermodynamic state and the ones that a�ect the ow direction. Furthermanipulation of 4.18 reveals that NtN and NtS can be expressed in terms of a =[�u;�v]t and b = [�v; u]t which represent gradients of pressure and ow angle.NtN = 2664 q2 00 q2 3775 = ata + btb NtS = 2664 0 q2�q2 0 3775 = atb� btaTerms such as bta are outer products. It is then quite natural to split the updatematrix U in terms of contributions dedicated to p and those aimed at �.Ui;j = nqniqnjat + qtiqtjat + hqniqtj � qtiqnj ibto a+ update to pnqniqnjbt + qtiqtjbt � hqniqtj � qtiqnj i atob update to �4.4.2 Compressible Potential FlowThe compressible potential ow equations are just a non-linear version of theCauchy-Riemann system. Due to this similarity, the least-squares approach given forpotential ow equations is considerably shortened here because the details are iden-tical to those described in the previous section. The following system fully governspotential ow, with terms including Mach number to take in account compressibilitye�ects. ~Axs + ~Bxn = 0 , 2664 �2 00 �q2 3775 2664 p� 3775+ 2664 0 ��q21 0 3775 2664 p� 3775 = 0where �q2 = �1� � 12 q2� 1 �1 q2 and �2 = 1�M2 = 1� +12 q21� �12 q2The only changes to the procedure are a modi�ed transformation matrix T andweighting matrix D. This particular choice for scaling vorticity relative to continuity

85follows from the linearized analysis found in [64].T = @x@w = 2664 ��u ��v� vq2 uq2 3775 D = 2664 1 00 � 3775One then obtains the update matrix which can either be written in compact formor, alternatively, in terms of the gradients of pressure and ow angle a and b. Therelations di�er slightly from the Cauchy-Riemann case by some � terms: NtN =�2ata + btb and NtS = � (atb� bta).Ui;j = h�2qniqnj + qtiqtj iNtN� � hqniqtj � qtiqnj iNtS (4.19)= n�4qniqnjat + �2qtiqtjat + �2 hqniqtj � qtiqnj ibto a +n�2qniqnjbt + qtiqtjbt � �2 hqniqtj � qtiqnj i atobIn the limit of M ! 0, equation 4.19 becomes identical to 4.18 since � ! 1. Thisproperty is desired because at very low Mach number, we are essentially simulatingincompressible ow so our potential ow solver scheme should recover a solution tothe Cauchy-Riemann equations.Due to the purely advective behavior of the supersonic potential equations, least-squares is not an optimal scheme and instead one would recur to upwind uctuation-splitting. Nonetheless, it is still interesting to see how least-squares transitionsthrough M = 1. For supersonic ow, a similar procedure is followed with the ex-ception that the weighting matrix is now D = diag(1; �) and � must be rede�ned as� = pM2 � 1. Also N and S have slightly di�erent properties:NtN = StS = �2ata + btbNtS = StN = � (atb+ bta) (4.20)Having made these changes, one obtains an update matrix which is close enough to

86the subsonic one that both are expressed compactly using the signum function.S(x) = 8>><>>: 1 if x � 0�1 otherwise (4.21)In the limit of Mach number approaching 1 from above and below, the least-squaresupdate formula is identical so the sonic transition is smooth. It is clear from theupdate matrix below that at the sonic line only the ow angle � is changed. Thisindicates that very close to M = 1, pressure is invariant to any changes made to thevelocity components.Ui;j = n�4qniqnjat + S(�2)�2qtiqtjat + �2qniqtj � S(�2)�2qtiqnjbto a +nS(�2)�2qniqnjbt + qtiqtjbt � S(�2)�2qniqtj + �2qtiqnjatob

CHAPTER VASSEMBLING AN EULER SOLVER5.1 Conservative Linearization of the Euler equationsConservation is a necessary condition to be observed by any compressible Eulerscheme which aims at correctly capturing discontinuities, their strength and position.To maintain conservation at the discrete level, the cell residual has to be evaluatedas an integral of the ux divergence of the conservative variables over the cell. Forthe discrete gradient calculations to be consistent, we require that they be de�nedas averages over a cell.d~ru = 1ST Z ZT ~ru(z)dS bfx = 1ST Z ZT fx(z)dS cgy = 1ST Z ZT gy(z)dSThe particular variable z must be chosen such that the above integrations are exact.Then the residual may be written as�T = I@T ~f ~n dl = Z ZT (fx + gy) dS = ST h bfx + cgyi (5.1)Provided that the above integrations are exact, the ux contour integrals over thetriangles will telescope (see 3.6 for de�nition) over the domain regardless of theunderlying variable z having linear variation over the triangle. It was shown for the�rst time in [63] and later generalized to higher dimensions ([17]) that the choice87

88of z which leads to the exact linearization of 5.1 must be the parameter vectorz = p� [1; ~q; h]t. Since u, f and g are quadratic in the components of z, the Jacobianmatrices @u@z , @f@z and @g@z are linear in it, therefore making the integration over atriangle trivial. bfx + cgy = A(z)cux +B(z)cuywhered~ru = @u@zrz and z = z1 + z2 + z33 = 130BBBBBBBBBBB@

p�1 +p�2 +p�3p�1u1 +p�2u2 +p�3u3p�1v1 +p�2v2 +p�3v3p�1h1 +p�2h2 +p�3h31CCCCCCCCCCCAIt is important to note that any system of equations equivalent to the original equa-tions in conservative form can be used to produce a conservative linearization. Hold-ing the conservative variables u is too cumbersome for the sake of uctuation calcula-tion so we will start with another form of the Euler equations. The only requirementis that the transformation matrix which takes us from one formulation to anotherbe evaluated at the average state z, and all gradients be computed from rz. Inour case, for example, we �nd it convenient to represent the solution in terms of theparameter vector and �nd the conservative residual by integrating over the cell ofarea ST . �uT = 1ST Z ZT (Czx +Dzy) dS = 1ST I@T Czdy �Dzdxwhere the matrices C = @f@z and D = @g@z are locally constant because of the quadraticproperty and are evaluated at the average state z. Explicit expressions for C and Dare given section 2.1.1.Compared to previous work where residual-distribution was applied to some de-composition of the Euler equations, the novelty in our approach will be to split the

89 uctuation into its elliptic and hyperbolic parts. Our uctuation-splitting schemeof choice would then be applied to the resulting portions. The advantages behindthis methodology are accrued modularity and better control over the update pro-cedure. Indeed, we do not need to make any decisions on the choice of an ellipticsolver (Lax-Wendro�, least-squares or other) until very late stages in the design ofour Euler code. Also, it is at the residual level where one can incorporate many ofthe desired properties into the scheme, forcing perceived singularities such as sonicand stagnation regions to be benign.As expressed in 5.1, the conservative residual does not re ect the independenthyperbolic and elliptic parts of Euler. Although our aim is to reduce that residual,we must do so in some suitable norm which splits away the advective and elliptice�ects from each other. For the supersonic regime, the hyperbolic residuals arestraightforward because the system is fully diagonalizable so each scalar advectionequation produces a uctuation independent of all others. The treatment of theacoustic part of subsonic Euler is a more interesting problem. It is possible to extractthe elliptic part of the conservative residual since we have a unique decompositionof Euler when expressed in natural variables.5.2 The Hyperbolic ResidualsWe have already seen in section 2.1.4 that the supersonic Euler equations are fullydiagonalizable and equation 2.8 may be written in terms of the natural variables wwith spatial coordinates in the (x; y) direction.Awx +Bwy = 0 (5.2)

90wherew = 2666666666664

@S@h@p + �q2� @�@p� �q2� @�3777777777775 A = 2666666666664

u 0 0 00 u 0 00 0 u� � v 00 0 0 v� + u3777777777775 B = 2666666666664

v 0 0 00 v 0 00 0 u� + v 00 0 0 v� � u3777777777775The vector of four uctuations, each representing one of the scalar advection equa-tions is simply obtained by the standard rules of computing spatial derivatives on atriangular cell. Since the system in 5.2 is a direct diagonalization of the conservativeformulation, we are assured that any residual calculation following from it will alsobe conservative as long as matrices are evaluated at cell average z.

�T = 2666666666664�T S�T h�TW+�TW�

3777777777775 = 12ST 3Xi=1 [AT(�y)i �BT(�x)i] zi (5.3)where T = @w@z is the transformation matrix between w and the parameter vec-tor z. Each row of T essentially consists of the gradients of the natural variables(at;bt; ct;dt) expressed in the parameter vector space.T = 2666666666664

ctdtatbt3777777777775 =

2666666666664(1� 2 )z4 + z1 (z22 + z32) �z2 �z3 z1�z4 0 0 z1�( �1) z4 ��( �1) z2 � z3 ��( �1) z3 + z2 �( �1) z1�( �1) z4 ��( �1) z2 + z3 ��( �1) z3 � z2 �( �1) z1

3777777777775To split the conservative uctuation into its hyperbolic pieces, we de�ne a decom-position matrix D� which extracts the appropriate part. For example, the entropy uctuation is obtained as �T S = DS �T where DS = diag(1; 0; 0; 0). Similarly, uc-tuations for the enthalpy and acoustic Riemann invariants are established through

91Dh = diag(0; 1; 0; 0), DW+ = diag(0; 0; 1; 0), and DW� = diag(0; 0; 0; 1). The resid-uals are then distributed to vertices according to the upwind scheme chosen andthis process brings a certain change �w to each node. Actual adjustments are neverexplicitly calculated for the natural variables but only for the parameter vector, �z.�z = T�1�z / T�1�Twhere the proportionality coe�cient depends on the upwind method used. The aboveequation simply states that the net change �z actually consists on four independentcontributions, one from each uctuation. The portion, �zS, originating from theentropy residual is proportional to the outer product of the entropy gradient anda vector along which only entropy changes. Similar statements can be made forfor �zh, �zW+, �zW�. The proportionality coe�cients are such that updates aredimensionally consistent with each other.�zS / ct �rS�zh / dt �rh�zW+ / at �rW+�zW� / bt �rW�It is thus clear that, even at the update level, the entropy, enthalpy and acousticparts of the problem are decoupled from each other. This is a direct result of havingindependent (S; h;W+;W�) residuals. The updates resulting from each of thoseresiduals fall on directions where only the corresponding quantity varies. Thesevectors are de�ned by the quadruplet (�rS;�rh;�rW+;�rW�) which form a basis.�rS = 2666666666664

12 �q2z4z2z312 �q2z43777777777775 �rh = 2666666666664

0z2z3�q2z13777777777775 �rW+ = 2666666666664

M2z1(M2 � 2)z2 � 2�z3(M2 � 2)z3 + 2�z2M2z43777777777775 �rW� = 2666666666664

M2z1(M2 � 2)z2 + 2�z3(M2 � 2)z3 � 2�z2M2z43777777777775

92The procedure explained in this section is used without any modi�cations in ac-tual computation of supersonic ows. There are a variety of choices for the residualdistribution method but PSI seems to be the best because of its monotonicity, accu-racy and purely upwind character. Of course, the recipe presented here also holdsfor the entropy and enthalpy advection when M < 1. In the subsonic regime, how-ever, the acoustic part will produce an elliptic residual which will be distributed ina di�erent way. It is minimized in the least-squares sense, as described in the nextsection.5.3 The Elliptic ResidualWhen the ow is subsonic, we can easily split away the elliptic part of the con-servative residual 5.1 because a decomposition of the Euler equations at the partialdi�erential equation level is available. That breakup produces an elliptic part whichonly involves two unknowns, pressure and ow direction, and takes the form of aCauchy-Riemann system. Therefore, we may build upon the procedure analyzed inchapter IV. It turns out that the resulting update formulas are just generalizations ofthose for Cauchy-Riemann and potential ow equations. At the end of this section,we brie y look at a modi�cation to the elliptic residual which produces a third-orderdiscretization.Second-OrderWe saw in chapter II that the Euler equations can be cast in several di�erent setof variables and that each form exhibits a distinct aspect of the Euler equations. Thediscussion below builds on that and shows how a least-squares scheme for the acousticpart of the Euler equations can exploit the di�erent properties of the di�erent setof unknowns. Suppose that we only had access to the natural variables x and that

93those were the set of unknowns stored at the vertices. We would then compute thecell residual based on x and minimize it in the least-squares sense. In the cartesiancoordinates, the governing equations are~Axs + ~Bxn = 0 , �~Au� ~Bv�xx + �~Av + ~Bu�xy = 0x = 2666666666664

@S@h@p�q2@�3777777777775 ~A = 2666666666664

1 0 0 00 1 0 00 0 �2 00 0 0 13777777777775 ~B = 2666666666664

0 0 0 00 0 0 00 0 0 �10 0 1 03777777777775where � = p1�M2.Using standard methods to evaluate the derivatives, the four components of theresiduals �T x are given by

�xT = 12ST 3Xi=1 h~Aqni � ~Bqtiixi = 12ST 3Xi=12666666666664qnj 0 0 00 qnj 0 00 0 �2qnj qsj0 0 �qsj qnj

3777777777775xiwhere qni = [u(�y)i � v(�x)i] and qti = [u(�x)i + v(�y)i]. The entropy and en-thalpy uctuations are extracted using DS = diag(1; 0; 0; 0) and Dh = diag(0; 1; 0; 0),subsequently distributed via some upwind method. The last two components of �xTcomprise the elliptic part of the problem which is split away withDE = diag(0; 0; 1;p1�M2).As in the case of compressible potential ow, this weighting follows from linear anal-ysis performed in [66]. Hence, the quantity to be minimized is the purely ellipticpart of Euler. DE �xT = 12ST 3Xi=1 hDE ~Aqni �DE ~Bqtiixi

94This results in an update matrix which fully determines the adjustments made tothe natural variables,Ui;j = h~Aqni � ~BqtiitDEtDE h~Aqnj � ~Bqtj iIn fact, we do not store the natural variables because it is the parameter vector whichconservatively linearizes Euler. Also, we have seen that (S; h; p; �)t is di�cult to workwith, in particular with the treatment of ow direction near stagnation points. Theadvantage of the least-squares formulation is that, at this point, we have the choiceof introducing any set of unknowns as long as we know the transformation matrixwhich links it to x. Having already selected z as our variables, that matrix T isT = @x@z = 2666666666664

ctdtatbt3777777777775 =

2666666666664(1� 2 )z4 + z1 (z22 + z32) �z2 �z3 z1�z4 0 0 z1z4 �z2 �z3 z10 �z2 z3 0

3777777777775It is now straightforward to obtain the residual to be minimized.�ET = 12ST 3Xi=1 hD ~ATqni �D~BTqtii zi = 12ST 3Xi=1 [�Nqni � Sqti ] zi (5.4)A convenient way of writing it is in terms of N and S because of their interestingstructure. De�ning k = �1 , these matrices areN = 2666666666664

0 0 0 00 0 0 0�k�z4 k�z2 k�z3 �k�z10 z3 �z2 03777777777775 S = 2666666666664

0 0 0 00 0 0 00 �z3 z2 0�k�z4 k�z2 k�z3 �k�z13777777777775Note that N and S satisfy some special identities, more precisely, NtN = StS andStN = �NtS. With these substitutions, the least-squares minimization of 5.4 pro-

95duces the following update matrix.Ui;j = h�2qniqnj + qtiqtj iNtN� � hqniqtj � qtiqnjiNtSTo further gain insight into the update procedure, we introduce the gradients ofpressure and ow direction, a = [z4;�z2;�z3; z1]t and b = [0;�z3; z2; 0]t.NtN = k2�2ata+ btb NtS = k� �atb� bta�Just as in the potential ow case, the update matrix U explicitly written in termsof a and b is quite revealing (referring to equation 5.5). The contributions fromthe acoustic part of the problem originate only from changes in pressure and owangle. Interestingly, the second term of Ui;j largely dominates over the �rst in theneighborhood of a stagnation point. This con�rms that any e�ect in that region isdue to wide variations in the ow angle. On the other hand, in regions of parallel owwhere ow angle is almost constant, it is the �rst terms that takes over and velocitychanges are brought only by pressure gradients. Finally, at any sonic transition(� = 1), all terms in Ui;j vanish except for qtiqtjbtb, clearly showing that velocitychanges at the sonic line are insensitive to pressure gradients.Ui;j = nk2�4qniqnjat + k2�2qtiqtjat + k�2 hqniqtj � qtiqnj ibto a+n�2qniqnjbt + qtiqtjbt � k�2 hqniqtj � qtiqnji atob (5.5)A least-squares update formula similar to 5.5 can be derived for supersonic oweven though, in actual computation, multidimensional upwind methods are preferred.For M > 1 the acoustic part of Euler is advective so least-squares is not an optimalchoice. Nonetheless, it will be presented here just for the sake of completeness.The same approach is followed, this time with � = pM2 � 1 and slightly di�erentproperties for N and S, namely NtN = StS and StN = NtS. Using the same

96notation introduced in 4.21, the subsonic and supersonic updates can compactly bewritten in a single expression. Note that when M = 1, the supersonic and subsonicleast-squares update formulas are identical so the sonic transition is smooth.Ui;j = nk2�4qniqnjat + k2S(�2)�2qtiqtjat + k�2qniqtj � kS(�2)�2qtiqnjbto a+nS(�2)�2qniqnjbt + qtiqtjbt � kS(�2)�2qniqtj + k�2qtiqnjatobThird-OrderAs mentioned in 3.5, a third-order extension of uctuation-splitting schemes maybe obtained through a higher-order quadrature of the data. The only changes broughtare in the uctuation calculation, so this makes for a very modular code when usedwith di�erent distribution methods. It turns out that the new uctuation is simplythe old one with an added correction term which contains Ri. So the elliptic residualto minimize instead of 5.4 is�T E = 12ST 3Xi=1 [�Nqni � Sqti ] zi + 112ST 3Xi=1 [�Nqni � Sqti ] [Ri+1 �Ri�1]In the above expression, we use the rotated index notation where i� 1 lies clockwiseof i and i + 1 marks the counter-clockwise vertex. Since the distribution methodremains identical, the signals going to each node are weighted just as in the second-order case. This is equivalent to taking the gradients of the corrective term withrespect to nodal states to be zero. Thus, the block of our third-order update matrixwhich describes the e�ect of node j on i will beUi;j = h�Nqnj � Sqtj it �12 [�Nqni � Sqti ] zi + 112 [�Nqni � Sqti ] [Ri+1 �Ri�1]�= nh�2qniqnj + qtiqtj iNtN��hqniqtj � qtiqnj iNtSo �12zi + 112 [Ri+1 �Ri�1]�Fortunately many terms factorize and the only new term is Ri+1�Ri�1, making thethird-order extension quite inexpensive in terms of computational time. Each 'extra'

97term is associated to a certain triangle edge, and since each edge is visited twice, wemay store the quantity and use it again. Denoting ~si as the vector formed by theedge opposite node i and pointing in the counterclockwise direction, Ri+1 �Ri�1 isRi+1 �Ri�1 = 2666666666664

(rz1)i+1 � ~si � (rz1)i�1 � ~si(rz2)i+1 � ~si � (rz2)i�1 � ~si(rz3)i+1 � ~si � (rz3)i�1 � ~si(rz4)i+1 � ~si � (rz4)i�1 � ~si3777777777775

5.4 Forcing the Euler Solver to Preserve Potential FlowIf an Euler code were to behave correctly in the limit of irrotational ow, it wouldproduce potential ow results. In other words, the 2 � 2 elliptic subsystem shouldnot generate any spurious entropy or enthalpy, thus resulting in (S; h) �elds thatare constant in the steady state. The preservation of potential ow is automati-cally obtained in the supersonic regime where (S; h) residuals are una�ected by theacoustic part. However, that same property is not trivial in the subsonic solver.Despite a strict decomposition of the elliptic-hyperbolic parts at the residual level,the least-squares minimization of the elliptic uctuation does not completely restrictit from changing (S; h). As such, the process studied in the previous chapter shouldmerely be a �rst step. In a second stage, through a constrained minimization, onemust ensure that changes brought upon the (p; �) variables strictly leave entropy andenthalpy unchanged.5.4.1 A Constrained MinimizationThe question is how to modify the minimization described earlier such that theelliptic and hyperbolic parts be strictly decoupled at the numerical level. The elliptic

98residual 5.4 must be reduced as quickly as possible subject to the constraint that thispart of the procedure should not change the convected quantities. For the advectivevariables to remain independent of the elliptic residual, changes to (p; �) have to bein directions of constant (S; h). In the space of natural variables, de�ne �rS, �rh, �rpand �r� as vectors along which only S, h, p, and � vary respectively. For example, theresidual from the entropy part of the uctuation should not change enthalpy, pressureor ow direction, therefore it must be in a direction �rS. To �nd the coordinates of �rS,�rh, �rp and �r� in the parameter vector space, the following underdetermined equationshave to be satis�ed. At each location in z space, the locus of points on which only oneof (S; h; p; �) changes forms a surface, not a curved line. Hence there are in�nitelymany vectors that �t our description but we will look for ones with relatively simpleexpressions. With T = @x@z and � just representing a non-zero entry, the followingequations have to be satis�ed.T�rS = � � 0 0 0 �t T�rp = � 0 0 � 0 �tT�rh = � 0 � 0 0 �t T�r� = � 0 0 0 � �tWe �nd that for each natural variable, the direction of a vector along which onlythat variable changes is�rS = 2666666666664

12 z22+z32z4z2z312 z22+z32z13777777777775 �rh = 2666666666664

0z2z3z22+z32z13777777777775 �rp = 2666666666664

M2M2�2z1z2z3M2M2�2z43777777777775 �r� = 2666666666664

0�z3z203777777777775From here on, we will say that the above vectors de�ne the natural basis. If theupdate �zu resulting from an unconstrained minimization of the elliptic residual isthought of as a vector in the natural basis, then the constrained �zc update will be

99its projection onto surfaces of constant entropy and enthalpy. An arbitrary changein the parameter vector such as �zu can be expressed in this basis through the vectorof coe�cients � = (�S; �h; �p; ��)t�zu = �S�rS + �h�rh + �p�rp + ���r�= � �rS �rh �rp �r� � � �S �h �p �� �t= P�Knowing the direction matrix P, we can then solve for � = P�1�zu. If entropy andenthalpy �elds are to be invariant to changes brought upon by the elliptic residual,we must suppress �S;h. The overbar operator (�) will be de�ned to do just that, ittakes the last two components only.�zc = P(P�1�zu) (5.6)Following the same de�nitions introduced in previous sections, the solution at thenext iteration will again be written in terms of an update matrix. However, this timeit is the constrained update matrix.Ui;j = P nh�4qniqnj + �2qtiqtj i (P�1ata) + �2 hqniqtj � qtiqnj i (P�1bta)o+P nh�2qniqnj + qtiqtj i (P�1btb)� �2 hqniqtj � qtiqnj i (P�1atb)oAs mentioned earlier, the �rst line in Ui;j represents e�ects due to pressure gradientsand the second line to ow angle gradients. Carrying out the matrix multiplicationsreveals some simpli�cation. More importantly, it emphasizes that the constrainedminimization of the elliptic residual modi�es pressure and ow direction only, justas in potential ow. Indeed, the terms in the update formula associated with @� arealready parallel to �r� so they do not a�ect p, S or h. Equations 5.7 and 5.8 show

100that those terms are invariant with respect to the projection. In other words, the(P�1�) operator has no e�ect on them, it just returns its argument.(P�1atb) = � z4 �z2 �z3 z1 � � 0 �z3 z2 0 �t= at�r� (proportional to atb) (5.7)(P�1btb) = � 0 �z3 z2 0 � � 0 �z3 z2 0 �t= bt�r� (proportional to btb) (5.8)On the other hand, terms originating from pressure gradients do have an e�ect onadvective variables, so that is where the (P�1�) operator takes full e�ect. It discardsany components along �rS and �rh, and as a result, forces complete decoupling betweenthe acoustic and advective parts. Moreover, it's clear from 5.9 and 5.10 that updatesfrom the �rst line of Ui;j only contribute to pressure updates since they are parallelto �rp. (P�1ata) = l � z4 �z2 �z3 z1 � � M2M2�2z1 z2 z3 M2M2�2z4 �t= lat�rp (not proportional to ata) (5.9)(P�1bta) = l � 0 �z3 z2 0 � � M2M2�2z1 z2 z3 M2M2�2z4 �t= lbt�rp (not proportional to bta) (5.10)where l = �1 M2�2M2 at�abt�b . We denote bt � b to be an inner product whereas btb is anouter product.The least-squares minimization process under the constraint described above willlead to a full decoupling of the elliptic and hyperbolic parts at the numerical level.It should allow potential ow calculations at both high and low Mach number withthe same Euler solver. However, the limit M ! 0 represents an obstacle becausethe natural basis (�rS, �rh, �rp, �r�) degenerates and constraining the update leads to

101instabilities. We have determined that these problems which unveil themselves onlynear stagnation regions are related to conservation issues. Remember that the un-constrained least-squares minimization has the property that, in the steady state,changes inside any cell sum to zero, thereby ensuring conservation. To preserve thisproperty, all terms in Ui;j must be evaluated at the cell average z. Similarly in theconstrained minimization, as long as the local set of projection directions (�r�) areevaluated at z (cell-wise projection), the scheme will remain conservative. Since thischoice leads to instabilities, the alternative would be a vertex-wise projection, thenon-conservative version. Thus changes at each node are projected onto allowabledirections which are evaluated at the respective vertex.We try to gain some insight into why the cell-wise projection is prone to in-stabilities in stagnation regions whereas higher Mach numbers pose no problem.Meanwhile, to achieve convergence in the incompressible limit, we use the vertex-wise projection. The solver will not be strictly conservative in that limit but it maybe argued that issue is irrelevant. Indeed, elliptic ows being naturally very smooth,any conservation related errors are negligible compared to truncation errors. Weexperimentally con�rm this by �nding that in stagnation regions, the L1 norm stilldecreases with mesh re�nement in a second-order fashion (or third-order if usinghigher quadrature uctuations).Figure 5.1 shows a projection of the four-dimensional space (z1; z2; z3; z4) ontoplane A de�ned by (z2; z3). When we place ourselves in plane A, a surface on whichz1 and z4 are constant, it does not necessarily mean that the ow is incompressibleand homenthalpic. It just signi�es that we ignore the changes brought by (z1; z4) tothe natural variables. It is important to note how the natural basis collapses, causingit to form a degenerate basis. With the subscript (�)A indicating the image of (�)

102

z

z3

2

Plane A

constant r

S)( A r

h)( r )(A Ap

r )( A

z2

z3

+ = d2 2 2Figure 5.1: Projection of the natural basis onto the (z2; z3) plane.

z

z1

4

Plane B

r p)(

r )(

B

BS

r )( Bh

r )( B

( constant)

Figure 5.2: Projection of the natural basis onto the (z1; z4) plane.

103on plane A, (�rS)A, (�rh)A, and (�rp)A become co-linear and lie on a circle of radius d.As such, the projection on plane A creates a situation where (S; h; p) are only andidentically a�ected by changes in (z2; z3).(�rS)A = 2664 z2z3 3775 (�rh)A = 2664 z2z3 3775 (�rp)A = 2664 z2z3 3775 (�r�)A = 2664 �z3z2 3775Note that triangular cells very close to the stagnation point in physical space, whendrawn in the (z2; z3) space will end up very close to the origin. The stagnation pointitself will be at (0; 0) because the velocity magnitude vanishes there, and hence, thenatural basis is ill-de�ned.Consider now what happens when a similar projection is made onto plane Bde�ned by (z1; z4) (�gure 5.2). On this plane, the ow direction � is invariant to allchanges, and enthalpy is only modi�ed by variations in z4. The natural basis againbreaks down since (�rS)B and (�rp)B are co-linear. Note that the stagnation point is aspecial limit because, in its vicinity, the natural basis degenerates and all directionvectors become parallel.(�rS)B = 2664 12 z32+z32z412 z32+z32z1 3775 (�rh)B = 2664 0z32+z32z1 3775 (�rp)B = 2664 M2M2�2z1M2M2�2z4 3775 (�r�)B = 2664 00 3775The projections considered above give some insight about the behavior of thenatural basis but simplify matters more than they should. To simplify the problemas much as possible but still maintain its necessary complexity, we will project thenatural basis on the z4 axis (�gure 5.3). The enthalpy level is essentially given by z4and for a uniform enthalpy ow, the contribution of this variable is justi�ably ignored.As a generalization of the plane A projection, all vectors (�rS)A, (�rh)A, and (�rp)A inthis reduced dimensional space lie on a cylinder de�ned by z22 + z22 = d2. Narrowcylinders simply imply that the point at which we are considering the projection is

104

z z23

Projection onto (z ,z ,z ) (h constant)1 2 3

z1

r )( CS

r )( Ch

r )( Cp

T1

T3

T2

r )( C

cylinder corresponding tocell-wise projection

cylinder corresponding tovertex-wise projection

d d

Figure 5.3: Projection onto a three dimensional space.

105close to a stagnation point whereas wide cylinders indicate the opposite. Given anycell, whether close (triangle T3) or far (triangle T1) from a stagnation region, thecylinders corresponding to cell-wise and vertex-wise projections will be concentric(see �gure 5.3). These concentric cylinders will only have small di�erences in radius�d, but what really matters is the relative disparity between �d and d. That disparityis a good analogy to errors introduced by cell-wise projections. It is clear that forcells far away from a stagnation point, we have a very small relative error between acell-wise and vertex-wise projection, �(�rS;p;�. An equivalent statement is�(�rS;p;�)k�rS;p;�k � 0However, near stagnation points, the cell-wise projection direction can be very dif-ferent from the projection direction evaluated at a vertex.�(�rS;p;�)k�rS;p;�k � 1At each iteration, the error introduced in the new update is of the same order as themagnitude of the projection itself. Therefore the scheme is inconsistent and will notconverge.5.4.2 Treatment at DiscontinuitiesIn shock-free regions, the elliptic part of the residual must be completely de-coupled from the advective portion, therefore providing a way to perfectly preservepotential ow with an Euler solver. The projection of the elliptic updates alongcertain preferred directions e�ectively prevents it from producing spurious entropyand enthalpy. However, in the proximity of shocks, the contributions from the el-liptic residual to the entropy variable is genuine and not just numerical. In thatregion only, pressure and ow angle gradients must be able to a�ect entropy. Unless

106M < 1

M > 1shock

s > sl r

ab

c

Figure 5.4: Cells where the constrained minimization will be neglected.the constrained minimization is disabled in the direct neighborhood of shocks, thescheme will produce an incorrect shock strength. Indeed, this would be the result ofan incorrect entropy jump across the discontinuity. For cells adjacent to a shock re-gion such as the shaded cells in �gure 5.4, the projection will be neglected altogether.These cells have the peculiar property that they sit right on the fence between theelliptic and hyperbolic regions. Such cells are recognized by evaluating the Machnumber at all three vertices and checking that at least one is supersonic while theother two are subsonic and vice-versa. For example cell (a; b; c) would be suspectbecause the following condition is satis�ed,(Ma � 1)(Mb � 1) < 0 and=or (Mb � 1)(Mc � 1) < 0That test by itself is a necessary but not a su�cient condition for the presence ofa shock. Indeed, in expansion regions when the ow seamlessly transitions fromsubsonic to supersonic, at least one of the above inequalities holds. Therefore, we

107

Ta

c

b

qc

qa

qb

Case of shock

S

TS T+dS

a

c

b

qa

qb

Case of expansion

qb

TS T+dS

ST

Figure 5.5: The velocity vectors evaluated at each node translate the triangle. andeither produce a contracted cell (left) or an in ated one (right). Thistest accompanied with 5.4.2 is a robust way of di�erentiating betweenentropy producing shocks and isentropic expansions.need to di�erentiate between isentropic rarefactions and entropy-producing shocks.An elegant method of making that distinction is presented in [53] and we make useof those ideas here. The entropy variable gets advected at the characteristic speed~q. Across expansion waves the entropy remains constant and the ow accelerates,whereas across shocks the entropy increases and the uid slows down. If ~q evalu-ated at each cell vertex is considered as a nodal translation operator, then cells aredeformed as illustrated in �gure 5.5. With such an interpretation for the character-istic speed of entropy advection, cells will shrink across shocks and will in ate acrossexpansion waves. Since velocity always increases through rarefactions and decreasesacross shocks, the change in area of a cell which rides on a discontinuity is a veryrobust indication of the nature of that discontinuity. In [53], the area change formulais given by dSTdt = 12 3Xi=1~qi � ~ni

108The sign of the area change will tell if the characteristics are converging or divergingwhich reveals respectively a shock or an expansion.dSTdt < 0 : shockdSTdt > 0 : expansion

CHAPTER VIIMPLEMENTATION AND RESULTS6.1 Pseudo-Time IterationThe explicit time integration is not a high priority in this work, but we havemade some e�ort to improve convergence rates since slow convergence is a hindranceto test calculations. Most of the code optimization will come from better iterativealgorithms. Note that when it comes to time savings, uctuation and gradient cal-culations have a �xed number of oating point operations for each cell so furtherimprovements there are slim. Eventually, one would want to implement multigridacceleration methods since these are known to produce the most e�cient conver-gence. The residual decomposition into elliptic and hyperbolic parts makes evenmore sense when used in conjunction with multigrid. Indeed, multigrid takes com-pletely di�erent forms for those two types of problems, and our decomposition helpsin making that distinction.Applying the least-squares scheme to the Cauchy-Riemann equations results ina minimization problem at each interior node of the domain. Using the notationintroduced in chapters III and IV, there are a number 2NVi of linear equations to be

109

110solved. When paired up per node, we have,@@ui 24 1ST XT2TiWtRtQRW35 = 0 (6.1)@@vi 24 1ST XT2TiWtRtQRW35 = 0where i = 1 � � �NVi . Combining these 2NVi equations with 2NVb equalities emergingfrom the boundary conditions, the full problem may be be rewritten as a large2NV � 2NV system. The entries bi=1���2NV on the right hand side are all zero exceptfor the equations expressing boundary conditions.u

v1

1

u

vN

NV

V

.

.

.

.

.

.

a a

a

1,1 1,2NV

2N ,1V

a2N ,2N

V V

. . . . .

. . . .

.

.

.

.

.

.

.

.

.

.

.

.

2NV

2NV

.

.

.

.

.

.

.

.

b

b1

2NVThe above is more concisely written using the positive de�nite matrixA = ai=1���2NV ;j=1���2NVand vector b = bi=1���2NV AW = b (6.2)Solving equation 6.2 is equivalent to minimizing the functional of W =W tAW �W tb (6.3)Satisfying either of equations 6.2 or 6.3 enforces local equilibrium on each patch ofcells surrounding a node. It ensures that a minimum has been reached for �i.The linear system 6.2 could be solved using some direct method, especially ifmatrix A has structure. For polar grids, A will be banded but for arbitrary meshes

111which we commonly use, it will be sparse and unstructured. With a given meshconnectivity, there exist techniques to rearrange the node ordering such as to turnA into banded form ([60] and [10]). One can then use the banded LU factorizationor other e�cient methods to solve the linear system. Direct techniques may beappropriate for the Cauchy-Riemann equations alone but they are not when solvingthe full Euler equations which have an advective component in space or time. Thehyperbolic problem is solved using multidimensional upwinding by marching in theadvection direction. Therefore if the Cauchy-Riemann solver is to be blended in thewhole Euler scheme, it must be of iterative type.Among the many iterative linear system solvers, the algorithm commonly used inour code is Successive Over-Relaxation (SOR) which provides signi�cant convergenceacceleration over Jacobi, Gauss-Seidel or other iterations of the typeMxn+1 = Nxn+b where A =M�N. For i = 1 � � �2NVWin+1 = !aii 24bi � i�1Xj=1aijWjn+1 � 2NVXj=i+1aijWjn35+ (1� !)Win (6.4)The relaxation factor ! controls the convergence speed of the iterative method and itsoptimal value is the one that minimizes the spectral radius �(M�1N). The spectralradius is de�ned as the maximum absolute value of the eigenvalues of M�1N, andits analysis for various iterative methods is extensively found in the literature (inparticular [29] and [1]). For each grid that we work with, an elaborate eigenvalueanalysis would be necessary in order to determine an appropriate ! but this processis computationally expensive. We chose a less rigorous option which is to determine! by experience.An alternative method to iteratively solving system 6.2 would be to apply gradient-based techniques to 6.3, the most simple one being steepest-descent, for which a brief

112explanation is given here. Given any point Wc in the solution space, our next guessto the converged solution will be in the search direction rc = b�AWc. As long as rcis not the null vector then there exists a scalar � such that (Wc + �rc) < (Wc).This process is repeated until a minimum for is reached. The steepest-descentmethod is not a particularly e�cient method to step towards convergence. Indeed,if we imagine the isolines of forming a valley, steepest descent travels back andforth across the valley rather than down the valley. This is directly related to thechoice of search directions which end up being almost identical to each other fromone iteration to the other.A more e�cient gradient-based scheme that we have implemented and use quiteoften is the Quasi-Newton method. It aims at modifying the choice of �, and is viewedas an approximation to Newton's method. Using the same notation as before, weconsider the system of non-linear equations@�i(W)@Wj=1���2NV = 0 , r�i(W) = 0 (6.5)The typical Newton method used for root-�nding would lead to the following updateWn+1 =Wn �Hi(Wn)�1r�i(Wn) (6.6)with the (j; k)th component of the Hessian matrix of �i denoted by[Hi(Wn)]j;k = @2�i(W)@Wj@Wk (6.7)The Quasi-Newton method uses and approximation ~Hi(Wn) to the Hessian matrixin 6.7. The approximation that we chose is inspired by [24]. It consists of using onlythe diagonal terms of the original Hessian and setting the rest to zero, that ish ~Hi(Wn)ijk = 8>><>>: 0 j 6= k@2�i(W)@Wj@Wj j = k (6.8)

113Just for illustration purposes, we show here what the approximate Hessian matrixlooks like for the potential ow solver discussed in section 4.4.2, using the samenotation. Note that a similar expression may be obtained for the supersonic regime.diag h ~Hii = @2�i(W)@Wj@Wj (6.9)= XT2Tih�2qniqni + qtiqtiiNtN� � [qniqti � qtiqni ]NtS= XT2Tih�2qniqni + qtiqtiiNtN= XT2Ti h�2qniqni + qtiqtii h�2ata + btbiFor the purposes of convergence acceleration, there are many sophistications to beadded to the above iterative schemes. For example, in gradient-based optimizationschemes, better search direction must be chosen, leading maybe to the conjugategradient method. On the other hand, to improve the convergence of the iterativelinear system solver, we may �rst chose to reorder the grid nodes such as to makematrix A structured. We may then apply more e�cient algorithms on the resultingsystem.6.2 Boundary ConditionsAccurate calculation of the global coe�cients such as lift and drag are very im-portant for aerodynamic design. These quantities depend not only on the accuracyof the scheme used in the interior domain, but also on the wall boundary treatment.Various ways of establishing the wall boundary conditions for the Euler equationshave been studied [34]. We have implemented the ones that suited our cell-vertexapproach and our need for very low level of numerical entropy generation the best.Boundary conditions can either be satis�ed with strong formulations or weak for-mulations. No matter which formulation we use, the state variables are �rst updated

114i

qn

S,h advected

through the solid

surface

tangency

condition

F1

( )n

F3

( )n

F2

( )n fluctuation

as a

sum of

edge

fluxes

Figure 6.1: Boundary treatmentat all the interior nodes using whatever formula is dictated by the interior scheme.Then the state variables on the boundary are updated using the new information butwith additional physical constraints. The di�erence between the strong and weakformulation is that the physical constraints are implemented in an absolute sense forthe former and in an integral sense for the latter.6.2.1 Solid WallsFor a solid wall, we only have to impose enough conditions to satisfy the acous-tic part of the Euler equations. Remember indeed that entropy and enthalpy areadvected along streamlines which are always tangent to solid boundaries. Near su-personic wall regions, only one acoustic characteristic enters the ow domain, hencewe provide only one condition. For subsonic wall regions, the acoustic part is an ellip-tic subsystem so a single fact restraining the velocity components would be enough.The common practice is to make this constraint the tangency (or slip) boundaryconditions, naturally based on physical considerations.If ~q is velocity vector, qt its tangent to wall component, qn its normal to wall

115component, ! vorticity and R wall radius of curvature, then at a node i, one mustenforce the �rst equation of 6.10. The other three are not necessary since the second isalready programmed in the interior scheme and the last two are simply for 'cleaning'around the solid wall. As described in [2], the normal gradient of entropy andenthalpy will be zero at any solid boundary, given the absence of shocks.~q � ~ni = 0@p@n = � qt2R@S@n = � �1� �1 qt!@h@n = 09>>>>>>>>>>>=>>>>>>>>>>>;

inviscid subsonic ow w=odiscontinuities=) 8>>>>>>>>>>><>>>>>>>>>>>:~q � ~ni = 0@p@n = � qt2R@S@n = 0@h@n = 0 (6.10)

In the strong formulation, the usual way of implementing the condition is simplyto ignore the normal velocity component contributed by the update procedure. Incell-vertex schemes, slip conditions can easily be enforced in a strong way becausenodes lie on the wall surface. At each iteration, simply ignore any contributionto the normal component of the velocity at the wall. To determine the normal toa curved surface at each vertex, use the average of the scaled normals to the twofaces which share that node. The scaling is done with respect to the length of thecorresponding face. This approach is rigorously justi�ed in [66] in terms of a least-squares minimization. The standard norm for the governing equations are modi�edto include the above tangency rule multiplied by Lagrange multipliers. This normis then minimized to produce updates not along the gradient path, but along aprojection of the gradient path. The projection is onto a surface where velocitycomponents normal to the wall are zero.The second equation corresponds to the normal momentum equation for rigidstationary walls. As mentioned above, this numerical condition is not implementedsince it is equivalent to the second equation of the 2 � 2 elliptic subsystem. The

116interior scheme already takes advantage of the curvature of the wall boundary becauseon any solid surface @�@s = 1R .The last two equations in 6.10 are enforced more as a luxury than necessity.Remember that zero normal gradients in entropy and enthalpy, at solid boundaries,only hold in inviscid subsonic ow without discontinuities ([2]). They are easilyimplemented since they boil down to taking all the excess entropy and enthalpyproduced at the wall and 'sweeping' them through the solid surface. Just as in theinterior domain, where entropy and enthalpy are already advected in the streamlinedirection, in the cells adjacent to the solid boundary, those same quantities areadvected normal to the wall. We implement the following conditions only at thesolid boundaries, for subsonic ow free of shocks.along streamlinez }| {uSx + vSy = 0uhx + vhy = 0 normal to wallz }| {vSx � uSy = 0vhx � uhy = 0The main advantage of the strong formulation is that it is less dissipative and caneasily be implemented for cell-vertex schemes. Its main drawback is that the schemeused for the boundary points is not the same as the one used for the interior points,therefore it violates global conservation laws. Conservation errors are large enoughto be noticeable in numerical experiments. In �gure 6.2, the computations are madefor a free stream Mach number of 0:85 and 0� angle of attack and should producea symmetric transonic ow even though the unstructured mesh generated aroundthe airfoil is asymmetric. This is an interesting test case because it illustrates howthe use of non-conservative boundary treatment in computing ows with shocks canintroduce asymmetry in the ow. The Mach contours obtained in 6.2 show that theupper and lower shocks are slightly o�set. A good boundary condition should not

117

Figure 6.2: NACA0012 at M1 = 0:85; � = 0�.be a�ected by grid irregularity. A detailed study of various boundary conditions isdone in [2] and the same trend was found. Results presented in that reference revealthat strong boundary conditions are sensitive to grid induced asymmetry. This is aninconvenience not found in the weak formulation, but the latter has its own drawback,it is excessively dissipative. The authors design a hybrid boundary condition whichborrows ideas from strong and weak formulations and has the advantages of both.Although believed to improve the boundary treatment, those hybrid conditions werenot implemented in the context of this work.An alternative to the explicit way of establishing tangency is the weak formula-tion. It is not a boundary condition that we implement but only discuss here forcompleteness. The essential element of a weak wall boundary condition is that theslip requirement is enforced only in an integral sense, through ux functions. Con-sequently, tangency is not satis�ed exactly by the state variables on the boundary.The uxes are determined on the cell interfaces coinciding with the wall boundary.Each cell uctuation can be interpreted as the sum of uxes normal to the three

118faces (see �gure 6.2), where the uxes are calculated under the assumption of linearvariation of the parameter vector. Physically, there is no ow through cell faces thatare adjacent to a wall boundary, so the edge that lies there should not contributeto the uctuation of the cell to which it belongs. Hence, the only ux through theboundary face is due to the pressure force at the wall. One must use arti�cial cellsor ghost-cells that are adjacent to the regular boundary cells and extrapolate thepressure from the interior. Once pressure gradients are known, the ux throughthe boundary face is then calculated using a numerical integration procedure. Fi-nally, the three edge uxes are added to produce an altered uctuation which is thendistributed in a standard uctuation-splitting way.6.2.2 In ow/Out owIn ow and out ow boundary conditions are determined depending on the owregime. Recall that the Euler equations are either purely advective (in the supersoniccase) or share elliptic and hyperbolic properties (in subsonic ow). Accordingly, onewill apply boundary conditions to satisfy incoming and outgoing characteristics.For supersonic in ow, all four advective variables are imposed since they all runfrom the exterior into the domain.For supersonic out ow no ow variables are imposed since no characteristic in-formation is owing into the domain from the outside. The state variables on theboundary are purely determined by the interior scheme.In the case of subsonic in ow two hyperbolic characteristics are incoming, bothstreamlines on which entropy and enthalpy are carried. In addition, enough informa-tion must be given for the elliptic part of the problem to be well-posed. It follows thatfour conditions are required at the subsonic in ow boundary. If the in ow is taken

119far enough from the body (� 30 radii), disturbances in the far-�eld are negligible sowe can just impose the state variables there.p�1 [1; u1; v1; h1]tHowever, for computational tractability, we would prefer a smaller domain that ex-tends roughly 10 radii. For this, we assume that pressure and density disturbancesare negligible in the far-�eld and impose the quantities p1 and �1. We then developRobin conditions for the velocity components. Note that velocity components canbe expressed in terms of a disturbance on top of a constant value, both of whichsatisfy the same set of linear equations. According to small disturbance transonic ow theory, this system is of Cauchy-Riemann type and is given by(1�M12)ux + q2vy = 0 (6.11)uy � vx = 0Since the disturbances u and v are conjugate functions of each other, they may bewritten as Laurent expansions.u = 1Xn=1 1rn (an cosn� + bn sinn�)v = 1Xn=1 1rn (an sinn�+ bn cosn�)where r = qx2 + (1�M12)y2. The pair (r; �) forms a stretched cylindrical coordi-nate system, but as Minfty ! 0 we recover the regular cylindrical coordinates. A�rst-order expansion of the above series produces the following requirements at theboundary @u@r = �ur (6.12)@v@r = �vr (6.13)

120These conditions, in addition to the knowledge of angle of attack (�) and constantfar-�eld values (u1; v1), all form the Robin boundary conditions. They completelyspecify the boundary treatment for the elliptic part of the problem.For subsonic out ow, no entropy or enthalpy information is needed since both arecarried out. The acoustic part of the problem is again handled using Robin conditionswritten in 6.13. In practice, the far-�eld is taken roughly ten to �fteen characteristiclengths (chord or diameter) away. At those distances, changes in thermodynamicvariables are insigni�cant compared to velocity components, so we need to supplyinformation for those two only. Since we hold the parameter vector z as unknown,this means supplying conditions on z2 and z3. The acoustic subsystem involves twovariables and we impose Robin type conditions as in equation 6.13, so the subsonicout ow boundary should be well-posed. It's worth noting that we don't always havepurely supersonic or subsonic out ow, sometimes patches of one regime and the othercoexist. If we suspect mixed out ow, the subsonic treatment takes precedence sinceit provides more information. This might lead to over-constraining the supersonicacoustic variables which are outgoing. Since the acoustic and advective parts aremaximally decoupled, entropy and enthalpy are safe from any changes in pressureand ow direction. Empirically, we have found not had any problems with such atreatment.6.3 Numerical ResultsThe numerical examples presented in this section illustrate the capabilities ofthe techniques described in the previous chapters. They mainly cover various ex-ternal ow �elds, around cylinders, ellipses and airfoils. All calculations are directimplementations of schemes described in sections 4.4.2 and 5.3, which respectively

121describe our potential ow solver and our Euler solver. There will be many cases inthe following sections where we apply the Euler solver to potential ows to evalu-ate its performance on those problems. Unless indicated, all computations are doneusing the third-order accurate version of both the potential and Euler solvers. Onemay argue that for the purpose of aerodynamic lift and drag coe�cients such higherorder accuracy may not be necessary. Nonetheless, for a given desired accuracy, itpermits computations on relatively coarse meshes at a very small additional cost.Whenever an exact solution is known, accuracy is measured by the Lp norm ofthe discrete cell error, de�ned byLp = 0@ 1NT Xi=1::NT j�vi � vexact(�xi)jp1A 1pwhere v is the velocity vector. The order of accuracy is estimated by evaluatingthe behavior of the error norms with increasing mesh re�nement. When graphingthe logarithm of the Lp norms against the logarithm of characteristic cell size 1pNT ,the slope of the plot indicates the order of the truncation error. Throughout thissection, results will be given in terms of L2 to measure error uniformity and L1 formaximum error.6.3.1 Ringleb FlowOne of the few known exact solutions to the steady isentropic irrotational plane ow equations was given by F. Ringleb [85], through the hodograph transformationof the Euler equations. Here it is used to assess the accuracy of our scheme on anunstructured mesh. The solution is parametrized in terms of the velocity magnitudeq and the streamline constant ,x(q; ) = 12� 2 2 � 1q2!� J2 (6.14)

122y(q; ) = � 1 �qvuut1� q !2a = s1� � 12 q2� = a 2 �1� = 2� � sin q !�1J = 1a + 13a3 + 15a5 � 12 log�1 + a1� a�All thermodynamic quantities are non-dimensionalized by their stagnation valuesand all speeds by the stagnation speed of sound. See chapter II for details on non-dimensionalization.One of the attractive features of Ringleb ow is smooth transition from subsonic ow to supersonic and back to subsonic. This feature makes it an ideal candidate fora �rst pass validation of an Euler solver restricted to isentropic irrotational assump-tions. Ringleb ow is symmetric with respect to the horizontal axis and has a clearregion of supersonic ow delimited by what looks like a circle but is not. A physicalsolution is not everywhere well-de�ned since there is a region bordered by the limitline where the solution is multivalued, corresponding to the part of the hodographsolution which folds onto itself. Although Ringleb ow is an imaginary one, solelycreated for its closed form solutions to the Euler equations, it may be interpretedin a couple of di�erent ways. Considering �gure 6.3, it can be seen as ow aroundan in�nitely sharp at plate containing a supersonic pocket, or alternatively as anaccelerating channel ow delimited by any two streamlines.The computations were performed on a domain shown in �gure 6.6. It is borderedon the left by streamline value = 1:2, on the upper right by = 0:6, and on thelower-right (below the cusp) by the limit line. We use a constant-pressure line on the

123

−2 −1 0 1 2 3 4 5 6−3

−2

−1

0

1

2

3

Figure 6.3: Streamlines in Ringleb ow. The limit line is de�ned by the locus ofpoints where the streamlines form cusps.

M < 1

M > 1

sonic

line

limit line

M = 1

Figure 6.4: Zoom on the sonic line and supersonic region.

124

−2.5 −2 −1.5 −1−8

−7.5

−7

−6.5

−6

−5.5

−5

−4.5

log(1/N0.5)

log

(L2(e

rro

r))

u: y=2.0442x−2.6661

v: y=1.9884x−2.6218

Figure 6.5: Accuracy of second-order Euler solverupper border and the symmetry axis on the lower border. The resulting ow has asubsonic in ow and a mixed supersonic/subsonic out ow.The results in 6.6 are produced using the second-order least-squares implementa-tion for the acoustic part of the Euler equations, both in the supersonic and subsonicregimes. The scalar advection equations for entropy and enthalpy were solved usingany higher order uctuation-splitting scheme. That choice is not important herebecause Ringleb ow being shock-free, it does not demand the monotonicity prop-erty. The boundary conditions that were imposed were tangency conditions on thestreamlines, a Neumann type condition imposing symmetry on the horizontal axis,constant pressure along the top boundary and characteristic conditions on the limitline. The isolines of Mach number and ow direction show that the ow is extremelysmooth so second-order accuracy (see 6.5) is not suprising. Although Ringleb owis a good test case to show the behavior of an Euler solver in the limit of potential ow, the next step is to move to less benign problems that include singularities and

125discontinuities.6.3.2 CylindersMeshesWe choose to only simulate ows around external bodies because they containthe necessary di�culties. In particular, cylinder ow is one of the best problems withwhich to test elliptic equation solvers since any result less than perfectly symmetricis incorrect. Indeed, at low enough Mach numbers and for uniform entropy andenthalpy in ow, the ow will be irrotational with perfect fore and aft symmetry.Numerical results were generated on both uniform and arbitrary triangular grids(�gure 6.9) with mesh sizes varying from 1600 cells for the coarsest to roughly 20000for the �nest. On the body itself, the number of nodes varied from 40 to over300 where the high end is excessive even when very accurate solutions are required.Calculations are made for incompressible (M1 = 0:01), subcritical (M1 = 0:41) andtransonic (M1 = 0:60) regimes.M1 = 0:01; � = 0�To study the accuracy of the least-squares scheme in low-speed calculations, wepresent results of cylinder ow in the incompressible limit, M1 = 0:01. Note thatin �gures 6.10 and 6.12, the numerical solution from our Euler and potential owcalculations are superimposed. Please refer to section 4.4.2 for a description of ourpotential ow solver. The �gures help visualize how our Euler solver captures po-tential ow in the low-Mach number limit with great �delity. All results presentedhere were obtained with the third-order version of the elliptic discretization. TheLDA and SUPG schemes were used to separately advect entropy and enthalpy forthe hyperbolic part of the Euler solver but di�erences in the �nal solutions were

126Inflow Inflow

Figure 6.6: Pressure isolines (left), ow direction contours (right).Inflow Inflow

Figure 6.7: First and second Riemann invariants.Inflow

Figure 6.8: Ringleb streamlines (left) and grid used for this set of Euler calculations(right).

127

Figure 6.9: Sample of triangular grids used for cylinder ow calculations. Cylinder isof unit radius, far-�eld boundary taken at 10 radii. The structured mesh(left) forms the basis for polar coordinates and is stretched to capturethe high gradients close to the body. The unstructured mesh (right) isgenerated by a frontal Delaunay method as described in [50].negligible.The Mach number and ow direction contour plots in 6.10 show perfect fore andaft symmetry about the cylinder. The dotted contours are for Euler whereas the solidones are for potential ow. The computations were performed on an arbitrary gridso its clear that mesh irregularities do not a�ect the desired symmetry. In addition,as shown in �gure 6.12, the cylinder surface pressure coe�cient resulting from bothEuler calculations (dotted line) and potential ow (solid line) fall nearly atop eachother. For a grid containing 60 nodes on the body, di�erences at the suction peakwere less than 0:23%. The numerical solution produced by our potential ow solvercoincides with the exact solution that is available for irrotational incompressible ow(not labeled). Naturally, with top-bottom and left-right symmetry in static pressure,the lift and drag coe�cients were correctly computed to be machine zero.This result is in sharp contrast with many conventional Eulers codes which pro-

128

X

Y

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

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0

1

2

3

4

5

0.01049540.010023

0.00970787

0.00891667

0.0112878

0.0120625

0.008125

Mmin = 0.001Mmax = 0.02dM = 0.000543

Cylinder M=0.01, superimposed Mach contour lines for potential flow and Euler

Figure 6.10: Incompressible cylinder ow, Mach isolines. Potential ow shown insolid lines, Euler ow shown in dashed.

X

Y

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

5

Thetamin = -1.47Thetamax = 1.49dTheta = 0.1565

Cylinder M=0.01, superimposed flow direction contour lines for potential flow and Euler

Figure 6.11: Incompressible cylinder ow, ow direction isolines. Potential owshown in solid lines, Euler ow shown in dashed.

129

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−2

−1

0

1

2

3

4

5

downstream distance: x/c

pre

ssu

re c

oe

ffic

ien

t: −

Cp

Cylinder M=0.01, presure distribution on upper and lower surfaces

− dashed line: upper and lower Cp for Euler solver

− solid line: upper and lower Cp for potential flow solver

Figure 6.12: Coe�cient of pressure distribution on cylinder surface.duce an incorrect wake behind the cylinder, therefore corrupting the CD calculation.Numerical wake e�ects are associated to excessive entropy generation which then ad-vects downstream along the ow path. As mentioned in section 6.3.3, when it comesto lifting bodies, spurious entropy will also a�ect CL calculations. Throughout thiswork, levels of numerical entropy are measured by� = ����� p=p1(�=�1) � 1�����except in �gures 6.17 and 6.19 where we use the de�nition � = ��� p=p1(�=�1) ��� instead.Figure 6.14, shows a plot of � distribution over the cylinder surface and compares itto the numerical entropy produced by the Lax Wendro�-PSI (LW-PSI) scheme foundin either [3], [47] or [56]. As presented in those references, LW-PSI is also based on anEuler decomposition into hyperbolic-elliptic parts which, however, remain somewhatcoupled once discretized. It is clear that with � in the order of 1:0 � 10�8, our

130entropy level are several orders of magnitude lower than those found in state of theart Euler solvers. The smallest spurious entropy level reported for a similar test caseis 9:3� 10�5 in [3].A study of the L2 norm of the velocity error (�gure 6.13) revealed it to decreaseas third-order versus the characteristic cell size, for both potential ow and Euler.Results depend somewhat on grid irregularity, since accuracy diminishes with greatermesh randomness and reaches below third-order. Recall that higher-order accuracywas obtained simply by assuming cubic variation of the data on edges and appro-priately modifying the uctuation calculation. This was done with both advectiveand elliptic operators, and the results presented here show that their combinationpreserves the desired outcome. Figure 6.13 also shows L1 error, the maximum er-ror found in the computational domain, usually found in the stagnation region forblunt-body ows. The maximum error shows close to third-order behavior as well.M1 = 0:41; � = 0�We now move on to a set of numerical experiments performed for the cylinderat the near-critical ow Mach number of M1 = 0:41. Here again, the hyperbolicpart of the problem was handled with either LDA or SUPG upwind advection, andthe elliptic part used least-squares. The Mach number distribution on the cylindersurface (�gure 6.15) shows that the ow is at the verge of becoming transonic. Wechose to superimpose potential ow and Euler results to clearly state that the twoshould match, as long as compressible e�ects do not produce discontinuities. BothEuler and potential ow results display near-perfect symmetry, as indicated by theMach isolines in �gure 6.16. The corresponding pressure distribution produces aero-dynamic coe�cients CL = 4:3� 10�12 and CL = 1:6� 10�10. Since the acoustic and

131

−2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1−8

−7

−6

−5

−4

−3

−2

−1

log(1/N0.5)

log

(L2(e

rro

r))

v: y=2.9943+1.1816

u: y=2.9929+0.9510

Figure 6.13: Log-log plot of L2 and L1 error norms versus characteristic cell sizeshows third-order accuracy.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−3

−2

−1

0

1

2

3

4x 10

−4for dashed, x 10−8 for solid

downstream distance: x

en

tro

py

ge

ne

ratio

n le

vel,

Σ

Spurious entropy generation from two different approaches. Note that the y−scale for the two plots are different:− dashed: LW−PSI− solid: LS−PSI

Figure 6.14: Numerical entropy generation on cylinder surface. Note that the least-squares-PSI scheme produces entropy levels several orders of magnitudelower than Lax-Wendro�-PSI (version found in earlier Euler solversbased on hyperbolic-elliptic decomposition). The above calculationsare done with double precision.

132advective parts of the solver decouple completely, it should be expected that Eulerreduce to potential ow in the irrotational limit. This simply re ects the extremelylow levels of numerical entropy generation by the Euler solver.In conventional Euler codes, the high level of spurious entropy generated at thesolid boundaries is concentrated at stagnation points where the dissipation e�ectis highest. Intuitively, one can attribute the problem to the upwinding that takesplace at the stagnation points even though no real waves exist. In Godunov-typeschemes, convergence of steady-state problems is achieved by the continuous sweep-ing of unsteady waves. Therefore, the stagnation point experiences compression andrarefaction waves which eventually erode the solution, even if the upwind methodhas good (very small) dissipation properties. As an alternative, we propose to solvean elliptic-type equation at the stagnation point since those are the equations thatcorrectly model the physics in that region. However, the Euler equations inherentlycontain a hyperbolic (entropy, enthalpy) component which we have been able to max-imally decouple from the elliptic part. Our low values of (CL; CD) are a consequenceof negligible entropy generation which leads to believe that our decoupling of theEuler residuals and the constrained minimization have been successful. Figure 6.17shows that arti�cial entropy levels from our code are in the 10�7 range while thebest results reported in literature ([3], [47]) is 4:0� 10�4 (a three order of magnitudedi�erence) for a similar near-critical cylinder ow.In any case, it seems that the present formulation provides a solid framework fortreating some of the more subtle features of Euler ow. For instance, not shownin the �gures but of certain importance is that enthalpy stays everywhere constantand equal to its initial far-�eld value. This property is not designed into most Eulercodes based on exact or approximate Riemann solvers. Also, the pressure non-

133

X

Y

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

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-2

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0

1

2

3

4

5

0.402788

0.386651

0.329963

0.278992

0.423997

0.443082

0.500569

0.559799

X

Y

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

5

Cylinder M=0.41, superimposed contour lines for potential flow and Euler

Mmin = 0.0623Mmax = 0.9872dM = 0.04853

Figure 6.15: Near-critical cylinder ow, Mach isolines. Potential ow (solid line) andEuler (dotted).

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

downstream distance: x/c

Ma

ch n

um

be

r o

n b

od

y

Cylinder M=0.41, Mach number distribution on upper surface

(lower) Euler result

(upper) potential flow result

Figure 6.16: Mach number distribution on the cylinder surface. Potential ow (uppercurve) and Euler (lower curve).

134

Distance along AA’ cut-4 -2 0 2 4

0.999999

0.999997

0.999995

0.999993

0.999991

0.999989

Entropy Level

0.98000.9850

0.99000.9950 1.00000

.975

0

0.9

70

0

0.9

65

0

0.9

60

0

0.9

55

0

0.9500

0.9450

0.9400

Pressure isolines instagnation point region.

Figure 6.17: Numerical entropy production on cylinder surface (left). Close-up ofthe leading edge, showing a non-dimensionalized pressure of exactly 1.0at the stagnation point (right).dimensionalized by its stagnation value reaches exactly 1:0 at both the front andrear stagnation points. A close up view of the leading edge is provided in 6.17.We speci�cally picked the Mach number to be 0.41 because ow around a cylinderat these speeds is on the verge of becoming transonic. Actually, for cylinders, thecritical Mach number of potential ow is a much contested topic of discussion in uidmechanics. Perturbation series analysis performed by several authors including vanDyke in [82] places the critical Mach number atMc = 0:4026, plus or minus a certainuncertainty range which depends on the extent of the series expansion. However,the same article points out that there is a controversy hanging over when the �rstsigns of a shock appear in the ow. It is generally agreed that there exists a rangeof Mach number between M? = 0:3982 and Mc where the ow is super-critical butshock-free, although the physical explanation is not yet clear. More recent results,both numerical and analytical, cast a shadow on the exact critical number. Thisopposing view is also taken by the present work, where contours shown in �gure 6.15

135A A’

M=1

Figure 6.18: Super-critical ow around cylinder, M1 = 0:60.reveal no sign of shock.M1 = 0:60; � = 0�The case of a transonic cylinder is included in this set of examples because itdemonstrates the behavior of the elliptic discretization in the presence of strongdiscontinuities. On the subsonic side of the shock, we update the solution withthe non-monotone least-squares scheme so post-shock oscillations are expected eventhough none are noticed. The shock is relatively strong as shown by the entropyjump (�gure 6.19), but least-squares has enough damping to contain overshoots. Onthe supersonic side of the shock, PSI (a non-linear version of the N scheme) is used forall four advective quantities so monotonicity is expected, and actually obtained. Dueto its limiting, PSI is everywhere second-order accurate except in the neighborhoodof shocks where it reverts back to a regular N scheme, this �rst-order.

136

Distance along cut AA’

Pre

ssu

re

-2 0 2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

sonic region

Distance along cut AA’

En

tro

py

-2 0 2

1

1.1

1.2

sonic region

Figure 6.19: Pressure distribution (left) and entropy levels (right) along cut 'AA'.Note the jump in pressure due to lack of 'smoothing' at the sonic line,it could be considered minor given the magnitude of the correspondingentropy change of 6:7�10�4. Note the absence of post-shock oscillationeven though least-squares in not monotone. This seems to indicate thatleast-squares has satisfactory damping properties in all directions.Another signi�cance of this test case is the handing of the sonic line, or lackthereof. Observe in �gure 6.19 that, at the sonic line, the pressure experienceswhat's considered a minor jump as measured by the very small entropy change (6:7�10�4). The ow transitions from subsonic to supersonic in a relatively smooth wayconsidering that no special treatment is applied at the sonic line. This result issurprising given the fact that two schemes of completely di�erent nature are meetingat the interface without any type of blending.With top and bottom symmetry in the pressure coe�cient, it is no surprise thatthe lift coe�cient is very close to zero, actually 3:5� 10�7. The drag coe�cient, onthe other hand, was calculated to be 0:804.

137Non-uniform enthalpy in owA good test case for evaluating the extent of hyperbolic-elliptic residual decom-position is non-uniform enthalpy in ow. Moreover, exact solutions exist for specialcases of such a problem. For the ow described in �gure 6.21, analytical equations areprovided in [25] which studies non-zero vorticity ows subject to corners or angles.The main di�culty in computing such a ow is capturing the recirculation zones inthe front and back of the cylinder. These features are extremely sensitive to the localenthalpy levels, so any small error will either smear or erase the recirculation. Forbest results, enthalpy must be carried along with a minimum amount of di�usion,and with little or no coupling from the rest of the system. The acoustic part can beparticularly corrupting because of its smoothing e�ects. Compressible ow solversgenerally fail to sharply capture the recirculation zones because they have no wayof preventing contributions from the acoustic portion to enthalpy. Even of Ta'asan'snon-conventional approach [76] of expressing the Euler equations in canonical formstill couples the enthalpy with the elliptic part of the problem.We use that problem as a source of validation for the constrained minimizationprocedure that maximally decouples advective and acoustic updates. Performingthe decomposition at the residual level and then making the appropriate projectionsin the natural basis seems to have the desired e�ect. As shown in �gure 6.21, theadvection of enthalpy is not corrupted by the smoothing e�ect of least-squares inthe neighborhood of the stagnation points, therefore accurately rendering the recir-culation zones. From these results, it appears that the projection ideas discussed inchapter V are successful in decoupling the elliptic operator from the advective part.Although at the wall boundaries we only imposed simple tangency conditions,the far-�eld boundary conditions are not completely straightforward. We proceed

138M = 0.15

= 0

inflow velocity profile

constant vorticity field oneach side of the horizontalaxis of symmetry, = const

streamlines

Figure 6.20: Cartoon of non-uniform enthalpy in ow problem. Due to the linearin ow velocity pro�le, the vorticity is constant everywhere in the ow-�eld.

Three stagnation points on front and rear of bodyFigure 6.21: Mach contours around cylinder and some streamlines showing circula-tion zones around the trailing-edge stagnation point.

139Non-Uniform Enthalpy Inflow, M=0.15

−2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2−7.5

−7

−6.5

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

log(1/N0.5)

log

(L2(e

rro

r))

u: y=3.0018x+0.8331

v: y=2.9937x+0.7299

Figure 6.22: Streamlines superimposed with contour lines of enthalpy (left). Sinceenthalpy is advected along streamlines, the two should coincide. Thefore-aft symmetry of the ow is well captured as indicated by the cir-culation zones. Order of accuracy of the Euler solver on this problemshows third-order behavior.to make use of the exact solution available, it gives the stream-function (x; y) interms of the constant vorticity !. According to Crocco's theorem~q � (r� ~q) = rh� TrSwhere ~q is velocity vector, T temperature, h total enthalpy and S entropy. For theproblem at hand, we assume homentropic ow so entropy gradients vanish every-where, rS = 0. Also, vorticity is constant every, ! = (r � ~q)=constant, thereforeCrocco's relation reduces to rh = ~q � ~! = ! � ( ~r ) (6.15)With each quantity non-dimensionalized by its stagnation state and using the appro-priate boundary values, equation 6.15 integrates to h= �1�! . Since h= �1 p�+ 12q2,it follows that �=1= �1 + 1 (! + 12q2)�. All this provides enough information to fullyconstrain the problem at the far-�eld boundary.

140

Figure 6.23: Grid used for ow calculations around ellipse.6.3.3 EllipseWe performed Euler calculations in a test case around an ellipse to illustratethe high resolution of the fore and aft stagnation points and suction peaks, even ata high degree of pitch. Note that the ow is at the verge of becoming transonicand although the peak Mach number in the front and back of the ellipse are notexactly alike, they are very close. Such symmetry in the Cp pro�le would not havebeen obtained if we had excessive arti�cial dissipation. The lift and drag coe�cientscalculated for the ellipse are 4:63� 10�3 and 2:25� 10�2 respectively which revealsthat the arti�cial viscosity generated from discretization errors is very small.The main purpose of this test case is to show that the scheme presented heremeets the challenge pointed out in [61]. This paper notes that Euler codes appliedaround cylinders and ellipses at angle of attack are prone to produce lifting solutions.The arti�cial lift in poor solutions is associated with the circulation generated by thearti�cial viscosity, which in turn is due to discretization errors. The correct solutionis zero CL and CD as long as there is no net circulation around the body. Any

141deviation from that is not acceptable. Our scheme consistently produces zero liftsolutions in these cases and this is independent of the mesh de�nition (see �gure6.25). We satisfy the discrete entropy relations directly to the same order of thetruncation error of the discrete conservation laws. Hence, entropy stays constantalong streamlines. In addition, entropy does not a�ect the rest of the equations sothere is no mechanism to break the symmetry inherent in the elliptic operator. Inthis manner, we avoid the wake (and therefore circulation) that is usually generateddue to excessive numerical viscosity.6.3.4 AirfoilsMeshesFlow simulations around two-dimensional NACA 0012 airfoils have become astandard since many comparative studies can be done with other work. Again,computational results were generated on both uniform and irregular triangular grids.The O-type grid in 6.26 was generated using a least-squares minimization on theLaplacian norm of the coordinates (x; y). The procedure is identical to applying least-squares on Cauchy-Riemann except that the coordinates are taken as the solutionspace. The mesh size ranged from roughly 3500 to over 20000 cells. It was foundthat around 160 nodes on the wing body are necessary for accurate solutions, evenwith sharp suction peaks. The unstructured grid in 6.26 was generated using afrontal Delaunay method according to the algorithm described in [50]. It is usuallyinteresting to look at the dissipation produced by a scheme when executed on highlyirregular grids. Indeed, that sort of study reveals how grid-sensitive the solutionmight be or whether symmetric solutions will be preserved. With low levels ofarti�cial viscosity, we should expect potential and subcritical ows to have nearperfect symmetry, even on asymmetric grids.

1420.2871

0.3445

0.21640.3445

0.2871

0.2164

Figure 6.24: Streamlines and some Mach contours, clearly indicating no lift on theellipse.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

downstream distance: x/c

Ma

ch n

um

be

r o

n b

od

y

Ellipse M=0.35, α=45o, body Mach number

fore and aft stagnation points

fore and aft suction peaks

Figure 6.25: Mach number pro�le over the ellipse body shows good symmetry. Thetrailing edge suction peak is almost captured with same intensity as theleading-edge one.

143

Figure 6.26: Sample grids used for NACA 0012 ow calculations. Airfoil has unitchord and far-�eld boundary is usually taken at around 10 chord lengthsaway from the airfoil.M1 ! 0; � = 0�Testing our Euler solver for applicability in a large range of Mach numbers startsin the incompressible range. Experiments were performed at M1 = 0:0001, M1 =0:001, M1 = 0:01, and M1 = 0:1. Pressure coe�cient for all test cases are shownin �gure 6.28 and compared to the solution produced by panel method. The Machcontours in �gure 6.27 reveals good symmetry with respect to horizontal access. Thisis con�rmed by the symmetric distribution of pressure coe�cient on upper and lowersurfaces. The aerodynamic coe�cients are machine zero for CL and CD = 6:5�10�8.The numerical entropy generation for M1 = 0:1 is roughly 7:0 � 10�8, which isover three orders of magnitude lower than the best results found in the literature([47], [56], [3]). The low levels of entropy generation are again attributed to the factthat the elliptic and hyperbolic parts are maximally decoupled and the elliptic partis constrained not to contribute to entropy or enthalpy production.

144

X

Y

-0.1 0 0.1 0.2 0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.0101813

0.0104082

0.0107143

0.0110204

0.00979592

0.00968312

0.0100131

X

Y

0 0.5 1 1.5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Mmin = 0.000304Mmax = 0.01242dM = 0.000251

NACA 0012 M=0.01, Mach contour lines for Euler solver

Figure 6.27: NACA 0012 calculations in the incompressible regime. Mach contourplots are superimposed for comparison, showing that all solutions areself-similar in the incompressible limit.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.8

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0

0.2

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0.6

0.8

downstream distance: x/c

pre

ssu

re c

oe

ffic

ien

t: −

Cp

Incompressible NACA 0012, presure distribution on upper and lower surfaces

− solid line, M=0.001,M=0.01,M=0.1, Euler solver

− dotted line, panel method

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15

6

7

8

9

10

11

12x 10

−8

downstream distance: x/c

en

tro

py

leve

l: Σ

NACA 0012 M=0.1, numerical entropy generation on body

Figure 6.28: NACA 0012 calculations in the incompressible regime. Pressure co-e�ecient (left) and numerical entropy generation over body (right).

145

10−4

10−3

10−2

10−1

100

101

102

Mach number

nu

mb

er

of ite

ratio

ns

(no

rma

lize

d b

y va

lue

at M

=0

.1)

Convergence behavior of Euler solver in the incompressible limit

Figure 6.29: Comparison of convergence history between M1 = 0:1, M1 = 0:01,M1 = 0:001 and M1 = 0:0001 shows that number of iterations is notinversely proportional to Mach numberFigure 6.29 shows a study of convergence behavior as a function of Mach num-ber. A typical compressible ow solver would have very poor convergence as theMach number approaches zero because of the disparity between propagation speedsof acoustic and advective waves. In that case, number of iterations are inverselyproportional to the Mach number. Local preconditioning combined with multigridacceleration provides a remedy by achieving optimal convergence [15], where itera-tions are constant in Mach number and only depend on grid size. Our scheme doesnot implement multigrid but it shares some common ground with preconditioning inthat convergence only mildly depends on the Mach number. As seen in �gure 6.29,the number of iterations increases roughly twenty-four fold while the Mach numberdecreases by a factor of one thousand.M1 = 0:70; � = 1:65�In the supercritical case, our Euler solver behaves equally well (see �gure 6.30).The number of elements in the mesh is comparable to the same test case performed in

146[90]. Calculations of lift and drag coe�cients produce CL = 0:326 and CD = 0:0614which are well within the acceptable range suggested by that reference. The PSIupwind scheme is used for advection of entropy and enthalpy, as well as the acousticRiemann invariants in the supersonic regions. The pressure coe�cient pro�le isoscillation free in the pre-shock region due to the monotonicity of PSI. In the post-shock region, least-squares scheme has enough damping to prevent any oscillations.Unlike schemes based on Riemann solvers, the damping happens independent ofdirection and grid orientation. Again, note the very low levels of spurious entropy inregions away from shocks (�gure 6.32).The di�culty in computing transonic ows stems from the mixed mathematicalbehavior of the Euler equations. The acoustic part is elliptic in some regions, hyper-bolic in others. These regions are separated by sonic lines in expansion waves or shockdiscontinuities whose positions are unknown. The sonic line obstacle is re ected inour solver by an upwind scheme meeting, somewhat uneventfully, an omni-directionaldistributive type scheme. Empirical results indicate that the scheme is robust for awide variety of ows, even with a lack of special treatment at the interface. Contourplots show that the transition across the sonic line takes place relatively smoothly.There is a small jump in entropy inside the Prantl-Meyer expansion, of roughly9:0� 10�5 magnitude. Eventually, a more rigorous treatment of the sonic line mustbe implemented to guarantee a seamless transition and perfectly isentropic rarefac-tions.M1 = 0:85; � = 1:0�The �gures presented in 6.34 are the result of Euler calculations on a NACA 0012airfoil with a free-stream Mach number of M1 = 0:85 and � = 1:0� incidence. This

147

X

Y

0 0.5 1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Mmin = 0.0Mmax = 1.3dM = 0.0325

NACA 0012, M=0.70, 1.65 degree pitch, Mach contours

Figure 6.30: NACA 0012 calculations in the supercritical regime. Mach contours forM1 = 0:70; � = 1:65�

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

downstream distance: x/c

pre

ssu

re c

oe

ffic

ien

t: C

p

NACA 0012 M=0.70, α=1.65 degrees, Pressure coefficient

upper surface

lower surface

Figure 6.31: Pressure coe�cient distribution

148

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−3

downstream distance: x/c

en

tro

py

leve

l: Σ

NACA 0012 M=0.1, numerical entropy generation on body

sonic line

shock

numerical entropy generation on top surface

0.4776

0.4352

0.3929

0.3505

0.3081

0.22330.2657

Mmin = 0.0962Mmax = 1.3432dM = 0.0423

Figure 6.32: Distribution of entropy along the body surface (left) and close-up of theleading edge stagnation pointis a di�cult test case because (CL; CD) calculations are very sensitive to the shockstrength and position on the wing's upper and lower surfaces. Calculations of lift anddrag coe�cients produce CL = 0:361 and CD = 0:0528 which are within a percent ofthe values published in [90] for the same ow conditions and comparable mesh size.M1 = 0:05; � = 25�The �gures presented in 6.36 are the result of Euler calculations on a NACA 0012airfoil with a free-stream Mach number of M1 = 0:05 and � = 25� incidence. Theydemonstrate the capability of our scheme to accurately calculate a high-lift con�g-uration (CL = 2:84). The plot of Mach number distribution over the wing surfaceshows a small region of compressible ow (M � 0:3)amid a nearly incompressible ow �eld. This is just another instance where compressible e�ects (near suctionpeak) mix with an otherwise low speed ow.

149

X

Y

0 0.5 1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

NACA 0012, M=0.85, 1.0 degree pitch, Mach contours

Figure 6.33: Transonic ow around NACA 0012 atM1 = 0:85 with � = 1:0� angle ofincidence. The plot of Mach contours emphasizes the upper and lowersurface shock formation.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1

−0.5

0

0.5

1

downstrean distance: x/c

pre

ssu

re c

oe

ffic

ien

t: C

p

NACA 0012 M=0.85, α = 1.0 degree, Pressure coefficient.

upper surface

lower surface

Figure 6.34: Coe�cient of pressure distribution on upper and lower bodies of theairfoil.

1500.0592

0.0559

0.0699

0.0284

0.05

24

0.0350

0.0404

0.0430

0.0874

Figure 6.35: High-lift subsonic ow around NACA 0012: M1 = 0:05; � = 25�. Machcontours and some streamlines.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−30

−20

−10

0

10

20

30

40

50

downstream distance: x/c

pre

ssu

re c

oe

ffic

ien

t: −

Cp

−solid line: numerical solution−dashed line: panel method

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

downstream distance: x/c

Ma

ch n

um

be

r o

n b

od

y

NACA 0012, M=0.05, 25o angle of attack, body Mach number

suction peak

stagnation point

Figure 6.36: Pressure coe�cient (left) compares numerical solution obtained againstpanel method. Mach number distribution (right) shows strong suctionpeak.

CHAPTER VIICONCLUSIONThe prototype Euler scheme which was built here follows a lineage of solversbased on discretizations that closely follow physical phenomena ([47], [56]). It aimsat accurately computing compressible ows for a large range of Mach numbers whilepaying close attention to the usual di�culties, whether they are shock capturing,stagnation points or sonic lines. Although the scheme as a whole does not yet serveas a recipe for solving the Euler equations in practical applications, many of the ideasimplemented could be the seed for future developments. Indeed, this presentationdoes intend to illustrate some new paradigms in obtaining high-quality solutions tosystems of conservation laws.7.1 Contributions over Previous Fluctuation-Splitting WorkThere are two main pillars which guide our design. The �rst is decompositionof the steady Euler equations into homogeneous hyperbolic and elliptic parts, onlyinvolving the natural variables. The second is applying uctuation-splitting schemesspeci�cally adapted to each residual.The method developed in [47] and [56] also uses a unique decomposition of theEuler system, in conjunction with uctuation-splitting. However, the main contri-151

152bution of this research over previous work is in the treatment of the acoustic partof the steady Euler equations. Many aspects of an Euler solution, such as accuracyof incompressible and irrotational regions, the amount of numerical entropy gener-ation, are linked to the way we discretize the acoustic subsystem. In [47] and [56],the decomposition is applied to the spatial gradients. When the acoustic parts arethen collected to form the corresponding residual, they represent a discretizationof the time augmented Cauchy-Riemann system. Converting the acoustic systeminto its 'pseudo' time-dependent counterpart is not the correct treatment for ellipticsubsystems, no matter what symmetric scheme is employed. Instead, we rely ondecomposing the temporal gradients into hyperbolic and elliptic parts. Since tran-sient solutions do not matter in this approach, each operator may independently besolved towards steady-state at its own convergence rate. It achieves decoupling atthe numerical level and ensures that acoustic and advective parts, which naturallyre ect di�erent physical phenomena, do not interact.There are a number of discernible advantages that appear in decomposing at thelevel of the residuals, as opposed to the partial di�erential equations. An importantone is that it allows a symmetric scheme such as least-squares, which acts throughdi�usion only, to have maximum e�ect on the subsonic acoustic system. Yet anotheradvantage is the use of constrained minimization which guarantees minimal e�ect ofthe acoustic part on the advective quantities, and vice-versa. Bene�ts also extendto the actual design of the algorithm. First, the process of calculating decomposedresiduals is more automatic because it requires fewer transformations between x andz variables. The second bene�t is e�cient residual calculation, since a seeminglycomplex chain of matrix multiplications may be reduced to simple explicit formulae.

1537.2 Summary of our ApproachDecomposing the conservative Euler residual eventually leads to four indepen-dent residuals, two for entropy and enthalpy convected along streamlines and twofor acoustic Riemann invariants along Mach lines. More interesting is the caseof subsonic ow, where all else remaining the same, a single residual appears forthe acoustic part and represents a Cauchy-Riemann type system. The presence ofCauchy-Riemann and advective portions in the Euler residual justi�es the need forseparate treatment of each operator according to the physical phenomena it tries tomodel. For the advection part, this means using uctuation-splitting schemes to dis-tribute the residual in the direction of characteristics. Multidimensional upwindingfor scalars is well understood and provides desired properties such as narrow shockcapturing, accuracy and monotonicity. On the other hand, the ideal distributionfor the Cauchy-Riemann residual would be one that propagates information in anisotropic fashion. The least-squares scheme seems to answer that concern, and hasthe advantage that when cast in the language of uctuation-splitting, it nicely in-tegrates in the rest of the algorithm. It consists of minimizing the residual in theleast-squares sense over a patch of cells surrounding each node. The weighting matrixwhich enters in the choice of the norm has important consequences on the results ofthe minimization. Since contributions to nodes follow from a minimization process,the scheme has no advective component to it, unlike Lax-Wendro� or SUPG. On auniform grid, least-squares applied to the linear Cauchy-Riemann system resemblesthe central �nite-di�erence discretization of two heat equations.In the process of looking for a discretization customized for the Cauchy-Riemannequations, we realized that it is not as simple a task as one might imagine. Although

154least-squares seems like a good candidate, some questions remain unanswered. Inparticular, there is the issue of coupling, the discrepancy found in it between bound-ary conditions and the interior scheme. Remember that tangency condition couplesthe velocity components whereas least-squares tends to treat them as two separateheat equations. Our �x, which turns out to be very robust, consists of rede�ning anon-area weighted norm. More work is needed to better understand the underlyingreasons.As far as computational e�ciency is concerned, one important factor is the cal-culation of the uctuation. We saw that residuals and gradients present a �xed costat each iteration and that expense can be reduced by factorizing expressions as muchas possible, reusing previously computed terms and only working with a single set ofvariables. Even though the overall scheme is developed within the framework of nat-ural variables x = (S; h; p; �), we chose not use them in actual calculations becausethey are di�cult variables to work with in numerous aspects. Respecting our require-ment of only holding a single set of variables, the parameter vector z = p�(1; ~q; h) isselected since it conservatively linearizes the Euler equations. In order to obtain bet-ter than second-order results, z is represented as Hermite cubic instead of piecewiselinear elements. This allows a higher quadrature of the residual and third-order ac-curacy at very little additional cost compared to second-order. The solver is modularenough that no changes are made to the distribution step.7.3 Highlight of Computational ResultsComputational experiments reveal very satisfactory results for compressible owscontaining low-Mach number regions. Our transonic calculations show relativelysmooth transitions at sonic lines, where the magnitude of entropy increase is small

155enough to be considered negligible. Ultimately, some special treatment of sonicpoints could �nd its way into the solver, maybe in the form of smoothing or anentropy satisfying condition. Stagnation points, often considered as a singularity,are captured automatically. The solver is able to accurately recover incompressible ows, with a convergence behavior which only mildly depends on the Mach numberas it approaches zero. The best situation would be when the number of iterations isO(N) (N , number of cells) asM ! 0. Preconditioning achieves this by removing thesti�ness which appears in the unsteady Euler equations at very low Mach numbers.To test the decoupling between the acoustic and advective parts, numerical ex-periments were performed at low Mach numbers for ows with and without vorticity.In the irrotational limit, the solver correctly recovers potential ow with very lowlevel of entropy generation and exactly uniform enthalpy �eld. Computational re-sults with non-uniform enthalpy in ow are provided to strengthen our case somemore. The residual decomposition appears to be successful because, without it, theelliptic part would a�ect enthalpy and cause smearing of the recirculation zones nearthe stagnation points. We attribute part of this outcome to the constrained mini-mization ideas, which essentially consist of projecting updates to acoustic variablesonto surfaces of constant entropy and enthalpy. It is still too early to point to thatprocedure as a feature which should establish itself in future Euler solvers. However,numerical results all indicate that when acoustic and advective residuals decouple,the constrained minimization has bene�cial e�ects.7.4 Practical Issues for the FutureThe highlights provided in the previous section certainly indicate that we havestretched out the limits of compressible ow solvers. We have produced solutions

156which answer Pulliam's challenge ([61]), replicate potential ows to machine accu-racy, and robustly capture incompressible ows. Despite these accomplishments,one could question how our method �ts onto a more practical picture, one whichincludes viscous e�ects or three-dimensional geometries. In this section, we discusssome ideas for future work, and how the paradigms introduced in this thesis couldform the foundation for computational solutions in more complicated uid dynamicproblems.7.4.1 Multigrid AccelerationOne topic of improvement, which should follow relatively smoothly without con-siderable amount of research, is convergence acceleration. Multigrid techniques seembest �t towards that goal. However, multigrid speed-up of a hyperbolic solver andthat for an elliptic solver represent two completely di�erent tasks. It therefore helpsto have the two operators completely decoupled so that the appropriate accelera-tor can be applied to each independently. Reference [38] contains some introductorymaterial and a useful bibliography concerning multigrid methods for hyperbolic prob-lems. On the other hand, the article in [20] is a good source for multigrid for ellipticsolvers.We have seen that least-squares minimization through a gradient-based techniqueis, by itself a very slow way to solve nonlinear equations. Even when combined withmore e�cient iterative methods such as quasi-Newton, there is something left to bedesired. Much of this is associated to the fact that least-squares removes error bydamping only, with no mechanism to reduce errors through propagation. The ipside is that multigrid is likely to produce signi�cant improvements, since it worksbest with schemes that have a smoothing e�ect. When combining least-squares with

157multigrid, considerable savings in number of iterations should be expected, with aspeed-up comparable to the best cases found in other elliptic problems. One of thecomplaints expressed in [47] was that the extreme e�ciency of multigrid could not berecovered when applying Lax-Wendro� to the elliptic subsystem, mainly due to thepropagating behavior of that scheme. Errors would re ect o� the far-�eld bound-ary because characteristic inlet and outlet conditions were not designed otherwise.Almost certainly, one will not be confronted to these e�ects when implementingmultigrid techniques on least-squares for Cauchy-Riemann.7.4.2 Extension to Three-DimensionsAnalogous to 2D, residual decomposition and uctuation-splitting could well formthe foundation for future Euler solvers in three dimensions. We are still many stepsaway from an elegant formulation of the 3D equations into simpler homogeneoussubproblems. However, there is reason to believe that such a decomposition is thecorrect approach. Beyond 2D, simultaneous block diagonalization of three or morematrices is an unsolved problem, so even the 3D supersonic Euler equations do notcompletely decouple into separate advection equations.In both 3D supersonic and subsonic ow, characteristic equations can be ex-tracted only for entropy and enthalpy. The corresponding residuals would then beconvected using scalar upwind schemes which have a natural extension from 2D to3D. The remaining three equations, which describe the acoustic part of the sys-tem, don't completely decompose into recognizable model systems. For M > 1, theacoustic system is hyperbolic. In that case, one possibility is to write an equationdescribing advection of helicity (H, streamwise component of vorticity) along stream-lines. However, the last two equations which determine the Mach cone will then be

158coupled by H. For M < 1, the acoustic system is mixed hyperbolic-elliptic. Again,advection of helicity may be split away but the two equations in the elliptic partcouple through source terms which are gradients in H.The main issue then becomes how to discretize the acoustic subsystem such thatthe helicity residual is split away from the rest. Being able to resolve such a problemwould be key to correctly capturing helicity. Future work will determine whether thisis possible but the following point might lead us in the right direction [68]. Considera uid element traveling in natural coordinates at speed ~q, with unit velocity vector~t(uq ; vq ; wq ). In 2D the acoustic system becomes(1�M2)ps � �q2 div(~t) = 0�q2 curl(~t) + pn = 0whereas in 3D, we have(1�M2)ps � �q2 div(~t) = 0�q2 curl(~t)� ~t + grad p = 0 (two equations only)The analogy indicates that, if in 2D, our goal was to preserve potential ow ( curl(~q) =curl(q~t) = 0 ) then in 3D, we should be preserving Beltrami ow ( curl(~q) � ~q =curl(q~t) � (q~t) = 0 ). This is simply an instance of correctly capturing helicity, be-cause Beltrami ow essentially means that vorticity only exists in the streamwisedirection. Just as potential ow was for 2D, Beltrami ow could be the buildingblock for 3D Euler solvers. In any case, it is probably in 3D that we have the mostto gain by incorporating decomposition and residual-distribution.

1597.4.3 Some Comments on Navier-StokesThe Navier-Stokes (NS) equations can be written as the Euler equations withviscous uxes added. Thus, it is a matter of discretizing these viscous uxes andadding this discretization to the scheme for the Euler equations to make a scheme forthe NS equations. Of course, di�erent boundary conditions must be speci�ed for theNS equations, but that will not be discussed here. We can write the NS equationsas @u@t + @f@x + @g@y + @fvisc@x + @gvisc@y = 0 (7.1)where the �rst three terms in 7.1 are exactly the Euler equations in conservativeform. The viscous uxes are given byfvisc = 2666666666664

0��xx��xy�u�xx � v�xy + hx3777777777775 gvisc = 2666666666664

0��xy��yy�u�xy � v�yy + hy3777777777775where �xx = 2�ux + �(ux + vy)�xy = �(uy + vx)�yy = 2�vy + �(ux + vy)hx = ��Txhy = ��TyIn all the above expressions � is the viscous stress matrix, (hx; hy) are componentsof the heat ux, (�; �) are viscosity coe�cients, � is the thermal conductivity and Tis the absolute uid temperature.The viscous terms may be integrated over the median dual cell area to give thecorresponding portion of the residual. The NS residual is consistent as long as we

160discretize the equations using the parameter vector, which is assumed to vary linearlyover each cell. The viscous portion of the residual is given by�viscT = ZZT r � (fvisc; gvisc)dS = I@T (fvisc; gvisc) � d~nJust as in equation 3.4, the net update at node i is due to contributions from alltriangles surrounding it,un+1i = uni + �tSi XT2Ti �T;i ��EulerT + �viscT� (7.2)The time step constraint of the NS equations combines the constraint from the Eu-ler equations with a correction provided for the viscous uxes. When extending anEuler code to NS, the uctuation calculation changes by merely a correction term,and otherwise, the distribution methods remain identical to the ones introduced inchapter III. However, there are many subtle points that enter a NS solver, and herewe will look at some in the context of our decomposition. It is clear that for NS,the acoustic variables should not be restricted to leave entropy and enthalpy un-changed. In fact, the elliptic and hyperbolic operators which used to be independentin Euler, are now coupled by the viscous terms. This is simply a statement thatinertial and di�usive forces must interact through uid viscosity. As a consequence,quantities such as entropy, enthalpy or vorticity which are generated in areas whereviscous e�ects are dominant, the boundary layer for example, are constantly underthe in uence of di�usion and advection processes.Given that the physics dictates entropy and enthalpy to be coupled to the rest ofthe system, one question that rises is if we can isolate advection and di�usion e�ectson the natural variable x. To answer that, we must create quantities in our NSsolver, which would be used as vehicles to decouple viscous and inertial e�ects fromeach other. Such quantities actually exist and have a true physical interpretation.

161In the boundary layer for example, two mechanisms involve entropy (S), one isits generation and di�usion due to viscosity and the other is its advection. It ispossible to split away the portion of the viscous residual �viscT which only a�ectsentropy, it would be �rS �viscT . The same holds for the portion of the viscous residualwhich only changes enthalpy (h), �rh �viscT only a�ects enthalpy. As such, we havea clear distinction between (S; h) updates coming from the viscous residual andthose coming from the inviscid parts. This decomposition allows us to replicate theseparate physical phenomena that are taking place, that is (S; h) di�usion and (S; h)convection.One might even go further, by splitting the contributions of the viscous residualto acoustic variables (p; �). Using the same projection ideas introduced in section 5.4,we end up breaking down the viscous residual into its components in the natural basis(�rS;�rh;�rp;�r�). Throughout this thesis, our philosophy for the Euler solver has beento isolate away the natural variables (S; h; p; �). We believe that a successful Navier-Stokes should follow in the same path, it should isolate contributions to (S; h; p; �)coming from the viscous ux divergence residual.

BIBLIOGRAPHY

162

163BIBLIOGRAPHY

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ABSTRACTA RESIDUAL DISTRIBUTION APPROACH TO THE EULER EQUATIONSTHAT PRESERVES POTENTIAL FLOWbyMani RadChairperson: Professor Philip L. Roe

The main goal of this work is to solve the steady Euler equations governinginviscid compressible uid ow as cleanly and accurately as possible, focusing on thelimiting cases where the physical behavior of the ow changes. Incompressible ow,where either the free stream Mach number or the local Mach number approach zero,and potential ow which is appropriate to shock-free ow originating in a uniformstream both present di�culties for conventional Euler schemes and the symptomsare generally excessive numerical entropy generation and loss of accuracy. Howeverthese cases of zero Mach number or zero vorticity should simply be stepping stonestowards the design of an accurate Euler code with shock capturing capability.Following such a line of reasoning, this dissertation will present a numerical im-plementation of an Euler solver based on the unique decomposition of the steady

equations into their simplest homogeneous parts. In the supersonic regime, the de-composition takes the form of scalar advection for entropy, enthalpy and two acousticvariables. In subsonic ow, the acoustic part stays coupled and forms a 2� 2 systemof Cauchy-Riemann type. To further ensure strict decoupling between hyperbolicand elliptic parts at the discrete level, the advective and acoustic equations must beintegrated into completely independent cell residuals.Residual-distribution schemes are well suited for solving Euler equations as ex-pressed in the decomposed form above. In the uctuation-splitting approach, themaximally decoupled hyperbolic and elliptic residuals are split to nodes according toa physically sound distribution method, until reaching convergence. For those uc-tuations representing scalar advection, multidimensional upwinding is the optimalchoice whereas for residuals originating from the 2 � 2 elliptic subsystem, an accu-rate approach is least-squares (LS). Within the LS minimization process, updates tothe variables underlying the elliptic part, namely p and �, are constrained such thatentropy and enthalpy are una�ected.Computational results reveal high accuracy and robustness for a large range ofMach numbers, with very little sensitivity on mesh orientation or irregularity. How-ever, the convergence rate of the least-squares scheme is currently a hindrance andit should be considerably improved to make the whole method attractive in appli-cations. In regions of incompressible ow, the scheme behaves very well, quite anencouraging fact when knowing that M ! 0 is a di�cult limit for Euler solvers. Inthe case of uniform entropy and enthalpy in ow and if no shocks occur, potential ow is recovered with extremely low levels of numerical entropy generation.