APPROXIMATE RIEMANN SOLVERSFOR MOMENT MODELS OF DILUTEGASESbyShawn Lee BrownA dissertation submitted in partial ful�llmentof the requirements for the degree ofDoctor of Philosophy(Aerospace Engineering)in The University of Michigan1996Doctoral Committee:Professor Philip L. Roe, ChairpersonProfessor Tamas I. GombosiAssociate Professor Kenneth G. PowellProfessor Bram van LeerJoseph Shang, Senior Scientist, Wright-Patterson AFB

To the love of my life, Susan

ii

ACKNOWLEDGEMENTSFirst and foremost I would like to thank my wife, Susan, who provided unendingsupport in my endeavor here at Michigan. She knew just how to lighten me up whenI would take my research too seriously. I would also like to thank my parents whohad the wisdom and con�dence to let me choose my own path in life knowing I wouldmake the right choices.I would also like to thank my chairperson, Professor Phil Roe, for being oneof the driving forces behind this research and for graciously providing answers to,what I often realized in hindsight, my inane questions. I also thank Professor TamasGombosi who provided much of the inspiration for this research. Thanks also toProfessors Bram van Leer and Ken Powell for the insight they provided to manyof the problems encountered in this thesis. I am also grateful to Joe Shang for hisemotional support throughout my graduate studies. A special thanks to ClintonGroth who provided unending help and collaboration throughout this research andfor the many non-research related discussions.I would like to acknowledge the many members of the W. M. Keck CFD Labo-ratory, they not only provided help in my research but also made working in the laban enjoyable experience. A special thanks goes to Lisa Mesaros and Rob Lowrie forproviding many good discussions.Finally, I would like to acknowledge the �nancial support that I have receivedfrom the U. S. Air Force Palace Knight Program.iii

TABLE OF CONTENTSDEDICATION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iiACKNOWLEDGEMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : iiiLIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viiLIST OF TABLES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xiiLIST OF APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : xiiiCHAPTERI. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1II. HISTORICAL BACKGROUND . . . . . . . . . . . . . . . . . . 12III. ELEMENTS OF KINETIC THEORY . . . . . . . . . . . . . . 163.1 The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 163.1.1 Expressed in terms of particle velocity . . . . . . . . 163.1.2 Expressed in terms of random velocity . . . . . . . . 173.2 Velocity Moments of the Boltzmann Equation . . . . . . . . . 173.3 Maxwell's Equation of Change . . . . . . . . . . . . . . . . . 233.3.1 Non-conservative form . . . . . . . . . . . . . . . . 243.3.2 Conservative form . . . . . . . . . . . . . . . . . . . 253.4 Modelling of the Collision Operator . . . . . . . . . . . . . . 253.5 Closure of the Moment Transport Equations . . . . . . . . . . 283.6 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.7 Example { Euler Equations . . . . . . . . . . . . . . . . . . . 313.7.1 Non-conservative form . . . . . . . . . . . . . . . . 323.7.2 Conservative form . . . . . . . . . . . . . . . . . . . 343.7.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . 35IV. ELEMENTS OF A RIEMANN SOLVER . . . . . . . . . . . . 36iv

4.1 The Riemann Problem . . . . . . . . . . . . . . . . . . . . . . 364.2 Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Roe's Approximate Riemann Solver . . . . . . . . . . . . . . 394.4 Construction of Ac . . . . . . . . . . . . . . . . . . . . . . . . 424.4.1 Property U . . . . . . . . . . . . . . . . . . . . . . . 424.4.2 Approaches . . . . . . . . . . . . . . . . . . . . . . 434.5 Example { Euler Equations . . . . . . . . . . . . . . . . . . . 444.5.1 Conservative form . . . . . . . . . . . . . . . . . . . 444.5.2 Primitive form . . . . . . . . . . . . . . . . . . . . . 464.5.3 Roe-averaged variables . . . . . . . . . . . . . . . . 474.5.3.1 Parameter vector approach . . . . . . . 474.5.3.2 Corrected average approach . . . . . . . 504.5.4 Roe's approximate Riemann solver . . . . . . . . . . 54V. 10-MOMENT MODEL . . . . . . . . . . . . . . . . . . . . . . . . 575.1 Governing Equations { Non-conservative Form . . . . . . . . 595.2 Eigensystem Analysis . . . . . . . . . . . . . . . . . . . . . . 625.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.4 Dispersion Analysis . . . . . . . . . . . . . . . . . . . . . . . 68VI. A 10-MOMENT RIEMANN SOLVER . . . . . . . . . . . . . . 776.1 Governing Equations { Conservative Form . . . . . . . . . . . 776.2 Roe-Average . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.2.1 Parameter vector approach . . . . . . . . . . . . . . 836.2.2 Corrected average approach . . . . . . . . . . . . . 856.3 A Roe-Type Approximate Riemann Solver . . . . . . . . . . . 88VII. 35-MOMENT MODEL . . . . . . . . . . . . . . . . . . . . . . . . 947.1 Governing Equations { Non-conservative Form . . . . . . . . 967.2 Eigensystem Analysis . . . . . . . . . . . . . . . . . . . . . . 997.2.1 Gaussian limit . . . . . . . . . . . . . . . . . . . . . 1037.2.2 Adiabatic limit . . . . . . . . . . . . . . . . . . . . 1087.2.3 Non-equilibrium . . . . . . . . . . . . . . . . . . . . 1177.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.4 Dispersion Analysis . . . . . . . . . . . . . . . . . . . . . . . 129VIII. 35-MOMENT RIEMANN SOLVERS . . . . . . . . . . . . . . . 1368.1 Governing Equations { Conservation Form . . . . . . . . . . . 1368.2 Roe-Average . . . . . . . . . . . . . . . . . . . . . . . . . . . 142v

8.3 Roe-Type Approximate Riemann Solvers . . . . . . . . . . . 1488.3.1 Gaussian eigenstructure . . . . . . . . . . . . . . . . 1518.3.2 Adiabatic eigenstructure . . . . . . . . . . . . . . . 1548.3.3 Non-equilibrium eigenstructure . . . . . . . . . . . . 157IX. NUMERICAL ALGORITHM . . . . . . . . . . . . . . . . . . . 1599.1 Solution Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 1599.2 Operator-Splitting (OPS) Method . . . . . . . . . . . . . . . 1649.3 Initial and Boundary Conditions . . . . . . . . . . . . . . . . 1669.4 Grids and Time-Stepping . . . . . . . . . . . . . . . . . . . . 1719.5 Navier-Stokes Model . . . . . . . . . . . . . . . . . . . . . . . 1739.6 Direct Simulation Monte Carlo Model . . . . . . . . . . . . . 1789.7 Calculation of Wavespeeds/Determination of Hyperbolicity . 181X. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19110.1 Shock Structure . . . . . . . . . . . . . . . . . . . . . . . . . 19110.1.1 In ow Mach number = 1.2 . . . . . . . . . . . . . . 19410.1.2 In ow Mach number = 1.5 . . . . . . . . . . . . . . 21110.1.3 In ow Mach number = 2.0 . . . . . . . . . . . . . . 22310.1.4 In ow Mach number = 10.0 . . . . . . . . . . . . . 23710.2 E�ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241XI. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24811.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24811.2 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 25011.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 253APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 255BIBLIOGRAPHY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 296vi

LIST OF FIGURESFigure1.1 The hierarchy of solution techniques for the Boltzmann equation. . 51.2 Knudsen number limits on the mathematical models. . . . . . . . . 83.1 Geometric interpretation of the �rst normalized particle velocity mo-ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 The e�ect of the second central moment on the distribution function. 223.3 The e�ect of the third central moment on the distribution function. 233.4 The e�ect of the fourth central moment on the distribution function. 244.1 Diagram that shows the computation of the interface ux for a sys-tem composed of three waves. The cell interface is located at x = 0. 405.1 Dispersion diagram for the 10-moment transport equations. Thearrows indicate the direction of increasing Knudsen number fromthe equilibrium limit (Kn = 0) to the collisionless limit (Kn!1). 735.2 Wave mode damping diagram for the 10-moment transport equa-tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.1 Hyperbolicity region for the 35-moment transport equations. Thepoint G represents the Gaussian limit. . . . . . . . . . . . . . . . . 1027.2 Convexity of the two right-moving acoustic waves of the adiabaticlimit. Comparison is made with the right-moving acoustic waves ofthe Gaussian limit. The subscript g denotes the Gaussian limit whilethe subscript a denotes the adiabatic limit. . . . . . . . . . . . . . 1167.3 Sign of �01 in the hyperbolic region of the 35-moment transport equa-tions. The left-moving fast acoustic wave is convex for the entireregion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.4 Sign of �09 in the hyperbolic region of the 35-moment transport equa-tions. The right-moving fast acoustic wave is convex for the entireregion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123vii

7.5 Sign of �03 in the hyperbolic region of the 35-moment transport equa-tions. The left-moving slow acoustic wave is convex for the majorityof the region but there are two regions where the wave becomesconcave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247.6 Sign of �07 in the hyperbolic region of the 35-moment transport equa-tions. The right-moving slow acoustic wave is convex for the major-ity of the region but there are two regions where the wave becomesconcave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.7 Sign of �04 in the hyperbolic region of the 35-moment transport equa-tions. The entropy wave is linearly degenerate for Qxxx = 0. . . . . 1257.8 Close up of upper-left area of the hyperbolicity region showing thesign of �04. The line separating the positive and negative regions iswhere the entropy wave is linearly degenerate. . . . . . . . . . . . . 1267.9 Dispersion diagram for the 35-moment transport equations. Thearrows indicate the direction of increasing Knudsen number fromthe equilibrium limit (Kn = 0) to the collisionless limit (Kn!1). 1327.10 Damping diagram for the 35-moment transport equations. . . . . . 1339.1 Wave behaviour at the supersonic upstream boundary of the Eulercharacteristic equations. . . . . . . . . . . . . . . . . . . . . . . . . 1689.2 Wave behaviour at the subsonic downstream boundary of the Eulercharacteristic equations. . . . . . . . . . . . . . . . . . . . . . . . . 1699.3 The behaviour of the polynomial factors which make up the charac-teristic equation with A = 0 and B = 0. . . . . . . . . . . . . . . . 1849.4 The behaviour of the polynomial factors P4 and P5 with A = 0 andB = 0 highlighting that the roots of P4 correspond to the extremumof P5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18510.1 Grid convergence study for the 10-moment model. The in ow Machnumber was 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19510.2 Grid convergence study for the 35-moment model using the non-equilibrium eigenstructure. The in ow Mach number was 1.2. . . . 19610.3 Convergence history of the two moment models. The in ow Machnumber was 1.2 and 240 cells were used. The 35-moment modelshows the same convergence history for the three di�erent eigen-structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19610.4 Density pro�les of the various models with an in ow Mach numberof 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19710.5 Velocity pro�les of the various models with an in ow Mach numberof 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19910.6 Pxx pro�les of the various models with an in ow Mach number of 1.2. 20010.7 Pnn pro�les of the various models with an in ow Mach number of 1.2. 200viii

10.8 Pressure pro�les of the various models with an in ow Mach numberof 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20110.9 Temperature pro�les of the various models with an in owMach num-ber of 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20110.10 Shear stress pro�les of the various models with an in ow Mach num-ber of 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20210.11 Qxxx pro�les of the various models with an in ow Mach number of1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20310.12 Qxnn pro�les of the various models with an in ow Mach number of1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20410.13 Heat ux pro�les of the various models with an in ow Mach numberof 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20510.14 Kxxxx pro�les of the various models with an in ow Mach number of1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20610.15 Kxxnn pro�les of the various models with an in ow Mach number of1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20710.16 Knnnn pro�les of the various models with an in ow Mach number of1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20710.17 Relative speci�c entropy pro�les of the various models with an in owMach number of 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 20910.18 The phase plots for in ow Mach number of 1:2. . . . . . . . . . . . 21110.19 Density pro�les of the various models with an in ow Mach numberof 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21210.20 Density pro�le of the 35-moment model using the Gaussian eigen-structure with an in ow Mach number of 1.5. The solution is notconverged. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21310.21 Pro�les of Pxx for the various models with an in ow Mach numberof 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21510.22 Pro�les of Pnn for the various models with an in ow Mach numberof 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21510.23 Hydrostatic pressure pro�les of the various models with an in owMach number of 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 21610.24 Temperature pro�les of the various models with an in owMach num-ber of 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21710.25 Shear stress pro�les of the various models with an in ow Mach num-ber of 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21710.26 Pro�les of Qxxx for the various models with an in ow Mach numberof 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21810.27 Pro�les of Qxnn for the various models with an in ow Mach numberof 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218ix

10.28 Heat ux pro�les of the various models with an in ow Mach numberof 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21910.29 Pro�les of Kxxxx for the various models with an in ow Mach numberof 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22010.30 Pro�les of Kxxnn for the various models with an in ow Mach numberof 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22110.31 Pro�les of Knnnn for the various models with an in ow Mach numberof 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22110.32 Entropy pro�les of the various models with an in ow Mach numberof 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22210.33 The phase plots for in ow Mach number of 1:5. . . . . . . . . . . . 22210.34 Density pro�les of the various models with an in ow Mach numberof 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22410.35 Pxx pro�les of the various models with an in ow Mach number of 2.0. 22510.36 Pnn pro�les of the various models with an in ow Mach number of 2.0. 22510.37 Pressure pro�les of the various models with an in ow Mach numberof 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22610.38 Temperature pro�les of the various models with an in owMach num-ber of 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22610.39 Shear stress pro�les of the various models with an in ow Mach num-ber of 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22710.40 Qxxx pro�les of the various models with an in ow Mach number of2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22810.41 Qxnn pro�les of the various models with an in ow Mach number of2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22810.42 Heat ux pro�les of the various models with an in ow Mach numberof 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22910.43 Kxxxx pro�les of the various models with an in ow Mach number of2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23010.44 Kxxnn pro�les of the various models with an in ow Mach number of2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23010.45 Knnnn pro�les of the various models with an in ow Mach number of2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23110.46 Entropy pro�les of the various models with an in ow Mach numberof 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23110.47 The phase plots for in ow Mach number of 2:0. . . . . . . . . . . . 23210.48 The axial distribution function of F 35 shown at several locationsupstream and including the discontinuity. . . . . . . . . . . . . . . 23310.49 The axial distribution function of F 35 shown at several locationsdownstream and including the discontinuity. . . . . . . . . . . . . . 233x

10.50 The distribution function G shown at several locations throughoutthe shock structure. . . . . . . . . . . . . . . . . . . . . . . . . . . 23610.51 Density pro�les of the various models with an in ow Mach numberof 10.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23810.52 Pxx pro�les of the various models with an in ow Mach number of10.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23910.53 Pnn pro�les of the various models with an in ow Mach number of10.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23910.54 Pressure pro�les of the various models with an in ow Mach numberof 10.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24010.55 Temperature pro�les of the various models with an in owMach num-ber of 10.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24010.56 Shear stress pro�les of the various models with an in ow Mach num-ber of 10.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241A.1 Comparison of the ODE solution with respect to the PDE solutionfor M1 = 1:1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261A.2 Comparison of the ODE solution with respect to the PDE solutionfor M1 = 1:35. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262A.3 Comparison of the ODE solution with respect to the PDE solutionfor M1 = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263A.4 Comparison of the ODE solution with respect to the PDE solutionfor M1 = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

xi

LIST OF TABLESTable5.1 Summary of the wave modes for the 10-moment model where ax = 5p3� . 765.2 Damping rates of the various wave modes for the linearized 10-moment transport equations. . . . . . . . . . . . . . . . . . . . . . . 767.1 Summary of the wave modes for the 35-moment model in the Gaus-sian limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097.2 Summary of the wave modes for the 35-moment model in the adia-batic limit where B = Kxxxx=(�c4xx). . . . . . . . . . . . . . . . . . . 1177.3 Summary of the wave modes for the 35-moment model for non-equilibrium states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.4 Damping rates of the various wave modes for the linearized 35-moment transport equations. . . . . . . . . . . . . . . . . . . . . . . 13510.1 The number of iterations needed to reach convergence for the di�er-ent models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24310.2 The cell cost (10�5 seconds/cell/iteration) of the various models. . . 24410.3 The total CPU cost (in seconds) for the di�erent models. . . . . . . 246xii

LIST OF APPENDICESAppendixA. Ordinary Di�erential Equation Solution of 10-Moment Model . . . . . 256A.1 Reduction of the Transport Equations to a Single ODE . . . 256A.2 Numerical Solution of ODE . . . . . . . . . . . . . . . . . . . 260B. One-Dimensional 35-Moment Transport Equations . . . . . . . . . . . 264B.1 Non-conservative Formulation . . . . . . . . . . . . . . . . . . 264B.2 Conservative Formulation . . . . . . . . . . . . . . . . . . . . 280B.3 Evaluation of the Source Terms . . . . . . . . . . . . . . . . . 289

xiii

CHAPTER IINTRODUCTIONThere are several aerospace vehicles in existence or being designed that will havea signi�cant portion of their ight envelopes encompassing a regime known as thetransitional regime. These include all atmosphere entry and transatmospheric vehi-cles such as the Space Shuttle, the National Aerospace Plane (NASP), unmannedrockets, and interplanetary probes.For low-altitude ight, where most aerospace vehicles operate, the collisions be-tween molecules are numerous and the gas remains in thermodynamic equilibrium.This is the continuum regime and aerodynamicists are very familiar with this regime.At higher altitudes, there are insu�cient collisions between molecules to allow thegas to reach an equilibrium state and this gives rise to regions of non-equilibrium.This is referred to as the transitional regime. At yet higher altitudes, the collisionsamong molecules are so infrequent that to a very good approximation they can beneglected; however, collisions between molecules and solid boundaries must still betreated. This regime is referred to as the free-molecular regime.It is very important to be able to accurately predict the physical phenomenaoccurring around these vehicles in all ow regimes. For continuum ows the tradi-tional hydrodynamics equations (i.e., Euler and Navier-Stokes) provide the appro-1

2priate mathematical model for the gas. These systems of transport equations arewell understood. The only satisfactory description of free-molecular ows is on themicroscopic level where the Boltzmann equation must be solved directly or simulatedas in direct simulation Monte Carlo (DSMC) techniques. In the intermediate or tran-sitional regime there is no accurate model that is also computationally e�cient thatcan be used for the prediction of these ows. The traditional hydrodynamic equa-tions are inappropriate since the usual continuum hypotheses are violated; DSMCtechniques can be used in this regime but the computational work and storage neededcan be prohibitive.As an example of the importance of being able to accurately model the tran-sitional ow regime the following is o�ered. Koppenwallner [44] has shown thatthe nose-up pitching moment of the Space Shuttle was very poorly predicted in thisregime due to inaccurate modelling. This fortunately could be corrected by de ectingthe body ap to 15 degrees, twice the expected amount. If the United States or anyother country is to design vehicles that have ight paths at the upper reaches of theatmosphere, our predictive capabilities in the transitional regime must be furtherdeveloped and validated. This provides the motivation for the research containedin this thesis. We are interested in developing an accurate model for transitional ows that can be e�ciently solved using modern powerful numerical techniques. Itshould be mentioned that the modelling of dilute gases is not limited to aerospaceapplications; other applications include space weather and micro-manufacturing.The above applications have in common a dimensionless parameter called theKnudsen number which provides a relative measure of the diluteness of a gas and isde�ned as Kn = �L ; (1.1)

3where � is the mean free path of the particles and L is some characteristic dimensionof the problem. The mean free path is the average distance traveled by the moleculesbetween two successive collisions. For continuum regime ows the Knudsen numberis relatively small, i.e., Kn � 1, and therefore, the mean free path is much smallerthan the macroscopic length scale and the ow is collision dominated. A collision-dominated ow is one in which, by the time a molecule has traveled a distancecomparable to the characteristic dimension of the problem, it has participated in avery large number of collisions. For free-molecular ows, Kn � 1, the mean freepath of the particles is much larger than the macroscopic length scale and the e�ectsof intermolecular collisions are negligible. Transitional ows, for which Kn � 1, liein the most di�cult and unexplored regime, and the mean free path is on the sameorder of magnitude as the characteristic dimension.There are three fundamental questions that need to be answered when addressinghow transitional ows should be computed:1. What is the appropriate set of extended hydrodynamic equations that willreplace the Navier-Stokes equations?2. What are the boundary conditions to use with this system?3. What numerical scheme should be used to e�ciently solve the system of trans-port equations and the accompanying boundary conditions?This thesis is primarily concerned with the �rst and third questions and considersthe problem of one-dimensional shock structure of a monatomic gas. The solution ofshock structures has been the subject of much interest because it provides a rigoroustest of the accuracy of a set of transport equations. A useful feature of this problemis that the issue of determining a suitable set of boundary conditions is avoided; i.e.,

4the boundary conditions are not in question. For the shock problem the boundaryconditions are provided by the jump conditions for the Euler equations.To develop a model for use in the transitional regime the kinetic theory of gasesmust be utilized. The fundamental equation of kinetic theory is the Boltzmannequation. This non-linear integro-di�erential equation governs the time evolutionin phase-space of the distribution function F describing the number of particles atposition x� having velocity v� and is given as@F@t + v� @F@x� + a� @F@v� = �F�t ; (1.2)where �F�t = 1ZZZ�1 Z 2�0 Z �0 [F 0F 02 � FF2] g S sin�d�d� d3c2; (1.3)and a� is the acceleration due to external forces. The particle-velocity distribu-tion function describes the behaviour of the dilute gas. The left-hand side of theBoltzmann equation describes the transport of the distribution function whereas theright-hand side represents the time rate of change of the distribution function dueto binary interparticle collisions. The Boltzmann equation is based on two funda-mental assumptions: The principle of molecular chaos and the requirement that thegas is dilute. The molecular chaos assumption means that particles which undergoa collision will have many encounters with other particles before they collide again.In other words, the velocities of a pair of colliding particles are completely uncorre-lated. The dilute-gas assumption ensures that the range of the intermolecular forcesis small compared to the mean free path, and hence, ternary interparticle collisionsare negligible.The Boltzmann equation is valid in all ow regimes; however, it is very di�cult toobtain closed-form analytic non-equilibrium solutions of this equation except in the

5Direct or particulate- ow General Flow ProblemsApproachesDirect Simulation Chapman-EnskogExpansion Technique Grad-TypeMethod of MomentsExtended HydrodynamicsApproachesMonte Carlo Methods

Analytic: SimpliestFlow Problems Solutions of theBoltzmann Equation Approximate:Figure 1.1: The hierarchy of solution techniques for the Boltzmann equation.simplest ow situations. For more general problems, it is necessary to apply approx-imation techniques to this governing equation in order to obtain solutions. There aremany techniques utilized to solve the Boltzmann equation and they can be broadlycategorized into two distinct approaches: Direct or particulate- ow approaches andextended hydrodynamic approaches. The most popular particulate- ow approachis the aforementioned direct simulation Monte Carlo method. The DSMC methodis capable of providing more general solutions to the Boltzmann equation; however,as stated previously, for transitional ows with relatively large densities the DSMCmethod can be prohibitive with regards to computational work and storage require-ments. Even for more rare�ed transitional ows this method is quite expensive;and therefore, the development of a more computationally e�cient approach wouldbe highly desirable. This leads to the consideration of extended hydrodynamic ap-proaches. Figure 1.1 shows the hierarchy of solution techniques for the Boltzmannequation.The extended hydrodynamic approaches replace the problem of solving the Boltz-

6mann equation with one of solving a system of generalized transport equations. Thetwo most popular approaches are the Chapman-Enskog expansion technique andGrad's method of moments. Most of the extended hydrodynamic methods are basedon expansions for the distribution function in terms of a local Maxwellian distribu-tion. The Maxwellian distribution function corresponds to a gas in local thermody-namic equilibrium (LTE). A gas in LTE is de�ned as a ow in which the macroscopicgradients are su�ciently small and the collision rate su�ciently large that each ele-ment of the gas (large enough to contain many molecules) as it moves through the ow has a distribution function corresponding to the equilibrium state appropriateto the local macroscopic properties. Expanding about such a ow means that thedeviation from the equilibrium state must be minor; i.e., the shear stresses and heat ux terms must be small. As the solution departs from LTE these expansions oftenyield negative values for the distribution function in some regions of velocity spacewhich is of course unphysical.The Chapman-Enskog technique is based on a perturbative expansion in terms ofthe Knudsen number. The zeroth- and �rst-order systems are the traditional Eulerand Navier-Stokes equations, respectively, and are well known. Beyond �rst-orderthis expansion technique augments the constitutive relations present in the Navier-Stokes system with additional terms. These terms consist of higher-order derivativesof the hydrodynamic variables. An advantage of the Chapman-Enskog expansionis that no additional hydrodynamic variables and/or transport equations are intro-duced. Not as well known are the second- and third-order systems referred to asthe Burnett equations and Super-Burnett equations, respectively [11, 12]. Unfortu-nately, there are serious drawbacks to higher-order systems of transport equationsbased on Chapman-Enskog expansions. They are extremely complex and contain

7higher-order derivatives which are di�cult to model numerically, especially on ir-regular or unstructured grids. For systems higher than zeroth-order, disturbanceshave non-�nite wavespeeds [53]. The Burnett equations are unstable for small dis-turbances [8]; adding terms from the Super-Burnett system is a technique used tostabilize the system [75]. Yet another problem is that the Burnett equations arecapable of only modelling a single translational temperature. This is inaccuratefor strong shocks [38]. There are also unresolved issues concerning frame depen-dence [74], appropriate boundary conditions, and the fact that the Burnett equationscan potentially violate the second law of thermodynamics [18].Yet another approach is o�ered by Grad-type moment closures [26,29,31] and/orextended thermodynamics [53] where the transport equations are obtained by takingvelocity moments of the Boltzmann equation. This approach introduces anisotropicpressures, heat uxes, and higher-order moments of the particle-velocity distribu-tion function as new unknowns. Closure of the system is needed and Grad achievedthis by truncating the in�nite Hermite polynomial series expansion for the particle-velocity distribution function [29]. The systems of transport equations resulting fromthis method are hyperbolic near LTE, and therefore action at a distance is excluded.Moment methods are widely used in the study of dilute gases; however, there are alsodrawbacks associated with this approach. As with the Burnett and Super-Burnettsystems of transport equations, moment models are extremely complex. Also, com-putational di�culties arise when dealing with large systems of equations havingoften sti� non-linear source terms. The Euler and Navier-Stokes equations remainstable and are well-behaved for all solution frequencies; this is not guaranteed for thehigher-order moment models. When the departure from LTE is large, correspondingto solutions with high-frequency content, the approximate distribution functions can

80

8

0.01 0.1 1 10 100

DISCRETEPARTICLE ORMOLECULAR

MODEL

LOCAL KNUDSEN NUMBER

TRANSPORTEQUATIONS

BOLTZMANN EQUATION

EULEREQUATIONS

NAVIER-STOKESEQUATIONS

EXTENDEDHYDRODYNAMIC

EQUATIONS

COLLISIONLESSBOLTZMANN

EQUATION

CONTINUUMREGIME REGIME

FREE-MOLECULARREGIME

FREE-MOLECULARLIMIT

INVISCIDLIMIT

TRANSITIONALFigure 1.2: Knudsen number limits on the mathematical models.become negative and the systems of transport equations become non-hyperbolic. Itcan be argued that this breakdown occurs when the validity of the closure becomessuspect; and therefore, the breakdown is not severely limiting. However, it would bedesirable, from a computational standpoint, for the moment model to remain stableand well-behaved even when the closure may not be strictly valid. As the computa-tional grid is re�ned, high-frequency solution content is resolved, therefore, stabilityat all frequencies is needed.The limits of applicability in the di�erent ow regimes for the transport equationsare presented in Figure 1.2. The collisionless Boltzmann equation refers to Equation(1.2) with �F=�t = 0.There is still uncertainty as to which of the two, if either, extended hydrodynamicapproaches presented above is the best for modelling transitional ows. It is believedby this author that the method of moments is the preferable approach even thoughthere is uncertainty in the formulation of the closure; and hence the form of theparticle-velocity distribution function. The present research attempts to address

9this issue.In this thesis two systems of transport equations based on the method of momentsare presented as possible models for transitional ows. They are the 10-moment and35-moment systems. The 10-moment model is based on the Gaussian distributionwhich incorporates the pressure anisotropies in a non-perturbative function; i.e.,there are no limitations on the magnitudes of the stress components to maintain apositive distribution function [10, 48, 52]. It yields a set of ten transport equationswith the independent variables being density, velocity vector, and the six componentsof the generalized pressure tensor. An eigensystem analysis reveals that this system ofpartial di�erential equations is hyperbolic for all realistic ow situations. A linearizeddispersion analysis of the 10-moment system was also performed and shows that theEuler modes are recovered in the collision-dominated limit. The hyperbolicity ofthe transport equations lends itself to solution techniques that take advantage ofthe wave-like nature of the physics. In this thesis a Roe-type approximate Riemannsolver is developed for the 10-moment model. A limitation of the 10-moment modelis that it is incapable of representing heat transfer, and is, therefore, not in itself aviable option for modelling transitional ows, although its study may be instructive.This lead us to perform an expansion in terms of the Gaussian distribution yield-ing the 35-moment model distribution function which is capable of modelling heattransfer e�ects [27, 33]. The use of the Gaussian for the expansion is unlike moretraditional approaches which use the Maxwellian for the expansion. The 35-momentdistribution function yields a system of 35 transport equations with the indepen-dent variables being, in addition to the aforementioned variables for the 10-momentsystem, ten components of a third-order tensor which is related to the heat uxes,and 15 components of a fourth-order tensor. An eigensystem analysis performed on

10this model reveals that the partial di�erential equations are hyperbolic for a �niteregion of phase-space. The dispersive properties of this model are also explored andreveal that in the collision-dominated limit the usual Euler modes are recovered. Inthis thesis a Roe-type approximate Riemann solver for the 35-moment system is pre-sented. Due to the large number of transport equations and the fact that analyticexpressions for the components of the eigensystem in terms of the hydrodynamicvariables cannot be found, the computational expense in solving this system is quitehigh. With this in mind, simpli�cations were attempted in constructing a Riemannsolver. The eigensystem is simpli�ed by evaluating the model in the Gaussian andadiabatic limits where analytic eigensystems can be determined; then a `�x' is foundfor the eigensystem that will ensure conservation of the scheme. These two limitedeigenstructures, along with the non-equilibrium eigenstructure where the eigenvec-tors can be expressed as functions of the eigenvalues, are utilized in the constructionof three Riemann solvers for the 35-moment model.The �rst-order �nite volume scheme is extended to second-order by use of a two-step time integration procedure. These models, as was mentioned previously, areapplied to the prediction of one-dimensional shock structures for a monatomic gasfor various in ow Mach numbers. Comparisons are made to similar results obtainedfrom the Navier-Stokes equations and the DSMC model. This thesis provides the�rst extensive set of numerical solutions for the 10- and 35-moment models and alsodevelops e�cient schemes, including approximate solvers, for these models.An outline of the remaining chapters in the thesis is as follows: A basic discussionon some elements of kinetic theory is presented in Chapter III. This basic knowledgeprovides the framework needed to derive the two higher-order moment models. Atthe end of this chapter, as an example, the Euler equations are presented. In the

11next chapter, Chapter IV, the Riemann problem is introduced and a brief discussionon di�erent types of Riemann solvers is given. Then Roe's approximate Riemannsolver is presented for the Euler equations. In Chapter V the 10-moment model ispresented and the results of the eigensystem and linearized dispersion analyses arediscussed. Next, in Chapter VI, a Roe-type approximate Riemann solver for the10-moment model is derived. The other higher-order moment model, the 35-momentmodel, is introduced in Chapter VII. In this chapter an eigensystem analysis isperformed not only on the non-equilibrium eigenstructure but also on the two near-equilibrium eigenstructures. The results of the dispersion analysis on the linearizedsystem of 35-moment transport equations closes out the chapter. In Chapter VIII,three Roe-type approximate Riemann solvers are derived, each one utilizing a di�er-ent eigensystem introduced in the previous chapter. The numerical algorithm used tosolve these systems of moment equations is presented in Chapter IX. Also discussedin this chapter are ancillary issues such as the treatment of the initial and bound-ary conditions, computational grids and time-stepping, and the calculation of thewavespeeds for the 35-moment model. A brief discussion on the Navier-Stokes andDSMC models is also presented. Chapter X contains the results obtained by solvingthese higher-order moment models when applied to the problem of one-dimensionalshock structure. Included in this chapter is a presentation on the relative computa-tional e�ciencies of the models. The conclusions formulated from performing thisresearch and what the future may hold for higher-order moment models for solvingtransitional ows are given in Chapter XI.

CHAPTER IIHISTORICAL BACKGROUNDThis chapter provides a historical background of the two extended hydrodynamicapproaches to solving the Boltzmann equation approximately.Chapman and Enskog independently developed the Chapman-Enskog expansiontechnique in the years 1916-17 [13, 19]. Using this technique both the Euler andNavier-Stokes equations were \rediscovered" and they correspond to the zeroth- and�rst-order expansions, respectively. Previously these equations were derived frompurely continuum arguments.The Chapman-Enskog expansion technique proves Stokes' hypothesis for a gascomposed of monatomic molecules. It also allowed for a theoretical derivation of thedependence of the viscosity on temperature. Prior to this the relationship was onlyknown through empirical arguments. Burnett was the �rst to carry the Chapman-Enskog expansion technique to second-order [11, 12], and therefore, this system oftransport equations is referred to as the Burnett equations. There were early at-tempts made at solving the Burnett equations for the shock structure problem bySherman and Talbot [62, 65]. The Burnett equations were simpli�ed by assum-ing a steady-state and then the resulting system of ordinary di�erential equations(ODEs) were solved. They used a technique which Von Mises [72] and Gilbarg and12

13Paolucci [22] successfully used to solve the similarly reduced Navier-Stokes equations.However, for the Burnett equations there exist strong singularities at the upstreamand downstream boundary conditions. This behaviour prohibited solutions abovea Mach number of two, and for Mach numbers less than two there was only slightimprovement over the Navier-Stokes equations [65]. This caused the Burnett equa-tions to fall into disrepute with the scienti�c community and they were subsequentlyabandoned for many years.Recent research, done mostly at Stanford University, has focused on solving thetime-dependent Burnett equations to obtain steady-state solutions as opposed tosolving the ODE [20,51,75]. This technique of solving the system allowed a measureof success to be attained where the Burnett equations provide better solutions thanthe Navier-Stokes equations to transitional-type ows including shock structures athypersonic Mach numbers. Their success has motivated others to take a new lookat the Burnett equations (for example, see [43]). However, the problems as listed inthe previous chapter still exist with no answers to them apparent at this time.For further details on the theory of the Chapman-Enskog expansion techniquethe interested reader is referred to the monograph by Chapman and Cowling [14].In 1949, Grad published a paper [29] in which he presents the 13-moment model.His seminal paper formally introduced the Grad-type moment approach. In a sub-sequent paper [30] he solved the 13-moment model, which have as the 13 physicalquantities: density, velocity, pressure tensor, and heat ow vector. Grad assumedsteady-state conditions and numerically integrated the resulting ODEs. The resultsshow that the 13-moment results begin to di�er from the Navier-Stokes results atan upstream Mach number of about 1.2. An analysis of the system shows that theODEs break down at a Mach number of 1.65, and therefore, results for higher Mach

14numbers are unobtainable. This failure is directly related to an embedded shockwhich appears in the solution for Mach numbers greater than 1.65. If the partialdi�erential form of the equations are solved in a time-dependent fashion it is possibleto obtain solutions of the 13-moment model above this Mach number [32].Sherman and Talbot [62] solved the 13-moment system (along with the Burnettequations), as others did, by reducing the system to a set of ODEs assuming steady-state conditions. Their results con�rmed the �ndings of Grad, in that it is notpossible to obtain a solution to the ODEs above a Mach number of 1.65. Cheng hasnumerically solved the 13-moment system for a Maxwellian gas where the transportequations have been simpli�ed in a manner consistent with the thin shock-layerapproximation. This reduced system has been applied to the problem of ow behinda bow shock with moderate success [15,16]. To the best of our knowledge there hasnever been an attempt to solve any set of moment equations (without any simplifyingassumptions) using the time-dependent form, i.e., the partial di�erential equationform.Levermore, motivated by the deleterious features of previous techniques, has re-cently developed a novel hierarchy of closed systems of moment equations havingmany desirable properties [48]. Two of these properties, associated with each mem-ber of the hierarchy, are that they yield hyperbolic transport equations and they eachhave an entropy. Levermore's technique of closing the system is similar to that ofthe Grad-type moment methods. Levermore's closure is based not on a perturbativeexpansion of the distribution function but instead on a distribution that has an ex-ponential form which ensures positivity. The �rst two members of this hierarchy arethe Euler and 10-moment models. Unfortunately, it is not possible to obtain explicitforms for the transport equations of the hydrodynamic quantities for models above

15the 10-moment and these are the members which include heat transfer e�ects. Theelegant mathematical properties of Levermore's non-perturbative closure techniqueare contained in reference [48]. The history involved in the Gaussian and 35-momentdistribution functions is contained in Chapters V and VII, respectively.

CHAPTER IIIELEMENTS OF KINETIC THEORY3.1 The Boltzmann EquationIn classical gas kinetic theory a gas ow composed of monatomic molecules iscompletely described by the position and velocity of each of the molecules at aparticular instant. The number of molecules, even for a rare�ed or dilute gas, is soenormously large that such a complete description is unimaginable; the alternativeis to use a statistical description in terms of a particle velocity distribution function,F .3.1.1 Expressed in terms of particle velocityThe time evolution in phase space of the particle velocity distribution functionF (v;x; t) is governed by the Boltzmann equation and can be expressed in terms oftime t, the spatial coordinates x�, and the components of the particle velocity v� fora monatomic gas [9]: @F@t + v� @F@x� + a� @F@v� = �F�t : (3.1)where a� is the acceleration term due to external forces and in the present workis assumed to be zero. In Equation (3.1), tensor notation has been utilized. The16

17right-hand side of the Boltzmann equation, �F=�t, is the collision operator and rep-resents the time rate of change of the phase-space distribution function due to binaryinterparticle collisions. This governing equation for the distribution function is anon-linear integro-di�erential equation with seven independent variables.3.1.2 Expressed in terms of random velocityThe Boltzmann equation can be written alternatively in terms of the componentsof random velocity which are de�ned as c�(x; t)=v��u�(x; t) where u� is the bulkor average ow velocity. When this substitution is made into Equation (3.1) thefollowing form for the Boltzmann equation is obtained:@F@t + (u� + c�) @F@x� � "@u�@t + (u� + c�) @u�@x� � a�# @F@c� = �F�t : (3.2)The particle velocity is an independent variable; however, the bulk velocity is byde�nition a function of position and time; therefore, the random velocity is also afunction of position and time.3.2 Velocity Moments of the Boltzmann EquationExpressions for the macroscopic (observable) properties of a rare�ed gas ow canbe obtained by taking averages or moments of the phase-space distribution functionover the entire velocity space. The mass density �, can be obtained as follows:� = 1ZZZ�1 mF (x�; v�; t) d3v = 1ZZZ�1 mF (x�; c�; t) d3c = m hF i ; (3.3)where m is the molecular weight of the gas of interest. In general, a moment repre-sents the integration of a weight M and the distribution function over all possiblevalues of velocity. If the moment weight is a function of the particle velocity v, thenhMF i = 1ZZZ�1 M(v)F d3v: (3.4)

18For the mass-density derivation the moment weight is the (constant) molecularweight, m. Note that the compact notation hMF i will be used throughout the bal-ance of the present work. The �rst-order particle velocity moment of the phase-spacedistribution function with weight mvi gives the particle momentumm hviF i = 1ZZZ�1 mviF (x�; v�; t) d3v = �ui: (3.5)This can be used to obtain the de�nition of the bulk velocity,ui = hviF ihF i : (3.6)If moments are taken with respect to the random velocity components ci, thenthe moment weight M is a function of random velocity and the integration is takenover the entire random velocity space:hMF i = 1ZZZ�1 M(c)F d3c: (3.7)Obviously, the �rst-order random velocity moment ishciF i = 1ZZZ�1 mciF (x�; c�; t) d3c = 0: (3.8)The higher-order random velocity moments result in the following useful quantities:m hcicjF i = Pij; (3.9)m hcicjckF i = Qijk; (3.10)m hcicjckclF i = Rijkl; (3.11)m hcicjckclcmF i = Sijklm; (3.12)

19where Pij is the generalized pressure tensor and, due to symmetry, has six distinctcomponents. The hydrodynamic pressure p is proportional to the trace of the pres-sure tensor, p=P��=3. Repeated indices in tensor notation denotes summation overall the components; therefore p=(Pxx+Pyy+Pzz )=3 in a Cartesian coordinate system.The stress tensor �ij is a measure of the deviation of the pressure tensor from thehydrodynamic pressure which is the equilibrium state and is given by �ij=p�ij�Pij.The Kronecker delta �ij is equal to unity if the indices are the same and equal tozero if the indices are di�erent. A scalar temperature T can be obtained by utilizingthe perfect gas equation of state, p = �RT; (3.13)where R is the speci�c gas constant. Qijk represents a generalized heat ow tensor; itcontains 27 elements, of which ten are independent due to symmetry. The usual heat ux vector hi can be obtained from the third-order heat ow tensor Qijk by usinghi =Qi��=2. Rijkl and Sijklm represent the fourth- and �fth-order random velocitymoments, respectively. Notice that the hydrodynamic pressure and heat ux vectorcan be obtained directly from Equation (3.7) by using the moment weights, mc2=3and mc2ci=2, respectively, where c2=c�c�.The higher-order moments of the particle velocity arem hvivjF i = �uiuj + Pij ; (3.14)m hvivjvkF i = �uiujuk + uiPjk + ujPik + ukPij +Qijk; (3.15)m hvivjvkvlF i = �uiujukul + uiujPkl + uiukPjl + uiulPjk + ujukPil + ujulPik+ ukulPij + uiQjkl + ujQikl + ukQijl + ulQijk +Rijkl; (3.16)

20m hvivjvkvlvmF i = �uiujukulum + uiujukPlm + uiujulPkm + uiujumPkl + uiukulPjm+ uiukumPjl + uiulumPjk + ujukulPim + ujukumPil + ujulumPik+ ukulumPij + uiujQklm + uiukQjlm + uiulQjkm + uiumQjkl+ ujukQilm + ujulQikm + ujumQikl + ukulQijm + ukumQijl+ ulumQijk + uiRjklm + ujRiklm + ukRijlm + ulRijkm+ umRijkl + Sijklm: (3.17)The random-velocity moments are also called central moments and they providequalitative measures for the shape of the phase-space distribution function. To sim-plify the understanding, assume that the distribution function is one-dimensional ina Cartesian coordinate system with the x-axis the direction of interest. Also, di-vide the distribution function by the number density n to form a non-dimensionalphase-space distribution function, f : f = Fn : (3.18)This gives as the zeroth central moment the normalization criteria, hfi=1.The �rst central moment is still zero but the normalized �rst-order particle ve-locity moment gives the average velocity as shown earlier, hvxfi = ux. This resulthas a simple geometrical meaning | it \balances" the distribution function at uxas illustrated in Figure (3.1). In general, the average velocity does not divide thedistribution function into equal areas unless the distribution function is symmetricabout vx=0.The second-order random velocity moment or second central moment gives thevariance of particle velocity, Dc2xfE = Pxx� = �2: (3.19)

210

f(vx)vxvxcxuxFigure 3.1: Geometric interpretation of the �rst normalized particle velocity mo-ment.The square root of the variance, �, is the familiar standard deviation. This canbe recognized as the root-mean-square uctuation value of vx. The second centralmoment gives some indication of the relative width of the distribution function.Therefore, when Pxx is small, values of vx far from the bulk velocity ux are relativelyrare. On the other hand, larger values of Pxx means that values of vx far from ux aremore common. The distribution function f(vx) can be represented as the sum of anodd and even function, then Pxx depends only on the even parts. The e�ect of thesecond central moment on a distribution function is shown in Figure (3.2).The third central moment gives the \skewness" of the distribution function,s0 = Dc3xfE = Qxxx� : (3.20)If this moment is non-dimensionalized by the second central moment the e�ect ofthe width of the distribution function is removed, therefore let s=s0=�3. Physically,the third central moment gives some indication of the degree of asymmetry in thedistribution function, i.e. small skewness means the distribution function is roughlysymmetric while larger values of the skewness means there is more asymmetry present

220 vx

f(vx) small �large �Figure 3.2: The e�ect of the second central moment on the distribution function.in the distribution function. Therefore, when Qxxx = 0 the distribution function issymmetric about the origin, while if Qxxx>0 the distribution is \skewed" to the leftand if Qxxx< 0 it is \skewed" to the right. The value of Qxxx depends only on theodd part of f . Figure (3.3) shows a distribution function with a negative value of s.The fourth central moment, non-dimensionalized by �4, is called the \kurtosis"and gives a measure of the atness of the distribution function,k = 1�4 Dc4xfE = Rxxxx��4 : (3.21)Because the fourth central moment is non-dimensionalized by the second centralmoment it removes any information about the width of the distribution function.Physically, the \kurtosis" gives an indication of how much of the area is in the tails

230 vx

f(vx) s = 0s < 0Figure 3.3: The e�ect of the third central moment on the distribution function.of the distribution function. The kurtosis is large if the values of f in the tails of thedistribution function are relatively large. For larger values of Rxxxx there is more areaunder the tails of the distribution function, i.e., more likely to have a particle withlarger velocity. The e�ect of k on the distribution function is shown in Figure (3.4).Notice for large kurtosis the distribution function is \peaky" and for this particularexample the center of the distribution function is also more represented.3.3 Maxwell's Equation of ChangeGeneralized transport equations describing the transport of macroscopic quan-tities are obtained by taking velocity moments of the Boltzmann equation. Thisgeneral equation of change was �rst derived by Maxwell and it allows the introduc-tion of a hierarchy of moment equations.

240 vx

f(vx) small klarge kFigure 3.4: The e�ect of the fourth central moment on the distribution function.3.3.1 Non-conservative formMultiplying both sides of Equation (3.2) (the Boltzmann equation expressed interms of the random velocities) by the moment weight M(c) and integrating overthe entire random velocity space the following is obtained:@@thMF i + @@x� (u�hMF i) + @@x� hc�MF i+ *@M@c�F+ @u�@t + u� @u�@x�!+ *c� @M@c� F+ @u�@x� = �[M ] ; (3.22)This is referred to as the non-conservative form of Maxwell's equation of change. Theterm �[M ] represents the change of the macroscopic moment hMF i due to binaryparticle collisions.

253.3.2 Conservative formThe conservative form of Maxwell's equation of change is derived by multiplyingboth sides of Equation (3.1) by the moment weight M(v) and integrating over allpossible values of particle velocity. This leads to@@thMF i+ @@x� hv�MF i = �[M ] : (3.23)which is mathematically equivalent to the non-conservative form of Maxwell's equa-tion of change but is in divergence form which has some numerical advantages.3.4 Modelling of the Collision OperatorIn this thesis the collision operator was modelled using the BGK or relaxationtime approximation as �rst proposed by Bhatnagar, Gross, and Krook [3]. Thisis a simple mathematical model which states that the particle velocity distributionfunctions of gases in non-equilibrium states relax via collisional processes towardsthe Maxwellian velocity distribution function. There are more complex models ofthe collision operator; however, the BGK model avoids the detailed evaluation ofcomplicated collision operator while still recovering the correct collisional limits.The Maxwellian distribution function is the equilibriumsolution of the Boltzmannequation and corresponds to the distribution function of a gas in a local thermody-namic equilibrium state and is given byM = �m(2�p=�)3=2 exp �12 �c2p ! : (3.24)The BGK approximation of the collision operator can be expressed mathemati-cally by the following, �F�t = �F �M� ; (3.25)

26where � is the characteristic relaxation time for the collisional processes. Underequilibrium conditions F =M and all the source terms are zero. The relaxationtime, which is the inverse of the collision frequency � =1=�, can be obtained fromrelations of basic kinetic theory. It is related to the coe�cient of viscosity �, and thehydrostatic pressure by � = �p : (3.26)For argon, the monatomic gas considered in the present work, a relationship existsbetween the coe�cient of viscosity and the gas temperature T :� = CT (12+ 2a�1); (3.27)where C is a proportionality constant and a is a constant which characterizes themolecular interaction. This is an important result which follows from a Chapman-Enskog technique [7,14,70]. The interparticle force law is given byF = Kara ; (3.28)where Ka is a positive constant for repulsive forces and negative for attractive forcesand r is the distance between the two colliding particles. For argon, a good choicefor a which provides good agreement with experimental observations is a=10. Thisvalue for a corresponds to �/T 0:72. For hard sphere molecules a is in�nite, therefore�/T 1=2.A nice feature, from a mathematical viewpoint, of the BGK model is that thecollision terms of the transport equations can be directly evaluated. The collisionterms of the random velocity moments obtained from the BGK approximation for asingle species gas are as follows ���t = m*�F�t + = 0; (3.29)

27�ui�t = m*ci �F�t + = 0; (3.30)�Pij�t = m*cicj �F�t + = �1� (Pij � p�ij); (3.31)�Qijk�t = m*cicjck �F�t + = �1� Qijk; (3.32)�Rijkl�t = m*cicjckcl �F�t + = �1� "Rijkl � (�ij�kl + �ik�jl + �il�jk)p2� # : (3.33)Throughout a collisional process there are three collision invariants, i.e., quantitieswhich are conserved. These invariants are conservation of mass, momentum, andenergy. Equations (3.29) and (3.30) can be viewed as the statements of conservationof mass and momentum during an interparticle collision. The conservation of energycan be obtained by contraction of Equation (3.31). This leads to �p=�t = 0.In the present work we have also employed an extension of the standard BGKmodel to two relaxation time scales [32,48]. This two-scale model can be representedmathematically, for the models considered in this thesis, as�F�t = �F �M�1 � F � G�2 ; (3.34)where G is a generalized anisotropic Gaussian distribution function and is given asG = �m(2�)3=2�1=2 exp(�12��1��c�c�); (3.35)where � = det� and ��� is a symmetric tensor related to the pressure tensor, ��� =P��=�. This distribution function can be viewed as a generalization of the Maxwelliandistribution function and is discussed in further detail in the next chapter. The singlerelaxation time scale model implies a Prandtl number of one while the two time scale

28model allows a non-unity Prandtl number to be recovered in the Navier-Stokes limit.Physically, �1 is related to the pressure anisotropies and gives the rate at which thedistribution function F relaxes to the local equilibrium state while �2 is related tothe heat conduction and gives the slower rate at which F relaxes to the Gaussian.In the Navier-Stokes limit it can be shown (refer to the above references for details)that the two time scales are related by �2 = �1= (Pr � 1). Equation (3.34) becomes�F�t = �F �M� � (Pr � 1) F � G� ; (3.36)where the subscript 1 has been dropped from the time scale.3.5 Closure of the Moment Transport EquationsIf an n-order system of moment transport equations is obtained as outlined in theprevious sections it will be noticed that there is an (n + 1)-order velocity momentcontained in the system. As an example, consider the non-conservative transportequation for Qijk which is derived from Equation (3.22) using the moment weight,M= mcicjck:@Qijk@t + u�@Qijk@x� +Qijk @u�@x� +Qij� @uk@x� +Qik� @uj@x� +Qjk� @ui@x�� Pij� @Pk�@x� � Pik� @Pj�@x� � Pjk� @Pi�@x� + @Rijk�@x� = �Qijk�t : (3.37)There is a term containing the fourth-order momentRijkl and therefore, an equationis needed that governs the transport of this higher-order quantity. This is the problemof closure with moment models.The closure problem is remedied if the form of the particle velocity distributionfunction, F , of the non-equilibrium gas is speci�ed in terms of only the n-order mo-ments. Then the highest-order velocity moment contained in the system of transport

29equation can be related directly back to the lower-order moment quantities. For theabove example, Rijkl = m hcicjckclF i = fcn(�; ui; Pij ; Qijk); (3.38)would constitute the closing relation given a form for F . It should be pointed out thatthe closing relationship is not de�ned by a unique phase-space distribution function,i.e. there may be more than one distribution function that leads to the same closure.This leads to the following important fact: A system of moment transport equationsis not de�ned by a unique phase-space distribution function. A related topic is thatof the realizability of the desired set of velocity moments. Realizibility is concernedwith whether or not some positive-valued, and hence physically realistic, particle-velocity distribution function exists that leads to the same set of velocity moments.For further details on realizibility and its implication on moment models one shouldrefer to the paper by Levermore et. al. [49]Traditionally, the non-equilibrium phase-space distribution function is achievedby a perturbative expansion about the Maxwellian (equilibrium) distribution func-tion and has the general formF = F (0) + �F (1)+ �2F (2)+ � � � (3.39)where F (0) =M, � is a smallness parameter and is a measure of the Knudsen number,and F (k) are distribution functions of higher-order. Equation (3.39) can be rewrittenas F =M [1 + �1 + �2 + �3 + � � � ] ; (3.40)This technique is called the Chapman and Enskog method [13, 14, 19] and is basedon a perturbative expansion of increasing orders of accuracy.

30Another approach due to Grad [29, 31] is based on a power series expansiontechnique and can be expressed mathematically asF =M [1 +A�c� +B��c�c� +D�� c�c�c + E�� �c�c�c c� + � � � ] ; (3.41)where A�, B��, D�� , E�� �, etc. are coe�cients related to the �rst 35 moments ofthe distribution function. The above approximate distribution function is an in�niteseries and must be truncated at some point to achieve a �nite number of transportequations. In contrast to the Chapman-Enskog method, Grad's method is basedon a power series expansion and may contain the solution content of all orders. Ifeither of these methods are truncated at the lowest level, i.e. F=M, then the Eulerequations of gas dynamics are obtained.In the present work, an expansion similar to that of Grad is performed not aboutthe Maxwellian but instead, about an anisotropic Gaussian phase-space distributionfunction. This distribution function includes the hydrodynamic stresses in a non-perturbative manner and is discussed in Chapter V.3.6 EntropyFrom kinetic theory the speci�c entropy s of a gas is de�ned ass = �k�hF lnF i; (3.42)where k is Boltzmann's constant. It follows from the conservative form of the Boltz-mann equation, Equation (3.1), that the speci�c entropy satis�es the balance law@@t (�s) + @@xi (�uis� khciF lnF i) = ��t (�s) � 0; (3.43)

31for all solutions of the Boltzmann equation. Introducing the Boltzmann function H,which is de�ned as H = 1�hF lnF i; (3.44)it can be shown, for a particle-velocity distribution function which satis�es the Boltz-mann equation, that H is a monotonically decreasing function with respect to in-creasing time; that is, dHdt � 0: (3.45)This result is known as Boltzmann's H-theorem. Physically, since s = �kH, itmeans that the total entropy of a closed system increases monotonically with respectto increasing time. The Boltzmann function H does not decrease without limit to�1 but instead tends to a �nite value and thereafter remains constant. It can beshown that this limiting value corresponds to equilibrium conditions; therefore, atequilibrium conditions dH=dt = 0. It can also be shown that when the gas is inan equilibrium state the only distribution function which satis�es dH=dt = 0 is theMaxwellian distribution function M which was given in Equation (3.24).In the upcoming chapters, when the di�erent moment models used to investigateone-dimensional shock structures are presented, the relations for the speci�c entropyand the corresponding transport equations will be discussed. for further details referto Gombosi's [27] or Chapman and Cowling's [14] texts.3.7 Example { Euler EquationsAs an example of how to obtain a set of transport equations, the Euler gas dy-namic equations for a perfect gas composed of monatomic molecules will be derived.

32The Euler (or 5-moment) equations are obtained by assuming that F can be ap-proximated by the Maxwellian distribution function of Equation (3.24). This is thesimplest set of moment transport equations and corresponds to a gas everywhere inlocal thermodynamic equilibrium. The Maxwellian distribution has the formM = exp h� �A+B�c� +Dc2�i ; (3.46)where the �ve coe�cients A, B�, and D are determined by requiring thatmhMi = �; (3.47)mhciMi = 0; (3.48)mhc2Mi = P�� = 3p: (3.49)When the coe�cients are determined and substituted back into Equation (3.46),Equation (3.24) is obtained.3.7.1 Non-conservative formThe non-conservative form of the transport equations can be derived by usingthe moment weights M = fm;mci;mc2g, the Maxwellian distribution function, andthe non-conservative form of Maxwell's equation of change, Equation (3.22). Thelowest order equation is obtained when using M = m which when substituted intoEquation (3.22) gives@@t (mhMi) + @@x� (mu�hMi) + @@x� (mhc�Mi) + *@m@c�M+ @u�@t + u� @u�@x�!+ *c� @m@c�M+ @u�@x� = �[m] : (3.50)This can be simpli�ed by noting that hc�Mi = 0 and @m=@c� = 0. Note that� [m] = ��=�t. The BGK model is used to approximate the collision terms and

33since F = M all the source terms for the system are zero. This makes sense phys-ically because the source terms drive the non-equilibrium gas to equilibrium andthe Euler equations represent a gas already in thermal equilibrium. Simplifying andrearranging Equation (3.50) the following is obtained@�@t + u� @�@x� + �@u�@x� = 0: (3.51)This is the lowest-order transport equation and is the continuity equation. Next the�rst-order transport equation is derived which utilizes M = mci. Substituting thismoment weight into Equation (3.22) results in@@t (mhciMi) + @@x� (mu�hciMi) + @@x� (mhcic�Mi)+ * @@c� (mci)M+ @u�@t + u� @u�@x�!+ *c� @@c� (mci)M+ @u�@x� = �[mci] : (3.52)Note that m hcic�Mi = Pi� = p�i� � �i� = p�i�: (3.53)The gas is in equilibrium, therefore �i� = 0. Using Equation (3.53) along with@ci=@c� = �i� Equation (3.52) simpli�es to@ui@t + u� @ui@x� + 1� @p@xi = 0: (3.54)This is the momentum equation. The next higher-order transport equation is derivedusing the weight M = mc2. When used in conjunction with Equation (3.22) theresulting transport equation is obtained@@t �mhc2Mi�+ @@x� �mu�hc2Mi�+ @@x� �mhc�c2Mi�+ * @@c� �mc2�M+ @u�@t + u� @u�@x�!+ *c� @@c� �mc2�M+ @u�@x� = � hmc2i : (3.55)

34The term hc�c2Mi = 0 because M is an even function; therefore, the integrandis odd and when evaluated over the entire velocity space gives zero contribution.Physically, it can be reasoned that hc�c2Mi = Q��� = 2h� and the Euler equationsdescribe a model in equilibrium therefore h� = 0. In simplifying Equation (3.55) thefollowing is used: h@c2=@c�Mi = 2 hc�Mi = 0 which leads to@p@t + u� @p@x� + 53p@u�@x� = 0: (3.56)Equations (3.51), (3.54), and (3.56) are the usual Euler equations of hydrodynamicsin non-conserved form for a perfect gas with a ratio of speci�c heats of 5/3.3.7.2 Conservative formThe conserved form of the Euler equations can be obtained by performing theabove analysis with the following weights: M = fm;mvi;mv2g; along with theconserved form of the equation of change, Equation (3.23), this gives@�@t + @@x� (�u�) = 0; (3.57)@@t (�ui) + @@x� (�u�ui + p�i�) = 0; (3.58)@@t �12�u2 + 32p�+ @@x� �12�u�u2 + 52u�p� = 0: (3.59)Equations (3.57), (3.58), and (3.59) are the conserved form of the Euler equations.See the text by Gombosi [26] for further details.In the upcoming chapters the techniques discussed in this chapter will be utilizedin the derivation of the 10- and 35-moment systems of transport equations.

353.7.3 EntropyThe Maxwellian particle-velocity distribution functionM when inserted into thede�nition of speci�c entropy given by Equation (3.42), and after performing thenecessary integrations, yields the following results = 3k2m ln p�5=3! ; (3.60)where s is the relative entropy density. The equation which describes the transportof the relative speci�c entropy for the Euler equations is@@t (�s) + @@xi (�uis) = ��t (�s) � 0: (3.61)This follows from Equation (3.43) since hciM lnMi = 0. Equation (3.61) revealsthat the entropy is convected with the bulk ow velocity.

CHAPTER IVELEMENTS OF A RIEMANN SOLVER4.1 The Riemann ProblemConsider the initial-value problem for a one-dimensional system of hyperbolicconservation laws represented by@U@t + @F(U)@x = 0; (4.1)and U(x; t = 0) = Uo(x); (4.2)where U is the solution vector of conserved quantities and is a function of x and t,F is the ux vector, and Uo(x) is the initial data. Equation (4.1) can be rewrittenas @U@t +Ac@U@x = 0; (4.3)where Ac = @F=@U and is called the ux Jacobian matrix. The Riemann problem isa special form of the initial-value problem having two-state piecewise constant initialdata: U(x; t = 0) = 8>><>>: UL; x < 0;UR; x > 0; (4.4)36

37where UL and UR are the left and right states at the interface, respectively. Aself-similar solution (depending only on x=t) of the Riemann problem exists if theeigenvalues of Ac are all real and the eigenvectors are complete. Assuming thereare n independent conservation laws then the solution of the Riemann problem willin general consist of n centered waves separating n + 1 piecewise constant states,including the two initial states.4.2 Riemann SolversThe algorithm which solves the Riemann initial-value problem is called a Riemannsolver. Riemann solvers can be categorized into two types: exact and approximate.The solution of Riemann initial-value problems are key ingredients for many powerfulnumerical schemes and therefore e�cient solvers are needed.The algorithm pioneered by Godunov [24, 25] represents the initial data in eachcell on either side of the interface by piecewise constant states with a discontinuity atthe cell interface. At the interface the Riemann problem is solved exactly. The exactsolution in each cell is then replaced by a new piecewise constant approximation.Godunov's method is only �rst-order accurate. Van Leer [67] extended Godunov'smethod to second-order by approximating the data in each cell by piecewise linearsegments while allowing a discontinuity at the interface. Colella and Woodward [17]extended the method further by approximating the data by piecewise parabolic seg-ments once again allowing a discontinuity between the segments. A novel approachby Glimm [23] called the random-choice method still solves the problem exactly atthe interface but obtains the new approximation by a random sampling procedure.The advantage of this method is that initially sharp discontinuities remain sharpthroughout the calculation. Glimm's method is also the basis of some existence and

38uniqueness proofs for hyperbolic systems. Godunov's method is conservative (therandom-choice method is conservative only on the average) and satis�es an entropycondition. In a paper by Gottlieb and Groth [28] a relatively e�cient exact solverusing an iterative numerical technique is presented along with a survey of other exactRiemann solvers.It is computationally expensive to solve the Riemann problem exactly at theinterface (the solution procedure is iterative for many non-linear systems of inter-est). Moreover, the exact solution is to a simpli�ed problem since the initial data isapproximated in each cell. The above methods (except for Glimm's random-choicemethod) do not require all the information that is provided by the exact solution ofthe Riemann problem. Therefore, an approximate solution which retains the rele-vant features of the exact solution is desirable. This has led to the development ofapproximate Riemann solvers which are non-iterative and therefore, in general, moree�cient than the exact solvers.There have been a number of approximate Riemann solvers proposed; however,one of the most popular currently in use in the computational uid dynamics com-munity is due to Roe [57, 58] and is the solver of choice for the present work. Thebasic idea is to approximate locally the original non-linear system of partial di�eren-tial equations by a constant coe�cient linear system. Roe's approximate Riemannsolver is relatively e�cient and has the useful property of producing the exact so-lution to the Riemann problem for a single discontinuity. This technique is of the ux-di�erence-splitting type as opposed to ux-vector splitting techniques which arenot considered here. Interested readers should consult the papers by Van Leer [68]and Steger and Warming [63] for more information about the latter. Roe's approxi-mate Riemann solver ensures the conservative properties of the scheme; however, it

39does not have a built-in entropy condition and therefore an entropy �x must be usedto prevent expansion shocks.For details on the theory of the numerical techniques that are used to solve owproblems see the books by Hirsch [36,37] and Leveque [47].4.3 Roe's Approximate Riemann SolverIn the approximate Riemann solver developed by Roe a solution is sought for thelinearized form of Equation (4.3):@U@t + Ac@U@x = 0; (4.5)where Ac is an approximation of the ux Jacobian matrixAc. It is a locally constantmatrix dependent on some average state U which is a function of the left and rightstates, i.e. Ac = Ac (UL;UR).Roe's approximate Riemann solver is based on a characteristic decomposition ofthe ux di�erences which can always be uniquely expressed asFR � FL = nXk=1 �k�krck; (4.6)where �k is an eigenvalue of Ac and the approximate speed of the k-th wave, rckis the right eigenvector corresponding to the conserved formulation of the transportequations, �k is the wavestrength, and n is the total number of waves in the system.Each term in the summation represents the contribution from the k-th wave to thetotal change in ux { see Figure (4.1). The ux at the interface F? is needed andcould be computed using eitherF? (UL;UR) = FL + X�k<0 �k�k rck; (4.7)

40FL FR

tx0

�F1 �F2 �F3F?Figure 4.1: Diagram that shows the computation of the interface ux for a systemcomposed of three waves. The cell interface is located at x = 0.or F? (UL;UR) = FR � X�k>0 �k�k rck: (4.8)The interface ux calculated using Equation (4.7) starts at the left state and includesthe contributions made by the left moving waves (waves with negative speeds) whileEquation (4.8) starts from the right state and includes the contributions made bythe right moving waves (waves with positive speeds). Another form for the interface ux can be obtained by averaging Equations (4.7) and (4.8) leading toF? (UL;UR) = 12 (FL + FR)� 12 nXk=1 �kj�kjrck; (4.9)where the summation is over all the waves. This form of the interface ux can beinterpreted as the average ux plus a \dissipation" term. A nice feature of Roe'sscheme is that if a wavespeed of the linearized ux Jacobian vanishes then the dis-sipation term vanishes also { this leads to a crisp representation of discontinuities.

41However, an important drawback of this feature is that Roe's scheme cannot prop-erly identify an expansion fan that contains a sonic point and, therefore, an entropycorrection is needed to break up expansion shocks [46].The wavestrengths for the linearized system can be de�ned as� = Lc�U; (4.10)where �U = UR � UL and, in general, the notation � (�) = (�)R � (�)L is usedthroughout this thesis.. Denote the Jacobian matrix of the conserved variables Uwith respect to the vector composed of the primitive variables V by M:M = @U@V : (4.11)Using this transformation evaluated at the averaged state and the fact that theleft eigenvectors obtained from the primitive variable formulation of the transportequations are related to the left eigenvectors obtained from the conservative variableformulation by Lp = LcM; (4.12)the wavestrengths can be expressed in terms of Lp and the change in the primitivevariables, i.e. � = Lp�V: (4.13)This expression for the wavestrengths will prove computationally advantageous sincethe matrix Lp for the transport equations used in this work is more sparse than thematrix Lc.For large systems of equations such as the ones considered in this work, Equation(4.9) may not be the optimal choice for the calculation of the interface ux since the

42sum is over all k. If either Equation (4.7) or (4.8) is used the calculation will involvea summation over no more than half the total number of waves. This can reduce, byabout a half, the computational time spent in obtaining a solution to the Riemannproblem.By reconstructing the data segments within each cell using a polynomial functioninstead of approximating with piecewise constant states, Roe's ux function canachieve higher-order accuracy. This will be addressed in Chapter IX.4.4 Construction of AcThe key step in developing a Roe-type approximate Riemann solver is the con-struction of the locally constant matrix Ac. Roe [58] formulated a set of propertiesthat must be satis�ed and gave them the collective name Property U.4.4.1 Property UIn formulating an approximate Riemann solver based on Roe's scheme, a locallyconstant matrix Ac is sought which satis�es three properties as determined by Roe[58]:1. For any UR and UL the following must be true:�F = Ac(UR;UL) �U: (4.14)2. For UR = UL = U the matrix Ac (U;U) = Ac (U) � @F=@U.3. Ac has real eigenvalues with linearly independent eigenvectors.If UL and UR are connected by a single shock wave or contact discontinuity then the�rst property ensures that the solution obtained by the approximate Riemann solveragrees with the exact Riemann solver. As mentioned earlier, the method will not

43be able to properly identify a sonic expansion fan. The second property guaranteesthat the method behaves appropriately on smooth solutions while the last propertyis required so that the problem is hyperbolic and solvable. The averaged quantitieswill be collectively referred to as the Roe-averaged variables.Condition 1 does not lead directly to an average for density since the ux Jacobianmatrix Ac is not a function of density. The present work introduces another usefulcondition to be satis�ed when obtaining a Roe-average:4. For any UR and UL the following must be true:�U = M(UR;UL) �V: (4.15)This requirement will be referred to as Condition 4. The application of this conditionprovides a unique Roe-averaged density and for the higher-order moment modelsprovides a unique average state as will be shown in the later chapters. Equation(4.15) must be satis�ed if Lp is used in the calculation of the wavestrengths. Recallthat Equation (4.11) was used in deriving the wavestrengths in terms of the primitivevariables.4.4.2 ApproachesThere are several approaches in determining an appropriate matrix Ac. A methoddue to Roe [58] introduces a parameter vector z which allowsU and F to be expressedas quadratics in terms of z. It is then possible to construct a conservative linearizationaround the state z = 12 (zL + zR) where zL; zR are the input states to a Riemannproblem [69]. In general, this is not a feasible approach for higher-order momentequations. A parameter vector can be found for the Euler and 10-moment equationsbut not for the 35-moment equations. Another approach due to Vinokur [71] is

44to assume that the desired averaged quantity can be expressed as some weightedcombination of the left and right states. These averaged quantities are inserted intoEquation (4.14) and the unknown coe�cients are solved for. In the present workVinokur's approach is extended to assume that an averaged quantity can include\correction" terms consisting of changes between the left and right states of theprimitive variables. In general, Vinokur's approach does not lead to a unique average;however in this thesis another condition on the averaging is introduced which doesprovide a unique average. A third approach developed by Roe and Pike [61] isthrough the construction of Riemann tables. This method is not utilized here andthe interested reader is referred to the original paper. Vila [69] has presented ageneral framework for the construction of Roe-type matrices.4.5 Example { Euler EquationsIn this section the Roe approximate Riemann solver will be constructed for theone-dimensional Euler equations. This will serve as an example of how Roe-typeapproximate Riemann solvers are developed for the 10- and 35-moment transportequations. Both the parameter vector approach and the assumed form approach ofobtaining the Roe-averaged variables are described.4.5.1 Conservative formThe one-dimensional conservation form of the Euler or 5-moment equations fol-low from Equations (3.57) { (3.59) and for a Cartesian coordinate system can beexpressed as @U5@t + @F5@x = 0; (4.16)

45where U5 is the solution vector of conserved quantities and F5 is the ux vector.They are given asU5 = 0BBBBBBBBBBBBBBB@ ��ux�uy�uz12�u2 + 32p1CCCCCCCCCCCCCCCA ; F5 = 0BBBBBBBBBBBBBBB@ �ux�u2x + p�uxuy�uxuz12�uxu2 + 52uxp

1CCCCCCCCCCCCCCCA ; (4.17)where u2 = u2x + u2y + u2z. The ux Jacobian matrix A5c = @F5=@U5 is given byA5c = 0BBBBBBBBBBBBBBB@ 0 1 0 0 0�13 �2u2x � u2y � u2z� 43ux �23uy �23uz 23�uxuy uy ux 0 0�uxuz uz 0 ux 0�ux �16u2 + 52 p�� 12u2 � 23u2x + 52 p� �23uxuy �23uxuz 53ux1CCCCCCCCCCCCCCCA :(4.18)Introducing the internal energy E and speci�c enthalpy hE = 32p + 12� �u2x + u2y + u2z� ; (4.19)h = E + p� = 52 p� + 12 �u2x + u2y + u2z� ; (4.20)the solution vector and ux vector can be expressed more compactly asU5 = 0BBBBBBBBBBBBBBB@ ��ux�uy�uzE 1CCCCCCCCCCCCCCCA ; F5 = 0BBBBBBBBBBBBBBB@ �ux�u2x + p�uxuy�uxuz�uxh

1CCCCCCCCCCCCCCCA : (4.21)

46The ux Jacobian matrix becomesA5c = 0BBBBBBBBBBBBBBB@ 0 1 0 0 0�13 �2u2x � u2y � u2z� 43ux �23uy �23uz 23�uxuy uy ux 0 0�uxuz uz 0 ux 0�ux hh� 13 �u2x + u2y + u2z�i h� 23u2x �23uxuy �23uxuz 53ux1CCCCCCCCCCCCCCCA :(4.22)4.5.2 Primitive formThe Euler equations can also be expressed in primitive or non-conserved form.Using Equations (3.51), (3.54), and (3.56) the one-dimensional form of the transportequations can be expressed in vector form as@V5@t +A5p@V5@x = 0; (4.23)where V5 is the solution vector of primitive variables given byV5 = 0BBBBBBBBBBBBBBB@ �uxuyuzp 1CCCCCCCCCCCCCCCA ; (4.24)and A5p is the coe�cient matrix of the primitive variable uxes for the x-direction,hence the p subscript. This matrix is given byA5p = 0BBBBBBBBBBBBBBB@ ux � 0 0 00 ux 0 0 1=�0 0 ux 0 00 0 0 ux 00 53p 0 0 ux

1CCCCCCCCCCCCCCCA : (4.25)

47In deriving a Roe-average using the assumed form approach the Jacobian betweenthe conserved variables and the primitive variables will be needed { see Equation(4.11). For the Euler equation this is de�ned as M5 = @U5=@V5.M5 = 0BBBBBBBBBBBBBBB@ 1 0 0 0 0ux � 0 0 0uy 0 � 0 0uz 0 0 � 012 �u2x + u2y + u2z� �ux �uy �uz 321CCCCCCCCCCCCCCCA : (4.26)4.5.3 Roe-averaged variablesIn this section the Roe-average variables for the Euler equations are developedthrough the parameter vector approach and the assumed form approach. It is shownthat these two methods lead to the same set of Roe-averaged variables.4.5.3.1 Parameter vector approachTo derive an averaged ux Jacobian which satis�es Property U, the followingparameter vector is introducedz = p�0BBBBBBB@ 1uih 1CCCCCCCA = p�0BBBBBBBBBBBBBBB@ 1uxuyuzh 1CCCCCCCCCCCCCCCA : (4.27)The averaging is performed using the variables z,z = 12 (zL + zR) = 0BBBBBBB@ p�L +p�Rp�LuiL +p�RuiRp�LhL +p�RhR 1CCCCCCCA : (4.28)

48The parameter vector allows both �U5 and �F5 to be expressed in the form of somematrix times �z. This leads to�U5 = 0BBBBBBBBBBBBBBB@ 2z1 0 0 0 0z2 z1 0 0 0z3 0 z1 0 0z4 0 0 z1 035 z5 25 z2 25 z3 25 z4 35 z11CCCCCCCCCCCCCCCA�z = B�z; (4.29)

�F5 = 0BBBBBBBBBBBBBBB@ z2 z1 0 0 025 z5 85 z2 �25 z3 �25 z4 25 z10 z3 z2 0 00 z4 0 z2 00 z5 0 0 z21CCCCCCCCCCCCCCCA�z = C�z; (4.30)These two relations can be combined to give�F5 = CB�1�U5: (4.31)Condition 1 of Property U is satis�ed when A5c = CB�1. Performing this calculationthe locally constant matrix A5c is given asA5c = 0BBBBBBBBBBBBBBB@ 0 1 0 0 0�13 �2u2x � u2y � u2z� 43ux �23uy �23uz 23�uxuy uy ux 0 0�uxuz uz 0 ux 0�ux hh� 13 �u2x + u2y + u2z�i h� 23 u2x �23uxuy �23uxuz 53 ux

1CCCCCCCCCCCCCCCA :(4.32)

49The matrix A5c can be viewed as the ux Jacobian matrix A5c evaluated using theRoe-averaged variables. The averaged variables are de�ned asux = z2z1 = p�LuxL +p�RuxRp�L +p�R ; (4.33)uy = z3z1 = p�LuyL +p�RuyRp�L +p�R ; (4.34)uz = z4z1 = p�LuzL +p�RuzRp�L +p�R ; (4.35)h = z5z1 = p�LhL +p�RhRp�L +p�R : (4.36)In most literature the averaged density is usually obtained by assuming that thefollowing relation is true: � (�ui) = ��ui + ui��: (4.37)Everything in this relation is known except for � which when solved for gives� = p�L�R: (4.38)It should be noted that Equation (4.37) is the second, third and fourth componentof the vector equation obtained when Condition 4 is enforced on the Euler system.The �rst component of this vector equation is automatically satis�ed while the lastcomponent yields no new information once the substitution of Equation (4.38) ismade.The speed of sound a is de�ned as a2 = 5p3� . This is a useful quantity and itsRoe-average can be obtained from the de�nition of speci�c enthalpy, Equation (4.20):a = �23 h � 13 �u2x + u2y + u2z��1=2: (4.39)The Roe-average of pressure is also needed. It can be obtained by using the Roe-average of the speci�c enthalpy and substituting it into the de�nition of speci�c

50enthalpy and then solving for the pressure. This leads to the following, wherew = p=�w = p�LwL +p�RwRp�L +p�R + 15 p�L�R�p�L +p�R�2 h(�ux)2 + (�uy)2 + (�uz)2i :(4.40)4.5.3.2 Corrected average approachAnother possibly more direct approach in determining a suitable average is toassume the primitive variables (excluding density which is not needed to satisfyCondition 1 of Property U) have the following form:ui = �uiL + �uiR; (4.41)w = �wL + !wR + �(�ux)2 + �(�uy)2 + �(�uz)2; (4.42)where �, �, !, �, �, �, and � are dimensionless unknowns. It is important thatany correction (the � terms) be consistent with the averaged variable. Since w is aterm of dimension velocity squared the correction must also be a term of dimensionvelocity squared to remain consistent. Referring to Property U, the �rst condition,

51Equation (4.14) must be satis�ed. For the Euler equations this leads to0BBBBBBBBBBBBBBB@ �(�ux)� (�u2x + p)� (�uxuy)� (�uxuz)� (�uxh)1CCCCCCCCCCCCCCCA = 0BBBBBBBBBBBBBBB@ 0 1 0 0 013u2 � u2x 43ux �23uy �23uz 23�uxuy uy ux 0 0�uxuz uz 0 ux 0�ux �h� 13u2� h� 23 u2x �23uxuy �23uxuz 53 ux

1CCCCCCCCCCCCCCCA�0BBBBBBBBBBBBBBB@ ���(�ux)� (�uy)� (�uz)� �12�u2 + 32p�1CCCCCCCCCCCCCCCA :(4.43)The above is a vector equation with �ve components, each of which must be satis�ed.The �rst component is automatically satis�ed regardless of the assumed form of theaveraging. When Equations (4.41) and (4.42), are substituted into Equation (4.43)the second component consists of products of the velocity components and can bewritten as C1 �2u2xL + u2yL + u2zL�+ C2 (2uxLuxR � uyLuyR � uzLuzR)� C3 �2u2xR � u2yR � u2zR� = 0; (4.44)where the coe�cients C1, C2, and C3 areC1 = 13 h��2�R + (1 � �)2�Li ; (4.45)C2 = 23 [� (1 � �) �R � � (1 � �) �L] ; (4.46)C3 = 13 h(1 � �)2�R � �2�Li : (4.47)

52The left and right states can vary independently, therefore the products of the leftand right states are also independent. For Equation (4.44) to be satis�ed the threecoe�cients must be simultaneously equal to zero, i.e. C1 = C2 = C3 = 0. Thesolution of these three equations for � and � reveals two solutions with one solutionbeing � = p�Lp�R +p�L ; � = p�Rp�R +p�L ; (4.48)and the other � = � p�Lp�R �p�L ; � = p�Rp�R �p�L : (4.49)For �L = �R the second solution is meaningless and is discarded while the �rstsolution always makes sense. With the above solution for � and � substituted backinto Equation (4.43) it is found that the third and fourth components of the vectorequation provide no new information. The last component is not automaticallysatis�ed and gives a set of �ve equations to solve for the �ve remaining unknowns.These equations are � (1 � �)p�L + �p�R = 0; (4.50)!p�L � (1� !)p�R = 0; (4.51)5��L + (10� � 1)p�L�R + 5��R = 0; (4.52)5��L + (10� � 1)p�L�R + 5��R = 0; (4.53)5��L + (10� � 1)p�L�R + 5��R = 0: (4.54)

53There is only one solution for the unknowns and it is� = p�Lp�R +p�L ; ! = p�Rp�R +p�L ; (4.55)� = � = � = 15 p�L�R�p�L +p�R�2 : (4.56)Substituting the solution for the coe�cients back into the assumed form for theRoe-averaged variables they becomeui = p�LuiL +p�RuiRp�L +p�R ; (4.57)w = p�LwL +p�RwRp�L +p�R + 15 p�L�R�p�L +p�R�2 h(�ux)2 + (�uy)2 + (�uz)2i :(4.58)These agree with the results from the parameter vector approach, Equations (4.33) {(4.40). Note that using the above variables the Roe-average of enthalpy can beobtained, Equation(4.36).Enforcing Condition 4 the following vector equation must be satis�ed0BBBBBBBBBBBBBBB@ ���(�ux)� (�uy)� (�uz)� �12�u2 + 32p�1CCCCCCCCCCCCCCCA = 0BBBBBBBBBBBBBBB@ 1 0 0 0 0ux � 0 0 0uy 0 � 0 0uz 0 0 � 012 u2 �ux �uy �uz 32

1CCCCCCCCCCCCCCCA0BBBBBBBBBBBBBBB@ ���ux�uy�uz�p 1CCCCCCCCCCCCCCCA : (4.59)The �rst component of this vector equation is obviously automatically satis�ed whilethe second, third, and fourth components are just Equation (4.37) expanded fromtensor notation and leads to the same Roe-averaged density, Equation (4.38). Usingthis information the last component of Condition 4 is satis�ed. Condition 4 will playa more signi�cant role with the higher-order moment models in obtaining a uniqueaveraged state.

54In summary, the parameter vector approach and the corrected average approachlead to the same set of Roe-averaged variables:� = p�L�R (4.60)ui = p�LuiL +p�RuiRp�L +p�R (4.61)h = p�LhL +p�RhRp�L +p�R : (4.62)It should be pointed out that Conditions 2 and 3 of Property U can be veri�ed.4.5.4 Roe's approximate Riemann solverThe interface ux F? for the Euler equations can be determined from Equation(4.9): F? = 12 (FL + FR)� 12 5Xk=1 �kj�kjrck: (4.63)An eigensystem analysis performed on A5c reveals the following wavespeeds�1 = ux � a; (4.64)�2;3;4 = ux; (4.65)�5 = ux + a; (4.66)with the corresponding right eigenvectorsrc1 = 0BBBBBBBBBBBBBBB@ 1ux � auyuzh� uxa1CCCCCCCCCCCCCCCA ; rc2 = 0BBBBBBBBBBBBBBB@ 1uxuyuz12u2

1CCCCCCCCCCCCCCCA ; (4.67)

55rc3 = 0BBBBBBBBBBBBBBB@ 00a0uy a1CCCCCCCCCCCCCCCA ; rc4 = 0BBBBBBBBBBBBBBB@ 000auz a

1CCCCCCCCCCCCCCCA ; rc5 = 0BBBBBBBBBBBBBBB@ 1ux + auyuzh+ uxa1CCCCCCCCCCCCCCCA : (4.68)The �rst entry in the right eigenvectors are normalized to unity. The wavestrengthsare calculated using � = Lp�V; therefore, Lp is needed. The wavestrengths can beobtained from either the transformation of Lc or by an eigenstructure analysis onA5p. The left eigenvectors are normalized such that LcRc = I where I is the identitymatrix. For the Euler equations the wavestrengths are� = 0BBBBBBBBBBBBBBB@ (�p � �a�ux) = (2a2)(a2����p) =a2��uy=a��uz=a(�p+ �a�ux) = (2a2)

1CCCCCCCCCCCCCCCA : (4.69)Notice that all the components of the wavestrength vector have dimensions of density.It was shown in the previous chapter that the relative speci�c entropy for theEuler equations is s = 3k2m ln p�5=3! : (4.70)The in�nitesimal strength of the entropy wave isd�2 = d� � 3�5pdp: (4.71)Integration of this equation yields the result that d�2 / ds which, in agreement withphysical expectations, states that the strength of this convective wave is proportionalto the change in entropy across the wave front.

56As mentioned earlier Roe's approximate Riemann solver can not distinguish ashock and an expansion fan and therefore an entropy �x is needed. In the Eulersystem the waves associated with wavespeeds �1 and �5 are the acoustic waves.They are responsible for the formation of shocks and possible expansion shocks. Asmoothed value, j�kj�, is de�ned below to replace j�kj for the two non-linear acousticwaves, (k = 1; 5) [34,46]:j�kj� = 8>>><>>>: j�kj if j�kj � ��k2�2k��k + ��k4 if j�kj < ��k2 (4.72)��k = max(0; 4(�kR � �kL)) (4.73)

CHAPTER V10-MOMENT MODELThe 10-moment model is based on an anisotropic particle velocity distributionfunction, a discussion of which �rst appears in the early work on kinetic theory byMaxwell [52]. Maxwell recognized the existence of this non-equilibrium generaliza-tion of the Maxwellian solution and formulated an expression for it in terms of theprinciple stresses. The next published discussion of the model appears to have beengiven by Holway [40], who referred to the 10-moment solution of the Boltzmannequation as the ellipsoidal distribution function. He used the distribution functionin a modi�ed Mott-Smith approach to predict shock structure [41, 42]. He showed,using this distribution function, improvement in predicting shock structures at verylow Mach numbers when compared to the standard Mott-Smith approach. The so-lution is also cited in the work by Hertwick [35] and Oraevskii et. al. [54], who gavecredit to research done in the 1950's by Schl�uter. (Schl�uter evidently never publishedhis work as it relates to the 10-moment closure.) More recently, the model appearsas an important member of a hierarchy of moment closures in the theoretical workof Levermore [48]. Levermore refers to the 10-moment solution as the Gaussian clo-sure and the same terminology is used herein. The general form for the Gaussian57

58distribution function G is as follows:G = �m(2�)3=2�1=2 exp(�12��1��c�c�); (5.1)where, as mentioned in the previous chapter, � = det� and ��� is a symmetrictensor related to the pressure tensor, ��� = P��=�. For a Cartesian coordinatesystem, � = 1� 0BBBBBBB@ Pxx Pxy PxzPxy Pyy PyzPxz Pyz Pzz 1CCCCCCCA ; (5.2)therefore,��1 = 1�2� 0BBBBBBB@ PyyPzz � P 2yz PxzPyz � PxyPzz PxyPyz � PxzPyyPxzPyz � PxyPzz PxxPzz � P 2xz PxyPxz � PyzPxxPxyPyz � PxzPyy PxyPxz � PyzPxx PxxPyy � P 2xy 1CCCCCCCA ; (5.3)� = 1�3 �PxxPyyPzz + 2PxyPxzPyz � PxxP 2yz � PyyP 2xz � PzzP 2xy� : (5.4)For the conditions � > 0 and � > 0, Levermore has shown [48] that � is a positivede�nite matrix and remains so under the evolution of the ow. The inverse of apositive de�nite matrix is also a positive de�nite matrix; from this it follows that forc not equal to zero ��1��c�c� > 0. This leads to the conditions that for all physicallyrealistic values of � and P the Gaussian phase-space distribution function is �niteand positive valued. It improves on the usual equilibriumMaxwellian solution of theBoltzmann equation in that it is able to represent large departures from pressureisotropy.The Gaussian distribution function yields a closing relationship ofQijk = m hcicjckGi = 0: (5.5)

59As all third-order random velocity moments of G are identically zero, the 10-momentmodel is de�cient in that it predicts zero heat ux.As mentioned in the previous chapter, when discussing closure techniques, theremay be more than one distribution function which leads to the same set of trans-port equations. If a Chapman-Enskog-like expansion technique is used with theMaxwellian as the base function then the following distribution function will lead tothe same set of transport equations:F 10 =M 1� �2p2 ���c�c�! : (5.6)The Gaussian distribution function has the advantage over F 10 in that it is non-perturbative and therefore has no limitations on the magnitude of the stress tensorcomponents to maintain positivity of the distribution function. It is obvious fromEquation (5.6) that no matter how small the components of the shear stress are thereare always regions in random velocity space where F 10 is negative.5.1 Governing Equations { Non-conservative FormUtilizing G and kinetic theory as discussed in the previous chapter a closed setof transport equations can be derived. These equations can be expressed in non-conserved form using tensor notation as@�@t + u� @�@x� + �@u�@x� = 0; (5.7)@ui@t + u� @ui@x� + 1� @Pi�@x� = 0; (5.8)@Pij@t + u�@Pij@x� + Pij @u�@x� + Pi� @uj@x� + Pj� @ui@x� = �1� (Pij � 13P���ij): (5.9)

60The above yields a set of ten partial di�erential equations (PDEs) with the hydro-dynamic quantities being the density, �, the velocity vector, ui, and the pressuretensor, Pij. As mentioned in the previous chapter, the pressure tensor is related tothe hydrostatic pressure and the shear stresses by p = P��=3 and �ij = p�ij�Pij. Thesource terms represent interparticle collisional processes that drive non-equilibriumstates to equilibrium.For a Cartesian coordinate system (x; y; z), the 10-moment transport equationscan be re-expressed compactly in vector form:@V10@t +A10p @V10@x +B10p @V10@y +C10p @V10@z = S10p ; (5.10)where V10 is the solution vector of primitive variables and S10p is the source vectorgiven byV10 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@�uxuyuzPxxPxyPxzPyyPyzPzz

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; S10p = �1�0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

0000(2Pxx � Pyy � Pzz)=3PxyPxz(2Pyy � Pxx � Pzz)=3Pyz(2Pzz � Pxx � Pyy)=31CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; (5.11)

and A10p is the coe�cient matrix of the primitive variable uxes for the x-direction,

61hence the p subscript. This matrix is given byA10p =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@ux � 0 0 0 0 0 0 0 00 ux 0 0 1=� 0 0 0 0 00 0 ux 0 0 1=� 0 0 0 00 0 0 ux 0 0 1=� 0 0 00 3Pxx 0 0 ux 0 0 0 0 00 2Pxy Pxx 0 0 ux 0 0 0 00 2Pxz 0 Pxx 0 0 ux 0 0 00 Pyy 2Pxy 0 0 0 0 ux 0 00 Pyz Pxz Pxy 0 0 0 0 ux 00 Pzz 0 2Pxz 0 0 0 0 0 ux

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA : (5.12)The coe�cient matrices B10p and C10p have similar forms.The primitive variable form of the Euler equations can be obtained from the aboveset of transport equations in the collision-dominated limit, i.e., � ! 0. The transportequations for � and ux are identical for both systems. The equation governing thetransport of Pxx can have its source term eliminated by adding to it the transportequations for Pyy and Pzz . This leads to@@t (Pxx + Pyy + Pzz) + (3Pxx + Pyy + Pzz) @ux@x + 2Pxy @uy@x + 2Pxz @uz@x+ ux@Pxx@x + ux@Pyy@x + ux@Pzz@x = 0: (5.13)In the collision-dominated limit the source terms yield Pxy = Pxz = Pyz = 0 andPxx = Pyy = Pzz = p, and therefore, the transport equations for uy and uz reduce tothe corresponding Euler equations. Equation (5.13) simpli�es to@p@t + 53p@ux@x + ux @p@x = 0; (5.14)which is the primitive form of the Euler energy equation.

625.2 Eigensystem AnalysisHyperbolicity of Equation (5.10) can be investigated by examining the eigenstruc-ture of A10p , B10p , and C10p . The transport equations are hyperbolic if the eigenval-ues of the ux coe�cient matrices are all real and their corresponding eigenvectorsare distinct. This is a less restrictive requirement than Lax [45] makes which isthat the eigenvalues must also be distinct. For in�nitesimal disturbances, an eigen-value describes the characteristic wavespeed of propagation, the corresponding righteigenvector gives the hydrodynamic quantities transported by the wave, and the lefteigenvector prescribes the strength of the disturbance. A characteristic analysis ofA10p discloses the following polynomialdet �A10p � �I� = P 41P 22P3 = 0; (5.15)where � is an eigenvalue, I is the identity matrix, and P1, P2, and P3 are the followingfactors P1 = Z; (5.16)P2 = Z2 � c2xx; (5.17)P3 = Z2 � 3c2xx; (5.18)where Z = ux� � and c2xx = Pxx=�. In general, the notation c2ij = Pij=� will be usedherein. The roots of the polynomial factors lead to the following eigenvalues:�1 = ux �p3cxx; (5.19)�2;3 = ux � cxx; (5.20)�4;5;6;7 = ux; (5.21)�8;9 = ux + cxx; (5.22)�10 = ux +p3cxx: (5.23)

63A complete set of distinct eigenvectors can be found even though some of the eigen-values are repeated. The ten right eigenvectors, rpk, satisfying A10p rpk = �rpk arerp1 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1�p3cxx=��p3c2xy=(�cxx)�p3c2xz=(�cxx)3c2xx3c2xy3c2xz(c2xxc2yy + 2c4xy)=c2xx(c2xxc2yz + 2c2xyc2xz)=c2xx(c2xxc2zz + 2c4xz)=c2xx

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp2 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

00�cxx=�00c2xx02c2xyc2xz01CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp3 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@000�cxx=�00c2xx0c2xy2c2xz

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA;(5.24)rp4 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@10000000001CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp5 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@0000000c2xx001CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp6 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@00000000c2xx01CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp7 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@000000000c2xx1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; (5.25)

64rp8 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@000cxx=�00c2xx0c2xy2c2xz

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp9 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

00cxx=�00c2xx02c2xyc2xz01CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp10 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1p3cxx=�p3c2xy=(�cxx)p3c2xz=(�cxx)3c2xx3c2xy3c2xz(c2xxc2yy + 2c4xy)=c2xx(c2xxc2yz + 2c2xyc2xz)=c2xx(c2xxc2zz + 2c4xz)=c2xx

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA: (5.26)The corresponding left eigenvectors, lpk, satisfying lpkA10p = �lpk arelp1 = " 0 ; �p3�6cxx ; 0 ; 0 ; 16c2xx ; 0 ; 0 ; 0 ; 0 ; 0 #; (5.27)lp2 = " 0 ; �c2xy2c3xx ; � �2cxx ; 0 ; � c2xy2c4xx ; 12c2xx ; 0 ; 0 ; 0 ; 0 #; (5.28)lp3 = " 0 ; �c2xz2c3xx ; 0 ; � �2cxx ; � c2xz2c4xx ; 0 ; 12c2xx ; 0 ; 0 ; 0 #; (5.29)lp4 = " 1 ; 0 ; 0 ; 0 ; � 13c2xx ; 0 ; 0 ; 0 ; 0 ; 0 #; (5.30)lp5 = " 0 ; 0 ; 0 ; 0 ; 4c4xy � c2xxc2yy3c6xx ; �2c2xyc4xx ; 0 ; 1c2xx ; 0 ; 0 #; (5.31)lp6 = " 0 ; 0 ; 0 ; 0 ; 4c2xyc2xz � c2xxc2yz3c6xx ; �c2xzc4xx ; �c2xyc4xx ; 0 ; 1c2xx ; 0 #; (5.32)lp7 = " 0 ; 0 ; 0 ; 0 ; 4c4xz � c2xxc2zz3c6xx ; 0 ; �2c2xzc4xx ; 0 ; 0 ; 1c2xx #; (5.33)

65lp8 = " 0 ; ��c2xz2c3xx ; 0 ; �2cxx ; � c2xz2c4xx ; 0 ; 12c2xx ; 0 ; 0 ; 0 #; (5.34)lp9 = " 0 ; ��c2xy2c3xx ; �2cxx ; 0 ; � c2xy2c4xx ; 12c2xx ; 0 ; 0 ; 0 ; 0 #; (5.35)lp10 = " 0 ; p3�6cxx ; 0 ; 0 ; 16c2xx ; 0 ; 0 ; 0 ; 0 ; 0 #: (5.36)The preceding right and left eigenvectors are normalized such that lpirpj = �ij.Inspection of Equations (5.19) { (5.36) reveals that the eigenvalues are real andtheir eigenvectors are distinct for c2xx > 0. Recall that c2xx = Pxx=�; therefore for� > 0 and Pxx > 0 the eigenvalues are real and their eigenvectors are distinct. Theeigenvectors form a complete non-degenerate set. As the eigenstructure of B10p andC10p are similar in structure the 10-moment transport equations are of the hyperbolictype. It is also worth mentioning that the wavespeeds depend only on �, ux, andPxx; therefore planar disturbances propagating in the x-direction have wavespeedswhich are independent of the transverse stress components.The eigensystem reveals much about the physics of the 10-moment transportequations. The waves with speeds �1;10 = ux �p3cxx are the acoustic waves and byinspection of rp1 and rp10, they propagate changes to all components of the primitivesolution vectorV10. A wave is genuinely non-linear and convex and therefore permitsthe formation of shocks if �0k = rpk � @�k@V ; (5.37)is positive for waves moving to the right and negative for waves moving to the left.The right eigenvector is normalized by the entry of the right eigenvector associatedwith density and it is assumed that the ow density increases across a shock. In the

66case of the 10-moment acoustic waves:�0k = �2p3cxx� : (5.38)Here k is either 1 or 10 and the \minus" sign is associated with the left-going acousticwave (k = 1) and the \plus" sign corresponds to the right-going acoustic wave(k = 10). Both of these non-linear waves are convex and can lead to the formationof shocks. For further details refer to the monograph by Lax [45].The waves which correspond to the wavespeeds �2;9 = ux � cxx are associated,when in a Cartesian coordinate system, with the uy shear wave while the �3;8 =ux � cxx characteristic �elds correspond to the uz shear wave. These waves arelinearly degenerate [45], �0k = 0; (5.39)and a�ect the transverse velocities and the shear stress components.The convective wave associated with the �4 = ux wavespeed is labelled the en-tropy wave, also linearly degenerate, and by inspection of the rp4 and lp4 eigenvectorsadvects variations in density. It should be noted that the eigenvector lp4, in its presentform, is not proportional to the jump in entropy across the wave front and this isaddressed in the next section.The three linearly degenerate waves related to the wavespeeds �5;6;7 = ux areconvective waves associated with the pressures Pyy , Pyz , and Pzz , respectively, andherein are referred to as transverse pressure waves. These waves produce changesonly in their respective pressures and are fully damped in the equilibrium limit. Thewaves corresponding to the eigenvectors rp5 and rp7 are related to the transversenormal pressures Pyy and Pzz while the wave associated with rp6 is related to the

67shear stress Pyz . This is why the left eigenvector lp6 does not display symmetry withthe eigenvectors lp5 and lp7.5.3 EntropyFor the Gaussian particle-velocity distribution function G a relative speci�c en-tropy can be derived using Equation (3.42). The resulting expression for the relativespeci�c entropy is s = 3k2m ln0@�01=3�5=3 1A ; (5.40)where �0 = �3�. Recall that � = det�. It can be shown that for the Gaussian dis-tribution function hciG lnGi = 0; therefore, the equation which governs the transportof the relative speci�c entropy is identical to that derived for the Euler equations:@@t (�s) + @@xi (�uis) = ��t (�s) � 0: (5.41)Physically, this transport equation shows that entropy is convected with the bulk ow velocity. This is in agreement with the results of the eigensystem analysis; i.e.,�4 = ux.Using the 10-moment model's eigenstructure as found in the previous section, theform of the entropy can be inferred, as was done for the Euler equations. In general,the strength of a wave is speci�ed byd�k = lpk � dV; (5.42)where dV represents an in�nitesimal change in the primitive variables. The primitiveleft eigenvector for the entropy wave as found earlier can be linearly combined withthe transverse pressure waves since all four of the waves have identical speeds. This

68can be done to obtain the following form for the entropy wave eigenvectorlp4 = � 1 ; 0 ; 0 ; 0 ; �15��1xx ; �25��1xy ; �25��1xz ; �15��1yy ; �25��1yz ; �15��1zz �:(5.43)Using this form of the entropy wave eigenvector the in�nitesimal change in the en-tropy wavestrength is d�4 = lp4 � dV10 = d� � 15�ijdPij ; (5.44)where V10 = (d�; dux; duy; duz; dPxx; dPxy; dPxz; dPyy ; dPyz ; dPzz )T . d�4 / ds whichagain states that the strength of the entropy wave is proportional to the jump inentropy across the wave.It should be noted that in the equilibrium limitPij = p�ij and the speci�c entropyderived for the 10-moment model reduces to that obtained for the Maxwellian closure.5.4 Dispersion AnalysisThe solution of the 10-moment transport equations exhibits dispersive wave be-haviour. This is an important feature of the 10-moment system and other sets of gen-eralized transport equations of this type. It means that the wavespeeds and dampingrates of disturbances are dependent on the frequency or wavenumber of the prop-agating signal. At the high frequency and high wavenumber limit the propagatingdisturbances will have signal velocities which correspond to the frozen wavespeeds.This is the collisionless limit; therefore, the e�ects of the collisions are negligibleand for the 10-moment system the frozen wavespeeds are the eigenvalues which weredescribed in the previous section. At the other extreme, the low frequency and lowwavenumber limit, the signal velocities of the disturbances propagate with the equi-librium wavespeeds. At this limit, the ow is collision-dominated and for a \proper"model the equilibrium wavespeeds should correspond to the Euler wavespeeds.

69To study the dispersive wave behaviour of the one-dimensional 10-moment trans-port equations the system is linearized about the Maxwellian limit with a quiescentreference state V10o = [�o; 0; 0; 0; po; 0; 0; po; 0; po]T such thattto = t� ; xLo = x� ; ��o = 1 + ��; (5.45)uiao = u�i ; Pijpo = �ij + P �ij ; (5.46)and where to = Lo=ao, a2o = 5po=3�o, Lo = (16=5p2�)(�o=p�opo) is the mean freepath, � = � ��o=po is the characteristic relaxation time, �o is the viscosity of thereference state, and T = (25� �=8)q�=30. The 10-moment system can then bewritten as @V10�@t� +A10p �@V10�@x� = 1TQ10�V10� ; (5.47)where V10� is the perturbed solution vector and is given asV10� = [��; u�x; u�y; u�z; P �xx; P �xy; P �xz; P �yy ; P �yz ; P �zz ]T; (5.48)A10p � is the primitive variable ux matrix for the linearized system, and Q10� is amatrix associated with the linearized source terms of the BGK collision operator,

70S10p = Q10V10. The perturbed primitive variable ux matrix A10p � is given asA10p � =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@0 1 0 0 0 0 0 0 0 00 0 0 0 3=5 0 0 0 0 00 0 0 0 0 3=5 0 0 0 00 0 0 0 0 0 3=5 0 0 00 3 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 00 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; (5.49)and Q10� is given as

Q10� =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 �2=3 0 0 1=3 0 1=30 0 0 0 0 �1 0 0 0 00 0 0 0 0 0 �1 0 0 00 0 0 0 1=3 0 0 �2=3 0 1=30 0 0 0 0 0 0 0 �1 00 0 0 0 1=3 0 0 1=3 0 �2=31CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA : (5.50)

The perturbative solution is assumed to have the formV10� =[V10� exp[i(!t� � �x�)] ; (5.51)

71where ! is the frequency, � is the wavenumber, and i = p�1. This allows the disper-sive properties of the set of linearized PDEs to be explored. When this perturbativesolution is inserted into Equation (5.47) it yields the homogeneous system of linearalgebraic equations�i!I� i�A10p � � 1TQ10��[V10� = H10�[V10� = 0 : (5.52)The condition for the existence of non-trivial solutions of Equation (5.52) is that thedeterminant of H10� must be equal to zero, i.e.,det�i!I� i�A10p � � 1TQ10�� = � 1125T 5D1D2D3D4 = 0; (5.53)D1 =!;D2 =(T! � i)2;D3 =(5T!2 � 5i! � 3T�2)2;D4 =(5T!3 � 5i!2 � 9T�2! + 5i�2): (5.54)The above relation is the di�erential wave operator for the linearized 10-momenttransport equations.Of interest in this work is the initial-value problem (IVP); therefore, � is a realwavenumber, i.e., � = �R, and ! is a complex frequency, i.e., ! = !R + i!I . Thecondition for stability of the linearized 10-moment equations is that !I > 0. Let�R = 2�Lo=` = 2�Kn where ` is the solution wavelength and Kn is the Knudsennumber. Then �R will de�ne the frequency content of the perturbative solution. Low-frequency solutions (small Knudsen number) will correspond to local thermodynamicequilibrium states while high-frequency solution content (large Knudsen number)

72will correspond to non-equilibrium states. The roots, !=�R, of the di�erential waveoperator de�ne the wavespeeds and damping rates of the various types of wave motionor solution modes present in the system as a function of the Knudsen number. Forthe 10-moment system the di�erential wave operator has ten roots, of which sevenare distinct. The non-dimensional wavespeeds of the solution modes are given by�� = !R=�R where a value of unity corresponds to the equilibrium wavespeed ofacoustical disturbances, s5po3�o . The non-dimensional damping rate is !I=�R and thesolution modes are stable for !I=�R > 0.There are two types of diagrams that can be constructed which will provide insightinto the dispersive wave properties of the linearized system of transport equations.Both diagrams provide useful information about the transient response of the systemof transport equations to in�nitesimal disturbances. One is a wave dispersion dia-gram, which can be generated by plotting the non-dimensional damping rate !I=�Rversus the non-dimensional wavespeed �� for each wave mode present in the sys-tem, where the Knudsen number is taken as a free parameter. The other diagram isa damping-rate diagram, plotting the resulting attenuation rate exp(�!I) for eachwave versus the Knudsen number. Figure 5.1 shows the wave dispersion diagramfor the 10-moment system for � � = 1. This diagram illustrates the various typesof wave modes present, with each curve associated with a di�erent mode. Noticethat the labeling of the modes is done according to their frozen wavespeeds. Recallthat the frozen wavespeeds correspond to the propagation velocities evaluated in thecollisionless limit and therefore �R ! 1. For the 10-moment system these are thewavespeeds as given in the previous section. The types of wave motion present in the10-moment system are the acoustic, entropy, shear, and transverse pressure modes.The non-dimensionalized frozen wavespeeds for these modes are ��1;10 = �3=p5,

73�1 0 101 �� = !R�R

!I�R Shear waves, ��8;9���Shear waves, ��2;3@@@Acoustic wave, ��1 Acoustic wave, ��10Entropy and transversepressure waves, ��4;5;6;7���? 6 @@R�� 6 ?Figure 5.1: Dispersion diagram for the 10-moment transport equations. The arrowsindicate the direction of increasing Knudsen number from the equilibriumlimit (Kn = 0) to the collisionless limit (Kn!1).��2;3;8;9 = �q3=5, and ��4;5;6;7 = 0. A path for a particular wave mode shows howthat mode changes as the frequency content of the perturbative solution passes fromthe collision-dominated limit to the collisionless limit. The wavespeeds at the twolimits are interleaved. That is, the equilibrium wavespeeds are subcharacteristics ofthe frozen wavespeeds. Also notice that for all wave modes the damping rate is al-ways greater than or equal to zero. These two properties are necessary conditions forlinear stability and are to be expected of stable dispersive hyperbolic systems withreal eigenvalues [50,73].The wave-mode damping diagram for the 10-moment system is given in Figure5.2. This �gure shows the damping characteristics of each of the wave modes, witheach mode corresponding to a particular curve. Notice that, if the damping rate in-creases for a given mode, then its attenuation rate decreases. When the equilibriumlimit is approached, i.e., the Knudsen number approaches zero, the transverse pres-

74

0:0 0:2 0:4 0:6 0:8 1:00:00:20:40:60:81:0

Knudsen number, KnAttenuationrateexp (�!I) Transverse pressure wave, ��6;7@@@ Transverse pressure wave, ��5@@@ Shear waves, ��2;3;8;9��� Acoustic waves, ��1;10�� Entropy wave, ��4QQ

Figure 5.2: Wave mode damping diagram for the 10-moment transport equations.sure modes are purely damped, as is one pair of shear waves. The other pair of shearwaves have zero damping (attenuation rate of exp(�!I) = 1) along with the acousticand entropy modes. In the equilibrium limit, the dispersion diagram shows that theundamped modes take on the Euler wavespeed values as expected, i.e., ��1;10 = �1and ��4;8;9 = 0. In the collisionless limit, the entropy wave is undamped while all theother modes exhibit �nite damping. In fact, the entropy mode is undamped for allsolution frequencies. The transverse pressure waves associated with Pyz and Pzz alsohave constant attenuation for all solution frequencies. The damping is relatively largefor these two waves. The acoustic waves and the transverse pressure wave associ-

75ated with Pyy asymptotically approach constant attenuation rates for large Knudsennumbers.The dispersive wave behaviour of the shear modes of the 10-moment system isvery similar to the dispersive properties of the hyperbolic heat equations studiedby Roe and Arora [60]. All the shear waves do not propagate, i.e., ��2;3;8;9 = 0for Knudsen numbers less than (1=2�T )q5=12. In this regime the shear wave isadvected with the ow. For Knudsen numbers larger than the above value the shearwaves no longer propagate with the ow and they undergo constant attenuation.The 10-moment equations possess the phenomenon of second sound for this rangeof Knudsen number [53]. For � � = 1, the limiting value becomes Kn := 0:1017.This is below the value of Knudsen number beyond which the continuum hypothesisare believed to be invalid. The results of the dispersion analysis of the 10-momentsystem of transport equations �rst appeared in the papers by Brown, et. al. [10] andGombosi, et. al. [27].The above eigenstructure and dispersion analyses of the various wave modes ofthe 10-moment closure are summarized in Tables 5.1 and 5.2.

76Mode Wave Description Frozen Equilibrium Wave TypeNumber Wavespeed Wavespeed1 Acoustic wave ux �p3cxx ux � ax Non-linear2,3 Shear waves ux � cxx ux Linear4 Entropy wave ux ux Linear5-7 Transverse pressure waves ux ux Linear8,9 Shear wave ux + cxx ux Linear10 Acoustic wave ux +p3cxx ux + ax Non-linearTable 5.1: Summary of the wave modes for the 10-moment model where ax = 5p3� .Mode Wave Description Equilibrium CollisionlessNumber Damping Rate Damping Rate(!IT ) (!IT )1,10 Acoustic waves 0 2/92,3,8,9 Shear waves 0,1 1/24 Entropy wave 0 05-7 Transverse pressure waves 1 5/9,1Table 5.2: Damping rates of the various wave modes for the linearized 10-momenttransport equations.

CHAPTER VIA 10-MOMENT RIEMANN SOLVERAs was shown in the previous chapter, the 10-moment model is of hyperbolictype. This makes it amenable to powerful numerical schemes which take advantageof the wave-like nature of the physics. In Chapter IV the approximate Riemannsolver as developed by Roe [57, 58] for the Euler equations was described and thissame approach is adopted to derive a Roe-type approximate Riemann solver for the10-moment model.6.1 Governing Equations { Conservative FormUsing the conservative form of Maxwell's equation of change, the 10-momenttransport equations can be derived in conservation form. The system can be ex-pressed in tensor notation as @�@t + @@x� (�u�) = 0; (6.1)@@t (�ui) + @@x� (�uiu� + Pi�) = 0; (6.2)@@t (�uiujPij) + @@x� (�uiuju� + uiPj� + ujPi� + u�Pij) = �1� (Pij � 13P���ij);(6.3)77

78where the source terms have been derived using the BGK model as described inChapter III.Expanding from the tensor form and simplifying to one dimension, the 10-momenttransport equations take on the following vector form@U10@t + @F10@x = S10c ; (6.4)where U10 is the solution vector of conserved quantities, F10 is the ux vector, andS10c is the source vector for the conserved form of the transport equations. Thesevectors are given asU10 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@��ux�uy�uz�u2x + Pxx�uxuy + Pxy�uxuz + Pxz�u2y + Pyy�uyuz + Pyz�u2z + Pzz

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; F10 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

�ux�u2x + Pxx�uxuy + Pxy�uxuz + Pxz�u3x + 3uxPxx�u2xuy + 2uxPxy + uyPxx�u2xuz + 2uxPxz + uzPxx�uxu2y + uxPyy + 2uyPxy�uxuyuz + uxPyz + uyPxz + uzPxy�uxu2z + uxPzz + 2uzPxz1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ;(6.5)

79S10c = �1�

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@0000(2Pxx � Pyy � Pzz)=3PxyPxz(2Pyy � Pxx � Pzz)=3Pyz(2Pzz � Pxx � Pyy)=3

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA : (6.6)Notice that the conserved source terms are identical to the primitive source terms,i.e., S10c = S10p . This also follows from the general relationshipSc =MSp: (6.7)The conservative form of the Euler equations can be recovered from the conservative10-moment system for the limit � ! 0 as was done for the primitive form of the10-moment system in the previous chapter.The ux Jacobian matrix for the 10-moment model, A10c = @F10=@U10, is given

80byA10c =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0 00 0 0 0 0 0 1 0 0 0u3x �3u2x 0 0 3ux 0 0 0 0 0�3uxWxx +3Wxxu2xuy �2uxuy �u2x 0 uy 2ux 0 0 0 0�2uxWxy +2Wxy +Wxx�uyWxxu2xuz �2uxuz 0 �u2x uz 0 2ux 0 0 0�2uxWxz +2Wxz +Wxx�uzWxxuxu2y �u2y �2uxuy 0 0 2uy 0 ux 0 0�uxWyy +Wyy +2Wxy�2uyWxyuxuyuz �uyuz �uxuz �uxuy 0 uz uy 0 ux 0�uxWyz +Wyz +Wxz +Wxy�uyWxz�uzWxyuxu2z �u2z 0 �2uxuz 0 0 2uz 0 0 ux�uxWzz +Wzz +2Wxz�2uzWxz

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA;

(6.8)where Wij = Pij=�. The Jacobian of the conserved variables with respect to theprimitive variables is de�ned for the 10-moment model as M10 = @U10=@V10. This

81transformation matrix is given asM10 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1 0 0 0 0 0 0 0 0 0ux � 0 0 0 0 0 0 0 0uy 0 � 0 0 0 0 0 0 0uz 0 0 � 0 0 0 0 0 0u2x 2�ux 0 0 1 0 0 0 0 0uxuy �uy �ux 0 0 1 0 0 0 0uxuz �uz 0 �ux 0 0 1 0 0 0u2y 0 2�uy 0 0 0 0 1 0 0uyuz 0 �uz �uy 0 0 0 0 1 0u2z 0 0 2�uz 0 0 0 0 0 1

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA : (6.9)An internal energy tensor and a speci�c enthalpy tensor for the 10-moment modelcan be de�ned as Eij = 12Pij + 12�uiuj; (6.10)hij = Eij + Pij� = 32 Pij� + 12uiuj; (6.11)and introduced into Equation (6.5). These tensors simplify the transport equationsand make easier the development of the Roe-averaged variables, via the parametervector approach. If the transport equations for the pressure tensor are then divided

82by two, the solution vector of conserved quantities and the ux vector becomeU10 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@��ux�uy�uzExxExyExzEyyEyzEzz

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; F10 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

�ux�u2x + Pxx�uxuy + Pxy�uxuz + Pxz�uxhxx�(2uxhxy + uyhxx)=3�(2uxhxz + uzhxx)=3�(uxhyy + 2uyhxy)=3�(uxhyz + uyhxz + uzhxy)=3�(uxhzz + 2uzhxz)=31CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; (6.12)

where the source vector becomes S10c ) S10c =2. Notice the similarity with the Eu-ler equations. This would be expected since the 10-moment model generalizes thehydrostatic pressure to a second-order tensor.6.2 Roe-AverageThe next step in the development of an approximate Riemann solver for the10-moment model is to �nd an appropriate Roe-average for the primitive variables.For the 10-moment model, much like with the Euler equations, a Roe-average canbe obtained from either the parameter vector approach or the corrected averageapproach. It will be seen that the Roe-averaged quantities obtained for the 10-moment transport equations are very similar in form to those obtained for the Eulerequations.

836.2.1 Parameter vector approachOur purpose in making the substitution of Equations.(6.10) and (6.11) into theconserved form of the governing equations is to simplify the construction of theapproximate Riemann solver. A parameter vector z can be introduced,z = p�0BBBBBBB@ 1uihij 1CCCCCCCA = p�

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1uxuyuzhxxhxyhxzhyyhyzhzz

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; (6.13)in terms of which all components of U10 and F10 are bilinear. As with the Eulerequations, it is possible to construct a conservative linearization around the state z =12 (zL + zR) where zL; zR are the input states to a Riemann problem [69]; thereforez = 0BBBBBBB@ p�L +p�Rp�LuiL +p�RuiRp�LhijL +p�RhijR 1CCCCCCCA : (6.14)The ordering of the individual components in z is identical to that in z. Just as with

84the Euler equations both �U10 and �F10 can be expressed as some matrix times�z:�U10 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@2z1 0 0 0 0 0 0 0 0 0z2 z1 0 0 0 0 0 0 0 0z3 0 z1 0 0 0 0 0 0 0z4 0 0 z1 0 0 0 0 0 0z5 43 z2 0 0 z1 0 0 0 0 0z6 23 z3 23 z2 0 0 z1 0 0 0 0z7 23 z4 0 23 z2 0 0 z1 0 0 0z8 0 43 z3 0 0 0 0 z1 0 0z9 0 23 z4 23 z3 0 0 0 0 z1 0z10 0 0 43 z4 0 0 0 0 0 z1

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA�z = B�z; (6.15)�F10 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@z2 z1 0 0 0 0 0 0 0 0z5 43 z2 0 0 z1 0 0 0 0 0z6 23 z3 23 z2 0 0 z1 0 0 0 0z7 23 z4 0 23 z2 0 0 z1 0 0 00 3z5 0 0 3z2 0 0 0 0 00 2z6 z5 0 z3 2z2 0 0 0 00 2z7 0 z5 z4 0 2z2 0 0 00 z8 2z6 0 0 2z3 0 z2 0 00 z9 z7 z6 0 z4 z3 0 z2 00 z10 0 2z7 0 0 2z4 0 0 z2

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA�z = C�z; (6.16)where � = (�)R�(�)L. When CB�1 is evaluated the ux Jacobian matrix is obtainedwith the elements evaluated at the averaged state. This leads to the following set of

85Roe-averaged variables ui = p�RuiR +p�LuiLp�R +p�L ; (6.17)hij = p�RhijR +p�LhijLp�R +p�L : (6.18)The averaged density is obtained using the same relation as in the Euler equationsexample { see Equation (4.37), therefore,� = p�L�R: (6.19)Recall that c2ij = Pij=� =Wij which leads from Equation (6.11) tocij = (13 hij � 13 uiuj)1=2: (6.20)Equations (6.17) and (6.18) inserted into Equation (6.20)along with the de�nitionof the enthalpy tensor, Equation (6.11), leads to the Roe-average for the pressuretensor components in terms of the left and right states:Wij = p�LWijL +p�RWijRp�L +p�R + 13 p�L�R�p�L +p�R�2�ui�uj: (6.21)6.2.2 Corrected average approachThe preferred method for obtaining a Roe-average set of variables is the correctedaverage approach as it does not involve the \�nding" of a suitable parameter vector.For the 10-moment model the Roe-averaged quantities are assumed to have thefollowing form ui = �uiL + �uiR; (6.22)Wij = �WijL + !WijR + ��ui�uj; (6.23)where �, �, �, !, and � are unknowns. The correction to the pressure tensor term isconsistent withWij being a term with the dimensions of velocity squared. Condition

861 of Property U for the 10-moment system is given as�F10 = A10c �U10: (6.24)When Equations (6.5) and (6.8) are inserted into this vector equation it yields tencomponents which must be satis�ed. The �rst four components of this vector equa-tion are ���ux� = ���ux�; (6.25)�h��u2x +Wxx�i = �h��u2x +Wxx�i; (6.26)�h��uxuy +Wxy�i = �h��uxuy +Wxy�i; (6.27)�h��uxuz +Wxz�i = �h��uxuz +Wxz�i; (6.28)and are all automatically satis�ed regardless of the assumed form of the averaging.The �fth component of the vector equation is�h��u3x + uxWxx�i = �u3x � 3uxWxx���+ ��3u2x + 3Wxx����ux�+ �3ux��h��u2x +Wxx�i: (6.29)The assumed forms for the averaging, Equation (6.22) and (6.23) are inserted. Sincethe left and right input states to the Riemann problem can vary independently, theproducts of these states are also independent. Therefore the coe�cients of theseproducts must all vanish simultaneously. The solution of the �fth component for theunknowns yields the following� = 1 � �; (6.30)� = ��L��L + (1� �) �R ; (6.31)! = (1� �) �R��L + (1� �) �R ; (6.32)� = 13 �3�L + (1� �)3�R��L + (1� �)�R : (6.33)

87The last �ve components of Equation (6.24) are all satis�ed if the above values for �,�, !, and � are inserted and hence, for the 10-moment transport equations, Condition1 does not on its own provide enough information to complete specify the unknowncoe�cients of the assumed form. In other words, if this was the only condition to besatis�ed then there will be a single degree of freedom (for example, the speci�cationof �) in choosing a Roe-averaged set of variables.Condition 4 for the 10-moment system is�U10 = M10(UR;UL) �V10: (6.34)When Equations (6.5), (6.9), and (5.11) are inserted, the �rst component of thisvector equation gives the identity �� = �� which is independent of the form of theaveraging. The second, third, and fourth components provide the Roe-average fordensity: � (�ui) = ��ui + ui��: (6.35)This leads to the following Roe-average for density� = p�L�R: (6.36)The �fth component of the vector equation is�h��u2x +Wxx�i = u2x��+ 2�ux�ux +���Wxx�: (6.37)Utilizing the previously found information, this component yields two solutions for�: � = p�Rp�R +p�L ; (6.38)and � = p�Rp�R �p�L : (6.39)

88Only the �rst solution makes sense for �L = �R. With this solution for � the last�ve components of the vector equation, Equation (6.34), are satis�ed. Using theseresults for the unknown coe�cients �, �, �, !, and � and inserting back into theassumed form it is found that the Roe-averaged variables are identical to the setobtained using the parameter vector approach. In summary,� = p�L�R; (6.40)ui = p�RuiR +p�LuiLp�R +p�L ; (6.41)Wij = p�LWijL +p�RWijRp�L +p�R + 13 p�L�R�p�L +p�R�2�ui�uj: (6.42)Conditions 2 and 3 are satis�ed for the 10-moment transport equations.It should also be pointed out that if the wavestrengths are calculated using theeigensystem of the conservative form then Condition 4 is not required. However, asmentioned earlier, using the more sparse left eigenvectors obtained from the primitiveform of the transport equation will be computationally more e�cient.6.3 A Roe-Type Approximate Riemann SolverThe interface ux F? for the 10-moment system of transport equations is givenas F? = 12 (FL + FR)� 12 10Xk=1 �kj�kjrck; (6.43)where the summation takes place over all ten waves present in the 10-moment system.To develop a Roe-type approximate Riemann solver for the 10-moment transportequations information concerning the eigensystem of the averaged ux Jacobian A10c

89is required. This eigensystem analysis of A10c yields the following eigenvalues�1 = ux �p3cxx; (6.44)�2;3 = ux � cxx; (6.45)�4;5;6;7 = ux; (6.46)�8;9 = ux + cxx; (6.47)�10 = ux +p3cxx: (6.48)Note that, as expected, the eigenvalues obtained from the matrixA10c are the same asthose obtained from the matrixA10p in Chapter V, but evaluated at the Roe-averagedstate.The ten right eigenvectors satisfying A10c rck = �k rck arerc1 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1ux �p3cxxuy �p3c2xy=cxxuz �p3c2xz=cxx(ux �p3cxx)2(ux �p3cxx)(uy �p3c2xy=cxx)(ux �p3cxx)(uz �p3c2xz=cxx)(uy �p3c2xy=cxx)2 + (c2yy � c4xy=c2xx)(uy �p3c2xy=cxx)(uz �p3c2xz=cxx) + (c2yz � c2xyc2xz=c2xx)(uz �p3c2xz=cxx)2 + (c2zz � c4xz=c2xx)

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; (6.49)

90rc2 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@00�cxx00�(ux � cxx)cxx0�2(uy � c2xy=cxx)cxx�(uz � c2xz=cxx)cxx0

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; rc3 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

000�cxx00�(ux � cxx)cxx0�(uy � c2xy=cxx)cxx�2(uz � c2xz=cxx)cxx1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; (6.50)

rc4 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1uxuyuzu2xuxuyuxuzu2yuyuzu2z1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; rc5 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@0000000c2xx001CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; rc6 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@00000000c2xx01CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; rc7 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@000000000c2xx1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ;(6.51)

91rc8 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@000cxx00(ux + cxx)cxx0(uy + c2xy=cxx)cxx2(uz + c2xz=cxx)cxx

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; rc9 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

00cxx00(ux + cxx)cxx02(uy + c2xy=cxx)cxx(uz + c2xz=cxx)cxx01CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; (6.52)

rc10 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1ux +p3cxxuy +p3c2xy=cxxuz +p3c2xz=cxx(ux +p3cxx)2(ux +p3cxx)(uy +p3c2xy=cxx)(ux +p3cxx)(uz +p3c2xz=cxx)(uy +p3c2xy=cxx)2 + (c2yy + c4xy=c2xx)(uy +p3c2xy=cxx)(uz +p3c2xz=cxx) + (c2yz + c2xyc2xz=c2xx)(uz +p3c2xz=cxx)2 + (c2zz + c4xz=c2xx)1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA : (6.53)

It should be pointed out that the eigenstructure using the conservative form of thetransport equations can be obtained fromrck = Mrpk: (6.54)

92where rpk are the right eigenvectors obtained from the primitive form of the transportequations as derived in Chapter V evaluated using the Roe-averaged variables.The wavestrengths, �k = lpk ��V are as follows� =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

16c2xx ��Pxx �p3�cxx�ux�12c4xx hc2xy (�Pxx � �cxx�ux)� c2xx (�Pxy � �cxx�uy)i12c4xx hc2xz (�Pxx � �cxx�ux)� c2xx (�Pxz � �cxx�uz)i��� 13c2xx�Pxx� 13c6xx hc2xx �c2yy�Pxx � 3c2xx�Pyy�� 2c2xy �2c2xy�Pxx � 3c2xx�Pxy�i� 13c6xx hc2xx �c2yz�Pxx � 3c2xx�Pyz�� c2xy �2c2xz�Pxx � 3c2xx�Pxz��c2xz �2c2xy�Pxx � 3c2xx�Pxy�i� 13c6xx hc2xx �c2zz�Pxx � 3c2xx�Pzz�� 2c2xz �2c2xz�Pxx � 3c2xx�Pxz�i12c4xx h�c2xz (�Pxx + �cxx�ux) + c2xx (�Pxz + �cxx�uz)i12c4xx h�c2xy (�Pxx + �cxx�ux) + c2xx (�Pxy + �cxx�uy)i16c2xx ��Pxx +p3�cxx�ux�

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA:

(6.55)where lpk are left eigenvectors obtained from the primitive form of the transportequations as derived in the previous chapter evaluated at the Roe-averaged state. Thevectors rck are fairly sparse, and because many entries are repeated, the numericalcalculation of the ux is less expensive than might be feared. The left eigenvectorsobtained from the conservative form of the transport equations can be determined

93from lpk using the relation lck = lpkM�1: (6.56)It should be pointed out that as long as the Roe-averaged set of variables for a systemof transport equations satis�es Condition 4, as de�ned in Chapter IV, calculating thewavestrengths using either the primitive or conservative variables will lead to sameresult.Note that because the two acoustic waves are genuinely nonlinear, as was shownin the previous chapter, there is a need for an entropy �x to break up possiblerarefaction shocks. A smoothed value, j�kj�, is de�ned below to replace j�kj for thetwo nonlinear acoustic waves, (k = 1; 10) [34,46]:j�kj� = 8>>><>>>: j�kj if j�kj � ��k2�2k��k + ��k4 if j�kj < ��k2 (6.57)��k = max(0; 4(�kR � �kL)) : (6.58)

CHAPTER VII35-MOMENT MODELIn this chapter the 35-moment model is presented and its eigenstructure for near-equilibrium and non-equilibrium ows is discussed. The 35-moment distributionfunction is a perturbative model that uses the Gaussian distribution as its basefunction. This is fundamentally di�erent from classical distributions which use theMaxwellian distribution function as the base. The advantage of using the Gaussiandistribution is that the stresses are incorporated into the model in a non-perturbativemanner, thereby allowing virtually arbitrarily large values for the components of thestresses. This model was �rst proposed by Gombosi [27,33]. The 35-moment model,unlike the 10-moment model, is capable of representing heat transfer.The 35-moment model is based on the following particle velocity distributionfunctionF 35 = G "1 +D�� c� c�c � 3P� � !+ E�� � c�c�c c� � 6P �� c�c� + 3P��P ��2 !# ;(7.1)where G is the 10-moment distribution function { see Equation (5.1). The third-ordertensor D�� is related to the generalized heat- ow tensor by the expressionQijk = 6�2Pi�Pj�Pk D�� : (7.2)94

95Likewise, the fourth-order tensor E�� � is related to the fourth-order moment tensorKijkl by the relation Kijkl = 24�3Pi�Pj�Pk Pl�E�� �: (7.3)Equations (7.1) { (7.3) provide the description for the 35-moment phase-space dis-tribution function. It should be re-iterated that this model is an improvement overclassical distribution functions which employ the Maxwellian distribution functionas the base function in that there are no limits on the magnitudes of the stresscomponents.The tensor Kijkl is related to Rijkl, which was introduced in Chapter III, by thefollowing relation Kijkl = Rijkl � 1� (PijPkl + PikPjl + PilPjk) : (7.4)Note that Kijkl, like Qijk, is a perturbative quantity in that it is a measure of thedeviation from equilibrium. When the system is in an equilibrium state both Q andK vanish while R is non-zero.The 35-moment particle distribution function yields a closing relationship ofSijklm =m DcicjckclcmF 35E=1� (PlmQijk + PkmQijl + PklQijm + PjmQikl + PjlQikm + PimQjkl+ PilQjkm + PjkQilm + PijQklm + PikQjlm) : (7.5)Notice that this closing relationship relates the �fth-order tensor to the quantities�, P, and Q. Evaluation of the fourth-order tensor is not needed for the closingrelation.

967.1 Governing Equations { Non-conservative FormUtilizing the non-conserved form of Maxwell's equation of change and the abovedistribution function the 35-moment transport equations can be derived using thetechniques as discussed in Chapter III. In tensor notation this system can be ex-pressed as @�@t + u� @�@x� + �u�x� = 0; (7.6)@ui@t + u� @ui@x� + 1� @Pi�@x� = 0; (7.7)@Pij@t + u�@Pij@x� + Pij @u�@x� + Pi� @uj@x� + Pj� @ui@x� + @Qij�@x� = �1� (Pij � 13P���ij);(7.8)@Qijk@t + u�@Qijk@x� +Qijk @u�@x� +Qij� @uk@x� +Qik� @uj@x� +Qjk� @ui@x�+ Pi� @@x� Pjk� !+ Pj� @@x� Pik� !+ Pk� @@x� Pij� !+ @Kijk�@x�= �1� Pr Qijk; (7.9)@Kijkl@t +u�@Kijkl@x� +Kijkl@u�@x� +Kijk� @ul@x� +Kijl� @uk@x� +Kikl� @uj@x� +Kjkl� @ui@x�+Pi� @@x� Qjkl� !+ Pj� @@x� Qikl� !+ Pk� @@x� Qijl� !+ Pl� @@x� Qijk� !+Qij� @@x� Pkl� !+Qik� @@x� Pjl� !+Qil� @@x� Pjk� !+Qjk� @@x� Pil� !+Qjl� @@x� Pik� !+Qkl� @@x� Pij� !=� 1� "PrKijkl � 1� �Pij � 13P���ij��Pkl � 13P���kl��1� �Pik � 13P���ik��Pjl � 13P���jl��1� �Pil � 13P���il��Pjk � 13P���jk�# : (7.10)

97This system of transport equations contains 35 partial di�erential equations with theindependents variables being �, ui, Pij , Qijk, and Kijkl. The source terms have beenderived using the two-time-scale BGK model.For a Cartesian coordinate system, the 35-moment transport equations can beexpressed compactly in vector form as@V35@t +A35p @V35@x +B35p @V35@y +C35p @V35@z = S35p : (7.11)where V35 is the solution vector of primitive variables, A35p , B35p , and C35p are thecoe�cient matrices of the primitive variable uxes for the x-, y-, and z-directionsrespectively, and S35p is the source vector for the 35-moment system.This set of governing equations is quite large. The previous set of coupled par-tial di�erential equations provides a complete description of fully three-dimensional ows; however, this thesis is primarily concerned with one-dimensional ows and,in particular, one-dimensional shock structure and for these ows the 35-momentsystem reduces to a system of nine transport equations. For completeness the fullone-dimensional system is presented in Appendix A in both a conservative and non-conservative formulation. The reduced system can be formulated as@V35@t +A35p @V35@x = S35p : (7.12)

98where the primitive solution vector V35 and the primitive source vector S35p areV35 = 0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

�uxPxxPnnQxxxQxnnKxxxxKxxnnKnnnn1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; S35p = �1�

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@0023(Pxx � Pnn)13(Pnn � Pxx)Pr QxxxPr QxnnPr Kxxxx � 43(Pxx � Pnn)2=�Pr Kxxnn + 29(Pxx � Pnn)2=�Pr Knnnn � 13(Pxx � Pnn)2=�

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; (7.13)where the subscript n is to denote normal to the x-direction. As an example, Pnn =Pyy = Pzz .

99The coe�cient matrix of the primitive variable uxes for the x-direction isA35p =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@ux � 0 0 0 0 0 0 00 ux 1� 0 0 0 0 0 00 3Pxx ux 0 1 0 0 0 00 Pnn 0 ux 0 1 0 0 0�3P 2xx�2 4Qxxx 3Pxx� 0 ux 0 1 0 0�PxxPnn�2 2Qxnn 0 Pxx� 0 ux 0 1 0�10PxxQxxx�2 5Kxxxx 6Qxxx� 0 4Pxx� 0 ux 0 0�3PxxQxnn + PnnQxxx�2 3Kxxnn Qxnn� Qxxx� 0 2Pxx� 0 ux 06PnnQxnn�2 Knnnn 0 6Qxnn� 0 0 0 0 ux

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA:

(7.14)The primitive form of the Euler transport equations is recovered when the abovesystem is evaluated in the limit of � ! 0.7.2 Eigensystem AnalysisIn the eigensystem analysis to be presented in this section the full one-dimensional35-moment system is investigated initially. Then the characteristic equation whichcorresponds to the reduced system is found and the nine eigenvalues associated withplanar disturbances in the x-direction are determined. The full characteristic equa-tion plays an essential role in the numerical solution of the reduced 35-moment systemof transport equations; and this role will be discussed in Chapter X.As with the 10-moment transport equations, the hyperbolicity of the 35-moment

100transport equations can be investigated by an eigensystem analysis of A35p , B35p , andC35p . The transport equations are hyperbolic if the eigenvalues of the coe�cientmatrices are all real and their corresponding eigenvectors are distinct.In an analysis of the coe�cient matrix of the primitive variable uxes for the fullsystem, i.e., using A35p as given in Appendix B, the following characteristic equationis obtained det �A35p � �I� = P 51P 42P 33P 24P5 = 0; (7.15)where � is an eigenvalue, I is the identity matrix, and P1, P2, P3, P4, and P5 are thefollowing factorsP1 = Z; (7.16)P2 = Z2 � c2xx; (7.17)P3 = Z3 � 3c2xxZ + Qxxx� ; (7.18)P4 = Z4 � 6c2xxZ2 + 4Qxxx� Z + 3 � Kxxxx� ; (7.19)P5 = Z5 � 10c2xxZ3 + 10Qxxx� Z2 + 15c4xxZ � 5Kxxxx� Z � 10c2xxQxxx� ; (7.20)where Z = ux � � and c2xx = Pxx=�. The roots of this characteristic equation yieldthe eigenvalues of the 35-moment system. Note that the characteristic polynomialis a function only of �, Pxx, Qxxx, and Kxxxx and not of any of the transversevelocity moments. Physically, this means that the transverse components have noe�ect on the propagation speeds of planar disturbances. Another interesting featureof the factors that make up the characteristic polynomial is that the lower-orderpolynomial factors are proportional to the derivatives, with respect to Z, of the

101higher-order factors: Pk = k!n! @n�k@Zn�kPn; n > k: (7.21)Consequently, the region of hyperbolicity for the 35-moment transport equationscorresponds directly to where the polynomial P5 has all real roots. Along the bound-ary of this region both P5 and dP5=dZ must vanish. To map this region onto the(Qxxx;Kxxxx)-phase plane, Z is taken as the free parameter and two equations arederived for Qxxx and Kxxxx using the conditions P5 = 0 and dP5=dZ = 0:Qxxx = �25�Z3 (Z2 � 5c2xx)Z2 + c2xx ; Kxxxx = 35�Z6 � 5c2xxZ4 + 5c4xxZ2 � 5c6xxZ2 + c2xx :(7.22)Figure 7.1 shows the resulting hyperbolic region for the 35-moment transport equa-tions for planar wave propagations in the x-direction. The point in the �gure labeledG denotes the Gaussian limit which is de�ned as Q = 0 and K = 0 but the com-ponents of the stress tensor are allowed to be large. This is a near-equilibrium limitas opposed to the equilibrium limit where the stress components also must vanish.As the �gure shows the hyperbolic region contains the Gaussian limit and a �nitearea surrounding this point. It should also be mentioned that, along with this regionof the (Qxxx;Kxxxx)-phase plane, the conditions � > 0 and � > 0, as shown in thediscussion of the 10-moment model, must still be met.As was mentioned earlier, in this work only the nine transport equations whichprovide insight into one-dimensional shock structures are of interest; therefore, thecharacteristic equation reduces todet �A35p � �I� = P1P3P5 = 0; (7.23)where the polynomial factors P1, P3, and P5 are as de�ned above. This characteristic

102�1:5 �1 �0:5 0 0:5 1 1:5�2�10123 �GHYPERBOLIC REGION

Qxxx�c3xxKxxxx�c4xx

Figure 7.1: Hyperbolicity region for the 35-moment transport equations. The pointG represents the Gaussian limit.equation can be non-dimensionalized to giveZ �Z3 � 3Z +A� �Z5 � 10Z3 + 10AZ2 + 15Z � 5BZ � 10A� = 0; (7.24)where Z = Z=cxx, A = Qxxx=(�c3xx), and B = Kxxxx=(�c4xx). This form of the char-acteristic equation is a function of only two variables, A and B. There is one obviousroot to the characteristic equation, Z = 0, which corresponds to an eigenvalue of�1;1 = ux: (7.25)The �rst subscript on � denotes the order of the particular polynomial of interestand the second subscript denotes which root of that polynomial. It is possible to get

103analytic expressions for the roots of the cubic polynomial P3 and they are given as�3;1 = ux � 2 cos "13 cos�1 �Qxxx2�c3xx!# cxx; (7.26)�3;2 = ux + 12 (ux � �3;1)�s3c2xx � 34(ux � �3;1)2; (7.27)�3;3 = ux + 12 (ux � �3;1) +s3c2xx � 34(ux � �3;1)2: (7.28)The condition on the wavespeeds, �3;1, �3;2, and �3;3, to obtain real roots is�����Qxxx�c3xx ����� � 2: (7.29)There are no analytic expressions for the roots of the quintic polynomial and theymust be solved for numerically. The fact that the lower-order polynomial factors areproportional to the derivatives of the higher-order polynomial factors will play anessential role in the development of an e�cient numerical technique for the solutionof the roots of the characteristic equation and will be discussed in the next chapter.7.2.1 Gaussian limitIn the Gaussian limit, which was de�ned in the previous section, the non-equilibriumcharacteristic equation for the reduce 35-moment model, Equation (7.24), simpli�esto Z3 �Z2 � 3� �Z4 � 10Z2 + 15� = 0: (7.30)Analytic expressions for the eigenvalues of this characteristic equation can be ob-tained. They can be expressed as�k = ux + �kcxx; (7.31)

104where �1 = �q5 +p10; (7.32)�2 = �p3; (7.33)�3 = �q5 �p10; (7.34)�4;5;6 = 0; (7.35)�7 = q5 �p10; (7.36)�8 = p3; (7.37)�9 = q5 +p10: (7.38)A complete set of distinct eigenvectors can be found in the Gaussian limit. Thenine primitive right eigenvectors satisfying A35p rpk = �krpk arerp1 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1�1cxx��21c2xxc2nn(�21 � 3) �1c3xx04 (�21 � 3) c4xx00

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp2 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

000c2xx0�2c3xx0(�22 � 1) c4xx01CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp3 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1�3cxx��23c2xxc2nn(�23 � 3) �3c3xx04 (�23 � 3) c4xx00

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA;(7.39)

105rp4 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@100c2nn003c4xx00

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp5 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

000c2xx000�c4xx01CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp6 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@00000000c4xx1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; (7.40)

rp7 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1�7cxx��27c2xxc2nn(�27 � 3) �7c3xx04 (�27 � 3) c4xx001CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp8 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@000c2xx0�8c3xx0(�28 � 1) c4xx0

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp9 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1�9cxx��29c2xxc2nn(�29 � 3) �9c3xx04 (�29 � 3) c4xx001CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA:(7.41)The corresponding primitive left eigenvectors which satisfy lpkA35p = �klpk arelp1 = � 320 (�21 � 3)" 1 ; �� �1 (�21 � 7)3cxx ; �(�21 � 4)3c2xx ; 0 ; � �13c3xx ; 0 ;� 13c4xx ; 0 ; 0 #; (7.42)lp2 = " � c2nn6c2xx ; 0 ; 0 ; 16c2xx ; 0 ; �26c3xx ; 0 ; 16c4xx ; 0 #; (7.43)

106lp3 = � 320 (�23 � 3)" 1 ; �� �3 (�23 � 7)3cxx ; �(�23 � 4)3c2xx ; 0 ; � �33c3xx ; 0 ;� 13c4xx ; 0 ; 0 #; (7.44)lp4 = " 45 ; 0 ; � 415c2xx ; 0 ; 0 ; 0 ; 115c4xx ; 0 ; 0 #; (7.45)lp5 = " �2c2nn3c2xx ; 0 ; 0 ; 23c2xx ; 0 ; 0 ; 0 ; � 13c4xx ; 0 #; (7.46)lp6 = " 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1c4xx #; (7.47)lp7 = � 320 (�27 � 3)" 1 ; �� �7 (�27 � 7)3cxx ; �(�27 � 4)3c2xx ; 0 ; � �73c3xx ; 0 ;� 13c4xx ; 0 ; 0 #; (7.48)lp8 = " � c2nn6c2xx ; 0 ; 0 ; 16c2xx ; 0 ; �86c3xx ; 0 ; 16c4xx ; 0 #; (7.49)lp9 = � 320 (�29 � 3)" 1 ; �� �9 (�29 � 7)3cxx ; �(�29 � 4)3c2xx ; 0 ; � �93c3xx ; 0 ;� 13c4xx ; 0 ; 0 #: (7.50)As with the 10-moment system the eigenvectors are normalized such that lpirpj = �ij.It should be pointed out that the primitive variable matrixA35p used in the derivationof the right and left eigenvectors is evaluated in the Gaussian limit.Inspection of Equations (7.31) { (7.50) reveals that in the Gaussian limit the 35-moment eigensystem the eigenvalues are real and the corresponding eigenvectors aredistinct for c2xx > 0, i.e. � > 0 and Pxx > 0. This is the same set of conditions that wasimposed on the 10-moment eigensystem for hyperbolicity. It follows that the reduced

107system of 35-moment transport equations is also hyperbolic in the Gaussian limit.The full one-dimensional 35-moment system of transport equations when evaluatedin the Gaussian limit also has real eigenvalues and distinct eigenvectors for the sameset of conditions. Refer to the papers by Gombosi et. al. [27] and Groth et. al. [33]for details on the full one-dimensional eigensystem in the Gaussian limit.There are two sets of acoustic waves present in the 35-moment model. A pair offast acoustic waves with speeds �1;9 = ux+ �1;9cxx and a pair of slow acoustic waveswith speeds �3;7 = ux+ �3;7cxx. Inspection of the right eigenvectors shows that thesewaves a�ect all components of the primitive solution vector V35 except Qxnn, Kxxnn,and Knnnn. The convexity of the acoustic waves can be analyzed as was done for the10-moment model. A calculation shows that in the Gaussian limit�0k = 2�k 3�4k � 14�2k + 9�4k � 6�2k + 3 cxx� ; (7.51)where k 2 f1; 3; 7; 9g. Inserting the appropriate �k's, it is found that�cxx�01;9 = �26:18; �cxx�03;7 = �3:85; (7.52)where the minus signs correspond to the left-moving waves while the plus signscorrespond to the right-moving waves. Therefore, all four of these non-linear wavesare convex and can lead to the formation of shocks. It is interesting to note that thepair of fast acoustic waves are about seven times more compressive than the slowpair of acoustic waves.The wave which corresponds to the wavespeed �4 = ux, by inspection of the righteigenvector rp4, propagates changes in the variables �, Pnn, and Kxxxx. This wave iscalled the entropy wave, is associated with a contact discontinuity, and is convectedat the ow velocity in the Gaussian limit. As in the 10-moment system this wavein its present form is not proportional to the jump in entropy across the wave front.

108This eigenvector can be manipulated using linear combinations of rp5 and rp6 toremove the e�ect that this wave has on the transverse pressure component. This isattainable because they have identical wavespeeds. The entropy wave would thenonly a�ect density and the three components of the fourth-order tensor. As will beshown in a later section the corresponding left eigenvector yields the correct changein entropy. This wave can be shown to be linearly degenerate, i.e., �04 = 0.The convective wave with speed �6 = ux a�ects only the component Knnnn of thesolution vector. This wave will be called the transverse fourth-moment wave and islinearly degenerate. The eigenvalue and right eigenvector of this wave are valid awayfrom the Gaussian limit. The assumption of the Gaussian limit is not necessarysince this wave corresponds to a known root of the non-equilibrium characteristicequation.The waves with speeds �2;8 = ux+�2;8cxx directly a�ect the Pnn, Qxnn, and Kxxnncomponents of the solution vector while the wave corresponding to the wavespeed�5 = ux a�ects Pnn andKxxnn. These three waves will be referred to as the transversewaves and are linearly degenerate.The above wave modes for the 35-moment closure evaluated in the Gaussian limitare summarized in Table 7.1.7.2.2 Adiabatic limitThe adiabatic limit is de�ned asQ = 0. In this near-equilibrium limit there are norestrictions on the stress components, as in the Gaussian limit, but now K is allowedto be non-zero. A complete set of analytic expressions for the eigenvalues and theeigenvectors can also be de�ned for the adiabatic limit. The non-equilibrium char-acteristic equation of the one-dimensional form of the 35-moment model simpli�es

109Mode Wave Description Frozen Wavespeed Wave TypeNumber1 Fast acoustic wave ux �q5 +p10cxx Non-linear2 Transverse wave ux �p3cxx Linear3 Slow acoustic wave ux �q5 �p10cxx Non-linear4 Entropy wave ux Linear5 Transverse wave ux Linear6 Transverse fourth-moment wave ux Linear7 Slow acoustic wave ux +q5�p10cxx Non-linear8 Transverse wave ux +p3cxx Linear9 Fast acoustic wave ux +q5 +p10cxx Non-linearTable 7.1: Summary of the wave modes for the 35-moment model in the Gaussianlimit.to Z3 �Z2 � 3� �Z4 � 10Z2 + 15 � 5B� = 0: (7.53)Recall that B = Kxxxx=(�c4xx). The roots of this equation lead to the following setof eigenvalues �k = ux + �kcxx; (7.54)where �1 = �vuut5 +s10 + 5Kxxxx�c4xx ; (7.55)�2 = �p3; (7.56)�3 = �vuut5 �s10 + 5Kxxxx�c4xx ; (7.57)�4;5;6 = 0; (7.58)

110�7 = vuut5�s10 + 5Kxxxx�c4xx ; (7.59)�8 = p3; (7.60)�9 = vuut5 +s10 + 5Kxxxx�c4xx : (7.61)Notice that in the adiabatic limit of the nine waves present only the four acousticwavespeeds di�er from those obtained in the Gaussian limit. A complete set ofdistinct eigenvectors can be found for the 35-moment model in the adiabatic limit.The nine primitive right eigenvectors satisfying A35p rpk = �krpk arerp1 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1�1cxx��21c2xxc2nn + 3Kxxnn(�21 � 3) �c2xx(�21 � 3) �1c3xx3�1Kxxnn(�21 � 3) �cxx4 (�21 � 3) c4xx + 5Kxxxx�3(�21 � 1)(�21 � 3)Kxxnn�Knnnn�

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; (7.62)

111rp2 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@000c2xx0�2c3xx0(�22 � 1) c4xx0

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp3 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1�3cxx��23c2xxc2nn + 3Kxxnn(�23 � 3) �c2xx(�23 � 3) �3c3xx3�3Kxxnn(�23 � 3) �cxx4 (�23 � 3) c4xx + 5Kxxxx�3(�23 � 1)(�23 � 3)Kxxnn�Knnnn�1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; (7.63)

rp4 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

100c2nn � Kxxnn�c2xx003c4xxKxxnn�Knnnn�1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp5 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@000c2xx000�c4xx0

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp6 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

00000000c4xx1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; (7.64)

112rp7 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1�7cxx��27c2xxc2nn + 3Kxxnn(�27 � 3) �c2xx(�27 � 3) �7c3xx3�7Kxxnn(�27 � 3) �cxx4 (�27 � 3) c4xx + 5Kxxxx�3(�27 � 1)(�27 � 3)Kxxnn�Knnnn�

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp8 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

000c2xx0�8c3xx0(�28 � 1) c4xx01CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; (7.65)

rp9 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1�9cxx��29c2xxc2nn + 3Kxxnn(�29 � 3) �c2xx(�29 � 3) �9c3xx3�9Kxxnn(�29 � 3) �cxx4 (�29 � 3) c4xx + 5Kxxxx�3(�29 � 1)(�29 � 3)Kxxnn�Knnnn�1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA: (7.66)

The corresponding primitive left eigenvectors which satisfy lpkA35p = �klpk arelp1 = � 320 (�21 � 3)" 1 ; �� �1 (�21 � 7)3cxx ; �(�21 � 4)3c2xx ; 0 ; � �13c3xx ; 0 ;� 13c4xx ; 0 ; 0 #; (7.67)

113lp2 =" � c2nn6c2xx � 32 Kxxnn6�c4xx + 5Kxxxx ; �2 �2�Kxxnn(6�c4xx + 5Kxxxx) cxx ;12 Kxxnn(6�c4xx + 5Kxxxx) c2xx ; 16c2xx ; �12 �2Kxxnn(6�c4xx + 5Kxxxx) c3xx ; �26c3xx ;�12 Kxxnn(6�c4xx + 5Kxxxx) c4xx ; 16c4xx ; 0 #; (7.68)lp3 = � 320 (�23 � 3)" 1 ; �� �3 (�23 � 7)3cxx ; �(�23 � 4)3c2xx ; 0 ; � �33c3xx ; 0 ;� 13c4xx ; 0 ; 0 #; (7.69)lp4 =" 15 12�c4xx � 5Kxxxx3�c4xx �Kxxxx ; 0 ; �45 �c2xx3�c4xx �Kxxxx ; 0 ; 0 ; 0 ;15 �3�c4xx �Kxxxx ; 0 ; 0 #; (7.70)lp5 = " �13 2�c2nnc2xx � 3Kxxnn�c4xx ; 0 ; 0 ; 23c2xx ; 0 ; 0 ; 0 ; � 13c4xx ; 0 #; (7.71)lp6 = " �Knnnn�c4xx ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 0 ; 1c4xx #; (7.72)lp7 = � 320 (�27 � 3)" 1 ; �� �7 (�27 � 7)3cxx ; �(�27 � 4)3c2xx ; 0 ; � �73c3xx ; 0 ;� 13c4xx ; 0 ; 0 #; (7.73)

114lp8 =" � c2nn6c2xx � 32 Kxxnn6�c4xx + 5Kxxxx ; �2 �8�Kxxnn(6�c4xx + 5Kxxxx) cxx ;12 Kxxnn(6�c4xx + 5Kxxxx) c2xx ; 16c2xx ; �12 �8Kxxnn(6�c4xx + 5Kxxxx) c3xx ; �86c3xx ;�12 Kxxnn(6�c4xx + 5Kxxxx) c4xx ; 16c4xx ; 0 #; (7.74)lp9 = � 320 (�29 � 3)" 1 ; �� �9 (�29 � 7)3cxx ; �(�29 � 4)3c2xx ; 0 ; � �93c3xx ; 0 ;� 13c4xx ; 0 ; 0 #: (7.75)The eigenvectors are again normalized such that lpirpj = �ij. Furthermore, itis important to remember that the primitive eigenvectors in the adiabatic limit arederived using the primitive variable ux coe�cient matrixA35p evaluated in this limit.From inspection of Equations (7.54) { (7.75) it is revealed that the eigenvaluesare real if the condition �2 � Kxxxx�c4xx � 3; (7.76)is satis�ed, along with the conditions that � and Pxx be positive. The eigenvectorsfor this limiting case are distinct for the same conditions as in the Gaussian limit,i.e., for � > 0 and Pxx > 0. Referring to Figure 7.1 for the region of hyperbolicity ofthe 35-moment model the inequality gives the limits on Kxxxx for Qxxx = 0 betweenwhich the system remains hyperbolic.The wavespeeds for the two pairs of acoustic waves, in this limit, are a functionof Kxxxx. The right eigenvectors reveal that these acoustic waves actually a�ect allcomponents of the solution vector. This di�ers from the acoustic right eigenvectorsfound in the Gaussian limit. The convexity of these waves can be analyzed and

115reveals that in the adiabatic limit�0k = 2�k 3�4k � 14�2k + 9 �B�4k � 6�2k + 3�B cxx� ; (7.77)where k 2 f1; 3; 7; 9g. The convexity of the acoustic waves depends on Kxxxx. Equa-tion (7.76) is the condition on Kxxxx for hyperbolicity; therefore, using these limitsa plot can be obtained of �0k versus Kxxxx. This is shown in Figure 7.2 and for com-parison the Gaussian limit values are also shown. The convexity is shown only forthe right-moving waves. The plot for the left-moving waves is re ection with respectto the convexity axis.The plot is symmetric about the Kxxxx=(�c4xx) axis for the left-moving and right-moving waves. For convexity the right-moving acoustic waves must have �0k > 0.This �gure reveals that the fast right-moving acoustic wave in the adiabatic limitis convex for all Kxxxx inside the hyperbolic range. The �gure also shows that theslow right-moving acoustic wave in the adiabatic limit is convex only for a portion ofthe hyperbolicity range of Kxxxx. The slow acoustic wave is convex for Kxxxx=(�c4xx)from about -1.67 to 1.91. It will be seen in the results section that Kxxxx never movesoutside of this range for the study of one-dimensional shock structures.The wavespeed in the adiabatic limit of the mode called the entropy wave isthe same as in the Gaussian limit; and therefore, this wave is convected with the ow velocity. As with the Gaussian limit entropy wave the three right eigenvectorsrp4, rp5, and rp6 have identical wavespeeds. Therefore, the right eigenvector rp4 canbe manipulated using the right eigenvectors rp5 and rp6 to remove the a�ect thatthe entropy wave in the adiabatic limit has on the pressure component Pxx. In theadiabatic limit the entropy wave is linearly degenerate, i.e., �04 = 0.The transverse fourth-moment wavespeed and right eigenvector are the same as

116

�2 �1 0 1 2 3�40�20020406080

Kxxxx�c4xx�0k �07g�09g�07a�09a

Figure 7.2: Convexity of the two right-moving acoustic waves of the adiabatic limit.Comparison is made with the right-moving acoustic waves of the Gaus-sian limit. The subscript g denotes the Gaussian limit while the subscripta denotes the adiabatic limit.

117Mode Wave Description Frozen Wavespeed Wave TypeNumber1 Fast acoustic wave ux �q5 +p10 +Bcxx Non-linear2 Transverse wave ux �p3cxx Linear3 Slow acoustic wave ux �q5�p10 +Bcxx Non-linear4 Entropy wave ux Linear5 Transverse wave ux Linear6 Transverse fourth-moment wave ux Linear7 Slow acoustic wave ux +q5 �p10 +Bcxx Non-linear8 Transverse wave ux +p3cxx Linear9 Fast acoustic wave ux +q5 +p10 +Bcxx Non-linearTable 7.2: Summary of the wave modes for the 35-moment model in the adiabaticlimit where B = Kxxxx=(�c4xx).in the Gaussian limit since this is a known root of the non-equilibrium characteristicequation; however, the left eigenvector now has a non-zero component for density inthe adiabatic limit. This wave is still linearly degenerate.The transverse waves have the same wavespeeds and right eigenvectors as in theGaussian limit. The strength of the two waves which are not convected with the owvelocity are a�ected by all the primitive variables except Knnnn.Table 7.2 shows the various wave modes of the 35-moment system of transportequations evaluated in the adiabatic limit.7.2.3 Non-equilibriumEven though the non-equilibrium characteristic equation, Equation (7.24), cannot be completely solved to obtain analytic expressions for the wavespeeds, a com-

118plete set of right and left eigenvectors can be found as a function of the wavespeeds.Assuming the following form for the non-equilibrium wavespeeds�k = ux + �kcxx; (7.78)the eigenvectors can be expressed as functions of �k where the �k's must be de-termined numerically. It should be pointed out that when the ow is in a non-equilibrium state there is, in general, no multiplicity of wavespeeds for the reducednine-component one-dimensional system. The nine primitive right eigenvectors sat-isfying A35p rpk = �krpk, where A35p is the non-equilibrium coe�cient matrix of theprimitive variable uxes, are given asrpi =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1�icxx��2i c2xxc2nn + 3(�2i � 1) cxxQxnn + �iKxxnn�i (�2i � 3) �c3xx �Qxxx cxx�i (�2i � 3) c3xx3�i (�2i � 1) cxxQxnn + �iKxxnn�i (�2i � 3) �c3xx �Qxxx c2xx(�4i � 6�2i + 3) c4xx � 4�icxxQxxx�(�4i + 3) �c3xxQxnn + 2�iQxxxQxnn + 3�i (�2i � 1) �c2xxKxxnn�i (�2i � 3) �c3xx �Qxxx cxxKnnnn� � 18(�2i � 1) cxxQxnn + �iKxxnn�i (�2i � 3) �c3xx �Qxxx cxxQxnn

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA;(7.79)

119rpj =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@000c2xx0�jc3xx0��2j � 1� c4xx6cxxQxnn��j

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; rp6 =0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

00000000c4xx1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA; (7.80)

where i = 1; 3; 4; 7; 9 and j = 2; 5; 8. The subscript i corresponds to the eigenvectorsobtained using the eigenvalues of the �fth-order polynomial P5 while the subscriptj corresponds to the eigenvectors obtained using the eigenvalues of the third-orderpolynomial P3. The primitive left eigenvectors which satisfy lpkA35p = �klpk arelpi =" 15 (�4i � 10�2i + 12) �c4xx � 10�icxxQxxx � 5Kxxxx(�4i � 6�2i + 3) �c4xx � 4�icxxQxxx �Kxxxx ;15 � [�i (�2i � 7) �c3xx + 6Qxxx](�4i � 6�2i + 3) �c4xx � 4�icxxQxxx �Kxxxx ;15 (�2i � 4) �c2xx(�4i � 6�2i + 3) �c4xx � 4�icxxQxxx �Kxxxx ; 0 ;15 �i�cxx(�4i � 6�2i + 3) �c4xx � 4�icxxQxxx �Kxxxx ; 0 ;15 �(�4i � 6�2i + 3) �c4xx � 4�icxxQxxx �Kxxxx ; 0 ; 0#; (7.81)

120lpj =" � ��2j � 2� c2nn3 ��2j � 1� c2xx� h��4j � 7�2j � 12� �c4xx + 5KxxxxiQxnn � �j �10�2j � 27� �c3xxKxxnn�c3xx ��2j � 1� h�j ��4j � 10�2j + 15� �c4xx � 10 ��2j � 1� cxxQxxx � 5�jKxxxxi ;� 13�j h6 ��4j � 3�2j + 3� �c4xx � 5KxxxxiQxnn � 3�j �5�2j � 11� �c3xxKxxnnc4xx ��2j � 1� h�j ��4j � 10�2j + 15� �c4xx � 10 ��2j � 1� cxxQxxx � 5�jKxxxxi ;��4j � 5�2j + 4� cxxQxnn + �j ��2j � 4�Kxxnnc2xx ��2j � 1� h�j ��4j � 10�2j + 15� �c4xx � 10 ��2j � 1� cxxQxxx � 5�jKxxxxi ;�2j � 23 ��2j � 1� c2xx ; �j ��2j � 1�Qxnn + �2jKxxnnc3xx ��2j � 1� h�j ��4j � 10�2j + 15� �c4xx � 10 ��2j � 1� cxxQxxx � 5�jKxxxxi ;�j3 ��2j � 1� c3xx ; ��2j � 1�Qxnn + �jKxxnnc4xx ��2j � 1� h�j ��4j � 10�2j + 15� �c4xx � 10 ��2j � 1� cxxQxxx � 5�jKxxxxi ;13 ��2j � 1� c4xx ; 0#; (7.82)lp6 =" �Knnnn�c4xx � 6 �2�c2xxc2nn + 3Kxxnn� QxnnQxxx � 95 �12�c4xx � 5Kxxxx� Q2xnnQ2xxx ;� 245 Q2xnnc4xxQxxx ; �365 Q2xnnc2xxQ2xxx ; 12 Qxnnc2xxQxxx ; 0 ; 0 ;95 Q2xnnc4xxQ2xxx ; �6 Qxnnc4xxQxxx ; 1c4xx #: (7.83)The eigenvectors have been normalized as in the previous eigensystems. Expressingthe eigenvectors as functions of the wavespeeds will prove very advantageous whensolving for the non-equilibrium eigenstructure. Computationally it is more e�cientto evaluate Equations (7.79) { (7.83) than to determine the eigenvectors numerically.

121The criterion for a hyperbolic system is as discussed previously: � > 0, Pxx > 0,and the solution must lie in the hyperbolic region as shown in Figure 7.1.As in the adiabatic limit, the acoustic waves when in a non-equilibrium statea�ect all components of the primitive variable solution vector. Recall that in theGaussian limit the acoustic waves do not a�ect all the components. The convexity ofthe nine waves will be discussed at the end of this section. The mode correspondingto i = 4 is called the entropy wave and a�ects all components of the primitivevariable solution vector when the gas is in a non-equilibrium state. Unlike in thenear-equilibrium limits this wave cannot be combined with linear combinations ofrp5 and rp6 since the wavespeeds, when in a non-equilibrium state, are unique. Thetransverse waves are associated with the third-order polynomial P3 and rpj. Thesewaves a�ect all the transverse components, i.e., Pnn, Qxnn, Kxxnn, and Knnnn whichis why all three waves are called the transverse waves. As stated in the previousdiscussion of the two limits for the 35-moment model, the wavespeeds and righteigenvectors for the fourth-order moment wave are unchanged in non-equilibriumstates. The left eigenvector reveals that the strength of this wave is una�ected bythe heat ux components.It should be pointed out that care must be exercised when trying to obtain theGaussian or adiabatic limit eigenvectors from the non-equilibrium eigenvectors. TheGaussian or adiabatic wavespeeds can not simply be inserted and the correspondinglimit assumptions applied. There are some components of the non-equilibrium eigen-vectors for which this procedure will lead to obtaining an unde�ned result. As anexample, the last component of lpj is 6cxxQxnn=(��j). In either of the two limitsQxnnvanishes along with �j for j = 5. This leads to an unde�ned entry in the eigenvectorand suggests that a careful limiting procedure is required. This will pose a problem

122when trying to use the non-equilibrium eigenstructure in regions where the ow isin a near-equilibrium state.The convexity can be analyzed in the general non-equilibrium situation and itis found that the transverse waves and transverse fourth-moment wave are linearlydegenerate for all ow situations. The convexity for the acoustic and entropy wavesis given as �0k = 23�5k � 14�3k + 9�k �B�k + 2A�4k � 6�2k + 3 �B cxx� ; (7.84)where k 2 f1; 3; 4; 7; 9g and recall that A = Qxxx=(�c3xx). In general the �k's are notknown; however, the region of hyperbolicity is known and was given in Figure 7.1. Ata particular location in the hyperbolicity region the roots of P5 can be determinednumerically and then used to obtain �0k. Therefore, �0k can be mapped onto thehyperbolic region of the 35-moment system for each k. Note that in the analysis ofnon-equilibrium convexity the �k's may change sign within the hyperbolicity planeand therefore, a wave may change from left-moving to right-moving or vice versa.Figure 7.3 shows the sign of �01 for the always left-moving fast acoustic wave. It isdesired that for this wave �01 < 0 for the entire region of hyperbolicity; therefore,the wave would be convex. The �gure shows that indeed this left-moving wave isconvex for all hyperbolic conditions. The sign of �09 for the always right-moving fastacoustic wave is shown in Figure 7.4. Since it is a right-moving wave it is desired that�09 > 0 for convexity. This wave is also convex for the entire region of hyperbolicityas would be expected due to symmetry of the waves. The fast acoustic waves arestrictly convex and therefore may always lead to the formation of shocks.The non-linearity of the slow pair of acoustic waves is shown in Figures 7.5 and7.6. The slow acoustic mode corresponding to �03 is always a left-moving wave and

123�1:5 �1:0 �0:5 0:0 0:5 1:0 1:5�2�10123

Qxxx�c3xxKxxxx�c4xx �

Figure 7.3: Sign of �01 in the hyperbolic region of the 35-moment transport equations.The left-moving fast acoustic wave is convex for the entire region.�1:5 �1:0 �0:5 0:0 0:5 1:0 1:5�2�10123

Qxxx�c3xxKxxxx�c4xx +

Figure 7.4: Sign of �09 in the hyperbolic region of the 35-moment transport equations.The right-moving fast acoustic wave is convex for the entire region.

124�1:5 �1:0 �0:5 0:0 0:5 1:0 1:5�2�10123

Qxxx�c3xxKxxxx�c4xx �+ +�Figure 7.5: Sign of �03 in the hyperbolic region of the 35-moment transport equations.The left-moving slow acoustic wave is convex for the majority of theregion but there are two regions where the wave becomes concave.the mode corresponding to �07 is always a right-moving wave for the hyperbolicityregion. These �gures show that this pair of waves is not convex for the entire regionof hyperbolicity for the 35-moment transport equations. There are two regions ineach of the �gures where the waves are actually concave. The slow acoustic waves arenot strictly convex and there are regions in the hyperbolicity plane in which shockformation is not permitted. Along the Qxxx = 0 axis the information regardingconvexity in the adiabatic limit of the four acoustic waves can be recovered. Theoccurrence of the non-convex regions for the slow acoustic modes may indicate someimportant physical feature; however, as the concave regions are near the boundaries ofthe hyperbolic region, they are probably more indicative of the impending breakdownof the model at the boundary where the equation system becomes non-hyperbolic.Figure 7.7 shows the behaviour of �04 for the entropy wave. For the entropy

125�1:5 �1:0 �0:5 0:0 0:5 1:0 1:5�2�10123

Qxxx�c3xxKxxxx�c4xx +�� -Figure 7.6: Sign of �07 in the hyperbolic region of the 35-moment transport equations.The right-moving slow acoustic wave is convex for the majority of theregion but there are two regions where the wave becomes concave.

�1:5 �1:0 �0:5 0:0 0:5 1:0 1:5�2�10123

Qxxx�c3xxKxxxx�c4xx �+

Figure 7.7: Sign of �04 in the hyperbolic region of the 35-moment transport equations.The entropy wave is linearly degenerate for Qxxx = 0.

126wave �4 changes sign depending on the value of Qxxx. For Qxxx < 0, �4 > 0, andthe entropy wave is a right-moving wave, and conversely, for Qxxx > 0, �4 < 0,and the entropy wave is a left-moving wave. If Qxxx = 0 then �4 = 0 and theentropy wave is advected with the ow velocity. This wave is linearly degeneratefor Qxxx = 0. This agrees with the the results found in the Gaussian and adiabaticlimits. Figure 7.7 reveals that �04 > 0 when Qxxx < 0 and since the entropy waveis right-moving it is convex in this region. When Qxxx > 0, �04 < 0, and since theentropy wave is left-moving, the wave is again convex. Also along the line Qxxx = 0inside the hyperbolicity region the entropy wave is linearly degenerate. Figure 7.8is an enlargement of the upper-left corner of the hyperbolicity region and clearlyshows that �04 changes sign from the adjacent region, and therefore, the entropymode changes from a convex wave to a concave wave. Along this line separating the�1:15 �1:10 �1:05 �1:00 �0:95 �0:90 �0:851:11:21:31:41:5

Qxxx�c3xxKxxxx�c4xx +�����

Figure 7.8: Close up of upper-left area of the hyperbolicity region showing the signof �04. The line separating the positive and negative regions is where theentropy wave is linearly degenerate.

127Mode Wave Description Frozen Wavespeed Wave TypeNumber1 Fast acoustic wave ux + �1cxx Non-linear2 Transverse wave ux + �2cxx Linear3 Slow acoustic wave ux + �3cxx Non-linear4 Entropy wave ux + �4cxx Non-linear5 Transverse wave ux + �5cxx Linear6 Transverse fourth-moment wave ux Linear7 Slow acoustic wave ux + �7cxx Non-linear8 Transverse wave ux + �8cxx Linear9 Fast acoustic wave ux + �9cxx Non-linearTable 7.3: Summary of the wave modes for the 35-moment model for non-equilibriumstates.convex and concave regions the entropy wave is linearly degenerate. This type ofbehaviour is repeated in the upper-right corner of the hyperbolicity region. Hencethe entropy wave will in general be convex for most of the hyperbolicity plane. Thechange in convexity near the upper boundary may again be indicative of impendingbreakdown in the set of transport equations.The non-linearity of the acoustic and entropy waves will be monitored when the35-moment system of transport equations is solved numerically and the �ndings willbe presented in the results chapter.Table 7.3 shows the various wave modes for the 35-moment system of transportequations for non-equilibrium states.

1287.3 EntropyIt is not possible to obtain an analytic expression for the speci�c entropy usingthe 35-moment particle-velocity distribution function which was given in Equation(7.1). However, in the Gaussian limit, F 35 = G, and the speci�c entropy and itsrelated transport equation for the 35-moment model are identical to those obtainedfor the 10-moment model as presented in Chapter VII. By combining the primitiveleft eigenvector lp4 with the a multiple of lp5 the entropy wave eigenvector can beexpressed aslp4 = " 1 ; 0 ; � �5Pxx ; � 2�5Pnn ; 0 ; 0 ; 120 �2P 2xx ; 15 �2PxxPnn ; 0 #: (7.85)The strength of the entropy wave for the 35-moment model in the Gaussian limitcan be shown to bed�4 = lp4 � dV35 = 1�d� � 15 � 1PxxdPxx + 2Pnn dPnn� ; (7.86)where dV35 = (d�; dux; dPxx; dPnn; 0; 0; 0; 0; 0)T . Notice that dQijk = 0 and dKijkl =0 in the Gaussian limit. If the wavestrength for the 10-moment model, Equation(5.44), is simpli�ed to the reduced system; that is, Pyy = Pzz = Pnn and Pxy = Pxz =Pyz = 0, then Equation (7.86) is obtained.When the gas is under general non-equilibrium conditions hciF 35 lnF 35i 6= 0 andit follows from the general entropy transport equation, Equation (3.43), that theentropy is not advected with the bulk ow velocity. This result is supported by theeigensystem analysis which showed that under general non-equilibrium conditions�4 6= ux. However, when the gas in a near-equilibrium state, the eigensystem analysissupports the �ndings of the 10-moment entropy transport equation, which revealedthat entropy is advected with the bulk ow velocity; that is, �4 = ux. It should also

129be pointed out that the non-equilibrium eigenvector lp4 evaluated in the Gaussianlimit is not proportional to the jump in entropy across the wave.It can also be shown that to �rst-order s35 = s10. The details are contained inthe paper by Gombosi et. al. [27].7.4 Dispersion AnalysisAs with the 10-moment system of transport equations, investigation of the 35-moment system can reveal much information about the response of the set of trans-port equations to in�nitesimal disturbances. Linearization is performed about theMaxwellian reference state V35o = [�o; 0; po; po; 0; 0; 0; 0; 0]T similar to the lineariza-tion used for the 10-moment system. For the 35-moment system two additionalassumptions are needed for the linearization and they are Qijk=(poao) = Q�ijk and�oKijkl=p2o = K�ijkl. The 35-moment linearized system can then be expressed as@V35�@t� +A35p �@V35�@x� = 1TQ35�V35� ; (7.87)where the perturbed solution vector isV35� = [��; u�x; P �xx; P �nn; Q�xxx; Q�xnn;K�xxxx;K�xxnn;K�nnnn]T; (7.88)A35p � is the coe�cient matrix of the primitive variable uxes for the linearized system,and Q35� is a matrix associated with the linearized source terms of the BGK collisionoperator, S35p = Q35V35. The perturbed coe�cient matrix of the primitive variable

130 uxes for the 35-moment linearized system isA35p � = 0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

0 1 0 0 0 0 0 0 00 0 3=5 0 0 0 0 0 00 3 0 0 1 0 0 0 00 0 1 0 0 1 0 0 0�9=5 0 9=5 0 0 0 3=5 0 0�3=5 0 0 3=5 0 0 0 3=5 00 0 0 0 4 0 0 0 00 0 0 0 0 2 0 0 00 0 0 0 0 0 0 0 01CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; (7.89)

and Q35� is given asQ35� = 0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 �2=3 2=3 0 0 0 0 00 0 1=3 �1=3 0 0 0 0 00 0 0 0 �1 0 0 0 00 0 0 0 0 �1 0 0 00 0 0 0 0 0 �1 0 00 0 0 0 0 0 0 �1 00 0 0 0 0 0 0 0 �11CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA : (7.90)

It should be noted that for the purposes of the dispersion analysis, a Prandtl numberof unity has been assumed. The perturbative solution for the 35-moment linearizedsystem has a form similar to that used in the 10-moment dispersion analysis and canbe expressed as V35� =[V35� exp[i(!t� � �x�)] : (7.91)

131When the perturbative solution is inserted into Equation (7.87) it yields the homo-geneous system of linear algebraic equations�i!I� i�A35p � � 1TQ35��[V35� = H35�[V35� = 0 : (7.92)This leads to the following di�erential wave operator for the 35-moment systemdet�i!I� i�A35p � � 1TQ35�� = i125T 6D1D2; (7.93)whereD1 =T! � i;D2 =125T 5!8 � 625iT 4!7 � �1250T 3 + 975T 5�2�!6+ i �1250T 2 + 3400T 4�2�!5 + �625T + 4475T 3�2 + 2025T 5�4 + 1215T 5�6�!2+ i �125 + 2775T 2�2 + 4245T 4�4�!3 � �850T�2 + 3090T 3�4 + 1215T 5�6� !2+ i �125�2 + 945T 2�4 + 1134T 4�6�! + 75T�4 + 207T 3�6: (7.94)As in the 10-moment linearized dispersion analysis, a dispersion and dampingdiagram can be developed from the di�erential wave operator for the 35-momentsystem of transport equations. There are nine distinct roots of the ninth-order waveoperator and eight occur in pairs, therefore the dispersion diagram is symmetric withrespect to the damping axis. The frozen wavespeeds of the various wave modes are��k = q3=5�k where the �k's correspond to the eigenvalues obtained in the Gaussianlimit and were given in Equations (7.31) { (7.38). Figure 7.9 shows that the dampingof all the various wave modes is greater than or equal to zero and, as with the 10-moment dispersion analysis, the 35-moment analysis reveals the desirable interleavingproperty of the equilibrium and frozen wavespeeds. These two properties are strongindicators of stability for the 35-moment system.

132

�2 �1 0 1 20:00:51:01:52:02:5

�� = !R�R!I�R Slowacousticwave, ��7@@@Slowacousticwave, ��3��� Fastacousticwave, ��9��Fastacousticwave, ��1@@

Coupledtransversepressure-heatwave, ��2���� Coupledtransversepressure-heatwave, ��8@@@@Convectivewaves, ��4;5;6����? ?

?? ?

6? 6 ?Figure 7.9: Dispersion diagram for the 35-moment transport equations. The arrowsindicate the direction of increasing Knudsen number from the equilibriumlimit (Kn = 0) to the collisionless limit (Kn!1).

133The damping characteristics of each of the 35-moment wave modes for the reducedsystem are further illustrated in Figure 7.10 by plotting the attenuation rate versusthe Knudsen number. As with the dispersion diagram, this �gure shows that the0:0 0:2 0:4 0:6 0:8 1:00:20:40:6

0:81:0Knudsen number, Kn

Attenuationrateexp (�!I) Entropy wave, ��4@@Slow acoustic waves, ��3;7@@@@@@Fast acoustic waves, ��1;9 @@@@@Coupled transversepressure-heat waves, ��2;8���Transverse pressure wave, ��5@@Transverse fourth-moment wave, ��6����Figure 7.10: Damping diagram for the 35-moment transport equations.damping for all the present modes is greater than or equal to zero.The dispersion diagram shows that the transverse fourth-moment wave, whichis associated with the polynomial D1, does not propagate (��6 = 0). The dampingdiagram reveals that this wave mode is purely damped with a relatively large constantrate of attenuation of exp(�1=T ) for all Knudsen numbers.The rest of the waves have their dispersive characteristics represented by theeighth-order polynomial D2. One of the three transverse waves, ��5, does not prop-agate for all solution frequencies, i.e., ��5 = 0 as shown in the dispersion diagram

134while the other two transverse waves propagate in the frozen limit at velocities of��2;8 = �3=p5. The damping diagram reveals that in the collision-dominated limitall three of these waves are heavily damped with an attenuation rate of exp(�1=T ).When the system is in a more non-equilibrium state, the damping rate diminishesfor these three waves, i.e., the attenuation rate increases. In the collisionless limitthese attenuation rates are exp[�8=(9T )] for ��2;8 and exp[�(21 + 4p6)=(45T )] for��5. The damping diagram reveals that when considering all the wave modes theentropy wave is the least damped mode and in the limit of vanishing Knudsen numberthe entropy wave becomes undamped. The latter is to be expected in the collision-dominated limit. The �nite damping of the entropy wave in the 35-moment systemis unlike the entropy wave in the 10-moment system where it is undamped for allsolution frequencies. In the limit of large Knudsen number the attenuation rate ofthe entropy wave approaches the value of exp[(21� 4p6)=(45T )].The damping diagram reveals that the slow pair of acoustic waves degenerateto the usual Euler sound modes. The slow pair of acoustic waves experience nodamping in the collision-dominated limit while the fast pair of acoustic waves areheavily damped in this limit with an attenuation rate of exp(�1=T ). For highersolution frequencies, and therefore larger Knudsen numbers, both sets of acousticwaves experience �nite damping; however, the damping is still greater for the fast pairof acoustic waves. In the collisionless limit the fast modes approach an attenuationrate of exp[�(103 + 13p10)=(180T )] while the slow modes have an attenuation rateof exp[�(103� 13p10)=(180T )]. It is quite interesting to note that even though theslow acoustic modes reduce to the usual Euler sound modes this is the pair of wavesthat is not convex for all hyperbolic ow situations while the fast acoustic modes

135Mode Wave Description Equilibrium CollisionlessNumber Damping Rate Damping Rate(!IT ) (!IT )1,9 Fast acoustic waves 1 � 0.80062,8 Transverse waves 1 � 0.88893,7 Slow acoustic waves 0 � 0.34384 Entropy wave 0 � 0.24895 Transverse wave 1 � 0.68446 Transverse fourth-moment wave 1 1Table 7.4: Damping rates of the various wave modes for the linearized 35-momenttransport equations.are purely damped in the equilibrium limit but are convex for all hyperbolic owsituations.Table 7.4 summarizes the damping rates of the wave modes for the linearized35-moment model. Further details of the linearized dispersion analysis for thefull one-dimensional form of the 35-moment transport equations are contained in thepapers by Gombosi et. al. [27] and Groth et. al. [33].

CHAPTER VIII35-MOMENT RIEMANN SOLVERSIt was shown in the previous chapter that the 35-moment transport equationspossess hyperbolic qualities for a �nite region of the (Qxxx;Kxxxx)-phase plane. Thismakes it suitable to develop Roe-type approximate Riemann solvers based on thissystem of transport equations. In a numerical calculation the system will have tobe monitored to ensure hyperbolicity. In this chapter three Riemann solvers will bedeveloped. The �rst will take advantage of the Gaussian eigenstructure as describedin Chapter VII. In this limit the eigenvalues are real and therefore, the set of equa-tions will always be hyperbolic. The second Riemann solver will use the adiabaticeigenstructure which has a condition on Kxxxx for real eigenvalues. The last solverto be developed and presented in this chapter will not make any assumption withregards to the eigenstructure. The eigenvectors will be expressed as a function of theeigenvalues as was done in the previous chapter and the eigenvalues will be obtainednumerically.8.1 Governing Equations { Conservation FormTo derive the 35-moment transport equations in conservation form F 35 is usedin conjunction with Maxwell's conservative equation of change as outlined in the136

137chapter on kinetic theory. The conserved 35-moment system can be expressed intensor notation as @�@t + @@x���u�� = 0; (8.1)@@t��ui�+ @@x���uiu� + Pi�� = 0; (8.2)@@t��uiuj + Pij�+ @@x���uiuju� + uiPj� + ujPi� + u�Pij +Qij��= �1� �Pij � 13P���ij�; (8.3)@@t��uiujuk + uiPjk + ujPik + ukPij +Qijk�+ @@x���uiujuku� + uiujPk�+ uiukPj� + ujukPi� + uiu�Pjk + uju�Pik + uku�Pij + 1�PijPk� + 1�PikPj�+ 1�PjkPi� + u�Qijk + ukQij� + ujQik� + uiQjk� +Kijk��= �1� hui�Pjk � 13P���jk�+ uj�Pik � 13P���ik�+ uk�Pij � 13P���ij�+ Pr Qijki;(8.4)

138@@t��uiujukul + uiujPkl + uiukPjl + ujukPil + uiulPjk + ujulPik + ukulPij+ 1�PijPkl + 1�PikPjl + 1�PjkPil + ulQijk + ukQijl + ujQikl + uiQjkl +Kijkl�+ @@x���uiujukulu� + uiujukPl� + uiujulPk� + uiukulPj� + ujukulPi�+ uiuju�Pkl + uiuku�Pjl + uiulu�Pjk + ujuku�Pil + ujulu�Pik + ukulu�Pij+ 1�u�PijPkl + 1�u�PikPjl + 1�u�PilPjk + 1�ulPijPk� + 1�ulPikPj�+ 1�ulPi�Pjk + 1�ukPijPl� + 1�ukPilPj� + 1�ukPi�Pjl + 1�ujPikPl�+ 1�ujPilPk� + 1�ujPi�Pkl + 1�uiPjkPl� + 1�uiPjlPk� + 1�uiPj�Pkl+ uiujQkl� + uiukQjl� + uiulQjk� + uiu�Qjkl + ujukQil�+ ujulQik� + uju�Qikl + ukulQij� + uku�Qijl + ulu�Qijk+ 1�PijQkl� + 1�PikQjl� + 1�PilQjk� + 1�Pi�Qjkl + 1�PjkQil�+ 1�PjlQik� + 1�Pj�Qikl + 1�PklQij� + 1�Pk�Qijl + 1�Pl�Qijk+uiKjkl� + ujKikl� + ukKijl� + ulKijk� + u�Kijkl�= �1� huiuj�Pkl � 13P���kl�+ uiuk�Pjl � 13P���jl�+ uiul�Pjk � 13P���jk�+ ujuk�Pij � 13P���ij�+ ujul�Pik � 13P���ik�+ ukul�Pij � 13P���ij�+ 1��PijPkl + PikPjl + PilPjk�� 19�P��P����ij�kl + �ik�jl + �il�jk�+ Pr uiQjkl + Pr ujQikl + Pr ukQijl + Pr ulQijk + Pr Kijkli: (8.5)The source terms were derived using the two-time scale BGK model as discussed inChapter III.Of interest in this work is the reduced one-dimensional form of the 35-momentsystem which is composed of only the transport equations that provide insight intothe structure of one-dimensional shocks. Contracting the above tensor form and

139simplifying, the transport equations can be expressed in vector form as@U35@t + @F35@x = S35c ; (8.6)where U35 is the solution vector of conserved quantities, F35 is the ux vector, andS35c is the source vector for the conserved form of the transport equations. Thesevectors are given asU35 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@��ux�u2x + PxxPnn�u3x + 3uxPxx +QxxxuxPnn +Qxnn�u4x + 6u2xPxx + 3�P 2xx + 4uxQxxx +Kxxxxu2xPnn + 1�PxxPnn + 2uxQxnn +Kxxnn3�P 2nn +Knnnn

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ; (8.7)

140F35 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@�ux�u2x + Pxx�u3x + 3uxPxx +QxxxuxPnn +Qxnn�u4x + 6u2xPxx + 3�P 2xx + 4uxQxxx +Kxxxxu2xPnn + 1�PxxPnn + 2uxQxnn +Kxxnn�u5x + 10u3xPxx + 15� uxP 2xx + 10u2xQxxx + 10� PxxQxxx + 5uxKxxxxu3xPnn + 3�uxPxxPnn + 3u2xQxnn + 1�PnnQxxx + 3�PxxQxnn + 3uxKxxnn3�uxP 2nn + 6�PnnQxnn + uxKnnnn

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ;(8.8)S35c = �1�

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@0023(Pxx � Pnn)13(Pnn � Pxx)2ux(Pxx � Pnn) + Pr Qxxx13ux(Pnn � Pxx) + Pr Qxnn13� (8Pxx + 4Pnn)(Pxx � Pnn) + 4u2x(Pxx � Pnn)+4ux Pr Qxxx + Pr Kxxxx19�(Pxx � 4Pnn)(Pnn � Pxx) + 3u2x(Pnn � Pxx)+2ux Pr Qxnn + Pr Kxxnn13�(Pxx + 5Pnn)(Pnn � Pxx) + Pr Knnnn

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA : (8.9)The full one-dimensional form of the conserved 35-moment transport equations ispresented in Appendix B. The conservative form of the Euler equations can berecovered in the limit of � ! 0.

141The ux Jacobian matrix for the 35-moment model, A35c = @F35=@U35, is givenbyA35c =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

0 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 0u5x �5u4x 10u3x 0 �10u2x 0 5ux 0 0�10Wxxu3x +30Wxxu2x �30Wxxux +10Wxx+10Xxxxu2x �20Xxxxux +10Xxxx+15W 2xxux �15W 2xx�5Yxxxxux +5Yxxxx�10XxxxWxx�Wnnu3x 3Wnnu2x �3Wnnux u3x Wnn �3u2x 0 3ux 0+3Xxnnu2x �6Xxnnux +3Xxnn �3Wxxux +3Wxx+3WnnWxxux �3WnnWxx +Xxxx�3Yxxnnux +3Yxxnn�WnnXxxx�3WxxXxnn3W 2nnux �3W 2nn 0 �6Wnnux 0 6Wnn 0 0 ux�Ynnnnux +Ynnnn +6Xxnn�6WnnXxnn

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA;

(8.10)where Wij = Pij=�, Xijk = Qijk=�, and Yijkl = Kijkl=�. In addition to A35c beingneeded in the derivation of a suitable set of Roe-averaged variables, which is to followin the next section, the Jacobian of the conserved variables U35 with respect to theprimitive variables V35 is also needed. This transformation matrix is denoted by

142M35 and is given asM35 =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1 0 0 0 0 0 0 0 0ux � 0 0 0 0 0 0 0u2x 2�ux 1 0 0 0 0 0 00 0 0 1 0 0 0 0 0u3x 3�u2x 3ux 0 1 0 0 0 0+3�Wxx0 �Wnn 0 ux 0 1 0 0 0u4x 4�u3x 6u2x 0 4ux 0 1 0 0�3W 2xx +12�Wxxux +6Wxx+4�Xxxx�WnnWxx 2�Wnnux Wnn u2x 0 2ux 0 1 0+2�Xxnn +Wxx�3W 2nn 0 0 6Wnn 0 0 0 0 1

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA:(8.11)

8.2 Roe-AverageIn this section a Roe-average will be obtained for the 35-moment transport equa-tions using only the assumed form approach as discussed in previous chapters. Asit does not appear possible to develop a parameter vector for this system. It shouldbe pointed out that for the Riemann solvers developed in this chapter using theGaussian and adiabatic eigenstructures, the systems are not hyperbolic in all param-eter space; therefore, Condition 3 of Property U is not satis�ed for all of parameterspace. The reader is referred to Chapter VII for the discussion of hyperbolicity ofthe 35-moment model. The assumed form of the Roe-averaged primitive variables

143for the full 35-moment system isui = �uiL + �uiR; (8.12)Wij = �WijL + !WijR + ��ui�uj; (8.13)Xijk = �XijkL + �XijkR + ��ui�Wjk + ��uj�Wik + ��uk�Wij+ ��ui�uj�uk; (8.14)Yijkl = �YijklL + �YijklR + ��ui�Xjkl + ��uj�Xikl + ��uk�Xijl + ��ul�Xijk+ ��ui�uj�Wkl + ��ui�uk�Wjl + ��ui�ul�Wjk + ��uj�uk�Wil+ ��uj�ul�Wik + ��uk�ul�Wij + ��ui�uj�uk�ul+ �Wij�Wkl + �Wik�Wjl + �Wil�Wjk; (8.15)where �, �, �, !, �, �, �, �, �, �, �, �, �, � , and are unknowns. The correctionsto Xijk are consistent with the average having the dimensions of velocity raised to apower of three, while the corrections to Yijkl are consistent with an average havingthe dimensions of velocity raised to a power of four. For the reduced 35-momentsystem only the non-trivial transport equations are retained; therefore, the above setof Roe-averaged quantities reduces toux = �uxL + �uxR; (8.16)Wxx = �WxxL + !WxxR + �(�ux)2; (8.17)Wnn = �WnnL + !WnnR; (8.18)Xxxx = �XxxxL + �XxxxR + 3��ux�Wxx + �(�ux)3; (8.19)Xxnn = �XxnnL + �XxnnR + ��ux�Wnn; (8.20)Yxxxx = �YxxxxL + �YxxxxR + 4��ux�Xxxx + 6�(�ux)2�Wxx + � (�ux)4+ 3 (�Wxx)2; (8.21)Yxxnn = �YxxnnL + �YxxnnR + 2��ux�Xxnn + �(�ux)2�Wnn

144+ �Wxx�Wnn; (8.22)Ynnnn = �YnnnnL + �YnnnnR + 3 (�Wnn)2: (8.23)For the 35-moment system, Condition 1 of Property U is given as�F35 = A35c �U35: (8.24)Equations (8.7), (8.8), and (8.10) are inserted into this vector equation and yieldsnine components which must be simultaneously satis�ed. The �rst six componentsof this vector equation are ���ux� = ���ux�; (8.25)�h��u2x +Wxx�i = �h��u2x +Wxx�i; (8.26)�h��u3x + 3uxWxx +Xxxx�i = �h��u3x + 3uxWxx +Xxxx�i; (8.27)�h��uxWnn +Xxnn�i = �h��uxWnn +Xxnn�i; (8.28)�h��u4x + 6u2xWxx + 3W 2xx + 4uxXxxx + Yxxxx�i= �h��u4x + 6u2xWxx + 3W 2xx + 4uxXxxx + Yxxxx�i; (8.29)�h��u2xWnn +WxxWnn + 2uxXxnn + Yxxnn�i= �h��u2xWnn +WxxWnn + 2uxXxnn + Yxxnn�i: (8.30)These six equations are automatically satis�ed regardless of the assumed form of theaveraging. The seventh component of the vector equation is not satis�ed. Recallthat the left and right states to the Riemann problem can vary independently and,

145therefore, the products of these states are also independent. The coe�cients of theseproducts must all vanish simultaneously. The solution of set of coe�cients in theseventh component provides the following ten relations:� = 1 � �; ! = 1� �; (8.31)� = 12 (�� + �2 � 2� + 2�� + 1 � �) �L�R[(1� �) �L + ��R] [��L + (1� �) �R]; (8.32)� = (1� �) �L(1 � �) �L + ��R ; � = ��R(1 � �) �L + ��R ; (8.33)� = ��L��L + (1� �) �R ; � = (1� �)�R��L + (1 � �)�R ; (8.34)� = � 13 [��L + (1 � �)�R] h���3� + 3�� + �3 + � � 3�2�+ 3��� � 3�� � ��� �L+ ��� + 3�2�+ �3� + 3�+ 3�� � 3�2� � 3��� + 3�� � 6��� 3��+ ��� �Ri ;(8.35)� = � 15 (��L + (1� �) �R) h��10�2� + 10�� � 15��2 + 10�3� � �5� �L+ ��10�� � 20�� � 10�2 + 10� + 5� � 10�3� � 30�� + 15��2 + 10�2�+10�3 + 10� � 1 � 5�4 � 15�2 + �5 + 30�2�� �Ri ; (8.36) = � 1��L + (1� �) �R h��� � ��2 � 2�� + 2�+ 2��� �L+ ���2 � �2 + 2��� �Ri : (8.37)With these relations substituted into the last two components of the vector equationprovided from Condition 1, no new information on the form of the averaging for the35-moment system is obtained. If Condition 1 was the only condition enforced in the

14635-moment system then there would still be �ve degrees of freedom (�; �; �; �; �) inspecifying a Roe-average set of variables and the average is not uniquely determined.Condition 4 also needs to be satis�ed to use the left eigenvectors obtained fromthe primitive form of the transport equations in the calculation of the wavestrengths.It can be expressed for the reduced 35-moment system as�U35 = M35�V35: (8.38)This is a vector equation with nine components. When Equations (8.7), (8.11), and(7.13) are inserted, the �rst component gives the identity �� = ��. This is obviouslyindependent of the form of the averaging. The solution of the second component leadsto the Roe-average for density � = p�L�R: (8.39)The third component gives two solutions for � with only one making sense when theleft and right states of density are the same:� = p�Rp�R +p�L : (8.40)Recall that in the 10-moment system there were also two solutions obtained for �.The solution of the �fth component of the vector equation, Equation (8.38) provides� and � � = p�Lp�R +p�L ; � = 13 p�L�R�p�L +p�R�2 : (8.41)Utilizing these two coe�cients in the remaining unsatis�ed components of the vec-tor equation leads to the sixth component not providing any new information inthe determination of a Roe-averaged state for the 35-moment system. The seventh

147component provides the solution for the last two unknown coe�cients� = 16p�L�R �p�L �p�R��p�L +p�R�3 ; � = 13 p�L�R�p�L +p�R�2 : (8.42)All the coe�cients are known and the last two components of the vector equationare satis�ed. By applying both Condition 1 of Property U and Condition 4, a uniqueset of Roe-averaged variables is obtained for the 35-moment system of transportequations which can be summarized as� = p�L�R; (8.43)ux = p�RuxR +p�LuxLp�R +p�L ; (8.44)Wxx = p�LWxxL +p�RWxxRp�L +p�R + 13 p�L�R�p�L +p�R�2 (�ux)2; (8.45)Wnn = p�LWnnL +p�RWnnRp�L +p�R ; (8.46)Xxxx = p�LXxxxL +p�RXxxxRp�L +p�R + p�L�R�p�L +p�R�2�ux�Wxx+ 16p�L�R �p�L �p�R��p�L +p�R�3 (�ux)3; (8.47)Xxnn = p�LXxnnL +p�RXxnnRp�L +p�R + 13 p�L�R�p�L +p�R�2�ux�Wnn; (8.48)Yxxxx = p�LYxxxxL +p�RYxxxxRp�L +p�R + 43 p�L�R�p�L +p�R�2�ux�Xxxx+ p�L�R �p�L �p�R��p�L +p�R�3 (�ux)2�Wxx + 245 �2�L � 7p�L�R + 2�R��p�L +p�R�4 (�ux)4+ p�L�R�p�L +p�R�2 (�Wxx)2; (8.49)

148Yxxnn = p�LYxxnnL +p�RYxxnnRp�L +p�R + 23 p�L�R�p�L +p�R�2�ux�Xxnn+ 16p�L�R �p�L �p�R��p�L +p�R�3 (�ux)2�Wnn; (8.50)Ynnnn = p�LYnnnnL +p�RYnnnnRp�L +p�R + p�L�R�p�L +p�R�2 (�Wnn)2: (8.51)With this unique set of Roe-averaged variables, Roe-type approximate Riemannsolvers can be developed utilizing the various eigenstructures.8.3 Roe-Type Approximate Riemann SolversThe interface ux F? for the reduced 35-moment system of transport equationsis given as F? = 12 (FL + FR)� 12 9Xk=1 �kj�kjrck; (8.52)where the summation takes place over the nine waves present in the 35-momentsystem. In the subsections to follow, three di�erent Roe-type approximate Riemannsolvers are presented. The �rst makes use of the eigenstructure obtained in theGaussian limit, the second solver uses the adiabatic eigenstructure, and the last andmost accurate, but also the most computationally expensive, solver does not use anyapproximations for the 35-moment eigenstructure.For a general system of hyperbolic equations the change in the ux F across allthe waves present in the system can be expressed as�F = Ac�U= Rc�Lc�U; (8.53)

149where Rc is the matrix composed of the right eigenvectors obtained from the con-servative form of the transport equations where the columns of the matrix are theindividual eigenvectors, Lc is the matrix composed of the left eigenvectors obtainedfrom the conservative form of the transport equations where the rows of the matrixare the individual eigenvectors, � is the diagonal matrix of all the eigenvalues; thatis � = 0BBBBBBBBBBBBBBB@ �1 �2 � � �n1CCCCCCCCCCCCCCCA ; (8.54)where n is the number of waves in the hyperbolic system, and � (�) = (�)R� (�)L. Inthe Roe-type approximate Riemann solvers to be presented in the following subsec-tions, where the Gaussian and adiabatic eigenstructures are used, the total ux jumpacross the waves must be preserved to ensure that the scheme remains conservative.The right eigenvectors Rc are used as the basis along with the eigenvalues � bothobtained from the either limit. A set of left eigenvectors denoted L0c, consisting of theleft eigenvectors obtained from the limit and a set of \amendatory" left eigenvectors,are determined such that the correct ux jumps are obtained. These ux jumps canbe expressed mathematically as �F = Rc�L0c�U: (8.55)Comparison with Equation (8.53) yieldsRc�L0c = Ac: (8.56)

150Solving this equation for L0c, L0c = ��1R�1c Ac: (8.57)The left eigenvectors needed to ensure conservation can be decomposed into the lefteigenvectors obtained from a limit Lc and an amendatory set of left eigenvectors Lca;that is, L0c = Lc + Lca. Therefore, the amendatory eigenvectors can be calculatedfrom Lca = ��1R�1c Ac � Lc; (8.58)where everything on the right-hand side is known. The reader should be remindedthat Rc, Lc, and � are the components of the eigenstructure obtained from thelimit while Ac is the non-equilibrium ux Jacobian. Using this set of amendatoryleft eigenvectors along with the eigenstructure obtained from a particular limit, thecorrect ux jump across the wave is ensured.In the 35-moment system, Lp is more sparse than Lc and, therefore, it is de-sired to use them in the calculation of the wavestrengths. The relation betweenleft eigenvectors of the two sets was given in Chapter IV and is repeated here forconvenience: L0c = L0pM�1: (8.59)Recall that M is the Jacobian matrix of the conserved variables U with respect tothe vector composed of the primitive variables V. Therefore,L0p = ��1R�1c AcM= Lp + Lpa; (8.60)where Lp is the left eigenvectors obtained from a particular limit and Lpa is theamendatory left eigenvectors which together will give the correct ux jumps to en-

151sure the desired conservation properties. The amendatory left eigenvectors can bedetermined from Lpa = ��1R�1c AcM �Lp: (8.61)As with Equation (8.58), Rc, Lc, and � are the components of the eigenstructureobtained from the limit while Ac and M are the non-equilibrium Jacobians.8.3.1 Gaussian eigenstructureAn eigensystem analysis of A35c evaluated in the Gaussian limit yields the follow-ing eigenvalues �1 = ux �q5 +p10 cxx; (8.62)�2 = ux �p3 cxx; (8.63)�3 = ux �q5�p10 cxx; (8.64)�4;5;6 = ux; (8.65)�7 = ux +q5�p10 cxx; (8.66)�8 = ux +p3 cxx; (8.67)�9 = ux +q5 +p10 cxx: (8.68)These eigenvalues, as expected, agree with those obtained in the Gaussian limiteigensystem analysis of A35p when evaluated at the Roe-averaged state as presented

152in Chapter VII. The Gaussian right eigenvector matrix satisfying A35c Rc = �Rc isRc =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1 0 1 1 0 0 1 0 1�1 0 �3 ux 0 0 �7 0 �9�21 0 �23 u2x 0 0 �27 0 �29c2nn c2xx c2nn c2nn c2xx 0 c2nn c2xx c2nn�31 0 �33 u3x 0 0 �37 0 �39c2nn�1 c2xx�2 c2nn�3 c2nnux c2xxux 0 c2nn�7 c2xx�8 c2nn�9�41 0 �43 u4x 0 0 �47 0 �49c2nn�21 c2xx�22 c2nn�23 c2nnu2x c2xxu2x 0 c2nn�27 c2xx�28 c2nn�293c4nn 6c2xxc2nn 3c4nn 3c4nn 6c2xxc2nn c4xx 3c4nn 6c2xxc2nn 3c4nn

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ;(8.69)where the individual right eigenvectors make up the columns of the matrix. Thewavestrengths � for the Gaussian approximate Riemann solver are calculated from� = �Lp + Lpa��V; (8.70)where the Gaussian left eigenvectors were given in the previous chapter and are tobe evaluated at the Roe-averaged state. The amendatory left eigenvectors for theGaussian eigenstructure as determined from Equation (8.61) arelpa1 = 52(�21 � 3)�c4xx�1 � � c2xxQxxx ; 25 ��1Qxxx + 12 �Kxxxx ; Qxxx ; 0 ; 0 ;0 ; 0 ; 0 ; 0 �; (8.71)lpa2 = 16�c4xx�2 � � c2nnQxxx � 3c2xxQxnn ; 2��2Qxnn + 3�Kxxnn ; 3Qxnn ; Qxxx ;0 ; 0 ; 0 ; 0 ; 0 �; (8.72)

153lpa3 = 52(�23 � 3)�c4xx�3 � � c2xxQxxx ; 25 ��3Qxxx + 12 �Kxxxx ; Qxxx ; 0 ; 0 ;0 ; 0 ; 0 ; 0 �; (8.73)lpa4 = 23�c4xx�4 � � c2xxQxxx ; 25 ��4Qxxx + 12 �Kxxxx ; Qxxx ; 0 ; 0 ; 0 ;0 ; 0 ; 0 �; (8.74)lpa5 = � 13�c4xx�5 � � c2nnQxxx � 3c2xxQxnn ; 2��5Qxnn + 3�Kxxnn ; 3Qxnn ; Qxxx ;0 ; 0 ; 0 ; 0 ; 0 �; (8.75)lpa6 = 6�c4xx�6 � �c2nnQxnn ; �Knnnn ; 0 ; Qxnn ; 0 ; 0 ; 0 ; 0 ; 0 �; (8.76)lpa7 = 52(�27 � 3)�c4xx�7 � � c2xxQxxx ; 25 ��7Qxxx + 12 �Kxxxx ; Qxxx ; 0 ; 0 ;0 ; 0 ; 0 ; 0 �; (8.77)lpa8 = 16�c2xx�8 � � c2nnQxxx � 3c2xxQxnn ; 2��8Qxnn + 3�Kxxnn ; 3Qxnn ; Qxxx ;0 ; 0 ; 0 ; 0 ; 0 �; (8.78)lpa9 = 52(�29 � 3)�c4xx�9 � � c2xxQxxx ; 25 ��9Qxxx + 12 �Kxxxx ; Qxxx ; 0 ; 0 ;0 ; 0 ; 0 ; 0 �: (8.79)Notice the wavespeeds in the denominators of the amendatory left eigenvectors;therefore, in a numerical calculation the individual amendatory eigenvectors are notdetermined, but instead, the quantity �Lpa is calculated. This avoids the problemassociated with vanishing wavespeeds.

154For the 35-moment system of transport equations the eigensystem analysis showedthat in the Gaussian limit there are four genuinely nonlinear acoustic waves. Toprevent expansion shocks an entropy �x must be utilized. Just like with the 10-momentmodel a smoothed value, j�kj�, is de�ned to replace j�kj for the four nonlinearacoustic waves, (k = 1; 3; 7; 9) [34,46]:j�kj� = 8>>>><>>>>: j�kj if j�kj � ��k2��k�2��k + ��k4 if j�kj < ��k2 (8.80)��k = max(0; 4(�kR � �kL)) (8.81)8.3.2 Adiabatic eigenstructureTo obtain the adiabatic eigenstructure A35c is analyzed at this limit. This yieldsthe following set of eigenvalues�1 = ux �vuuut5 +vuut10 + 5Kxxxx�c4xx cxx; (8.82)�2 = ux �p3 cxx; (8.83)�3 = ux �vuuut5�vuut10 + 5Kxxxx�c4xx cxx; (8.84)�4;5;6 = ux; (8.85)�7 = ux +vuuut5 �vuut10 + 5Kxxxx�c4xx cxx; (8.86)�8 = ux +p3 cxx; (8.87)�9 = ux +vuuut5 +vuut10 + 5Kxxxx�c4xx cxx: (8.88)These wavespeeds agree with those obtained from the adiabatic eigensystem anal-ysis of the coe�cient matrix of the primitive variable uxes when evaluated at the

155linearized state. The adiabatic right eigenvectors satisfying A35c Rc = �Rc can beexpressed in matrix form asRc =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1 0 1 1 0 0 1 0 1�1 0 �3 ux 0 0 �7 0 �9�21 0 �23 u2x 0 0 �27 0 �29r41 c2xx r43 r44 c2xx 0 r47 c2xx r49�31 0 �33 u3x 0 0 �37 0 �39r41�1 c2xx�2 r43�3 r44ux c2xxux 0 r47�7 c2xx�8 r49�9�41 0 �43 u4x 0 0 �47 0 �49r41�21 c2xx�22 r43�23 r44u2x c2xxu2x 0 r47�27 c2xx�28 r49�29r91 6c2xxc2nn r93 r94 6c2xxc2nn c4xx r97 6c2xxc2nn r99

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ;(8.89)where r4i = c2nn + 3 Kxxnn� �Z2i � 3c2xx� ; (8.90)r9i = 3c4nn + Knnnn�+ 18 c2nnKxxnn� �Z2i � 3c2xx� ; (8.91)where Zi = ux = �i and i = 1; 3; 4; 7; 9. As was done for the Gaussian Roe-type ap-proximate Riemann solver, the amendatory left eigenvectors need to be determined.By use of Equation (8.61) this set of eigenvectors islpa1 = 52(�21 � 3)�c4xx�1 � � c2xxQxxx ; 25 ��1Qxxx ; Qxxx ; 0 ; 0 ;0 ; 0 ; 0 ; 0 �; (8.92)

156lpa2 = 16�c4xx�2 � � c2nnQxxx � 3c2xxQxnn + 30 c2xxQxxxKxxxx6�c4xx + 5Kxxxx ;2��2Qxnn � 12 ��2QxxxKxxnn6�c4xx + 5Kxxxx ; 3Qxnn � 30 QxxxKxxnn6�c4xx + 5Kxxxx ;Qxxx ; 0 ; 0 ; 0 ; 0 ; 0 �; (8.93)lpa3 = 52(�23 � 3)�c4xx�3 � � c2xxQxxx ; 25 ��3Qxxx ; Qxxx ; 0 ; 0 ;0 ; 0 ; 0 ; 0 �; (8.94)lpa4 = � 23 ��c4xx�4 � Kxxnn�� � c2xxQxxx ; 25 ��4Qxxx ; Qxxx ; 0 ; 0 ; 0 ;0 ; 0 ; 0 �; (8.95)lpa5 = � 13�c4xx�5 � � c2nnQxxx � 3c2xxQxnn ; 2��5Qxnn ; 3Qxnn ; Qxxx ;0 ; 0 ; 0 ; 0 ; 0 �; (8.96)lpa6 = 6�c4xx�6 � �c2nnQxnn ; 0 ; 0 ; Qxnn ; 0 ; 0 ; 0 ; 0 ; 0 �; (8.97)lpa7 = 52(�27 � 3)�c4xx�7 � � c2xxQxxx ; 25 ��7Qxxx ; Qxxx ; 0 ; 0 ;0 ; 0 ; 0 ; 0 �; (8.98)lpa8 = 16�c4xx�8 � � c2nnQxxx � 3c2xxQxnn + 30 c2xxQxxxKxxxx6�c4xx + 5Kxxxx ;2��8Qxnn � 12 ��2QxxxKxxnn6�c4xx + 5Kxxxx ; 3Qxnn � 30 QxxxKxxnn6�c4xx + 5Kxxxx ;Qxxx ; 0 ; 0 ; 0 ; 0 ; 0 �; (8.99)

157lpa9 = 52(�29 � 3)�c4xx�9 � � c2xxQxxx ; 25 ��9Qxxx ; Qxxx ; 0 ; 0 ;0 ; 0 ; 0 ; 0 �: (8.100)Due to the presence of the wavespeeds in the denominator the quantity �Lpa iscalculated in a numerical solution.As in the Gaussian limit eigenstructure it was found in the previous chapter thatthe adiabatic limit acoustic wavespeeds, �1, �3, �7, and �9, are genuinely non-linearand can lead to the formation of shocks. These four wavespeeds are replaced usingthe same entropy �x as was described in the previous subsection.8.3.3 Non-equilibrium eigenstructureThe determination of the wavespeeds for the full 35-moment eigenstructure mustbe done numerically since there are no analytic expressions available for all the rootsof the non-equilibrium characteristic equation. The determination of the wavespeedswill be discussed in the next chapter. For this subsection it is assumed that thesewavespeeds have been determined. The right eigenvectors are expressed in terms ofthe wavespeeds. This set of eigenvectors satis�es A35c Rc = �Rc and is given asRc =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@1 0 1 1 0 0 1 0 1�1 0 �3 �4 0 0 �7 0 �9�21 0 �23 �24 0 0 �27 0 �29r41 c2xx r43 r44 c2xx 0 r47 c2xx r49�31 0 �33 �34 0 0 �37 0 �39r41�1 c2xx�2 r43�3 r44�4 c2xx�5 0 r47�7 c2xx�8 r49�9�41 0 �43 �44 0 0 �47 0 �49r41�21 c2xx�22 r43�23 r44�24 c2xx�25 0 r47�27 c2xx�28 r49�29r91 r92 r93 r94 r95 c4xx r97 r98 r99

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA ;(8.101)

158where r4i = c2nn + 3 ZiKxxnn � (Z2i � c2xx)Qxnn�Z3i � 3�c2xxZi + Qxxx ; (8.102)r9i = 3c4nn + Knnnn�+ 18(�c2nnZi � Qxnn)[ZiKxxnn � (Z2i � c2xx)Qxnn]�Zi(�Z3i � 3�c2xxZi + Qxxx) ; (8.103)r9j = 6c2xx c2nn � Qxnn�Zj ! ; (8.104)where i = 1; 3; 4; 7; 9 and j = 2; 5; 8. The left eigenvectors were given in the previouschapter and since there were no approximations made in the derivation of this eigen-structure, i.e., the eigenstructure holds for non-equilibrium states, corrections to theleft eigenvectors are not required to ensure the conservation property �F = Ac�U.

CHAPTER IXNUMERICAL ALGORITHMIn this chapter the solution scheme used in the numerical calculation of the mo-ment models for one-dimensional shock structures is presented. In Chapters VI andVIII four Roe-type approximate Riemann solvers were developed: One for the 10-moment system of transport equations and three for the 35-moment system. Thesesolvers are key ingredients in the solution technique. Through the use of Hancock'spredictor/corrector scheme, a second-order upwind-based method is obtained [66].Also presented in this chapter are the initial and boundary conditions used for allmodels. The gridding of the computational domain is discussed along with the de-termination of an appropriate time step.9.1 Solution SchemeLet the computational domain be de�ned as follows: The spatial dimension isdiscretized uniformly, i.e., xj = j�x where j = 0; 1; 2; : : : and xj+1=2 = (j + 1=2)�xis the interface between cells j and j+1, and let the temporal dimension be discretizedas tn = n�t where n = 0; 1; 2; : : : . At time level n and in the j-th cell the averagevalue of the conserved solution vector U can be approximated byUnj � 1�x Z j+1=2j�1=2 U(x; tn)dx: (9.1)159

160The integration is performed on each component of the solution vector. There is acondition on the size of the time step which will be discussed later in this chapter.Only the interior points are considered; the treatment of the boundaries will also bepresented later in this chapter.Recall that the conserved form of the governing equation for the moment modelsis @U@t + @F@x = Sc; (9.2)where F is the ux vector and Sc is the conservative form of the source vector. It isdesired to determine the solution vector at time level n+1 denoted by Un+1j . Whenadvancing the solution over a full time step a second-order scheme can be derived byevaluating the uxes and the source terms at a half time step, i.e.,Un+1j �Unj�t + Fn+1=2j+1=2 � Fn+1=2j�1=2�x = Scn+1=2j : (9.3)Rearranging, the solution vector of conserved quantities advanced to the next timestep can be expressed asUn+1j = Unj � �t�x �Fn+1=2j+1=2 �Fn+1=2j�1=2 �+�tScn+1=2j : (9.4)The problem reduces to the determination of the ux and source vectors at timelevel n+ 1=2.To obtain the ux and source terms at this time level we approximate the prim-itive variable solution vector V in cell j by a linear function ~V(x; t) using a Taylorseries expansion ~Vnj (x; t) = Vnj + (x� xj)Xnj + (t� tn)Tnj ; (9.5)whereVnj is constant in cell j, xj�1=2 < x < xj+1=2, and tn < t < tn+1. Xnj = �xVnj =�xand is the �rst derivative in space of the primitive variable solution vector while

161Tnj = �tVnj =�t and is the �rst derivative in time of the primitive variable solutionvector.The primitive variable form of the moment transport equations can be expressedas @V@t +Ap@V@x = Sp; (9.6)where Ap is the coe�cient matrix of the primitive variable uxes and Sp is theprimitive formulation of the source terms. The �rst derivative in time Tnj can beeliminated from Equation (9.5) by use of Equation (9.6) which when rearranged givesTnj = �(Ap)nj �xVnj�x + (Sp)nj : (9.7)Inserting this into Equation (9.5) yields~Vnj (x; t) = Vnj + (x� xj)�xVnj�x � (t� tn)(Ap)nj �xVnj�x + (t� tn)(Sp)nj : (9.8)Evaluating the above formula at the two interfaces of cell j and at a half timestep the following is obtained~Vnj (xj�1=2; tn+1=2) = Vnj � 12 �I+ �t�x(Ap)nj � �xVnj + 12�t(Sp)nj ; (9.9)~Vnj (xj+1=2; tn+1=2) = Vnj + 12 �I� �t�x(Ap)nj � �xVnj + 12�t(Sp)nj ; (9.10)where I is the identity matrix. At the j + 1=2 interface there are two values for theprimitive variables, one from the reconstruction in cell j and one from the recon-struction in cell j + 1. From these two predicted values we can calculate the uxthrough the j + 1=2 interface at time level n + 1=2Fn+1=2j+1=2 = F? h~Vnj (xj+1=2; tn+1=2); ~Vnj+1(xj+1=2; tn+1=2)i ; (9.11)

162where F? is the interface ux obtained from the solution of the Riemann problemvia an approximate solution technique. Likewise, for the j � 1=2 interface at timelevel n+ 1=2 Fn+1=2j�1=2 = F? h~Vnj�1(xj�1=2; tn+1=2); ~Vnj (xj�1=2; tn+1=2)i : (9.12)As has been discussed in the previous chapters, for the models used in this thesis weemploy Roe-type approximate Riemann solvers to obtain the uxes normal to thecell interfaces. This is the step where the physics becomes important. The procedureof advancing the solution vector by a half time step without any interactions betweenthe cells was devised by Hancock [66].The quantity �xVnj is limited using the Van Albada limiter [66]�xVnj = ave(Vj �Vj�1;Vj+1 �Vj)= ave(a; b)= (b2 + �2)a+ (a2 + �2)ba2 + b2 + 2�2 : (9.13)The bias, �2, prevents smooth extrema from being clipped and avoids the possibilityof division by zero. In all the computations to be presented in the next chapter weused �2 = 0:0025. This gave good results over the range of Mach numbers studied.The results are found to be not very sensitive to the precise value of the bias.The Superbee limiter [59],�xVnj = max(0;min(a; 2b);min(2a; b)); (9.14)was also tried and produced excellent results at the lower Mach numbers but at thehigher Mach numbers caused undesirable oscillations. These oscillations could becaused by the discontinuous slopes of the Superbee limiter; the Van Albada limiteris smooth. When near these abrupt changes in slope the limited values may oscillate

163between two values. Therefore, all results presented in this work used the Van Albadalimiter.When calculating a numerical solution of models that contain a large numberof transport equations a goal is to calculate as e�ciently as possible the interface ux. Recall that the interface ux can be calculated using the contribution of all thewaves. This formula was given in Chapter IV and is repeated belowF? = 12 (FL + FR)� 12 nXk=1 �kj�kjrck: (9.15)This equation was obtained by averaging the two formulas which determined theinterface ux by either using the ux from the left cell and adding in the contributionfrom the right-moving waves or using the right cell ux and adding in the contributionmade by the left-moving waves. For convenience these equations are repeated belowF? = FL + X�k<0 �k�krck; (9.16)F? = FR � X�k>0 �k�k rck: (9.17)For a positive-valued bulk ow velocity the 10-moment model will have, upstreamof the shock where the velocity is supersonic, at most, a single left-moving wave.While downstream of the shock, where the velocity is subsonic, the 10-moment modelwill have no more than three left-moving waves and for the reduced system of 10-moment transport equations this becomes just a single wave. Likewise, for the 35-moment model, upstream of the shock since the ow velocity is supersonic, therewill be no more than three left-moving waves while on the downstream subsonic sidethere will be, at most, three left-moving waves. Therefore, using Equation (9.16)will be less costly computationally than either of the two alternative equations. Thiswill reduce the computational expense of the various models. It should be remarked

164that for the Euler equations upstream of the shock there are no left-moving waveswhile downstream there is at most a single left-moving wave. When the solution isbeing advanced in a cell in which there are no left-moving waves the interface ux isjust the ux from the left cell with no contribution from any of the waves.9.2 Operator-Splitting (OPS) MethodTransport equations containing source terms are in practice often decoupled intotwo parts. In one part the homogeneous problem@U@t + @F@x = 0; (9.18)is solved. In the other part the damping e�ects due to the source terms are solved;that is, @U@t = Sc: (9.19)De�ne the solution to the homogeneous problem as L1 and the damping operationas L2. To obtain second-order accuracy one of the following sequence of operationsmust be used [64]: L1L2L2L1 or L2L1L1L2; (9.20)where each operation is performed using a half time step. The separation of the over-all solution operator into more than one simpler operation is referred to as Operator-Splitting (OPS). For the solution of the homogeneous part of the problem Hancock'sscheme as presented in the previous section is used but with all the source termcontributions eliminated.A desirable feature of this solution technique is that, for the moment models pre-sented in this work, Equation (9.19) can be integrated to �nd the exact contribution

165of the source terms. For the 10-moment model and for an arbitrary time step �t theintegration of Equation (9.19) using the source term vector S10c yields�(tn +�t; xj) = �; (9.21)ui(tn +�t; xj) = ui; (9.22)Pik(tn +�t; xj) = Pik exp�� t� �+ 13P�� �1� exp�� t� �� �ik; (9.23)where all the quantities on the right-hand side are evaluated in cell j and at timelevel n. For the reduced 35-moment model the integration of Equation (9.19) usingthe source term vector S35c yields�(tn +�t; xj) = �; (9.24)ux(tn +�t; xj) = ux; (9.25)Pxx(tn +�t; xj) = 13 (Pxx + 2Pnn) + 23 (Pxx � Pnn) exp�� t� � ; (9.26)Pnn(tn +�t; xj) = 13 (Pxx + 2Pnn)� 13 (Pxx � Pnn) exp�� t� � ; (9.27)Qxxx(tn +�t; xj) = Qxxx exp��Pr t� � ; (9.28)Qxnn(tn +�t; xj) = Qxnn exp��Pr t� � ; (9.29)Kxxxx(tn +�t; xj) = "Kxxxx � 4(Pxx � Pnn)23� (Pr � 2) # exp��Pr t� �+ 4(Pxx � Pnn)23� (Pr � 2) exp��2 t� � ; (9.30)Kxxnn(tn +�t; xj) = "Kxxxx + 2(Pxx � Pnn)29� (Pr � 2) # exp��Pr t� �� 2(Pxx � Pnn)29� (Pr � 2) exp��2 t� � ; (9.31)Knnnn(tn +�t; xj) = "Kxxxx � (Pxx � Pnn)23� (Pr � 2) # exp��Pr t� �

166+ (Pxx � Pnn)23� (Pr � 2) exp��2 t� � : (9.32)As with the 10-moment model all the quantities on the right-hand side are evaluatedin cell j and at time level n.In the next chapter, the results of using the operator-splitting solution procedurewill be compared to the coupled solution procedure.9.3 Initial and Boundary ConditionsThe steady-state solution of the governing equations is obtained by marchingforward in time from a set of initial conditions until the change in the solution isless than some prescribed tolerance. The initial conditions used in this thesis werethe Rankine-Hugoniot jump conditions for the Euler equations at a particular in owMach number with the discontinuity placed at x = 0. The Euler jump equationsfor a stationary shock are derived from the relation [F] = 0 where F is the Euler ux function as given in Equation (4.17) and [F] represents the jump in the quantityF across the discontinuity; that is, [F] = F2 � F1 where the subscripts 1 and 2designate the upstream and downstream states, respectively. These jump relationsfor a monatomic gas are �2�1 = 4M213 +M21 ; (9.33)ux2ux1 = 3 +M214M21 ; (9.34)uy2uy1 = 1; (9.35)uz2uz1 = 1; (9.36)p2p1 = 1 + 54 �M21 � 1� ; (9.37)

167whereM is the Mach number. Initially, upstream and downstream of the discontinu-ity the ow is in equilibrium; therefore, for the moment models all the perturbativemoment quantities are set equal to zero.A problem with the solution scheme as presented in the previous sections is thedetermination of the interface uxes at the upstream and downstream boundaries ofthe computational domain. In the present work, the problem was resolved throughthe use of ghost cells introduced beyond the upstream and downstream boundaries.The boundaries were placed far enough upstream and downstream such that it isassumed the ow is in an equilibrium state; therefore at these boundaries, there areno pressure anisotropies, the components of Q and K have vanished, and the sourceterms are negligible. The solution vector in the two ghost cells must be determined.To determine the solution vector in the ghost cells the Euler characteristic equationswere utilized which can be expressed as@W5@t +�5@W5@x = 0; (9.38)where the characteristic quantities for the Euler equation are@W5 = 0BBBBBBBBBBBBBBB@ @p� �a@ux@p� a2@�@uy@uz@p+ �a@ux1CCCCCCCCCCCCCCCA : (9.39)

1680

n+ 11 2�1 �2;3;4 �5nFigure 9.1: Wave behaviour at the supersonic upstream boundary of the Euler char-acteristic equations.The wavespeed matrix for the Euler equation is�5 = 0BBBBBBBBBBBBBBB@ ux � a 0 0 0 00 ux 0 0 00 0 ux 0 00 0 0 ux 00 0 0 0 ux + a 1CCCCCCCCCCCCCCCA : (9.40)At the upstream boundary all the Euler characteristics are incoming for super-sonic in ow, therefore, the primitive solution vector is completely speci�ed in theupstream ghost cell using the in ow conditions, see Figure 9.1, where cell 0 is theupstream ghost cell. At the downstream boundary there are four outgoing char-acteristics and one incoming characteristic since the ow has become subsonic, seeFigure 9.2, where cellm+1 is the downstream ghost cell. At this boundary the Eulercharacteristic equations are solved in an upwind manner for the outgoing character-istics while the single incoming characteristic equation is replaced with a physicalcondition. The physical condition used in this work is the speci�cation of the down-

169 n+ 1�5 nm� 1 m m+ 1�1 �2;3;4Figure 9.2: Wave behaviour at the subsonic downstream boundary of the Euler char-acteristic equations.stream hydrostatic pressure as predicted by Equation (9.37). Discretizing explicitlyEquation (9.38) in an upwind manner the following is obtainedWn+1m+1 =Wnm+1 � �t�x(�)nm �Wnm+1 �Wnm� : (9.41)This vector equation is solved only for the four outgoing characteristics; that is, the�rst component of the vector equation, which corresponds to the incoming charac-teristic, is not solved. An implicit treatment of the characteristic equations did notprovide signi�cant improvement in convergence. The solution of the four outgoingdiscretized characteristic equations combined with the physical condition completelyspeci�es the state of the ghost cell m+ 1.Use of the 10-moment system of characteristic equations as boundary conditionswas also attempted. For the 10-moment model Equation (9.38) takes the form@W@t +�@W@x = LpSp: (9.42)By looking at the 10-moment characteristic speeds derived earlier, Equations (5.19-5.23), it is evident that for low supersonic ows, where the incoming velocity is less

170than p3cxx, there is an outgoing characteristic at the upstream boundary. Thisoutgoing wave is the slow acoustic wave, and initially, we attempted to handle thisstraightforwardly by solving the 10-moment characteristic equation for this wave and�xing the other nine quantities but this caused the solution to become unstable whenthis left-moving wave reached the upstream boundary. At the downstream boundary,we tried to implement the 10-moment characteristic equations with and without thesource terms but no improvement in convergence or accuracy was gained over theEuler characteristic system. This could be attributed to nearly equilibriumconditionsat the boundaries. Therefore, to reduce computational costs the Euler system wasused exclusively for all boundary calculations. We also tried several other treatmentsfor the boundaries but none proved to be better than the aforementioned technique.In all the implementations of di�erent boundary conditions it was observed that,to stabilize the shock at a speci�c location, there must be speci�cation of physicalquantities. For the outgoing characteristics, if an extrapolation was used, or if anon-re ecting condition of the form@p@t � �a@ux@t = 0; (9.43)was used for the ingoing characteristic there was no guarantee of the desired solutionsince physical quantities were allowed to ` oat'. The non-re ecting boundary con-dition is equivalent to setting �1 = 0 in the Euler wavespeed matrix and includingthis component of the characteristic vector equation in the advancement of Equation(9.41). A relaxation parameter could be incorporated into the non-re ecting bound-ary condition to ensure that the correct downstream hydrostatic pressure is obtainedat steady-state conditions; Equation (9.43) then becomes@p@t � �a@ux@t + � (p � �p) = 0: (9.44)

171where � is a relaxation parameter and �p denotes the exact downstream value of thehydrostatic pressure. The optimum choice of the relaxation parameter varies withchanging Mach number. The optimum � provided slightly better convergence times;however, this approach can be outweighed by having to `�nd' the optimum � for aparticular Mach number and it was not utilized for any of the results to be presentedin the next chapter.The Euler characteristic equations provides an adequate set of boundary condi-tions for this particular shock structure problem, as long as the boundaries are farfrom the shock transition, and since there was no improvement using the 10-momentsystem of characteristic equations there was no attempt to solve the 35-momentsystem of characteristic equations. Recall that for the 35-moment model the lefteigenvectors are known only as functions of wavespeeds; therefore, any savings initeration count would be countered by an increase in computational costs for solv-ing the 35-moment system of characteristic equations at the boundaries. For thesereasons the Euler characteristic equations were used for all the models.9.4 Grids and Time-SteppingThe computational domain consisted of evenly spaced cells with the cell size anddomain size being determined by trying to have approximately half the cells residein the shock structure. Therefore, the boundaries were placed far enough upstreamand downstream such that the ow is in a near-equilibrium state. Since the shockstructure is more compact at higher Mach numbers the cell size, relative to the meanfree path, decreases. For the results to be presented in the next chapter 240 cellswere used for all the models.The steady-state solution of the transport models is achieved by marching forward

172in time. The systems of equations we are solving are hyperbolic and therefore, themaximumtime step per iteration is limited. The maximumtime step must not exceedthe time it takes the fastest wave present in the system to traverse a computationalcell. This can be expressed mathematically asjqjj�t�x � 1; (9.45)where qj is the fastest wavespeed in the j-th cell. This is known as the Courant-Friedrichs-Lewy (CFL) stability condition. The time step is then calculated from�t = CFL�xjqjj ; (9.46)where CFL is the CFL (or Courant) number and for stability must be less than unity.For the 10-moment model the fastest wave is one of the acoustic waves (which onedepends on the sign of the ow velocity), therefore,jqjj = �juxj+p3cxx�j ; (9.47)was used in Equation (9.46) to calculate an adequate time step for this model withthe CFL number set to some value less than one. While for the 35-moment modelevaluated in the Gaussian limit the fastest wave is one of the fast acoustic waves,therefore, jqjj = �juxj+q5 +p10cxx�j ; (9.48)was used with Equation (9.46) when solving the 35-moment system of transport equa-tions. The above wavespeed was used for all three 35-moment Roe-type approximateRiemann solvers developed in Chapter VIII. For the adiabatic limit eigenstructureand the non-equilibrium eigenstructure, Equation (9.48) in general, is not equal to

173the fastest wave present in the system. However, numerical calculations using Equa-tion (9.48) have shown this wavespeed to be an adequate choice in calculating a timestep for all the 35-moment approximate Riemann solvers.Time accuracy is not required to obtain the steady-state solution of the stationaryshock structure problem; therefore, to accelerate convergence, local time steps wereutilized. Local time-stepping is the process of allowing the solution to advance in timetowards steady-state based on a locally determined time step. Each computationalcell is advanced using the maximum allowable time step based on information in thatcell. The solution will progress to a steady-state solution at a di�erent pace in eachof the cells. A steady-state solution was considered to be obtained when the RMSnorm for the density dropped below 10�7.9.5 Navier-Stokes ModelThe Navier-Stokes equations are a well-documented and well-understood systemof transport equations. While the Euler equations, which were presented in theprevious chapters, are the model for inviscid and non-heat-conducting ows, theNavier-Stokes equations model the non-equilibrium terms by second-order derivativesof velocity and temperature.The one-dimensional Navier-Stokes equations in a conservative formulation canbe written as the Euler equations plus a ux vector composed of the non-equilibriumcomponents; i.e., @U@t + @F@x + @Fv@x = 0; (9.49)where U is the solution vector of conserved quantities, F is the inviscid ux vector,and Fv is the ux vector containing the contributions of the viscous stresses and heat

174 uxes. These three vectors are given asU = 0BBBBBBBBBBBBBBB@ ��ux�uy�uzE 1CCCCCCCCCCCCCCCA ;F = 0BBBBBBBBBBBBBBB@ �ux�u2x + p�uxuy�uxuz�uxh1CCCCCCCCCCCCCCCA ;Fv = 0BBBBBBBBBBBBBBB@ 0��xx��xy��xz�ux�xx � uy�xy � uz�xz + qx

1CCCCCCCCCCCCCCCA ;(9.50)where �ij are the components of the viscous stress tensor, E is the internal energy,h is the speci�c enthalpy, and qx is the heat ux in the x - direction. The internalenergy and speci�c enthalpy for a monatomic gas areE = 32p+ 12� �u2x + u2y + u2z� ; (9.51)h = 52 p� + 12 �u2x + u2y + u2z� : (9.52)Assuming Stoke's hypothesis is valid, the shear stress components are given by�xx = 43�@ux@x ; (9.53)�xy = �@uy@x ; (9.54)�xz = �@uz@x ; (9.55)where � is coe�cient of viscosity. The heat ux term follows from Fourier's law andis given by qx = ��@T@x ; (9.56)where � is the heat conductivity, and T is the temperature. The temperature canbe obtained from the perfect gas equation of state,p = �RT: (9.57)

175The solution scheme used for the Navier-Stokes equations is similar to that usedfor the moment models as was presented in Chapter IX. In the Navier-Stokes equa-tions there are no source terms; however, there is an additional ux vector whichmodel non-equilibrium e�ects present in the ow. When advancing the Navier-Stokesequations over a full time step both ux vectors are evaluated at a half time step:Un+1j �Unj�t + Fn+1=2j+1=2 � Fn+1=2j�1=2�x + Fvn+1=2j+1=2 � Fvn+1=2j�1=2�x = 0: (9.58)Solving the above equation for the time advanced solution the following is obtainedUn+1j = Unj � �t�x h�Fn+1=2j+1=2 + Fvn+1=2j+1=2�� �Fn+1=2j�1=2 + Fvn+1=2j�1=2�i : (9.59)For the Navier-Stokes equations the problem reduces to the determination of boththe inviscid and viscous interface uxes at time level n + 1=2. As for the momentmodels, the primitive variable solution vector V is approximated in the j-th cellusing a Taylor series expansion~Vnj (x; t) = Vnj + (x� xj)Xnj + (t� tn)Tnj : (9.60)The non-conservative form of the Navier-Stokes equations can be written as@V@t +Ap @V@x +M�1@Fv@x = 0; (9.61)where recall that Ap is the coe�cient matrix of the primitive variable uxes forthe Euler equations and M is the transformation matrix of conserved variables withrespect to primitive variables, i.e.,M = @U=@V. Using the non-conservative formu-lation of the Navier-Stokes equation the time derivative Tnj can be determined:Tnj = �(Ap)nj �xVnj�x � (M�1)nj �xFvnj�x ; (9.62)

176and then used to eliminate Tnj from the Taylor series expansion. This leads to thefollowing approximation for the primitive variable solution vector in cell j~Vnj (x; t) = Vnj + (x� xj)�xVnj�x � (t� tn)(Ap)nj �xVnj�x � (t� tn)(M�1)nj �xFvnj�x :(9.63)Evaluating this approximation at the two interfaces of cell j and at a half timestep the following is obtained~Vnj (xj�1=2; tn+1=2) = Vnj � 12 �I+ �t�x(Ap)nj � �xVnj � �t2�x(M�1)nj �xFvnj�x ; (9.64)~Vnj (xj+1=2; tn+1=2) = Vnj + 12 �I� �t�x(Ap)nj � �xVnj � �t2�x(M�1)nj �xFvnj�x ; (9.65)where I is the identity matrix. The quantity �xVnj is limited using the Van Albadalimiter as discussed in the previous chapter. The quantity �xFvnj is the change in theviscous uxes across cell j; that is�xFvnj = Fvnj+1=2 �Fvnj�1=2 (9.66)= F?v (Vj+1;Vj)� F?v (Vj ;Vj�1) ; (9.67)where the interface viscous uxes F?v are obtained using central di�erences. As anexample of the central di�erencing across a cell interface, the second component ofthe viscous ux at time level n isFv2j+1=2 = �(�xx)j+1=2 = �43�j+1=2 @ux@x !j+1=2= �43 ��j+1 + �j2 ��uxj+1 � uxj�x � : (9.68)At the cell interface j + 1=2 there are two predicted values for the primitivevariable solution vector at time level n+1=2, one from cell j and one from cell j+1.

177Using these two predicted values we can calculate both the inviscid and viscous uxesthrough the j + 1=2 interface at the half time step; therefore,Fn+1=2j+1=2 = F? h~Vnj (xj+1=2; tn+1=2); ~Vnj+1(xj+1=2; tn+1=2)i ; (9.69)Fvn+1=2j+1=2 = F?v h~Vnj (xj+1=2; tn+1=2); ~Vnj+1(xj+1=2; tn+1=2)i ; (9.70)and likewise for the j � 1=2 interfaceFn+1=2j�1=2 = F? h~Vnj�1(xj�1=2; tn+1=2); ~Vnj (xj�1=2; tn+1=2)i ; (9.71)Fvn+1=2j�1=2 = F?v h~Vnj�1(xj�1=2; tn+1=2); ~Vnj (xj�1=2; tn+1=2)i : (9.72)The inviscid ux normal to the interface is evaluated using Roe's approximate Rie-mann solver for the Euler equations as discussed in Chapter IV, whereas, the viscous uxes are evaluated using central di�erences as described above.The Navier-Stokes system is closed by providing the two constitutive relationsfor � and �. The coe�cient of viscosity �, as described in Chapter III, is related tothe temperature by � = CT (12+ 2a�1); (9.73)where C is a proportionality constant and a is a constant which characterizes themolecular interaction. A good value of a for argon is a = 10 [1]. The heat conduc-tivity �, must also be prescribed. This is done by using the de�nition of the Prandtlnumber, Pr = Cp�=�, where Cp is the constant pressure speci�c heat. This leads to� = � 1R�Pr : (9.74)For a monatomic gas = 5=3 and Pr = 2=3.

178The initial and boundary conditions used for the Navier-Stokes calculations wereidentical to those used for the moment models and were discussed in the previouschapter. For a particular in ow Mach number the same computational grid was usedfor the Navier-Stokes, 10-moment, and 35-moment calculations. A suitable time stepfor the solution of the Navier-Stokes model is based on�t = CFL �min24�xjaj + 2��a2 ; jaj�x + 2���x2!�135 ; (9.75)where a is the local speed of sound.9.6 Direct Simulation Monte Carlo ModelA numerical method used to model ows in the transition regime is the DirectSimulation Monte Carlo (DSMC) method which belongs to a class of techniquesreferred to as probabilistic simulation methods. This popular method was initiallyproposed by Bird in 1963 who �rst used it to solve the homogeneous gas relaxationproblem [4]. The �rst application of this method to a ow problem was for thesolution of shock structures as carried out by Bird [5] in 1964. Over the years theDSMC method has been developed and brought to a level where it has gained wideacceptance in the scienti�c community. The DSMC technique does not explicitlysolve the Boltzmann equation, instead the technique directly simulates it. With asu�ciently large number of particles, Bird has claimed that this is fully equivalentto solving the Boltzmann equation [7].This method simulates the gas as a collection of thousands or millions of test par-ticles, each representative of a large number of real molecules, and thereby, recognizesthe true molecular nature of a gas. The motion of each test particle is computedexactly while the collisions between the test particles are handled in a probabilisticmanner. This decoupling of particle movement and their interactions is an essential

179approximation of the DSMC technique. To achieve this decoupling, a necessary con-dition is that the global incremental ow time be small when compared to the meantime between collisions at any point in the ow. All the test particles are moved dis-tances appropriate for this time step, followed by the calculation of a representativeset of interparticle collisions again appropriate for this time interval.The determination of the macroscopic ow quantities such as density and pressureis done by periodically sampling the particles until a su�cient number of sampleshave been obtained; that is, the ow features are determined as the large time scaleaverages of many isolated observations. The DSMC method is also capable of pre-dicting higher-order velocity moments of the particle velocity distribution functionF such as skewness and kurtosis. The particle velocity distribution function can alsobe completely determined at all points in the shock structure; however, to obtain thismicroscopic information to a reasonable level of accuracy requires a much larger sam-ple size and more involved sampling. The results produced by DSMC calculationsof normal shock structures, including particle-velocity distribution functions, agreequite well with experimental data [55]. These favorable comparisons give validity tothe claim of Bird that, at least for the prediction of normal shock structure, DSMCsimulations realistically capture the true physics [6].The DSMC computation is started using some initial condition and the solutionevolves in time until a steady-state is reached at large times. The time steps canbe directly related to physical times. As long as the time step is small compared tothe mean collision time the results obtained are independent of its actual value. Themolecules are indexed to sub-cells as well as cells to ensure that all collisions occurbetween near neighbors. The determination of a representative set of collisions, basedon probabilistic selection, is based directly on the basic relations of classical kinetic

180theory. Therefore, DSMC techniques have the same limitations as are inherent inall of classical kinetic theory, including the Boltzmann equation. The two principalassumptions are that of molecular chaos and the requirement that the gas is dilute.The molecular chaos assumption means that particles which undergo a collision willhave many encounters with other particles before they collide again. In other words,the velocities of a pair of colliding particles are completely uncorrelated. The dilutegas assumption is used to limit the ow to binary collisions, and therefore, the DSMCmethod cannot be applied to the solution of problems involving dense gases.To maintain realistic physics the computational cells used in a DSMC simulationcannot become larger than a mean free path. When approaching the limit of thecontinuum regime, the mean free path decreases and the density increases whichresults in the need for increasingly smaller computational cells. A su�cient numberof test particles must be present in the computational cells such that the collisionsare adequately simulated and to keep the statistical noise low which lends reliabilityto the mean macroscopic quantities obtained. For accuracy a certain number ofparticles per cell is required; therefore, the DSMC method becomes increasinglymore expensive as the continuum limit is approached. The net result is that accuratesimulations in the transitional regimewith relatively large densities and collision ratesusing the DSMC method is presently prohibitive in terms of computational work anddata storage requirements especially when compared to its continuum counterparts.To obtain steady-state shock waves numerically using the DSMCmethodmoleculesare placed into a shock tube and a piston is driven into the equilibrium gas. A shockis established which propagates down the tube. The shock wave propagation veloc-ity is a function of the piston velocity. The DSMC results presented in this chapteremployed 27,000 test particles in 400 computational cells with each cell divided into

181six sub-cells. The calculations were run until the highest-order velocity moment ofinterest achieved a steady-state solution. The higher the order of the velocity mo-ment the larger the number of collisional events necessary for the moment quantityto reach steady-state. Also, the weaker the shock, the number of collisions thatare required to reach a steady-state solution increases. All the DSMC calculationswere performed using a code written by Bird [7] with slight modi�cation to obtainhigher-order velocity moments.9.7 Calculation of Wavespeeds/Determination of Hyperbol-icityAn essential part of calculating numerical solutions using the 35-moment modeland the non-equilibrium eigenstructure is the determination of the wavespeeds. Re-call in Chapters VII and VIII that the eigenvectors belonging to the non-equilibriumeigenstructure are expressed as analytic functions of the wavespeeds. This will con-siderably decrease the computational work involved in obtaining solutions of the 35-moment model using this approximate Riemann solver when compared to solving forthe non-equilibrium eigenvectors numerically. The determination of the wavespeedsutilizes the information that the lower-order polynomials which make up the non-equilibrium characteristic equation are proportional to derivatives of the higher-orderpolynomials. As will be shown later, the calculation of the wavespeeds also providesa check on the hyperbolicity of the solution. The hyperbolicity of the solution variesfrom cell to cell and if only one cell anywhere in the computational domain is in astate where the 35-moment eigenstructure is not hyperbolic then the solution breaksdown.Some of the details to be presented in this section have already been discussed

182in Chapter VII; however, for completeness sake and for uidity of reading, they willbe repeated. Recall that in Chapter VII an analysis of the coe�cient matrix of theprimitive variable uxes for the full one-dimensional system, where A35p is given inAppendix B, yielded the following characteristic equationdet �A35p � �I� = P 51P 42P 33P 24P5 = 0; (9.76)where � is an eigenvalue, I is the identity matrix, and P1, P2, P3, P4, and P5 are thefollowing factors P1 = Z; (9.77)P2 = Z2 � 1; (9.78)P3 = Z3 � 3Z +A; (9.79)P4 = Z4 � 6Z2 + 4AZ + 3�B; (9.80)P5 = Z5 � 10Z3 + 10AZ2 + 15Z � 5BZ � 10A; (9.81)where the polynomial factors have been non-dimensionalized such that Z = Z=cxx =(ux � �)=cxx, A = Qxxx=(�c3xx), and B = Kxxxx=(�c4xx). The wavespeeds of the fullone-dimensional 35-moment system correspond to the roots of the above character-istic equation. Note that there is multiplicity of the roots, hence there are only�fteen unique roots that must be solved for in order to obtain a complete set ofwavespeeds. Also notice that the lower-order polynomial factors are proportional tothe derivatives, with respect to Z, of the higher-order factors:Pk = k!n! @n�k@Zn�kPn; n > k: (9.82)For the one-dimensional shock structure problem which is of interest in this workthere are only nine non-trivial transport equations and therefore, the characteristic

183equation reduces to det �A35p � �I� = P1P3P5 = 0; (9.83)where A35p is the coe�cient matrix of the primitive variable uxes for the reducedsystem and is given in Equation (7.14). In this reduced system there are no repeatedroots and hence there are nine unique roots that must be solved for in Equation(9.83).The roots of the polynomials P1, P2, and P3 can be expressed analytically. Thesingle root of P1 is obviously Z = 0 which corresponds to a wavespeed of�1;1 = ux: (9.84)The notation �i;k is used where the �rst subscript i denotes the order of the particularpolynomial of interest Pi while the second subscript k denotes which root of thatpolynomial the wavespeed corresponds to. The above wavespeed is obviously realfor all ow situations. The two roots of P2 yield the following wavespeeds�2;1 = ux � cxx; (9.85)�2;2 = ux + cxx: (9.86)The ordering of the wavespeeds for a particular polynomial will be from the left-most-going wave to the right-most-going wave for a positive valued bulk ow velocity. Forall physically realistic values of cxx these two wavespeeds are real. The polynomialP3 has three roots which can be expressed analytically and they yield the followingwavespeeds �3;1 = ux � 2 cos "13 cos�1 �Qxxx2�c3xx!# cxx; (9.87)�3;2 = ux + 12 (ux � �3;1)�s3c2xx � 34(ux � �3;1)2; (9.88)�3;3 = ux + 12 (ux � �3;1) +s3c2xx � 34(ux � �3;1)2: (9.89)

184These three wavespeeds may not always be real. The condition on Qxxx to ensurereal-valued speeds is �����Qxxx�c3xx ����� � 2: (9.90)There are analytic expressions for the roots of an arbitrary quartic polynomial butthey are very complicated; therefore, in this thesis the roots are solved for numeri-cally. There are no known analytic expressions for the roots of a quintic polynomialand therefore, these roots must be solved for numerically.Figure 9.3 shows the behaviour of the �ve non-dimensionalized polynomial factorsfor A = 0 and B = 0. Notice in the �gure that the extremum of a particular�3 �2 �1 0 1 2 3�30�20�1001020

30Z

Pk P5P4P3P2P1Figure 9.3: The behaviour of the polynomial factors which make up the characteristicequation with A = 0 and B = 0.polynomial correspond to the roots of the next lower-order polynomial. This is to beexpected since Pk�1 is directly proportional to @Pk=@Z. The relationship betweenthe polynomials is made more clear in Figure 9.4 which highlights the polynomials P4

185and P5. It should be stressed that this relationship between the roots and extremum�3 �2 �1 0 1 2 3�30�20�1001020

30Z

Pk P5P4Figure 9.4: The behaviour of the polynomial factors P4 and P5 with A = 0 and B = 0highlighting that the roots of P4 correspond to the extremum of P5.of the polynomials is true even in the general non-equilibrium situation; i.e., forQxxx 6= 0 and Kxxxx 6= 0 and this will be fully exploited when devising an iterativesolution procedure for the roots of the higher-order polynomials P4 and P5.Equation (9.82) leads directly to a convenient technique to check the hyperbolicityof the solution. For the full non-equilibrium characteristic equation only P3, P4, andP5 need to be checked for real roots. Since the extrema of a particular polynomial Pkbracket the roots (if they are real) of that polynomial, the product of neighbouringextrema provides the necessary check on the type of roots that will be obtained.Normally, this would be a problem since one does not know the location of theseextrema, and therefore, one must solve for them. This procedure is no better thansolving for the roots themselves. However, Equation (9.82), states that the extrema

186of Pk are located at the roots of Pk�1 and these have already been found. For thefull non-equilibrium characteristic equation we know Z2;1 and Z2;2, following thesubscript notation introduced for the wavespeeds; therefore ifP3(Z2;1)P3(Z2;2) � 0; (9.91)then real roots will be obtained for P3. The roots of P2 are �1 and leads to thecondition becoming P3(�1)P3(1) � 0: (9.92)Once the roots of P3 are known, the condition to obtain real roots from P4 isP4(Z3;1)P4(Z3;2) � 0 and P4(Z3;2)P4(Z3;3) � 0: (9.93)Finally, when the roots of P4 are known the condition for real roots on P5 isP5(Z4;1)P5(Z4;2) � 0 and P5(Z4;2)P5(Z4;3) � 0 and P5(Z4;3)P5(Z4;4) � 0:(9.94)To �nd a root of a polynomial numerically involves an iterative procedure. Thenumerical technique used to solve for the roots of the polynomials P3, P4, and P5 isthe Newton-Raphson method [2, 56]. The procedure starts with some approximatetrial solution and then improves on the solution until some prede�ned convergencecriterion is satis�ed. For smoothly varying functions and provided a good initialguess, the Newton-Raphson method will always converge. All �ve of the polynomialfactors that compose the full non-equilibrium characteristic equation are smoothlyvarying functions.The key to an e�ective Newton-Raphson method is �nding a good initial guessfor the root. The real roots of a polynomial Pk are bracketed by the extrema of that

187polynomial. A root is bracketed in the interval (a; b), if Pk(a) and Pk(b) have oppositesigns. The intermediate value theorem states that if a function is continuous thenthe interval must contain at least one root. Since the extrema of Pk are the roots ofthe polynomial Pk�1 we can use this information in the initial guess assuming thatthe roots of Pk�1 have already been found. An initial guess Z ik;R for a root of thepolynomial Pk could be written asZ ik;R = Zk�1;R + �k;R; (9.95)where the subscript R represents one of the roots of that polynomial, Zk�1;R is apreviously determined root of the polynomial Pk�1, and �k;R is a yet to be determinedquantity. A general � can be determined as follows: Performing a Taylor seriesexpansion on a polynomial Pk using an arbitrary � yieldsPk(Z + �) = Pk(Z) + �@Pk(Z)@Z + 12�2@2Pk(Z)@Z2 +O(�3): (9.96)Utilizing Equation (9.82), this equation can be recast to eliminate all the derivativesof the polynomial Pk,Pk(Z + �) � Pk(Z) + � k!(k � 1)!Pk�1(Z) + 12�2 k!(k � 2)!Pk�2(Z): (9.97)Taking the independent variable of the expansion to be the initial guess, the followingis obtainedPk(Zk�1;R + �k;R) � Pk(Zk�1;R) + 12�2k;Rk(k � 1)Pk�2(Zk�1;R); (9.98)since Pk�1(Zk�1;R) = 0. We want Z ik;R to be a good guess for a root of Pk, thereforeset it equal to a root; that is, Z ik;R = Zk;R. This gives Pk(Zk�1;R + �k;R) = 0 and

188then solving for �k;R yields�k;R = vuut� 2k(k � 1) Pk(Zk�1;R)Pk�2(Zk�1;R) : (9.99)As an example of the above, consider the polynomial P3. As long as P3(�1) P3(1) � 0the Newton-Raphson method will �nd the three real roots Z3;1, Z3;2, and Z3;3 ifstarted at �(1+ �3;1), 0, (1+ �3;3) where �3;1 and �3;3 are de�ned above and the initialguess for the middle root is (Z2;1 + Z2;2)=2. Using Equation (9.99), �3;1 and �3;3 are�3;1 = s2 +A3 �3;3 = s2�A3 : (9.100)Therefore the initial guesses for the roots of the polynomial P3 areZ i3;1 = �1�s2 +A3 ; (9.101)Z i3;2 = 0; (9.102)Z i3;3 = 1 +s2�A3 : (9.103)Real quantities are obtained for �3;1 and �3;3 as long as �2 < A < 2 which is identicalto the relation obtained by requiring real roots of P3. This makes sense since whenthe polynomial Pk reaches the brink of yielding imaginary roots, the extremumwhichis about to cross the Pk = 0 axis and the roots on either side of this extremum forthe polynomial are at the same location; therefore, at this location Zk�1;R = Zk;R.Inserting this result into Equation (9.99) yields �k;R = 0.To take advantage of the fact that the polynomial factors are related we solve forthe roots of all the polynomial factors instead of just the three present in the reducedsystem. This is not as disadvantageous as might �rst appear. There are analyticexpressions for the roots of the polynomial factors P1, P2, and P3. Therefore, only theroots of P4 are numerically solved for when they are not needed in the approximateRiemann solver since the roots of P5 are needed in both the full and reduced systems.

189For the polynomial P4, Equation (9.99) yields�4;R =vuut� P4(Z3;R)6P2(Z3;R) : (9.104)Therefore the starting guesses for the roots of P4 areZ i4;1 = Z3;1 �vuut� P4(Z3;1)6P2(Z3;1) ; (9.105)Z i4;2 = 12 (Z3;1 + Z3;2) ; (9.106)Z i4;3 = 12 (Z3;2 + Z3;3) ; (9.107)Z i4;4 = Z3;3 +vuut� P4(Z3;3)6P2(Z3;3) : (9.108)For the quintic polynomial P5, the following is obtained from Equation (9.99)�5;R = vuut� P5(Z4;R)10P3(Z4;R) ; (9.109)which leads to the initial guesses beingZ i5;1 = Z4;1 �vuut� P5(Z4;1)10P3(Z4;1) ; (9.110)Z i5;2 = 12 (Z4;1 + Z4;2) ; (9.111)Z i5;3 = 12 (Z4;2 + Z4;3) ; (9.112)Z i5;4 = 12 (Z4;3 + Z4;4) ; (9.113)Z i5;5 = Z4;4 +vuut� P5(Z4;4)10P3(Z4;4) : (9.114)If a check on hyperbolicity is performed using the quantity �2k;R then even thoughfor the initial guesses only two of these values are needed when solving for the rootsof the polynomials all the values of �k;R must be checked; i.e.,�2k;R � 0; (9.115)

190for k = 4; 5 and R = 1; 2; : : : ; k + 1. This is the hyperbolicity check used in theresults to be presented in the next section. A rather stringent convergence criterionwas used, � < 1x10�14, where � is the change in the answer from consecutiveiterations. Using this criterion it took no more than eight iterations to �nd a root.

CHAPTER XRESULTSIn this chapter the numerical solution of the stationary one-dimensional shockstructure problem for a monatomic gas using various models will be presented. Themoment models used are the 10-moment and 35-moment systems of transport equa-tions that have been described and analyzed in the previous chapters. In additionto these models the solution of the shock problem as predicted by the Navier-Stokesequations and the direct simulation Monte Carlo (DSMC) method will also be pre-sented.10.1 Shock StructureIn this section the numerical results obtained from the various models applied tothe problem of one-dimensional shock structure of a monatomic gas will be presentedand discussed. The in ow Mach numbers considered were 1.2, 1.5, 2, and 10. Themodels used are the 10-moment and 35-moment systems of transport equations asdescribed in the previous chapters and, for comparison purposes, the Navier-Stokesand DSMC models. In the results to follow, it will be assumed that the DSMCsolutions are providing the correct shock structure. For all the numerical results tobe presented in this chapter only the reduced system of transport equations were191

192solved. Of course, the solutions obtained are independent of whether the full orreduced one-dimensional system is solved but, by solving just the reduced systemof 35-moment transport equations, we need to numerically solve only nine transportequations. For the comparison of computational e�ciencies the reduced system ofthe Navier-Stokes equations and the 10-moment equations are also solved.In the results to be presented in this chapter, the spatial dimension has beennon-dimensionalized with respect to the mean free path of a gas composed of hardelastic spheres, which is given as � = 16�5(2��p)1=2 : (10.1)The actual mean free path of argon will be somewhat di�erent but this relationevaluated at the upstream conditions provides a suitable reference length scale.The pro�les to be presented have been normalized, unless otherwise indicated, asfollows �Q = Q�Q1Q2 �Q1 ; (10.2)where Q is the quantity of interest and the subscripts 1 and 2 represent the upstreamand downstream values, respectively. For the velocity pro�les the normalization hasbeen reversed, i.e., the upstream and downstream reference values are switched. Thepro�les have been centered by setting the point in the pro�le where �� = 1=2 equalto the location x=� = 0.An issue which arose when using the non-equilibrium eigenstructure to calculatea 35-moment solution of the shock problem was that the initial transients wouldcause the solution to become non-hyperbolic even though the steady-state solutionmay be hyperbolic everywhere. We are only interested in the steady-state solution;therefore, a work-around had to be found. It was determined that the simplest

193method to handle this problem was to use the adiabatic eigenstructure to start thesolution and then, after some of the initial transients have left the computationaldomain, to switch over to the non-equilibrium eigenstructure. Even though there isa hyperbolicity condition for the adiabatic eigenstructure, the initial transients didnot cause it to lose hyperbolicity. It was found that if for the �rst 400 iterationsthe adiabatic eigenstructure was used for all in ow Mach numbers the transient non-hyperbolicity problem was avoided. The solution is insensitive to the exact numberof iterations in which the adiabatic eigenstructure is utilized.When attempting to obtain a 35-momentmodel solution using the non-equilibriumeigenstructure another issue that must be dealt with is evaluating the eigenvectorsin the near-equilibrium limit. Recall that some of the components in the non-equilibrium eigenvectors are unde�ned in this limit; therefore, a hybrid approachwas used. Before the eigenvectors are calculated the solution in each cell is checkedand if a certain criterion, based on whether the solution in the cell is in a near-equilibrium state, is met, then the adiabatic eigenstructure is used in place of thenon-equilibrium eigenstructure. If however, this criterion is not met then the non-equilibrium eigenstructure is utilized. This criterion does not need to be `tuned' fromrun to run and the solution is relatively insensitive to its exact value.It was found that the code developed to solve these systems of moment transportequations is very robust. There are no tunable parameters required for the rangeof Mach numbers that the codes are capable of solving. The above parameter usedto avoid troublesome transients and the hybrid criterion are used for the completerange of Mach numbers that the 35-moment model using the non-equilibrium eigen-structure can solve. The only other parameter that must be set was that used inthe Van Albada limiter and when a suitable value was found it was not necessary to

194modify it.A sti�ness parameter for the 10- and 35-moment transport equations can bede�ned as the ratio of the time step, �t, to the characteristic collision time, � . For thestudy of shock structure the spatial and temporal scales are su�ciently resolved suchthat this quantity was small (� 1) for all the Mach numbers considered, therefore`sti�ness' was not a problem and a large CFL number (= 0:9) was possible for bothmoment models. The largest sti�ness ratio encountered for the 10-moment modelwas 0.21 for the M1 = 1:2 case and decreased as the Mach number increased for theparticular grids used in this study. For an in ow Mach number of 10 the sti�nessratio was 0.11. For the 35-moment model the sti�ness for an in ow Mach number of1.2 was only 0.15 while for a Mach number of 2 this ratio decreased to 0.08.10.1.1 In ow Mach number = 1.2The �rst shock structure to be analyzed is that of a weak wave with an in owMach number of 1.2. The �rst issue to be dealt with is that of grid convergence,i.e., whether the computational grid is su�ciently re�ned so that the solution isindependent of the cell size. Figure 10.1 shows the solution of 10-moment model foran in ow Mach number of 1.2 using 30, 60, 120, 240, and 480 cells. It is obvious thatexcept for the 30 and 60 cell cases the solutions are nearly identical. A convergedsolution would appear to be achieved using 240 cells. Figure 10.2 illustrates the gridconvergence for the 35-moment model using the non-equilibrium eigenstructure atthe same Mach number and using the same set of grids. As was the case with the10-moment model the solutions using 120, 240, and 480 cells are converged. Thesetwo �gures provide con�dence in that if 240 cells are used for the calculations thenthe solutions obtained are independent of cell size.

195�50�40�30�20�10 0 10 20 30 40 50�0:20:00:20:4

0:60:81:01:2x=�

�� 30 cells60 cells120 cells240 cells480 cellsFigure 10.1: Grid convergence study for the 10-moment model. The in ow Machnumber was 1.2.A second important issue concerns convergence of the solution to the steady-state result. Figure 10.3 shows the convergence histories of the 10-moment modeland the three 35-moment models for an in ow Mach number of 1.2 and using 240cells. The logarithm of the L2 norm of density is plotted versus the iteration count.The convergence histories using the three di�erent eigenstructures appear to haveidentical behaviour. As will be seen in the shock-pro�le plots, the solutions usingthe three di�erent eigenstructures of the 35-moment model produce, at this Machnumber, nearly identical results.The density pro�les as predicted by the various models for this weak shock waveare presented in Figure 10.4. The notation used in the legends of the �gures for the35-moment model solutions is that the `G' appended to `35-m' denotes the solutionobtained using the Gaussian eigenstructure and the appended `A' represents the

196�50�40�30�20�10 0 10 20 30 40 50�0:20:00:20:4

0:60:81:01:2x=�

�� 30 cells60 cells120 cells240 cells480 cellsFigure 10.2: Grid convergence study for the 35-moment model using the non-equilibrium eigenstructure. The in ow Mach number was 1.2.

0 500 1000 1500 2000 2500 3000 3500�8�7�6�5�4�3�2�101

iterationslog k�k2 10-m35-m

Figure 10.3: Convergence history of the two moment models. The in ow Mach num-ber was 1.2 and 240 cells were used. The 35-moment model shows thesame convergence history for the three di�erent eigenstructures.

197�50�40�30�20�10 0 10 20 30 40 50�0:20:00:20:4

0:60:81:01:2x=�

�� N-S10-m35-m-G35-m-A35-mDSMCFigure 10.4: Density pro�les of the various models with an in ow Mach number of1.2.solution obtained using the adiabatic eigenstructure. Nothing appended to `35-m'represents the full or complete non-equilibrium eigenstructure solution. The CFLnumber used for the moment models was 0.9 while for the Navier-Stokes equationsthe CFL number had to be decreased to 0.7. The number of iterations and the run-time of the various models will be discussed in a later section. The DSMC code wasevolved for over 165 million collisional events.The density pro�les as predicted by the 35-moment model using the three di�er-ent eigenstructures appear to be identical, judging from the �gure. The 35-momentsolutions are in excellent agreement with the solution of the DSMC model. In addi-tion, the Navier-Stokes solution agrees very well with all three 35-moment solutions.This would be expected for such a weak shock wave. It appears that there is onlyminor disagreement on the downstream side of the shock where the Navier-Stokes

198solution is less di�usive than the other heat-transfer-capable models. Notice thatthe 10-moment model is not in agreement with any of the other solutions. This isnot surprising since the 10-moment model is de�cient in that it can not model heattransfer; therefore, the pro�le as predicted by this model is not as di�use as themodels which do incorporate heat transfer. It would be expected that as the Machnumber increases, and the non-equilibrium e�ects become more important, this dis-agreement will increase. Notice that the 10-moment pro�le appears to be quite steepon the upstream side of the shock.The three 35-moment eigenstructures appear to be producing identical solutions.In addition, the three solutions take an identical number of iterations to reach thesame convergence criterion as was evident in Figure 10.3, although a comparisonof the actual data does show slight di�erences amongst the solutions. The threeeigenstructures producing nearly identical solutions is very desirable, because as willbe shown in the section concerning the computational e�ciency of the models, the35-moment model using the Gaussian eigenstructure takes considerably less compu-tational e�ort per iteration than the other two 35-moment models. Likewise, theadiabatic eigenstructure is less expensive than the non-equilibrium eigenstructure.However, it will be shown that at higher Mach numbers all is not well with the35-moment model when using the Gaussian eigenstructure.For completeness, the velocity pro�les for this shock are presented in Figure 10.5,even though as would be expected from the steady-state continuity equation, noadditional insight is provided.There are �ve quantities based on second-order velocity moments that will bepresented in the next series of �gures. These are the two non-trivial components ofthe pressure tensor Pxx and Pnn, the hydrostatic pressure p, the temperature T , and

199�50�40�30�20�10 0 10 20 30 40 50�0:20:00:20:4

0:60:81:01:2x=�

�ux N-S10-m35-m-G35-m-A35-mDSMCFigure 10.5: Velocity pro�les of the various models with an in ow Mach number of1.2.the shear stress component �xx. The two components of the pressure tensor are shownin Figures 10.6 and 10.7. As was the case for the density and velocity pro�les, theagreement between the heat-transfer-capable models is very good. The 10-momentsolution, as expected, does not agree very well with the other models. The hydrostaticpressure and temperature are shown in Figures 10.8 and 10.9, respectively. Recallthat the hydrostatic pressure for the moment and DSMC models are calculated fromthe trace of the pressure tensor; i.e., p = P��=3. While the temperature is determinedusing the perfect gas equation of state, T = p=(�R). The 10-moment system oftransport equations is incapable of modeling heat transfer, this does not however,preclude it from having a temperature pro�le. The pro�les of the hydrostatic pressureand temperature as predicted by the various models are quite similar qualitativelyto those pro�les presented earlier. The �nal second-order quantity, the shear stress,

200�50�40�30�20�10 0 10 20 30 40 50�0:20:00:20:4

0:60:81:01:2x=�

�Pxx N-S10-m35-m-G35-m-A35-mDSMCFigure 10.6: Pxx pro�les of the various models with an in ow Mach number of 1.2.

�50�40�30�20�10 0 10 20 30 40 50�0:20:00:20:40:60:81:01:2

x=��Pnn N-S10-m35-m-G35-m-A35-mDSMC

Figure 10.7: Pnn pro�les of the various models with an in ow Mach number of 1.2.

201�50�40�30�20�10 0 10 20 30 40 50�0:20:00:20:4

0:60:81:01:2x=�

�p N-S10-m35-m-G35-m-A35-mDSMCFigure 10.8: Pressure pro�les of the various models with an in ow Mach number of1.2.

�50�40�30�20�10 0 10 20 30 40 50�0:20:00:20:40:60:81:01:2

x=��T N-S10-m35-m-G35-m-A35-mDSMC

Figure 10.9: Temperature pro�les of the various models with an in ow Mach numberof 1.2.

202is presented in Figure 10.10. The shear stress for the moment models is calculated�50�40�30�20�10 0 10 20 30 40 50�:050�:045�:040�:035�:030�:025�:020�:015�:010�:005:000:005

x=���xx N-S10-m35-m-G35-m-A35-mDSMCFigure 10.10: Shear stress pro�les of the various models with an in ow Mach numberof 1.2.from �xx = p�xx � Pxx = 23 (Pnn � Pxx) : (10.3)It should be pointed out the quantity ��xx does not represent a normalized shear stressbut instead is the shear stress non-dimensionalized by the hydrostatic pressure at thein ow boundary p1; i.e., ��xx = �xx=p1. The heat-transfer-capable models agree verywell with one another. The 10-moment model, however, over-predicts signi�cantlythe shear stress. At �rst this was quite surprising since the 10-moment model canmodel shear stresses; however, the shear stress as determined using Equation (10.3),is the di�erence between the two pressure tensor components. Figures 10.6 and 10.7show that both of these pro�les are quite steep through the shock and since the risein Pxx precedes the rise in Pnn the di�erence between the components is quite large.

203There are three quantities based on the third-order velocity moments which areof interest for the shock problem: the two non-trivial components of the heat tensorQxxx and Qxnn and the only non-zero component of the heat ux vector hx. Thepro�les of the two heat tensor components as predicted by the di�erent models areshown in Figures 10.11 and 10.12. The 10-moment systems is incapable of modeling�50�40�30�20�10 0 10 20 30 40 50�:05�:04�:03�:02�:01:00:01

x=��Qxxx 35-m-G35-m-A35-mDSMCFigure 10.11: Qxxx pro�les of the various models with an in ow Mach number of 1.2.these quantities while the Navier-Stokes does not model these terms explicitly, theycan be determined from Qxxx = 6=5hx and Qxnn = 2=5hx; however, these results arenot shown. The third-order velocity moments are non-dimensionalized as follows�Qijk = Qijk�c3xx : (10.4)The heat ux vector is non-dimensionalized in the same manner as the heat tensor.Note that these quantities are non-dimensionalized with respect to local variables asopposed to the upstream reference quantities.

204�50�40�30�20�10 0 10 20 30 40 50�:016�:014�:012�:010�:008�:006�:004�:002:000:002:004:006

x=��Qxnn 35-m-G35-m-A35-mDSMCFigure 10.12: Qxnn pro�les of the various models with an in ow Mach number of 1.2.The agreement between the solutions of the 35-moment model and the DSMCmodel is very good qualitatively for these two heat tensor components. The heattensor components as predicted by the DSMC model are quite oscillatory whencompared to the lower-order quantities obtained from the DSMC model. This isa feature of the DSMC method, which needs a longer sampling time to reach alevel of accuracy with the higher-order moments comparable to that obtained forthe lower-order moments. To obtain velocity moments the DSMC method takesmoments about the particle velocities. The particle velocity is composed of the bulkvelocity component plus the random velocity component. For moments which areformed using the particle velocities in the direction of the ow, such as ux, Pxx, Qxxx,and Kxxxx, the random velocity does not contribute much to the particle's velocity.However, for the transverse components whose moments are derived using transverse

205velocity components, the particle velocity is just the random velocity since there isno bulk ow in this direction. Therefore, one would expect that the DSMC model'sprediction of the transverse moments to be more oscillatory than the pro�les of the ow-direction moments. This is supported in Figure 10.12 which is the pro�le of theQxnn component of the heat tensor. The pro�le as predicted by the DSMC model isconsiderable more oscillatory than the pro�le of Qxxx even taking into account thefact that they are plotted on di�erent scales.The heat ux hx is shown in Figure 10.13. For the 35-moment and DSMC models�50�40�30�20�10 0 10 20 30 40 50�:04�:03�:02�:01:00:01

x=��hx N-S35-m-G35-m-A35-mDSMCFigure 10.13: Heat ux pro�les of the various models with an in ow Mach numberof 1.2.the heat transfer is calculated usinghx = 12Qx�� = 12 (Qxxx + 2Qxnn) : (10.5)The Navier-Stokes equations provides an explicit expression for this quantity and itis, therefore, included in the �gure. As has been the case with all the ow quantities

206predicted by the heat-transfer-capable models at this low in ow Mach number, theagreement is excellent. All the models capture the extremum of the heat ux vector.There are three fourth-order velocity moments of interest and they are the threenon-zero components of the fourth-order tensorK. In Figures 10.14, 10.15, and 10.16the pro�le of the components Kxxxx, Kxxnn, and Knnnn are shown, respectively. The�50�40�30�20�10 0 10 20 30 40 50�:03�:02�:01:00:01:02

:03x=�

�Kxxxx 35-m-G35-m-A35-mDSMCFigure 10.14: Kxxxx pro�les of the various models with an in ow Mach number of1.2.fourth-order velocity moments are non-dimensionalized as follows�Kijkl = Kijkl�c4xx : (10.6)The only models capable of predicting these quantities are the 35-moment modeland the DSMC model. With respect to the DSMC model solutions the pro�lesare increasingly oscillatory as more transverse velocity components are used in thecalculation of the moment quantity; i.e., the pro�le of Kxxxx is formed by taking themoment about the particle velocity in the ow direction to the fourth power and is

207�50�40�30�20�10 0 10 20 30 40 50�:008�:006�:004�:002:000:002:004:006:008

x=��Kxxnn 35-m-G35-m-A35-mDSMC

Figure 10.15: Kxxnn pro�les of the various models with an in ow Mach number of1.2.�50�40�30�20�10 0 10 20 30 40 50�:015�:010�:005:000:005:010:015

x=��Knnnn 35-m-G35-m-A35-mDSMC

Figure 10.16: Knnnn pro�les of the various models with an in ow Mach number of1.2.

208less oscillatory than the other two components. While the componentKxxnn is formedusing the transverse particle velocity squared and is therefore more oscillatory thanKxxxx but less oscillatory then Knnnn which is formed using the transverse particlevelocity to the fourth power. Also, Kxxxx is more oscillatory relative to Qxxx.In terms of the quality of the pro�le prediction by the 35-moment model we seethat the agreement is as good as could be expected considering the oscillatory natureof the DSMC solution. The main characteristics of the pro�le of Kxxxx as given bythe 35-moment model agree with that obtained by using the DSMC method; likewise,with the component Kxxnn. The component Knnnn as predicted by the 35-momentmodel is zero for most of the structure except for a small rise in the middle of theshock. This feature is lost in the noise of the DSMC solution.As mentioned earlier, the DSMC calculations were run for 165 million collisionalevents which is adequate for the lower-order quantities; however, for the higher-orderquantities this may be inadequate for such a weak shock wave. The oscillatory be-haviour of the DSMC solution may be reduced by allowing more collisional eventsbefore stopping the simulation. However, for such a low in ow Mach number witha relatively large density, this may not be su�cient to suppress the oscillatory be-haviour and one may have to resort to using more simulated particles in the com-putational cells. The DSMC solution took several days to obtain on an 80486 classcomputer which is orders of magnitude more time required than the moment models.This illustrates the computational excesses of the DSMC model for near-equilibriumsolutions.Another variable that can be analyzed for the various models is the relativespeci�c entropy s which is shown in Figure 10.17. The relative speci�c entropy forthe DSMC model can be calculated if the velocity distribution function is known

209�50�40�30�20�10 0 10 20 30 40 50�:002:000:002:004:006:008:010:012:014:016:018:020

x=��s N-S10-m35-m-G35-m-A35-m

Figure 10.17: Relative speci�c entropy pro�les of the various models with an in owMach number of 1.2.through the use of Equation (3.42). However, the determination of the distributionfunction for the DSMC solution to a reasonable level accuracy requires a much largersample size and more involved sampling and was not done for this thesis. The �guredoes show the entropy as predicted by the moment models and the Navier-Stokesmodel. For the Navier-Stokes system the entropy is the same as that for the Eulersystem. This equation was given in Equation (3.60) and is repeated belows5 = 3k2m ln p�5=3! : (10.7)For the �gure above the reference value for the entropy is taken to be the entropyevaluated at the in ow boundary and is non-dimensionalized with respect to speci�cheat at constant volume Cv. Therefore,�s = s� s1Cv : (10.8)

210The moment models use the de�nition of entropy as determined for the 10-momentsystem s10 = 3k2m ln0@�01=3�5=3 1A ; (10.9)where �0 = �3� and � = det� = det(P=�). Recall that it was not possibleto determine an expression for the entropy of the 35-moment system; however, to�rst-order it is identical to the entropy of the 10-moment model. The �gure showsexcellent agreement between the Navier-Stokes model and the 35-moment model.The 10-moment model entropy prediction is not even qualitatively similar to theother models. The peak is characteristic of the entropy behaviour inside a shockwith heat conduction. The decrease in entropy is not violating any fundamental law.The local entropy can decrease in a system as long as the total entropy for the systemincreases. There is a total increase in entropy for this shock wave.In Figure 10.18, the 35-moment and DSMC solutions are mapped onto the(Qxxx;Kxxxx)-phase plane along with the extent of the hyperbolicity region. Forthis weak shock the paths traced out by these two models are very near the vicinityof the Gaussian point and well within the region of hyperbolicity; therefore, it is notsurprising that the the 35-moment solutions as predicted using the near-equilibriumeigenstructures is nearly identical to that predicted using the non-equilibrium eigen-structure.In summary, for this weak shock wave all the heat-transfer-capable models haveexcellent agreement with the DSMC solution whenever the DSMC solution is rea-sonably resolved to permit comparison. In regard to the 35-moment model, usingany one of the three eigenstructures produces nearly identical solutions. This is verydesirable since the 35-moment model using the Gaussian eigenstructure is relatively

211�1:5 �1:0 �0:5 0 0:5 1:0 1:5�2�10123 GQxxx�c3xx

Kxxxx�c4xx 35-mDSMCFigure 10.18: The phase plots for in ow Mach number of 1:2.inexpensive when compared to the other two eigenstructures. However, to furtherevaluate the 35-moment model and the use of the various eigenstructures to predictshock structure a stronger shock must be analyzed.10.1.2 In ow Mach number = 1.5In this subsection the shock structure for an in ow Mach number of 1.5 is pre-sented. For the moment models a CFL number of 0.9 was possible, while for theNavier-Stokes model the CFL number was limited to 0.5. To obtain better predic-tion of the higher-order velocity moments from the DSMC method this case was runfor over 800 million collisional events with a run-time of over two weeks, comparedto the 35-moment model, which had run-times of less than a minute. The densitypro�les as predicted by the di�erent models are presented in Figure 10.19.For this in ow Mach number it was not possible to obtain a converged solution of

212�20 �15 �10 �5 0 5 10 15 20�0:20:00:20:4

0:60:81:01:20x=�

�� N-S10-m35-m-A35-mDSMCFigure 10.19: Density pro�les of the various models with an in ow Mach number of1.5.the 35-moment model using the Gaussian eigenstructure. The solution was oscilla-tory in time and the L2 norm of density was consistently several orders of magnitudeabove the convergence criterion as speci�ed in the previous chapter. The densitypro�le of the non-converged solution at a particular time is shown in Figure 10.20.The problem appears to be associated with the left-moving slow acoustic wave. Atthe location of the dip this wave has a sonic point. Evidently the Gaussian eigen-structure is producing a non-physical solution. The entropy �x for this system asdescribed in Chapter VIII is insu�cient to correct this problem. This is an area offurther research which may bene�t other future moment models and provide addi-tional understanding into these systems of transport equations.A signi�cant feature of these results is that the 10-moment solution density pro�lehas now steepened into a small discontinuity on the upstream side. In fact, such

213�20 �15 �10 �5 0 5 10 15 20�0:20:00:20:4

0:60:81:01:20x=�

�� 35-m-GFigure 10.20: Density pro�le of the 35-moment model using the Gaussian eigenstruc-ture with an in ow Mach number of 1.5. The solution is not converged.a shock wave will appear whenever the in ow velocity is supersonic relative to thefastest frozen wavespeed. The change occurs whenM1 = 3=p5 = 1:3416. Additionalcalculations (not shown) were done for the 10-moment model with Mach numbersof 1.34 and 1.35 and a jump is clearly observed in the 1.35 case while there is asmooth transition for the 1.34 case. Partially dispersed shock structures of thiskind are of course typical of dispersive systems. A `frozen shock' is followed by arelaxation region [70]. The height of this jump can be analytically determined fromthe 10-moment jump equations; i.e., �[F10] = F10s � F101 = 0 where the subscripts represents the frozen jump value. For a monatomic gas the normalized jump

214equations for the 10-moment model are��s = (5M21 � 9)(3 +M21 )3(9 + 5M21 )(M21 � 1) (10.10)�uxs = 5M21 + 315(M21 � 1) (10.11)�Pxxs = 2(5M21 � 9)15(M21 � 1) (10.12)�Pnns = 4(5M21 � 9)5(9 + 5M21 )(M21 � 1) : (10.13)For M1 = 1:5, Equation (10.10) this relation predicts �s � 0:16. From Figure 10.19this appears to be low but as the grid is re�ned the predicted numerical jump valueapproaches the analytic frozen jump value. Notice that Equation (10.10) predicts anon-physical density when the in ow Mach number drops below 3=p5 which is thethreshold value for the appearance of a discontinuity.Notice that the 35-moment model using the adiabatic and the non-equilibriumeigenstructures agrees fairly well with the DSMC calculations. The most obviousdisagreement is in a portion of the interior of the shock structure. It appears thatthere is a discontinuity embedded in the 35-moment solutions. It is interesting thatat this Mach number the 35-moment model is in much better agreement downstreamof the shock when compared to the DSMC solution than is the Navier-Stokes solutionwhile the reverse is true upstream of the shock.The �ve quantities obtained from second-order velocity moments, Pxx, Pnn, p, T ,and �xx are shown in Figures 10.21{10.25, respectively. From Figures 10.21 and10.22 it is apparent that the strength of the discontinuity in the 10-moment solutionis considerable stronger for the Pxx component than for Pnn. This is predicted bythe jump equations for the pressure components as given in Equations (10.12) and(10.13). The ratio of these two equations evaluated at a Mach number of 1:5 revealsthat the magnitude of the jump is over three times larger for the Pxx component of

215�20 �15 �10 �5 0 5 10 15 20�0:20:00:20:4

0:60:81:01:2x=�

�Pxx N-S10-m35-m-A35-mDSMCFigure 10.21: Pro�les of Pxx for the various models with an in ow Mach number of1.5.

�20 �15 �10 �5 0 5 10 15 20�0:20:00:20:40:60:81:01:2

x=��Pnn N-S10-m35-m-A35-mDSMC

Figure 10.22: Pro�les of Pnn for the various models with an in ow Mach number of1.5.

216the pressure tensor.The pro�les of the hydrostatic pressure and the temperature show, as with thepreviously described pro�les, that the agreement among the heat-transfer-capablemodels is relatively good.�20 �15 �10 �5 0 5 10 15 20�0:20:00:20:4

0:60:81:01:2x=�

�p N-S10-m35-m-A35-mDSMCFigure 10.23: Hydrostatic pressure pro�les of the various models with an in owMachnumber of 1.5.The component of the shear stress ��xx is shown in Figure 10.25. It is evidentfrom this �gure that the 35-moment model solutions contain a discontinuity. Also,the 10-moment model is predicting a minimum in the shear stress which is much toosmall and can be attributed to the large di�erence in the pressure components aswas discussed for the case of the M1 = 1:2 shock.Figures 10.26 and 10.27 show the behaviour of the two components of the heat ux tensor Qxxx and Qxnn, respectively. The solutions shown are only for the 35-moment models and the DSMC model. Notice that the solutions obtained by the

217�20 �15 �10 �5 0 5 10 15 20�0:20:00:20:4

0:60:81:01:2x=�

�T N-S10-m35-m-A35-mDSMCFigure 10.24: Temperature pro�les of the various models with an in ow Mach num-ber of 1.5.

�20 �15 �10 �5 0 5 10 15 20�:30�:25�:20�:15�:10�:05:00:05x=�

��xx N-S10-m35-m-A35-mDSMCFigure 10.25: Shear stress pro�les of the various models with an in ow Mach numberof 1.5.

218�20 �15 �10 �5 0 5 10 15 20�0:3�0:2�0:10:00:1

x=��Qxxx 35-m-A35-mDSMCFigure 10.26: Pro�les of Qxxx for the various models with an in ow Mach number of1.5.

�20 �15 �10 �5 0 5 10 15 20�:06�:05�:04�:03�:02�:01:00:01x=�

�Qxnn 35-m-A35-mDSMCFigure 10.27: Pro�les of Qxnn for the various models with an in ow Mach numberof 1.5.

21935-moment system and the DSMC model agree only qualitatively. The Qxxx pro�leas predicted by the 35-moment system has a minimum value about 25% smallerthan that predicted using DSMC techniques. The DSMC solution is much smootherthan in the M1 = 1:2 calculation because of the higher Mach number and the factthat the particles in the ow underwent many more collisions. The pro�les of thetransverse component of the heat ux are in better quantitative agreement. TheDSMC pro�le of this component is more oscillatory than that obtained for Qxxxbecause, as mentioned earlier, this quantity is formed partially from the transversevelocity components.The heat ux hx predicted by the models is shown in Figure 10.28. If the DSMC�20 �15 �10 �5 0 5 10 15 20�:30�:25�:20�:15�:10�:05:00:05

x=��hx N-S35-m-A35-mDSMCFigure 10.28: Heat ux pro�les of the various models with an in ow Mach numberof 1.5.solution is taken as being `correct' then neither of the other two heat-transfer-capablemodels is very good. However, the qualitative trend of the heat ux vector is captured

220by these two models.The fourth-order velocity momentsKxxxx,Kxxnn, and Knnnn are shown in Figures10.29, 10.30, and 10.31, respectively. As with the heat ux pro�les the solutions�20 �15 �10 �5 0 5 10 15 20�:2�0:10:00:10:2

0:30:4x=�

�Kxxxx 35-m-A35-mDSMCFigure 10.29: Pro�les of Kxxxx for the various models with an in ow Mach numberof 1.5.predicted by the 35-moment model and DSMC model agree qualitatively. For theKxxxx and Kxxnn pro�les the 35-moment model is over-predicting compared to theDSMC results; while for the Knnnn pro�les the 35-moment system under-predicts.The entropy pro�les for this in ow Mach number are shown in Figure 10.32. Aswas the case for the entropy pro�les at the lower in ow Mach number, the Navier-Stokes and 35-moment entropy pro�les agree qualitatively while the 10-moment en-tropy fails to capture the peak.The phase plot of hyperbolicity for this case is represented in Figure 10.33. Thetrajectory traced in phase space by the 35-moment model and the DSMC model is

221�20 �15 �10 �5 0 5 10 15 20�:02�:01:00:01:02

:03:04:05x=�

�Kxxnn 35-m-A35-mDSMCFigure 10.30: Pro�les of Kxxnn for the various models with an in ow Mach numberof 1.5.

�20 �15 �10 �5 0 5 10 15 20�:01:00:01:02:03:04

x=��Knnnn 35-m-A35-mDSMC

Figure 10.31: Pro�les of Knnnn for the various models with an in ow Mach numberof 1.5.

222�20 �15 �10 �5 0 5 10 15 20�:02:00:02:04

:06:08:10:12x=�

�s N-S10-m35-m-A35-mFigure 10.32: Entropy pro�les of the various models with an in ow Mach number of1.5.

�1:5 �1:0 �0:5 0 0:5 1:0 1:5�2�10123 GQxxx�c3xx

Kxxxx�c4xx 35-mDSMCFigure 10.33: The phase plots for in ow Mach number of 1:5.

223easily discernible for this higher Mach number. Notice that the path takes the 35-moment model solution away from the Gaussian point which is the near-equilibriumlimit.In summary, it is interesting that the 35-moment model using the Gaussian eigen-structure fails to produce a steady-state solution at this Mach number while theadiabatic eigenstructure returns a solution nearly identical to that obtained usingthe non-equilibrium eigenstructure. The 35-moment model is in very good agree-ment with the DSMC model at this Mach number when the comparison is betweenlower-order quantities but for the higher-order quantities the agreement is not asgood.10.1.3 In ow Mach number = 2.0The CFL number used in this case for the moment models was again 0.9 while forthe Navier-Stokes model the CFL number was 0.5. The DSMC method at this Machnumber needed to undergo only 39 million collisions before a steady-state solutionwas reached. Figure 10.34 shows the density pro�le prediction by the models. The35-moment model pro�le has an embedded discontinuity in the interior of the shock.With the 10-moment model the discontinuity is from the upstream equilibrium stateto some analytically determinable frozen jump value. The 35-moment model beginsto relax to the downstream value and then it appears to realize that it is not relaxingfast enough and therefore it jumps and then relaxes to the downstream equilibriumvalues. The jump condition for this model can not be determined by analyticalmeans. The pro�le predicted by the 35-moment model on the downstream side of theshock is very good in comparison with the DSMC model; however, on the upstreamside of the shock the agreement is not nearly as good. At this Mach number the

224�15 �10 �5 0 5 10 15�0:20:00:20:4

0:60:81:01:20x=�

�� N-S10-m35-m-A35-mDSMCFigure 10.34: Density pro�les of the various models with an in ow Mach number of2.0.discontinuity in the density pro�le of the 10-moment solution is considerably strongerthan in the M1 = 1:5 case. The 10-moment jump equation predicts a normalizeddensity jump of 0.30 which is in fairly good agreement with the pro�le in Figure10.34.The two components of the pressure tensor are shown in Figures 10.35 and 10.36.It is evident from these two �gures that the discontinuity present in the 10-momentmodel solution is much stronger for Pxx than for Pnn. The 10-moment jump equationsyield the following frozen jump values: Pxxs = 0:49 and Pnns = 0:10.The hydrostatic pressure pro�les are shown in Figure 10.37, the temperaturepro�le in Figure 10.38, and the shear stress pro�les in Figure 10.39. The 10-momentmodel predicts a temperature rise farther downstream than the other models. This isnot too surprising since the 10-moment model cannot model heat uxes. In the shear

225�15 �10 �5 0 5 10 15�0:20:00:20:4

0:60:81:01:20x=�

�Pxx N-S10-m35-m-A35-mDSMCFigure 10.35: Pxx pro�les of the various models with an in ow Mach number of 2.0.

�15 �10 �5 0 5 10 15�0:20:00:20:40:60:81:01:20

x=��Pnn N-S10-m35-m-A35-mDSMC

Figure 10.36: Pnn pro�les of the various models with an in ow Mach number of 2.0.

226�15 �10 �5 0 5 10 15�0:20:00:20:4

0:60:81:01:20x=�

�p N-S10-m35-m-A35-mDSMCFigure 10.37: Pressure pro�les of the various models with an in ow Mach number of2.0.

�15 �10 �5 0 5 10 15�0:20:00:20:40:60:81:01:20

x=��T N-S10-m35-m-A35-mDSMC

Figure 10.38: Temperature pro�les of the various models with an in ow Mach num-ber of 2.0.

227�15 �10 �5 0 5 10 15�1:2�1:0�0:8�0:6�0:4�0:20:00:2

x=���xx N-S10-m35-m-A35-mDSMCFigure 10.39: Shear stress pro�les of the various models with an in ow Mach numberof 2.0.stress �gure the 10-moment over-predicts the minimum value while the heat-capablemodels approximate the minimum value fairly well. The discontinuity present in the35-moment solution is quite obvious in the shear stress pro�le.The components of the heat tensor Qxxx and Qxnn are shown in Figures 10.40 and10.41, respectively. In both of these �gures, the 35-moment model over-predicts thepro�les but maintains at least qualitative agreement. The heat ux hx is shown inFigure 10.42. This is an important quantity which, for a higher-order moment modelto be given consideration, must be able to predict accurately. The 35-moment modeldoes not do well with the heat ux vector. It predicts a minimum that is severaltimes larger than that produced by the DSMC model. The accurate prediction ofphysical quantities is very important for a system of moment transport equations. Ifa moment model is to be `deemed' a good model it must be able to predict accurately

228�15 �10 �5 0 5 10 15�0:8�0:7�0:6�0:5�0:4�0:3�0:2�0:10:00:1

x=��Qxxx 35-m-A35-mDSMCFigure 10.40: Qxxx pro�les of the various models with an in ow Mach number of 2.0.

�15 �10 �5 0 5 10 15�:24�:20�:16�:12�:08�:04:00:04x=�

�Qxnn 35-m-A35-mDSMCFigure 10.41: Qxnn pro�les of the various models with an in ow Mach number of 2.0.

229�15 �10 �5 0 5 10 15�1:6�1:4�1:2�1:0�0:8�0:6�0:4�0:20:00:2

x=��hx N-S35-m-A35-mDSMCFigure 10.42: Heat ux pro�les of the various models with an in ow Mach numberof 2.0.the density, pressure, temperature, heat ux, etc. For the heat ux the Navier-Stokesequations predict a pro�le which, while still poor, is in much better agreement withthe DSMC solution than does the 35-moment model.The three fourth-order tensor components are shown in Figures 10.43, 10.44, and10.45. This series of �gures shows that while the 35-moment model is able toproduce qualitative agreement with the DSMC models, the quantitative agreementis not as good. The 35-moment model over-predicts the componentsKxxxx and Kxxnnwhile it under-predicts Knnnn. The entropy pro�les as given by the various solutionsare shown in Figure 10.46.The trajectories traced in phase space by the 35-moment and DSMC solutionsare shown in Figure 10.47. This �gure shows that the 35-moment model is very nearthe limits of hyperbolicity. In fact, the maximum in ow Mach number in which a

230�15 �10 �5 0 5 10 15�0:6�0:4�0:20:00:20:40:60:8

1:01:21:4x=�

�Kxxxx 35-m-A35-mDSMCFigure 10.43: Kxxxx pro�les of the various models with an in ow Mach number of2.0.

�15 �10 �5 0 5 10 15�:05:00:05:10:15:20:25

x=��Kxxnn 35-m-A35-mDSMC

Figure 10.44: Kxxnn pro�les of the various models with an in ow Mach number of2.0.

231�15 �10 �5 0 5 10 15�:05:00:05

:10:15:20x=�

�Knnnn 35-m-A35-mDSMCFigure 10.45: Knnnn pro�les of the various models with an in ow Mach number of2.0.

�15 �10 �5 0 5 10 15�:05:00:05:10:15:20:25:30:35:40:45

x=��s N-S10-m35-m-A35-m

Figure 10.46: Entropy pro�les of the various models with an in ow Mach number of2.0.

232�1:5 �1:0 �0:5 0 0:5 1:0 1:5�2�10123 Ga; db cQxxx�c3xx

Kxxxx�c4xx 35-mDSMCFigure 10.47: The phase plots for in ow Mach number of 2:0.hyperbolic steady-state solution of the 35-moment system can be obtained was foundto be approximately 2.03. Referring to Figure 7.6 which was presented in ChapterVII, it is obvious that the solution obtained by the 35-moment model for this in owMach number of 2.0 is very near the region where the right-moving slow acousticwave becomes concave.In order to gain additional insight in to the moment closure solutions Figures10.48 and 10.49 present the 35-moment axial distribution function F 35 given inEquation (7.1) at several locations in the shock structure with the transverse ran-dom velocity components set equal to zero. The lower-case letters in parenthesesin the �gures correspond to speci�c locations labeled on the phase-plane plot. Thelocation x=� = �0:2 is where the discontinuity present in the 35-moment solutionfor an in ow Mach number of 2.0 is situated. Referring to either �gure shows that atthis location the distribution function is very close to taking on a negative value. It

233�5 �4 �3 �2 �1 0 1 2 3 4 5�:05:00:05:10

:15:20:25cx

F 35�1=( a31) x=� = �14:8 (a)x=� = �5:0x=� = �3:7x=� = �2:5x=� = �1:2x=� = �0:2 (b)Figure 10.48: The axial distribution function of F 35 shown at several locations up-stream and including the discontinuity.

�5 �4 �3 �2 �1 0 1 2 3 4 5�:05:00:05:10:15:20:25

cxF 35�1=( a31) x=� = �0:2 (b)x=� = 0:0 (c)x=� = 1:3x=� = 2:5x=� = 3:8x=� = 15:0 (d)

Figure 10.49: The axial distribution function of F 35 shown at several locations down-stream and including the discontinuity.

234should be pointed out that away from the Gaussian limit there is always some regionin velocity space where F 35 < 0. This usually occurs for large values of c whereG is very small; hence, F 35 is a very small but negative quantity. At the locationx=� = 0:0, shown in Figure 10.49, which is only two cells downstream of the discon-tinuity, the distribution function has almost relaxed to the downstream equilibriumdistribution function.Recall that the maximumMach number at which a hyperbolic solution of the 35-moment model can be obtained is 2.03. If the in ow Mach number is incrementallyincreased from 2 it is obvious from results not shown that the minimum in thedistribution pro�le at x=� = �0:2 approaches the F 35 = 0 axis. This suggests adirect link between the loss of positivity in this region and the lose of hyperbolicityin the solution.The reduced 35-moment distribution function with the transverse random veloc-ity components set to zero isF 35 = G "1 + �Kxxxx8P 2xx + �Kxxnn2PxxPnn + �Knnnn3P 2nn � �Qxxx2P 2xx + �QxnnPxxPnn! cx� �2Kxxxx4P 3xx + �2Kxxnn2P 2xxPnn! c2x + �2Qxxx6P 3xx c3x + �3Kxxxx24P 4xx c4x# : (10.14)Scrutiny of the numerical results reveals that when jcxj is in the region of 2.0 allterms involving (cx)p are about the same order of magnitude for each power p. Thisseems to be a hint that the form of the expansion assumed is breaking down.Notice that at the discontinuity the distribution function is bimodal. This is notunusual. Previous studies of the behaviour of distribution functions, from experi-mental and theoretical results, show that for the problem of normal shock structureof an argon gas the distribution function takes on a more pronounced bimodal shapein the interior of the shock [21,39]. It could be reasoned that through the shock the

235distribution function is trying to obtain a more pronounced bimodal shape; however,the 35-moment distribution function is not capable of representing this stronger fea-ture and this leads to a breakdown of the solution. This was also pointed out byCheng when solving the 13-moment system of transport equations [16].In Chapter III there was a discussion on the e�ect that the di�erent velocitymoments have on the distribution function. These e�ects can be seen in the two�gures showing the 35-moment distribution function. Since the transverse random-velocity components have been set equal to zero only the moments Pxx, Qxxx, andKxxxx have a direct e�ect on the distribution function pro�les shown in Figures 10.48and 10.49. Recall that Pxx is related to the standard deviation and as it increases thewidth of the distribution function also increases. Upstream of the discontinuity thisis not evident; however, when Pxx jumps across the discontinuity, the distributionfunction becomes signi�cantly wider.The heat tensor component Qxxx is related to the skewness of the distributionand for a negative Qxxx, which is the behaviour of this component through the shockstructure, the distribution function is skewed to positive values of cx. Referring to thetwo �gures this is precisely the behaviour of F 35. As Qxxx becomes more negative,it is quite obvious from the �rst �gure, that F 35 is being skewed to the right andthen when Qxxx vanishes downstream of the shock the distribution function returnsto being centered about cx = 0.The componentKxxxx of the fourth-order tensor is directly related to the kurtosisof the distribution function. This is a measure of the area under the tails of thedistribution function. Upstream of the discontinuity, Figure 10.43 shows that thiscomponent for the 35-moment model becomes quite large. Figure 10.48 reveals that,indeed, the area under the tail on the negative side of the distribution function is

236increasing, but on the downstream side this component is quite small and does nothave a noticeable e�ect on F 35.Figure 10.50 shows the 10-moment model distribution function G given in Equa-tion (5.1), at several locations throughout the shock structure including the discon-tinuity denoted by x=� = �0:7. The e�ect of the pressure tensor component Pxx on�5 �4 �3 �2 �1 0 1 2 3 4 5�:05:00:05:10

:15:20:25cx

F 10�1=( a31) x=� = �14:6x=� = �0:7x=� = �0:4x=� = �0:1x=� = 0:3x=� = 15:3Figure 10.50: The distribution function G shown at several locations throughout theshock structure.the distribution function is quite obvious. As this component increases through theshock the distribution function is widening. In fact the jump in this component andits e�ect on the distribution function is apparent when looking at the pro�le of Gat x=� = �14:6 and x=� = �0:7. The shape of the distribution function does notchange from the downstream boundary until it reaches the discontinuity.In summary, at this higher Mach number, where the 35-moment model solutionis almost no longer hyperbolic and the distribution function is almost becoming

237negative for a signi�cant portion of velocity space at the discontinuity, the pro�les ofthe higher-order quantities are unsatisfactory when compared to the DSMC results.There is better agreement of the lower-order quantities with respect to the DSMCsolution.10.1.4 In ow Mach number = 10.0For an in ow Mach number of 10 the 35-momentmodel can not produce a solutiondue to loss of hyperbolicity; however, there is no such unreasonable restriction on anyof the other models. The CFL number used for the 10-moment solution was again 0.9while for the Navier-Stokes solution it could only be 0.25. It is quite interesting thatas the in ow Mach number increases the CFL number for the Navier-Stokes modelhas to be decreased while for the moment models a CFL number of 0.9 can be usedfor all Mach numbers. The DSMC calculation was run for 23 million collisions. Thedensity pro�les of the models for this Mach number are shown in Figure 10.51. It isobvious that neither the 10-moment nor the Navier-Stokes solutions agree with theDSMC model. The Navier-Stokes equations are well outside their range of validityat this Mach number.The height of the discontinuity present in the 10-moment solution with respectto the normalized density is about the same as in the Mach 2 10-moment results. Ifnormalized density jump equation, Equation (10.10) is evaluated for a Mach numberapproaching in�nity then the normalized-density frozen-jump value approaches 1=3.A series of solutions was obtained for a Mach number of 50 with a re�nement inthe grid and the numerically predicted jump value does indeed approach this limit.Another interesting observation is that the extent of the shock structure downstreamas predicted by the 10-moment model and the full Navier-Stokes solutions agree quite

238�15 �10 �5 0 5 10 15�0:20:00:20:4

0:60:81:01:2x=�

�� N-S10-mDSMCFigure 10.51: Density pro�les of the various models with an in ow Mach number of10.0.well. This would be expected since the temperature transition occurs upstreamof the density rise; therefore, on the downstream side of the density increase thetemperature gradient is quite small and the absence of heat transfer in the 10-momentmodel is of no importance.The two pressure tensor components Pxx and Pnn are shown in Figures 10.52 and10.53, respectively. In the �gure containing the Pxx pro�les the 10-moment frozenjump value is large and Equation (10.12), in the limit of large Mach number, yieldsa Pxx value of 2=3. While Equation (10.13) shows that as M1 approaches in�nity thefrozen jump value of Pnn vanishes. Figure 10.53 supports this result.The hydrostatic pressure and temperature for the three models are shown inFigures 10.54 and 10.55, respectively. Notice that for this strong shock the DSMCmethod predicts an overshoot of about one percent in the temperature. Figure 10.56

239�15 �10 �5 0 5 10 15�0:20:00:20:4

0:60:81:01:2x=�

�Pxx N-S10-mDSMCFigure 10.52: Pxx pro�les of the various models with an in ow Mach number of 10.0.

�15 �10 �5 0 5 10 15�0:20:00:20:40:60:81:01:2

x=��Pnn N-S10-mDSMC

Figure 10.53: Pnn pro�les of the various models with an in ow Mach number of 10.0.

240�15 �10 �5 0 5 10 15�0:20:00:20:4

0:60:81:01:2x=�

�p N-S10-mDSMCFigure 10.54: Pressure pro�les of the various models with an in ow Mach number of10.0.

�15 �10 �5 0 5 10 15�0:20:00:20:40:60:81:01:2

x=��T N-S10-mDSMC

Figure 10.55: Temperature pro�les of the various models with an in ow Mach num-ber of 10.0.

241shows the shear stress pro�les. For this relatively large in ow Mach number, the�15 �10 �5 0 5 10 15�60�50�40�30�20�10010

x=���xx N-S10-mDSMCFigure 10.56: Shear stress pro�les of the various models with an in ow Mach numberof 10.0.magnitude of the shear stress greatly increased with the 10-moment model over-predicting the minimum value by at least a factor of two.In summary, the 35-moment model is incapable of solving the shock structureproblem above an in ow Mach number of about 2.03. The other models can predictshock structure for much larger Mach numbers; however, they do not reach goodagreement with the DSMC solutions.10.2 E�ciencyAlthough the shock structure computations met with only limited success it ispossible that the moment models will �nd more suitable application in other ar-eas. Therefore it is still of interest to examine the e�ciency of the various meth-ods. It should be pointed out that the following comparisons are valid only for

242one-dimensional ows. All of the Navier-Stokes and moment closure calculationspresented in this thesis were obtained using an HP 735/125 workstation. As a mea-sure of the e�ciency of the proposed method used to calculate the 35-moment so-lutions using the analytic non-equilibrium eigenstructure expressed in terms of thewavespeed, a comparison was made to calculating this eigenstructure numericallyusing standard eigensystem routines from the EISPACK library. The package re-turns the eigenvalues and eigenvectors for a given matrix which for the 35-momentmodel is the 9x9 ux Jacobian matrix. Then either the left eigenvectors obtainedfrom the conservative form of the governing equations along with the conserved so-lution vector can be used to determine the wavestrengths or these left eigenvectorscan be transformed to the left eigenvectors obtained from the primitive form of thegoverning equations and then along with the primitive solution vector used to cal-culate the wavestrengths. Either method of calculating the wavestrengths does notsigni�cantly change the times. The expensive part of the EISPACK routine is the nu-merical calculation of the eigenvectors from the eigenvalues. For these calculations,the numerical eigenstructure used the information that there are fewer left-movingwaves to calculate the interface ux as e�ciently as possible. The solutions obtainedusing the numerical non-equilibrium eigenstructure were identical to the solutionsobtained using the analytic non-equilibrium eigenstructure within machine accuracy.Table 10.1 summarizes the number of iterations the various models needed toachieve a steady-state solution. The entries that contain **** refer to caseswhere a solution could not be obtained. It is quite interesting that for the Navier-Stokes calculations as the Mach number increases the number of iterations requiredincreases signi�cantly. In comparison, the moment models require approximatelythe same number of iterations independent of the Mach number. This appears to

243M1 N-S 10-moment 35-moment 35-moment 35-moment 35-momentGaussian Adiabatic Analytic Numerical1.1 4792 2269 4441 4441 4441 44411.2 5357 2272 3190 3190 3190 31901.3 7505 2202 2985 2985 2985 29851.5 8008 2066 **** 2587 2548 25481.75 8960 1964 2928 2745 2742 27422 8164 1911 **** 2757 2755 27554 10458 2013 **** **** **** ****10 29059 2085 **** **** **** ****Table 10.1: The number of iterations needed to reach convergence for the di�erentmodels.be a remarkable and highly desirable feature of the moment models. The sameconvergence criterion was used for all models and for all in ow Mach numbers.The cell cost is de�ned as the CPU time it takes for a particular solver to advancethe solution in one cell for a single time step and for the various models is given in Ta-ble 10.2. It should be pointed out that these cell costs vary with the Mach numbersince the number of left-moving waves is dependent on the Mach number. One wouldexpect the cost to decrease as the Mach number increases since the number of left-moving waves decreases. It is quite interesting that this appears not to be the case. Apossible explanation is that the cost of calculating the interface ux is much smallerthan the most expensive part of the calculation which is the reconstruction stage.Therefore, one would not expect to see noticeable savings due to the decrease in thenumber of left-moving waves. The slight variations in the cost for the Navier-Stokes,10-moment, 35-moment using the Gaussian eigenstructure, and the 35-moment using

244M1 N-S 10-moment 35-moment 35-moment 35-moment 35-momentGaussian Adiabatic Analytic Numerical1.1 1.6 1.8 3.6 4.4 6.0 17.61.2 1.6 1.7 3.6 4.3 7.0 24.51.3 1.6 1.8 3.8 4.4 8.0 33.11.5 1.6 1.6 *** 4.2 9.0 40.01.75 1.6 1.6 3.6 4.2 9.2 43.62 1.6 1.7 *** 4.4 8.1 39.64 2.1 1.9 *** *** *** ****10 2.2 1.9 *** *** *** ****Table 10.2: The cell cost (10�5 seconds/cell/iteration) of the various models.the adiabatic eigenstructure systems can be attributed to the inaccuracies inherentin the technique that workstations use to measure CPU times. The variation appearsto be larger for the 35-moment model using the analytic non-equilibrium eigenstruc-ture. This could be attributed to the increased number of iterations needed to �nda wavespeed when the solution is in a more non-equilibrium state. The much largervariation in the numerically obtained non-equilibrium eigenstructure could be dueto the increased time it takes for the canned program to return the eigensystem asthe Mach number increases.The 10-moment calculations were only slightly more computationally intensivethan the Navier-Stokes solutions and for the high Mach number cases the 10-momentmodel was actually cheaper. If the code did not take advantage of the fact that thenumber of left-moving waves were far fewer than the number of right-moving wavesin the calculation of the interface ux then the 10-moment model becomes approxi-mately 2:4 times more computationally intensive than the Navier-Stokes system.

245Of the four di�erent techniques to solve the 35-moment system, the one whichutilizes the Gaussian eigenstructure is the cheapest. However, in its present form, itis incapable of solving the shock structure problem for the full range of Mach num-bers. It may be viable option in other applications. Calculating the non-equilibriumeigenstructure using the analytic expressions for the eigenvectors can be up to twiceas expensive as using the adiabatic eigenstructure. This is very important since therewas virtually no di�erence between the solutions obtained using these two eigenstruc-tures. The numerically generated non-equilibrium eigenstructure is very expensivewhen compared to any of the other techniques. It is three to �ve times more expen-sive than the analytical non-equilibrium eigenstructure and it is four to ten timesmore expensive than using the adiabatic eigenstructure. So it appears that the adi-abatic eigenstructure should be the method of choice when solving the 35-momentmodel. In comparison to the Navier-Stokes model the adiabatic eigenstructure isapproximately two and a half times as expensive.The overall CPU time needed by the various models to attain a steady-state isanother insightful measure of the e�ciency of a model. Table 10.3 shows the totalCPU time for the di�erent models with respect to the various Mach numbers. Atthe low Mach numbers the adiabatic eigenstructure model needs about three timesthe CPU time as the Navier-Stokes equations. However, at the higher Mach numbersthe 35-moment adiabatic eigenstructure is slightly cheaper than the Navier-Stokesequations which is due to the fact that the moment models require approximatelythe same number of iterations for all Mach number. The DSMC is very expensivecompared to all the above models. The run-time of a DSMC calculation varied from1x105 seconds to 1x106 seconds. This is several orders of magnitude larger thanany of the other models and illustrates the computational bene�t that higher-order

246M1 N-S 10-moment 35-moment 35-moment 35-moment 35-momentGaussian Adiabatic Analytic Numerical1.1 18.1 9.7 38.5 46.4 64.2 187.11.2 20.1 9.3 27.6 33.2 53.5 187.71.3 29.1 9.5 27.4 31.4 57.2 236.91.5 30.3 8.1 **** 26.3 55.2 244.61.75 33.4 7.7 25.6 27.9 60.8 286.62 30.7 7.6 **** 29.4 53.7 261.84 52.3 9.1 **** **** **** *****10 151.5 9.6 **** **** **** *****Table 10.3: The total CPU cost (in seconds) for the di�erent models.moment models provide.The convergence criterion used in the analytic non-equilibrium eigenstructurewas � < 1x10�14, where � is the change in the answer from consecutive iterations.This is a rather stringent criterion; however, it took no more than eight iterationsto �nd a root. If this is loosened to � < 1x10�4 the maximum number of iterationswas reduced to four. The total CPU time decreased by approximately 20% whileproducing nearly identical solutions.The above tables provide evidence, even though the 35-moment model is not theanswer to solving the shock structure problem, that higher-order moment modelsneed not necessarily be ruled out on the grounds of computational expense dueto an increase in the number of transport equations. If an accurate higher-ordermoment model can be found the savings could be tremendous compared to theDSMC technique which is the only accurate model for the full range of transitional

247 ows. As mentioned at the beginning of this section these comparisons are for one-dimensional ows. In general, for three-dimensional problems the full 35-momentsystem of transport equations must be utilized. Therefore, when compared to three-dimensional Navier-Stokes calculations which add only two more transport equations,the increase in CPU resources needed will be much greater for the 35-moment model.

CHAPTER XICONCLUSIONS11.1 SummaryTwo sets of transport equations based on the 10- and 35-moment models havebeen presented. The �rst system, the 10-moment model, incorporates the viscousstress terms while remaining hyperbolic for all physically realistic values of the hy-drodynamic quantities. However, the 10-moment transport equations make no rep-resentation of heat transfer and, therefore, are not a suitable model for transitional ows. The second system, the 35-moment model, is based on an expansion about theGaussian distribution function and is capable of modelling heat transfer; however,this model is strictly hyperbolic only in a �nite region of phase space. The hyperbol-icity of the sets of transports equations allows the use of powerful numerical schemeswhich take advantage of the wave-like nature of the physics. Dispersion analysesof the 10- and 35-moment systems have revealed that, in the collision-dominatedlimit, both models recover the usual Euler wave modes. Furthermore, the 10- and35-moment analyses of wave structures has provided new insight into the behaviourof non-equilibrium dilute gases.Once the physics present in these systems were well understood the developmentof e�cient numerical schemes was undertaken. It was decided that these systems248

249lend themselves well to using a Roe-type approximate Riemann solver. The �rststep in developing this solver was to �nd suitable Roe-averaged states. This the-sis formally introduced a fourth condition on Property U. This condition is veryimportant; not only did this lead to a proper determination of the wavestrengthsfor the model, if formulated in terms of the primitive variables, but it also leadsto a unique Roe-average. The eigensystem of the 10-moment transport equationscan be expressed analytically in terms of the hydrodynamic variables. Therefore, aRoe-type solver followed directly. Although the 35-moment eigensystem can not bedetermined analytically in terms of the hydrodynamic variables the eigensystem canbe determined in both the Gaussian and adiabatic limits. These eigensystems can beutilized in a Roe-type approximate Riemann solver, but in order to maintain the cor-rect ux jumps, `�xes' had to be incorporated into the left eigenvectors. In additionto these two solvers, a third solver was developed for the non-equilibrium eigenstruc-ture which expressed the eigenvectors as functions of the eigenvalues. This provedto be a very useful result that signi�cantly reduced the run-time when compared todetermining the non-equilibrium eigenstructure using the EISPACK routine. Thefact that the factors which make up the characteristic polynomial are inter-relatedwas fully exploited in developing a method to numerically determine the wavespeedsand at the same time provide a check on hyperbolicity.To determine the accuracy of the moment models the one-dimensional shockstructure for a monatomic gas as predicted by the models was compared to thecorresponding DSMC solution. Comparisons were also made with solutions of theNavier-Stokes model. For the 35-moment model, several cases were run from aweak shock wave up to a moderate shock wave with an in ow Mach number of two.Above this Mach number it was shown that it is not possible to obtain solutions

250for the 35-moment model as the steady-state solutions are non-hyperbolic and clo-sure breakdown occurs. The computational e�ciencies for the various models wasthen analyzed. To the best of the author's knowledge these solutions represent the�rst numerical solution of either the 10-moment or 35-moment systems of transportequations.11.2 Concluding RemarksNumerical calculations of shock structure have revealed that, like the Navier-Stokes equations but unlike other traditional generalized transport models basedon the method of moments [26], the 10-moment equations do not break down andare stable for all Mach numbers. This is highly desirable from a computationalstandpoint. The 35-moment equations, however, are not stable for all Mach numbersand the solutions become non-hyperbolic beyond an inlet Mach number of 2.03.The 35-moment model provides excellent agreement with the DSMC model forweak shock waves but su�ers once an embedded shock appears in the solution. Thesediscontinuities are characteristic of a hyperbolic system. Any hyperbolic model will,at large enough Mach numbers, have embedded shocks present. This leads the authorto believe, a mixed model, i.e., one that contains second-order derivatives warrantsconsideration. A better modelling of the source terms may provide additional di�u-sion of the discontinuity, however, it would not completely eliminate it.At lower Mach numbers the 35-moment model appears to be in better agreementwith the DSMC results than does the Navier-Stokes model. For higher Mach numbersthe 35-moment model agrees very well with DSMC results downstream of the shockwhere the ow speed has become subsonic while upstream the agreement is notas good. The 35-moment model may not be an adequate model for simulating

251 ows involving supersonic speeds because of the embedded discontinuities, but forproblems involving subsonic speeds the 35-moment model may be a viable option.Also, at the higher Mach numbers it was not possible to obtain solutions usingthe Gaussian eigenstructure. Evidently, this eigenstructure produces a non-physicalsolution. This may be related to inconsistencies between the assumed (Gaussian)wavespeed and the actual wavespeed.The behaviour of the distribution function through the shock structure appears toshow that the 35-moment distribution function is trying to take on a bimodal form.Experimental results show that indeed the particle-velocity distribution function isbimodal in the interior of the shock [55]. Evidently, the 35-moment distributionfunction is not capable of representing as strongly a bimodal shape as is necessaryto obtain correct shock structure. This de�ciency appears to drive the distributionto take on negative values causing hyperbolicity to be lost, and hence the solutionbreaks down. This observation suggests that a higher-order moment model, to beused for the prediction of shock structure, should be based on a distribution functionwhich is capable of representing a strongly bimodal shape and this would, hopefully,extend the range of applicabilityIn terms of computational e�ciency, it was shown that the moment models pro-vide an iteration savings over the Navier-Stokes equations. The number of iterationsneeded to reach convergence appears to be independent of the Mach number for themoment models, while for the Navier-Stokes model it increases as the Mach num-ber increases. The iteration savings o�sets the higher cell costs these models incurdue to the increase in the number of transport equations. For the reduced systemof transport equations, the 10-moment model requires less total CPU time thanthe Navier-Stokes equations. These comparisons are valid only for one-dimensional

252problems. If extended to two- and three-dimensions the computational workload ofthe 35-moment model would increase because all 35 transport equations must besolved, as opposed to the Navier-Stokes model, which requires only two additionaltransport equations. However, there is multiplicity of wavespeeds and a lot of theextra waves are very simple, so the increase in computational costs may not beall that large. The 35-moment model should still be considerable less expensive inhigher-dimensions than the DSMC technique.In regards to the 35-moment system, by utilizing an eigenstructure evaluated atthe adiabatic limit and incorporating a `�x', the computational cost of this modelis signi�cantly reduced while predicting nearly identical shock structure to the non-equilibrium eigenstructure. Calculating the non-equilibrium eigenstructure analyt-ically dramatically reduced the computational workload when compared to calcu-lating this eigenstructure numerically. With respect to the DSMC method, all thesystems of transport equations presented in this thesis require several orders of mag-nitude less CPU time. The run-time for a DSMC calculation is measured in unitsof days while for the transport equations the run-time is measured in units of sec-onds. It is hoped that the nice features of the 35-moment model can be carriedover to other moment models; e.g., the attractive relationship among the terms inthe characteristic polynomials and the expressing of the eigenvectors in terms of theeigenvalues.Recall from the introduction the three fundamental questions posed dealing withthe computation of transitional ows:1. What is the appropriate set of extended hydrodynamic equations which willreplace the Navier-Stokes equations?

2532. What are the boundary conditions to use with this system?3. What numerical scheme should be used to e�ciently solve the system of trans-port equations and the accompanying boundary conditions?This thesis was concerned only with the �rst and last items with the second itembeing avoided by considering the shock structure problem. As an answer to the �rstquestion, two higher-order moment models were put forward: a 10-moment modeland a 35-moment model; however, only the 35-moment model is a viable option. Thismodel does not fully answer the �rst question since there is a limitation on the Machnumber. It is hoped that for rare�ed ows involving subsonic velocities, such as occurin micro-manufacturing, that the 35-moment model will provide accurate solutionsat a considerable computational savings over the DSMC method. In regards to thethird question, the author believes a signi�cant step has been taken in applyingmodern upwinding techniques to systems of transport equations based on higher-order moment models.The ability to treat non-equilibrium problems by purely hyperbolic methods,with no requirement to evaluate higher than �rst-order derivatives, would prove veryadvantageous. The methods should be relatively insensitive to irregularities of thegrid, such as inevitably occur when adaptive grids are employed. This should allowfairly straightforward computation of those high-altitude cases where the real-gas ow di�ers from the ideal-gas ow largely by virtue of the altered shock structure.Even narrow shock structures should be resolvable by using adaptive grids and shouldresult in the correct non-equilibrium jump behaviour.11.3 Future WorkPossible paths for future work are as follows:

2541. Even though the 35-moment model does not appear to be a promising modelfor predicting shock structure, it may still be a suitable model for other owproblems which are not quite as severe and where the ow is everywhere sub-sonic and where DSMC techniques are prohibitively expensive. Work is alreadyunderway in using this model to predict micro-channel ows.2. The 35-moment model may be adequate for the prediction of shock structureup to a Mach number of two, if a better modelling of the collision terms isincorporated. The BGK approximation is a very simple model, and if a morephysically realistic model is used the discontinuity that appears embedded inthe shock structure may be di�used and then better agreement with the DSMCmodel may be possible throughout the shock. However, as was mentionedearlier in this chapter, these discontinuities are typical of hyperbolic systems oftransport equations. Therefore, it is believed that, in general, better modellingof the source terms will not completely eliminate the embedded shocks.3. Another area of further research is in the analysis of the non-physical solutiongiven by the 35-moment Gaussian eigenstructure at the higher Mach numbers.If a remedy can be found the cost in obtaining 35-moment model solutions willbe further reduced.4. Continue the investigation into a more accurate moment model for the pre-diction of shock structure for dilute gases; that is, one having the capacity torepresent a greater variety of distribution functions.

APPENDICES

255

256APPENDIX AOrdinary Di�erential Equation Solution of10-Moment ModelA.1 Reduction of the Transport Equations to a Single ODEFor the one-dimensional shock structure problem there are some equations in the10-moment system which are redundant or provide no insight into the problem, i.e., atransport equation for a quantity which initially is zero and remains zero throughoutthe calculation. For the shock problem the 10-moment set of transport equations canbe simpli�ed as follows: uy = uz = 0, Pxy = Pxz = Pyz = 0, and Pyy = Pzz = Pnn.With this in mind the 10-moment transport equations reduces to a system of fourpartial di�erential equations: @U100@t + @F100@x = S100c ; (A.1)where U100 is the reduced solution vector composed of the conserved quantities, F100is the reduced ux vector, and S100c is the reduced source vector. These vectors are

257given as U100 = 0BBBBBBBBBBB@ ��ux�u2x + PxxPnn 1CCCCCCCCCCCA ; F100 = 0BBBBBBBBBBB@ �ux�u2x + Pxx�u3x + 3uxPxxuxPnn 1CCCCCCCCCCCA ; (A.2)S100c = �1� 0BBBBBBBBBBB@ 0023(Pxx � Pnn)13(Pnn � Pxx) 1CCCCCCCCCCCA ; (A.3)where � is the characteristic time for the inter-particle collision processes. As men-tioned in the previous chapters the source terms drive the non-equilibrium states toequilibrium.Of interest here is the steady-state solution of Equation (A.1) which satis�esdF100dx = S100c : (A.4)Equation (A.4) is a system of four ordinary di�erential equations (ODEs) in whichthe four primitive variables (�; ux; Pxx; Pnn) are dependent only on x and not time t.The components of this ux vector equation can be expressed as�ux =M; (A.5)�u2x + Pxx = P; (A.6)ddx(�u3x + 3uxPxx) = � 23� (Pxx � Pnn); (A.7)ddx(uxPnn) = � 13� (Pnn � Pxx); (A.8)whereM and P are constants throughout the shock structure. Equations (A.7) and(A.8) can be combined to eliminate the source term in Equation (A.7) which then

258becomes �u3x + 3uxPxx + 2uxPnn = E; (A.9)where E is another quantity which remains constant throughout the shock. It istherefore evident that the 10-moment system may be reduced to a single ODE interms of three invariants.The relaxation time � needs to be expressed in terms of the primitive variables.The coe�cient of viscosity � and the hydrostatic pressure p can be used to determinea characteristic collision time � = �p ; (A.10)where p = Pii=3 = (Pxx + 2Pnn)=3. Note that the viscosity coe�cient appears onlyin the source terms of the 10-moment equations unlike the Navier-Stokes equationswhere the viscosity coe�cient appears in the di�usive terms. The coe�cient ofviscosity is proportional to the temperature T :� = �o� TTo�! = �T !; (A.11)where �o and To are a reference viscosity and reference temperature, respectively,� is a proportionality constant, and ! characterizes the molecular interaction ofthe particles. Even though the 10-moment model does not represent heat transfer atemperature can still be obtained by using the perfect gas equation of state, p = �RT ,where R is the real gas constant. The relaxation time can now be made a functionof the primitive variables, � = �(�R)�!(Pxx + 2Pnn)!�1: (A.12)The three invariant relations can be used to eliminate any three of the primitivevariables from the single ODE. If Pxx is the variable of choice for the ordinary dif-

259ferential equation (ODE) then the other three variables are given as functions of Pxxas shown below: � = M2P � Pxx ; (A.13)ux = P � PxxM ; (A.14)Pnn = 2P 2xx �PPxx �P2 +ME2(P � Pxx) : (A.15)When these equations are inserted into Equation (A.8) and the relaxation time ismade a function of Pxx using Equations (A.13) and (A.15) then after simpli�cationthe following ODE is obtaineddPxxdx = 1�3!�2M2!+1 (4P 2xx � 3PPxx �P2 +ME)(P 2xx �P2 +ME)1�!(4Pxx �P)(Pxx �P)2 :(A.16)This ODE involves Pxx only, i.e., it is decoupled from the other primitive variables.Insight can be gained by inserting the invariant quantities back into Equation (A.16).This leads to dPxxdx = 2�3!�2�ux2!+1 (Pxx � Pnn)(Pxx + 2Pnn)1�!(�u2x)!(�u2x � 3Pxx) : (A.17)This equation shows that at the upstream and downstream locations the right-handside is zero as should be expected because the ow has returned to an equilibriumstate (Pxx = Pnn = p). As was shown in Chapter X a frozen shock appears in thesolution of the 10-moment transport equations when the upstream Mach numberM1exceeds 3=p5 = 1:3416. At this threshold Mach number (ux = q3Pxx=�), the ODEsolution of Equation (A.17) attains an in�nite slope as expected.The solution of this ODE is an initial-value problem. It can be solved numericallyusing a standard algorithm such as the fourth-order Runge-Kutta scheme. After asolution is obtained then the pro�les of the other variables can be determined using

260Equations (A.13) { (A.15) where the constants M, P, and E are determined forconvenience at the upstream location. The initial conditions are at the upstreamlocation and the solution is marched downstream until a suitable equilibrium locationis reached.In solving this ODE, care must be observed since above the threshold Machnumber a discontinuity appears embedded in the solution. If the upstream Machnumber is above this threshold value then the initial value for Pxx becomes its frozenjump value. At Mach numbers below this threshold value the initial value is notthe upstream equilibrium because then the slope of Equation (A.17) is zero and thevariables will not evolve to the appropriate downstream equilibrium value. Instead,the initial value is slightly perturbed from equilibrium using the same value as theconvergence criteria downstream. The 10-moment jump equations are given as�s�1 = 2 M213 + M21 ; (A.18)uxsux1 = 3 + M212 M21 ; (A.19)PxxsPxx1 = M21 � 12 ; (A.20)PnnsPnn1 = 4 M213 + M21 ; (A.21)where the subscript s denotes the downstream frozen jump value and the subscript1 refers to the upstream equilibrium location.A.2 Numerical Solution of ODEAs was mentioned in the previous section the method adopted for the solutionof the ODE for Pxx was a fourth-order Runge-Kutta algorithm using a constantstep-size. Shown in the �gures are the shock structures for four di�erent Mach num-bers comparing the density pro�les obtained from the system of partial di�erential

261equations (PDEs) with that of the ordinary di�erential solution. The density is nor-malized with respect to the upstream and downstream densities and the pro�le iscentered about �� = 1=2 and x=� = 0. For the ODE solutions the number of pointsin the solution always numbered well over one thousand. The PDE solutions wereobtained with 200 cells; the �gures do not show the full extent of the computationaldomains. In Figure A.1 the upstream Mach number is 1:1. This Mach number isbelow the threshold Mach number therefore there is no embedded shock. The PDEsolution lies on top of the ODE solution. Figure A.2 shows the shock pro�le for an�40 �20 0 20 40�0:20:00:20:4

0:60:81:0x=�

�� ODEPDEFigure A.1: Comparison of the ODE solution with respect to the PDE solution forM1 = 1:1.incoming Mach number of 1:35 which is slightly above the threshold Mach number.The initial value of Pxx is its frozen jump value as obtained from Equation (A.20).Once again any di�erences between the two solutions are not noticeable. A solutionfor a M1 = 2 in ow is shown in Figure A.3. This incoming Mach number is well

262�15 �10 �5 0 5 10 15�0:20:00:20:4

0:60:81:0x=�

�� ODEPDEFigure A.2: Comparison of the ODE solution with respect to the PDE solution forM1 = 1:35.above the threshold value. As in the previous cases the agreement between the twosolutions is excellent. The PDE solution captures the frozen jump value quite nicely.Figure A.4 shows the solution for a much larger in ow Mach number of 10 and onceagain the agreement is excellent.

263�8 �4 0 4 8�0:20:00:20:4

0:60:81:0x=�

�� ODEPDEFigure A.3: Comparison of the ODE solution with respect to the PDE solution forM1 = 2.

�8 �4 0 4 8�0:20:00:20:40:60:81:0

x=��� ODEPDE

Figure A.4: Comparison of the ODE solution with respect to the PDE solution forM1 = 10.

264APPENDIX BOne-Dimensional 35-Moment Transport EquationsIn this appendix the full one-dimensional 35-moment transport equations are givenin primitive form and in conservative form with the source terms modelled using thetwo-time-scale BGK or relaxation time approximation as discussed in the chapter onkinetic theory.B.1 Non-conservative FormulationUsing the distribution function F 35 and the non-conservative form of Maxwell'sequation of change the non-conservative transport equations expressed in tensor no-tation are @�@t + u� @�@x� + �u�x� = ���t ; (B.1)@ui@t + u� @ui@x� + 1� @Pi�@x� = �ui�t ; (B.2)@Pij@t + u�@Pij@x� + Pij @u�@x� + Pi� @uj@x� + Pj� @ui@x� + @Qij�@x� = �Pij�t ; (B.3)@Qijk@t + u�@Qijk@x� +Qijk @u�@x� +Qij� @uk@x� +Qik� @uj@x� +Qjk� @ui@x�+ Pi� @@x� Pjk� !+ Pj� @@x� Pik� !+ Pk� @@x� Pij� !+ @Kijk�@x�= �Qijk�t � Pij �uk�t � Pik �uj�t � Pjk �ui�t ; (B.4)

265@Kijkl@t + u�@Kijkl@x� +Kijkl @u�@x� +Kijk� @ul@x� +Kijl� @uk@x� +Kikl� @uj@x� +Kjkl� @ui@x�+ Pi� @@x� Qjkl� !+ Pj� @@x� Qikl� !+ Pk� @@x� Qijl� !+ Pl� @@x� Qijk� !+Qij� @@x� Pkl� !+Qik� @@x� Pjl� !+Qil� @@x� Pjk� !+Qjk� @@x� Pil� !+Qjl� @@x� Pik� !+Qkl� @@x� Pij� != �Kijkl�t �Qijk �ul�t �Qijl �uk�t �Qikl �uj�t �Qjkl �ui�t : (B.5)Expanding this system of equations and assuming a one-dimensional ow theequations become @�@t + ux @�@x + �@ux@x = ���t ; (B.6)@ux@t + ux@ux@x + 1� @Pxx@x = �ux�t ; (B.7)@uy@t + ux@uy@x + 1� @Pxy@x = �uy�t ; (B.8)@uz@t + ux@uz@x + 1� @Pxz@x = �uz�t ; (B.9)@Pxx@t + ux@Pxx@x + 3Pxx@ux@x + @Qxxx@x = �Pxx�t ; (B.10)@Pxy@t + ux@Pxy@x + 2Pxy @ux@x + Pxx@uy@x + @Qxxy@x = �Pxy�t ; (B.11)@Pxz@t + ux@Pxz@x + 2Pxz @ux@x + Pxx@uz@x + @Qxxz@x = �Pxz�t ; (B.12)@Pyy@t + ux@Pyy@x + Pyy @ux@x + 2Pxy @uy@x + @Qxyy@x = �Pyy�t ; (B.13)@Pyz@t + ux@Pyz@x + Pyz @ux@x + Pxz @uy@x + Pxy @uz@x + @Qxyz@x = �Pyz�t ; (B.14)

266@Pzz@t + ux@Pzz@x + Pzz @ux@x + 2Pxz @uz@x + @Qxzz@x = �Pzz�t ; (B.15)@Qxxx@t + ux@Qxxx@x + 4Qxxx@ux@x + 3Pxx� @Pxx@x � 3P 2xx�2 @�@x + @Kxxxx@x= �Qxxx�t � 3Pxx �ux�t ; (B.16)@Qxxy@t + ux@Qxxy@x + 3Qxxy @ux@x +Qxxx@uy@x + 2Pxx� @Pxy@x + Pxy� @Pxx@x� 3PxxPxy�2 @�@x + @Kxxxy@x = �Qxxy�t � 2Pxy �ux�t � Pxx �uy�t ; (B.17)@Qxxz@t + ux@Qxxz@x + 3Qxxz @ux@x +Qxxx@uz@x + 2Pxx� @Pxz@x + Pxz� @Pxx@x� 3PxxPxz�2 @�@x + @Kxxxz@x = �Qxxz�t � 2Pxz �ux�t � Pxx �uz�t ; (B.18)@Qxyy@t + ux@Qxyy@x + 2Qxyy @ux@x + 2Qxxy @uy@x + 2Pxy� @Pxy@x + Pxx� @Pyy@x� 2P 2xy + PxxPyy�2 @�@x + @Kxxyy@x = �Qxyy�t � Pyy �ux�t � 2Pxy �uy�t ; (B.19)@Qxyz@t + ux@Qxyz@x + 2Qxyz @ux@x +Qxxz @uy@x +Qxxy @uz@x + Pxz� @Pxy@x+ Pxy� @Pxz@x + Pxx� @Pyz@x � 2PxyPxz + PxxPyz�2 @�@x + @Kxxyz@x= �Qxyz�t � Pyz �ux�t � Pxz �uy�t � Pxy �uz�t ; (B.20)@Qxzz@t + ux@Qxzz@x + 2Qxzz @ux@x + 2Qxxz @uz@x + 2Pxz� @Pxz@x + Pxx� @Pzz@x� 2P 2xz + PxxPzz�2 @�@x + @Kxxzz@x = �Qxzz�t � Pzz �ux�t � 2Pxz �uz�t ; (B.21)@Qyyy@t + ux@Qyyy@x +Qyyy @ux@x + 3Qxyy @uy@x + 3Pxy� @Pyy@x � 3PxyPyy�2 @�@x+ @Kxyyy@x = �Qyyy�t � 3Pyy �uy�t ; (B.22)

267@Qyyz@t + ux@Qyyz@x +Qyyz @ux@x + 2Qxyz @uy@x +Qxyy @uz@x + Pxz� @Pyy@x+ 2Pxy� @Pyz@x � PxzPyy + 2PxyPxz�2 @�@x + @Kxyyz@x= �Qyyz�t � 2Pyz �uy�t � Pyy �uz�t ; (B.23)@Qyzz@t + ux@Qyzz@x +Qyzz @ux@x +Qxzz @uy@x + 2Qxyz @uz@x + 2Pxz� @Pyz@x+ Pxy� @Pzz@x � 2PxzPyz + PxyPzz�2 @�@x + @Kxyzz@x= �Qyzz�t � Pzz �uy�t � 2Pyz �uz�t ; (B.24)@Qzzz@t + ux@Qzzz@x +Qzzz @ux@x + 3Qxzz @uz@x + 3Pxz� @Pzz@x � 3PxzPzz�2 @�@x+ @Kxzzz@x = �Qzzz�t � 3Pzz �uz�t ; (B.25)@Kxxxx@t + ux@Kxxxx@x + 5Kxxxx@ux@x + 4Pxx� @Qxxx@x + 6Qxxx� @Pxx@x� 10PxxQxxx�2 @�@x = �Kxxxx�t � 4Qxxx�ux�t ; (B.26)@Kxxxy@t + ux@Kxxxy@x + 4Kxxxy @ux@x +Kxxxx@uy@x + 3Pxx� @Qxxy@x + Pxy� @Qxxx@x+ 3Qxxx� @Pxy@x + 3Qxxy� @Pxx@x � 6PxxQxxy + 4PxyQxxx�2 @�@x= �Kxxxy�t � 3Qxxy �ux�t �Qxxx�uy�t ; (B.27)@Kxxxz@t + ux@Kxxxz@x + 4Kxxxz @ux@x +Kxxxx@uz@x + 3Pxx� @Qxxz@x + Pxz� @Qxxx@x+ 3Qxxx� @Pxz@x + 3Qxxz� @Pxx@x � 6PxxQxxz + 4PxzQxxx�2 @�@x= �Kxxxz�t � 3Qxxz �ux�t �Qxxx�uz�t ; (B.28)@Kxxyy@t + ux@Kxxyy@x + 3Kxxyy @ux@x + 2Kxxxy @uy@x + 2Pxx� @Qxyy@x+ 2Pxy� @Qxxy@x + Qxxx� @Pyy@x + 4Qxxy� @Pxy@x+ Qxyy� @Pxx@x � 3PxxQxyy + 6PxyQxxy + PyyQxxx�2 @�@x= �Kxxyy�t � 2Qxyy �ux�t � 2Qxxy �uy�t ; (B.29)

268@Kxxyz@t + ux@Kxxyz@x + 3Kxxyz @ux@x +Kxxxz @uy@x +Kxxxy @uz@x + 2Pxx� @Qxyz@x+ Pxy� @Qxxz@x + Pxz� @Qxxy@x + Qxxx� @Pyz@x + 2Qxxz� @Pxy@x + 2Qxxy� @Pxz@x+ Qxyz� @Pxx@x � 3PxxQxyz + 3PxyQxxz + 3PxzQxxy + PyzQxxx�2 @�@x= �Kxxyz�t � 2Qxyz �ux�t �Qxxz �uy�t �Qxxy �uz�t ; (B.30)@Kxxzz@t + ux@Kxxzz@x + 3Kxxzz @ux@x + 2Kxxxz @uz@x + 2Pxx� @Qxzz@x + 2Pxz� @Qxxz@x+ Qxxx� @Pzz@x + 4Qxxz� @Pxz@x+ Qxzz� @Pxx@x � 3PxxQxzz + 6PxzQxxz + PzzQxxx�2 @�@x= �Kxxzz�t � 2Qxzz �ux�t � 2Qxxz �uz�t ; (B.31)@Kxyyy@t + ux@Kxyyy@x + 2Kxyyy @ux@x + 3Kxxyy @uy@x + Pxx� @Qyyy@x + 3Pxy� @Qxyy@x+ 3Qxxy� @Pyy@x + 3Qxyy� @Pxy@x � PxxQyyy + 6PxyQxyy + 3PyyQxxy�2 @�@x= �Kxyyy�t �Qyyy �ux�t � 3Qxyy �uy�t ; (B.32)@Kxyyz@t + ux@Kxyyz@x + 2Kxyyz @ux@x + 2Kxxyz @uy@x +Kxxyy @uz@x + Pxx� @Qyyz@x+ 2Pxy� @Qxyz@x + Pxz� @Qxyy@x + 2Qxxy� @Pyz@x + Qxxz� @Pyy@x + Qxyy� @Pxz@x+ 2Qxyz� @Pxy@x � PxxQyyz + 4PxyQxyz + 2PxzQxyy + 2PyzQxxy + PyyQxxz�2 @�@x= �Kxyyz�t �Qyyz �ux�t � 2Qxyz �uy�t �Qxyy �uz�t ; (B.33)@Kxyzz@t + ux@Kxyzz@x + 2Kxyzz @ux@x +Kxxzz @uy@x + 2Kxxyz @uz@x + Pxx� @Qyzz@x+ Pxy� @Qxzz@x + 2Pxz� @Qxyz@x + Qxxy� @Pzz@x + 2Qxxz� @Pyz@x + 2Qxyz� @Pxz@x+ Qxzz� @Pxy@x � PxxQyzz + 2PxyQxzz + 4PxzQxyz + 2PyzQxxz + PzzQxxy�2 @�@x= �Kxyzz�t �Qyzz �ux�t �Qxzz �uy�t � 2Qxyz �uz�t ; (B.34)

269@Kxzzz@t + ux@Kxzzz@x + 2Kxzzz @ux@x + 3Kxxzz @uz@x + Pxx� @Qzzz@x + 3Pxz� @Qxzz@x+ 3Qxxz� @Pzz@x + 3Qxzz� @Pxz@x � PxxQzzz + 6PxzQxzz + 3PzzQxxz�2 @�@x= �Kxzzz�t �Qzzz �ux�t � 3Qxzz �uz� ; (B.35)@Kyyyy@t + ux@Kyyyy@x +Kyyyy @ux@x + 4Kxyyy @uy@x + 4Pxy� @Qyyy@x + 6Qxyy� @Pyy@x� 4PxyQyyy + 6PyyQxyy�2 @�@x = �Kyyyy�t � 4Qyyy �uy�t ; (B.36)@Kyyyz@t + ux@Kyyyz@x +Kyyyz @ux@x + 3Kxyyz @uy@x +Kxyyy @uz@x+ 3Pxy� @Qyyz@x + Pxz� @Qyyy@x + 3Qxyy� @Pyz@x + 3Qxyz� @Pyy@x� 3PxyQyyz + PxzQyyy + 3PyyQxyz + 3PyzQxyy�2 @�@x= �Kyyyz�t � 3Qyyz �uy�t �Qyyy �uz�t ; (B.37)@Kyyzz@t + ux@Kyyzz@x +Kyyzz @ux@x + 2Kxyzz @uy@x + 2Kxyyz @uz@x + 2Pxy� @Qyzz@x+ 2Pxz� @Qyyz@x + Qxyy� @Pzz@x + 4Qxyz� @Pyz@x + Qxzz� @Pyy@x� 2PxyQyzz + 2PxzQyyz + PyyQxzz + 4PyzQxyz + PzzQxyy�2 @�@x= �Kyyzz�t � 2Qyzz �uy�t � 2Qyyz �uz�t ; (B.38)@Kyzzz@t + ux@Kyzzz@x +Kyzzz @ux@x +Kxzzz @uy@x + 3Kxyzz @uz@x+ 3Pxz� @Qyzz@x + Pxy� @Qzzz@x + 3Qxzz� @Pyz@x + 3Qxyz� @Pzz@x� 3PxzQyzz + PxyQzzz + 3PzzQxyz + 3PyzQxzz�2 @�@x= �Kyzzz�t �Qzzz �uy�t � 3Qyzz �uz�t ; (B.39)@Kzzzz@t + ux@Kzzzz@x +Kzzzz @ux@x + 4Kxzzz @uz@x + 4Pxz� @Qzzz@x + 6Qxzz� @Pzz@x� 4PxzQzzz + 6PzzQxzz�2 @�@x = �Kzzzz�t � 4Qzzz �uz�t : (B.40)

270This set of equations can be put into vector form@V35@t +A35p @V35@x = S35p ; (B.41)whereV35 =(�; ui; Pij; Qijk; Kijkl)T ; (B.42)= (�; ux; uy; uz; Pxx; Pxy; Pxz; Pyy ; Pyz ; Pzz ; Qxxx; Qxxy; Qxxz; Qxyy; Qxyz;Qxzz ; Qyyy; Qyyz; Qyzz; Qzzz ; Kxxxx; Kxxxy; Kxxxz; Kxxyy ; Kxxyz ; Kxxzz ;Kxyyy ; Kxyyz ; Kxyzz ; Kxzzz ; Kyyyy ; Kyyyz ; Kyyzz ; Kyzzz ; Kzzzz )T : (B.43)The matrix composed of the primitive ux coe�cients are given below where onlythe non-zero elements are given:A35p (1; 1) = ux; A35p (1; 2) = �; (B.44)A35p (2; 2) = ux; A35p (2; 5) = 1�; (B.45)A35p (3; 3) = ux; A35p (3; 6) = 1�; (B.46)A35p (4; 4) = ux; A35p (4; 7) = 1�; (B.47)A35p (5; 2) = 3Pxx; A35p (5; 5) = ux; A35p (5; 11) = 1; (B.48)A35p (6; 2) = 2Pxy ; A35p (6; 3) = Pxx; (B.49)A35p (6; 6) = ux; A35p (6; 12) = 1; (B.50)A35p (7; 2) = 2Pxz; A35p (7; 4) = Pxx; (B.51)

271A35p (7; 7) = ux; A35p (7; 13) = 1; (B.52)A35p (8; 2) = Pyy ; A35p (8; 3) = 2Pxy; (B.53)A35p (8; 8) = ux; A35p (8; 14) = 1; (B.54)A35p (9; 2) = Pyz ; A35p (9; 3) = Pxz; A35p (9; 4) = Pxy; (B.55)A35p (9; 9) = ux; A35p (9; 15) = 1; (B.56)A35p (10; 2) = Pzz ; A35p (10; 4) = 2Pxz; (B.57)A35p (10; 10) = ux; A35p (10; 16) = 1; (B.58)A35p (11; 1) = �3P 2xx�2 ; A35p (11; 2) = 4Qxxx; (B.59)A35p (11; 5) = 3Pxx� ; A35p (11; 11) = ux; A35p (11; 21) = 1; (B.60)A35p (12; 1) = �3PxxPxy�2 ; A35p (12; 2) = 3Qxxy; (B.61)A35p (12; 3) = Qxxx; A35p (12; 5) = Pxy� ; (B.62)A35p (12; 6) = 2Pxx� ; A35p (12; 12) = ux; A35p (12; 22) = 1; (B.63)A35p (13; 1) = �3PxxPxz�2 ; A35p (13; 2) = 3Qxxz; (B.64)

272A35p (13; 4) = Qxxx; A35p (13; 5) = Pxz� ; (B.65)A35p (13; 7) = 2Pxx� ; A35p (13; 13) = ux; A35p (13; 23) = 1; (B.66)A35p (14; 1) = �PxxPyy + 2P 2xy�2 ; A35p (14; 2) = 2Qxyy ; (B.67)A35p (14; 3) = 2Qxxy; A35p (14; 6) = 2Pxy� ; (B.68)A35p (14; 8) = Pxx� ; A35p (14; 14) = ux; A35p (14; 24) = 1; (B.69)A35p (15; 1) = �PxxPyz + 2PxyPxz�2 ; A35p (15; 2) = 2Qxyz; (B.70)A35p (15; 3) = Qxxz; A35p (15; 4) = Qxxy; (B.71)A35p (15; 6) = Pxz� ; A35p (15; 7) = Pxy� ; (B.72)A35p (15; 9) = Pxx� ; A35p (15; 15) = ux; A35p (15; 25) = 1; (B.73)A35p (16; 1) = PxxPzz + 2P 2xz�2 ; A35p (16; 2) = 2Qxzz; (B.74)A35p (16; 4) = 2Qxxz; A35p (16; 7) = 2Pxz� ; (B.75)A35p (16; 10) = Pxx� ; A35p (16; 16) = ux; A35p (16; 26) = 1; (B.76)

273A35p (17; 1) = �3PxyPyy�2 ; A35p (17; 2) = Qyyy ; A35p (17; 3) = 3Qxyy;(B.77)A35p (17; 8) = 3Pxy� ; A35p (17; 17) = ux; A35p (17; 27) = 1; (B.78)A35p (18; 1) = �PxzPyy + 2PxyPyz�2 ; A35p (18; 2) = Qyyz; (B.79)A35p (18; 3) = 2Qxyz; A35p (18; 4) = Qxyy; A35p (18; 8) = Pxz� ; (B.80)A35p (18; 9) = 2Pxy� ; A35p (18; 18) = ux A35p (18; 28) = 1; (B.81)A35p (19; 1) = �PxyPzz + 2PxzPyz�2 ; A35p (19; 2) = Qyzz ; (B.82)A35p (19; 3) = Qxzz ; A35p (19; 4) = 2Qxyz; A35p (19; 9) = 2Pxz� ; (B.83)A35p (19; 10) = Pxy� ; A35p (19; 19) = ux; A35p (19; 29) = 1; (B.84)A35p (20; 1) = �3PxzPzz�2 ; A35p (20; 2) = Qzzz ; A35p (20; 4) = 3Qxzz ; (B.85)A35p (20; 10) = 3Pxz� ; A35p (20; 20) = ux; A35p (20; 30) = 1; (B.86)

274A35p (21; 1) = �10PxxQxxx�2 ; A35p (21; 2) = 5 Kxxxx + 3�P 2xx! ; (B.87)A35p (21; 5) = 6Qxxx� ; A35p (21; 11) = 4Pxx� ; A35p (21; 21) = ux; (B.88)A35p (22; 1) = �6PxxQxxy + 4PxyQxxx�2 ; A35p (22; 2) = 4 Kxxxy + 3�PxxPxy! ;(B.89)A35p (22; 3) = Kxxxx + 3�P 2xx; A35p (22; 5) = 3Qxxy� ; A35p (22; 6) = 3Qxxx� ;(B.90)A35p (22; 11) = Pxy� ; A35p (22; 12) = 3Pxx� ; A35p (22; 22) = ux; (B.91)A35p (23; 1) = �6PxxQxxz + 4PxzQxxx�2 ; A35p (23; 2) = 4 Kxxxz + 3�PxxPxz! ;(B.92)A35p (23; 4) = Kxxxx + 3�P 2xx; A35p (23; 5) = 3Qxxz� ; A35p (23; 7) = 3Qxxx� ;(B.93)A35p (23; 11) = Pxz� ; A35p (23; 13) = 3Pxx� ; A35p (23; 23) = ux; (B.94)A35p (24; 1) = �3PxxQxyy + 6PxyQxxy + PyyQxxx�2 ; (B.95)A35p (24; 2) = 3 Kxxyy + 1�PxxPyy + 2�P 2xy! ; (B.96)

275A35p (24; 3) = 2 Kxxxy + 3�PxxPxy! ; (B.97)A35p (24; 5) = Qxyy� ; A35p (24; 6) = 4Qxxy� ; A35p (24; 8) = Qxxx� ; (B.98)A35p (24; 12) = 2Pxy� ; A35p (24; 14) = 2Pxx� ; A35p (24; 24) = ux; (B.99)A35p (25; 1) = �3PxxQxyz + 3PxyQxxz + 3PxzQxxy + PyzQxxx�2 ; (B.100)A35p (25; 2) = 3 Kxxyz + 1�PxxPyz + 2�PxyPxz! ; (B.101)A35p (25; 3) = Kxxxz + 3�PxxPxz; A35p (25; 4) = Kxxxy + 3�PxxPxy; (B.102)A35p (25; 5) = Qxyz� ; A35p (25; 6) = 2Qxxz� ; A35p (25; 7) = 2Qxxy� ; (B.103)A35p (25; 9) = Qxxx� ; A35p (25; 12) = Pxz� ; A35p (25; 13) = Pxy� ; (B.104)A35p (25; 15) = 2Pxx� ; A35p (25; 25) = ux; (B.105)A35p (26; 1) = �3PxxQxzz + 6PxzQxxz + PzzQxxx�2 ; (B.106)A35p (26; 2) = 3 Kxxzz + 1�PxxPzz + 2�P 2xz! ; (B.107)

276A35p (26; 4) = 2 Kxxxz + 3�PxxPxz! ; (B.108)A35p (26; 5) = Qxzz� ; A35p (26; 7) = 4Qxxz� ; A35p (26; 10) = Qxxx� ; (B.109)A35p (26; 13) = 2Pxz� ; A35p (26; 16) = 2Pxx� ; A35p (26; 26) = ux; (B.110)A35p (27; 1) = �PxxQyyy + 6PxyQxyy + 3PyyQxxy�2 ; (B.111)A35p (27; 2) = 2 Kxyyy + 3�PxyPyy! ; (B.112)A35p (27; 3) = 3 Kxxyy + 1�PxxPyy + 2�P 2xy! ; (B.113)A35p (27; 6) = 3Qxyy� ; A35p (27; 8) = 3Qxxy� ; (B.114)A35p (27; 14) = 3Pxy� ; A35p (27; 17) = Pxx� ; A35p (27; 27) = ux; (B.115)A35p (28; 1) = �PxxQyyz + 4PxyQxyz + 2PxzQxyy + 2PyzQxxy + PyyQxxz�2 ;(B.116)A35p (28; 2) = 2 Kxyyz + 2�PxyPyz + 1�PxzPyy! ; (B.117)A35p (28; 3) = 2 Kxxyz + 1�PxxPyz + 2�PxyPxz! ; (B.118)

277A35p (28; 4) = Kxxyy + 1�PxxPyy + 2�P 2xy! ; (B.119)A35p (28; 6) = 2Qxyz� ; A35p (28; 7) = Qxyy� ; A35p (28; 8) = Qxxz� ; (B.120)A35p (28; 9) = 2Qxxy� ; A35p (28; 14) = Pxz� ; A35p (28; 15) = 2Pxy� ; (B.121)A35p (28; 18) = Pxx� ; A35p (28; 28) = ux; (B.122)A35p (29; 1) = �PxxQyzz + 2PxyQxzz + 4PxzQxyz + 2PyzQxxz + PzzQxxy�2 ;(B.123)A35p (29; 2) = 2 Kxyzz + 1�PxyPzz + 2�PxzPyz! ; (B.124)A35p (29; 3) = Kxxzz + 1�PxxPzz + 2�P 2xz! ; (B.125)A35p (29; 4) = 2 Kxxyz + 1�PxxPyz + 2�PxyPxz! ; (B.126)A35p (29; 6) = Qxzz� ; A35p (29; 7) = 2Qxyz� ; A35p (29; 9) = 2Qxxz� ; (B.127)A35p (29; 10) = Qxxy� ; A35p (29; 15) = 2Pxz� ; A35p (29; 16) = Pxy� ; (B.128)A35p (29; 19) = Pxx� ; A35p (29; 29) = ux; (B.129)

278A35p (30; 1) = �PxxQzzz + 6PxzQxzz + 3PzzQxxz�2 ; (B.130)A35p (30; 2) = 2 Kxzzz + 3�PxzPzz! ; (B.131)A35p (30; 4) = 3 Kxxzz + 1�PxxPzz + 2�P 2xz! ; (B.132)A35p (30; 7) = 3Qxzz� ; A35p (30; 10) = 3Qxzz� ; A35p (30; 16) = 3Pxz� ;(B.133)A35p (30; 20) = Pxx� ; A35p (30; 30) = ux; (B.134)A35p (31; 1) = �4PxyQyyy + 6PyyQxyy�2 ; A35p (31; 2) = Kyyyy + 3�P 2yy;(B.135)A35p (31; 3) = 4 Kxyyy + 3�PxyPyy! ; A35p (31; 8) = 6Qxyy� ; (B.136)A35p (31; 17) = 4Pxy� ; A35p (31; 31) = ux; (B.137)A35p (32; 1) = �3PxyQyyz + PxzQyyy + 3PyzQxyy + 3PyyQxyz�2 ; (B.138)A35p (32; 2) = Kyyyz + 3�PyyPyz ; A35p (32; 3) = 3 Kxyyz + 2�PxyPyz + 1�PxzPyy! ;(B.139)A35p (32; 4) = Kxyyy + 3�PxyPyy ; A35p (32; 8) = 3Qxyz� ; (B.140)

279A35p (32; 9) = 3Qxyy� ; A35p (32; 17) = Pxz� ; (B.141)A35p (32; 18) = 3Pxy� ; A35p (32; 32) = ux; (B.142)A35p (33; 1) = �2PxyQyzz + 2PxzQyyz + PyyQxzz + 4PyzQxyz + PzzQxyy�2 ;(B.143)A35p (33; 2) = Kyyzz + 1�PyyPzz + 2�P 2yz ; (B.144)A35p (33; 3) = 2 Kxyzz + 1�PxyPzz + 2�PxzPyz! ; (B.145)A35p (33; 4) = 2 Kxyyz + 2�PxyPyz + 1�PxzPyy! ; (B.146)A35p (33; 8) = Qxzz� ; A35p (33; 9) = 4Qxyz� ; A35p (33; 10) = Qxyy� ; (B.147)A35p (33; 18) = 2Pxz� ; A35p (33; 19) = 2Pxy� ; A35p (33; 33) = ux; (B.148)A35p (34; 1) = �PxyQzzz + 3PxzQyzz + 3PyzQxzz + 3PzzQxyz�2 ; (B.149)A35p (34; 2) = Kyzzz + 3�PyzPzz ; A35p (34; 3) = Kxzzz + 3�PxzPzz (B.150)A35p (34; 4) = 3 Kxyzz + 1�PxyPzz + 2�PxzPyz! ; (B.151)

280A35p (34; 9) = 3Qxzz� ; A35p (34; 10) = 3Qxyz� ; A35p (34; 19) = 3Pxz� ;(B.152)A35p (34; 20) = Pxy� ; A35p (34; 34) = ux; (B.153)A35p (35; 1) = �4PxzQzzz + 6PzzQxzz�2 ; A35p (35; 2) = Kzzzz + 3�P 2zz ; (B.154)A35p (35; 4) = 4 Kxzzz + 3�PxzPzz! ; A35p (35; 10) = 6Qxzz� ; (B.155)A35p (35; 20) = 4Pxz� ; A35p (35; 35) = ux: (B.156)B.2 Conservative FormulationDeriving the 35-moment transport equations from the conservative form of Maxwell'sequation of change the following system of equations expressed in tensor notation isobtained @�@t + @@x���u�� = ���t ; (B.157)@@t��ui�+ @@x���uiu� + Pi�� = ��t��ui�; (B.158)@@t��uiuj + Pij�+ @@x���uiuju� + uiPj� + ujPi� + u�Pij +Qij��= ��t��uiuj + Pij�; (B.159)@@t��uiujuk + uiPjk + ujPik + ukPij +Qijk�+ @@x���uiujuku� + uiujPk�+ uiukPj� + ujukPi� + uiu�Pjk + uju�Pik + uku�Pij + 1�PijPk� + 1�PikPj�+ 1�PjkPi� + u�Qijk + ukQij� + ujQik� + uiQjk� +Kijk��= ��t��uiujuk + uiPjk + ujPik + ukPij +Qijk�; (B.160)

281@@t��uiujukul + uiujPkl + uiukPjl + ujukPil + uiulPjk + ujulPik + ukulPij+ 1�PijPkl + 1�PikPjl + 1�PjkPil + ulQijk + ukQijl + ujQikl + uiQjkl +Kijkl�+ @@x���uiujukulu� + uiujukPl� + uiujulPk� + uiukulPj� + ujukulPi�+ uiuju�Pkl + uiuku�Pjl + uiulu�Pjk + ujuku�Pil + ujulu�Pik + ukulu�Pij+ 1�u�PijPkl + 1�u�PikPjl + 1�u�PilPjk + 1�ulPijPk� + 1�ulPikPj� + 1�ulPi�Pjk+ 1�ukPijPl� + 1�ukPilPj� + 1�ukPi�Pjl + 1�ujPikPl� + 1�ujPilPk� + 1�ujPi�Pkl+ 1�uiPjkPl� + 1�uiPjlPk� + 1�uiPj�Pkl + uiujQkl� + uiukQjl� + uiulQjk�+ uiu�Qjkl + ujukQil� + ujulQik� + uju�Qikl + ukulQij� + uku�Qijl + ulu�Qijk+ 1�PijQkl� + 1�PikQjl� + 1�PilQjk� + 1�Pi�Qjkl + 1�PjkQil� + 1�PjlQik�+ 1�Pj�Qikl + 1�PklQij� + 1�Pk�Qijl + 1�Pl�Qijk + uiKjkl� + ujKikl� + ukKijl�+ulKijk� + u�Kijkl� = ��t��uiujukul + uiujPkl + uiukPjl + ujukPil + uiulPjk+ ujulPik + ukulPij + 1�PijPkl + 1�PikPjl + 1�PjkPil + ulQijk + ukQijl + ujQikl+ uiQjkl +Kijkl� (B.161)Expanding the tensor notati0on and assuming a one-dimensional ow the systemcan be expressed in vector form as@U35@t + @F35@x = S35c ; (B.162)where the components of the conserved solution vector areU351 = �; (B.163)U352 = �ux; (B.164)U353 = �uy; (B.165)

282U354 = �uz; (B.166)U355 = �u2x + Pxx; (B.167)U356 = �uxuy + Pxy; (B.168)U357 = �uxuz + Pxz; (B.169)U358 = �u2y + Pyy ; (B.170)U359 = �uyuz + Pyz ; (B.171)U3510 = �u2z + Pzz ; (B.172)U3511 = �u3x + 3uxPxx +Qxxx; (B.173)U3512 = �u2xuy + 2uxPxy + uyPxx +Qxxy; (B.174)U3513 = �u2xuz + 2uxPxz + uzPxx +Qxxz; (B.175)U3514 = �uxu2y + uxPyy + 2uyPxy +Qxyy; (B.176)U3515 = �uxuyuz + uxPyz + uyPxz + uzPxy +Qxyz; (B.177)U3516 = �uxu2z + uxPzz + 2uzPxz +Qxzz; (B.178)U3517 = �u3y + 3uyPyy +Qyyy; (B.179)

283U3518 = �u2yuz + 2uyPyz + uzPyy +Qyyz ; (B.180)U3519 = �uyu2z + uyPzz + 2uzPyz +Qyzz ; (B.181)U3520 = �u3z + 3uzPzz +Qzzz ; (B.182)U3521 = �u4x + 6u2xPxx + 3�P 2xx + 4uxQxxx +Kxxxx; (B.183)U3522 = �u3xuy + 3uxuyPxx + 3u2xPxy + 3�PxxPxy + 3uxQxxy + uyQxxx +Kxxxy;(B.184)U3523 = �u3xuz + 3uxuzPxx + 3u2xPxz + 3�PxxPxz + 3uxQxxz + uzQxxx +Kxxxz;(B.185)U3524 = �u2xu2y + u2xPyy + 4uxuyPxy + u2yPxx + 1�PxxPyy + 2�P 2xy + 2uxQxyy+ 2uyQxxy +Kxxyy ; (B.186)U3525 = �u2xuyuz + u2xPyz + 2uxuyPxz + 2uxuzPxy + uyuzPxx + 1�PxxPyz + 2�PxyPxz+ 2uxQxyz + uyQxxz + uzQxxy +Kxxyz ; (B.187)U3526 = �u2xu2z + u2xPzz + 4uxuzPxz + u2zPxx + 1�PxxPzz + 2�P 2xz + 2uxQxzz+ 2uzQxxz +Kxxzz ; (B.188)U3527 = �uxu3y + 3uxuyPyy + 3u2yPxy + 3�PxyPyy + uxQyyy + 3uyQxyy +Kxyyy ;(B.189)U3528 = �uxu2yuz + 2uxuyPyz + uxuzPyy + u2yPxz + 2uyuzPxy + 2�PxyPyz + 1�PxzPyy+ uxQyyz + 2uyQxyz + uzQxyy +Kxyyz ; (B.190)

284U3529 = �uxuyu2z + uxuyPzz + 2uxuzPyz + 2uyuzPxz + u2zPxy + 2�PxzPyz+ 1�PxyPzz + uxQyzz + uyQxzz + 2uzQxyz +Kxyzz ; (B.191)U3530 = �uxu3z + 3uxuzPzz + 3u2zPxz + 3�PxzPzz + uxQzzz + 3uzQxzz +Kxzzz ;(B.192)U3531 = �u4y + 6u2yPyy + 3�P 2yy + 4uyQyyy +Kyyyy ; (B.193)U3532 = �u3yuz + 3uyuzPyy + 3u2yPyz + 3�PyyPyz + 3uyQyyz + uzQyyy +Kyyyz ;(B.194)U3533 = �u2yu2z + u2yPzz + 4uyuzPyz + u2zPyy + 1�PyyPzz + 2�P 2yz + 2uyQyzz+ 2uzQyyz +Kyyzz ; (B.195)U3534 = �uyu3z + 3uyuzPzz + 3u2zPyz + 3�PyzPzz + uyQzzz + 3uzQyzz +Kyzzz ;(B.196)U3535 = �u4z + 6u2zPzz + 3�P 2zz + 4uzQzzz +Kzzzz ; (B.197)and the components of the ux vector areF 351 = �ux; (B.198)F 352 = �u2x + Pxx; (B.199)F 353 = �uxuy + Pxy; (B.200)F 354 = �uxuz + Pxz; (B.201)

285F 355 = �u3x + 3uxPxx +Qxxx; (B.202)F 356 = �u2xuy + 2uxPxy + uyPxx +Qxxy; (B.203)F 357 = �u2xuz + 2uxPxz + uzPxx +Qxxz; (B.204)F 358 = �uxu2y + uxPyy + 2uyPxy +Qxyy; (B.205)F 359 = �uxuyuz + uxPyz + uyPxz + uzPxy +Qxyz; (B.206)F 3510 = �uxu2z + uxPzz + 2uzPxz +Qxzz; (B.207)F 3511 = �u4x + 6u2xPxx + 3�P 2xx + 4uxQxxx +Kxxxx; (B.208)F 3512 = �u3xuy + 3uxuyPxx + 3u2xPxy + 3�PxxPxy + 3uxQxxy + uyQxxx +Kxxxy;(B.209)F 3513 = �u3xuz + 3uxuzPxx + 3u2xPxz + 3�PxxPxz + 3uxQxxz + uzQxxx +Kxxxz;(B.210)F 3514 = �u2xu2y + u2xPyy + 4uxuyPxy + u2yPxx + 1�PxxPyy + 2�P 2xy + 2uxQxyy+ 2uyQxxy +Kxxyy ; (B.211)F 3515 = �u2xuyuz + u2xPyz + 2uxuyPxz + 2uxuzPxy + uyuzPxx + 1�PxxPyz + 2�PxyPxz+ 2uxQxyz + uyQxxz + uzQxxy +Kxxyz ; (B.212)F 3516 = �u2xu2z + u2xPzz + 4uxuzPxz + u2zPxx + 1�PxxPzz + 2�P 2xz + 2uxQxzz+ 2uzQxxz +Kxxzz ; (B.213)

286F 3517 = �uxu3y + 3uxuyPyy + 3u2yPxy + 3�PxyPyy + uxQyyy + 3uyQxyy +Kxyyy ;(B.214)F 3518 = �uxu2yuz + 2uxuyPyz + uxuzPyy + u2yPxz + 2uyuzPxy + 2�PxyPyz+ 1�PxzPyy + uxQyyz + 2uyQxyz + uzQxyy +Kxyyz ; (B.215)F 3519 = �uxuyu2z + uxuyPzz + 2uxuzPyz + 2uyuzPxz + u2zPxy + 2�PxzPyz+ 1�PxyPzz + uxQyzz + uyQxzz + 2uzQxyz +Kxyzz ; (B.216)F 3520 = �uxu3z + 3uxuzPzz + 3u2zPxz + 3�PxzPzz + uxQzzz + 3uzQxzz +Kxzzz ;(B.217)F 3521 = �u5x + 10u3xPxx + 15� uxP 2xx + 10u2xQxxx + 10� PxxQxxx + 5uxKxxxx;(B.218)F 3522 = �u4xuy + 4u3xPxy + 6u2xuyPxx + 3�uyP 2xx + 12� uxPxxPxy + 6u2xQxxy+ 4uxuyQxxx + 4�PxyQxxx + 6�PxxQxxy + uyKxxxx + 4uxKxxxy; (B.219)F 3523 = �u4xuz + 4u3xPxz + 6u2xuzPxx + 3�uzP 2xx + 12� uxPxxPxz + 6u2xQxxz+ 4uxuzQxxx + 4�PxzQxxx + 6�PxxQxxz + uzKxxxx + 4uxKxxxz; (B.220)F 3524 = �u3xu2y + u3xPyy + 6u2xuyPxy + 3uxu2yPxx + 6�uyPxxPxy + 3�uxPxxPyy+ 6�uxP 2xy + u2yQxxx + 3u2xQxyy + 6uxuyQxxy + 1�PyyQxxx + 6�PxyQxxy+ 3�PxxQxyy + 2uyKxxxy + 3uxKxxyy; (B.221)

287F 3525 = �u3xuyuz + u3xPyz + 3uxuyuzPxx + 3u2xuyPxz + 3u2xuzPxy + 3�uyPxxPxz+ 3�uzPxxPxy + 3�uxPxxPyz + 6�uxPxyPxz + uyuzQxxx + 3u2xQxyz+ 3uxuyQxxz + 3uxuzQxxy + 1�PyzQxxx + 3�PxxQxyz + 3�PxyQxxz+ 3�PxzQxxy + 3uxKxxyz + uyKxxxz + uzKxxxy; (B.222)F 3526 = �u3xu2z + u3xPzz + 6u2xuzPxz + 3uxu2zPxx + 6�uzPxxPxz + 3�uxPxxPzz+ 6�uxP 2xz + u2zQxxx + 3u2xQxzz + 6uxuzQxxz + 1�PzzQxxx + 6�PxzQxxz+ 3�PxxQxzz + 2uzKxxxz + 3uxKxxzz ; (B.223)F 3527 = �u2xu3y + u3yPxx + 6uxu2yPxy + 3u2xuyPyy + 6�uxPxyPyy + 3�uyPxxPyy+ 6�uyP 2xy + u2xQyyy + 3u2yQxxy + 6uxuyQxyy + 1�PxxQyyy + 3�PyyQxxy+ 6�PxyQxyy + 3uyKxxyy + 2uxKxyyy ; (B.224)F 3528 = �u2xu2yuz + 2uxu2yPxz + 2u2xuyPyz + u2yuzPxx + u2xuzPyy + 4uxuyuzPxy+ 1�uzPxxPyy + 2�uzP 2xy + 2�uxPxzPyy + 4�uyPxyPxz + 2�uyPxxPyz+ 4�uxPxyPyz + 4�uxPxyPyz + 2uxuzQxyy + 2uyuzQxxy + u2xQyyz + u2yQxxz+ 4uxuyQxyz + 2�PxzQxyy + 2�PyzQxxy + 1�PxxQyyz + 1�PyyQxxz+ 4�PxyQxyz + 2uxKxyyz + 2uyKxxyz + uzKxxyy ; (B.225)F 3529 = �u2xuyu2z + 2uxu2zPxy + 2u2xuzPyz + uyu2zPxx + u2xuyPzz + 4uxuyuzPxz+ 1�uyPxxPzz + 2�uyP 2xz + 2�uxPxyPzz + 4�uzPxyPxz + 2�uzPxxPyz+ 4�uxPxzPyz + 2uxuyQxzz + 2uyuzQxxz + u2xQyzz + u2zQxxy + 4uxuzQxyz+ 2�PxyQxzz + 2�PyzQxxz + 1�PxxQyzz + 1�PzzQxxy + 4�PxzQxyz+ 2uxKxyzz + uyKxxzz + 2uzKxxyz; (B.226)

288F 3530 = �u2xu3z + u3zPxx + 6uxu2zPxz + 3u2xuzPzz + 6�uxPxzPzz + 3�uzPxxPzz+ 6�uzP 2xz + u2xQzzz + 3u2zQxxz + 6uxuzQxzz + 1�PxxQzzz + 3�PzzQxxz+ 6�PxzQxzz + 3uzKxxzz + 2uxKxzzz ; (B.227)F 3531 = �uxu4y + 6uxu2yPyy + 4u3yPxy + 3�uxP 2yy + 12� uyPxyPyy + 6u2yQxyy+ 4uxuyQyyy + 6�PyyQxyy + 4�PxyQyyy + uxKyyyy + 4uyKxyyy ; (B.228)F 3532 = �uxu3yuz + u3yPxz + 3uxuyuzPyy + 3u2yuzPxy + 3uxu2yPyz + 3�uxPyyPyz+ 3�uzPxyPyy + 3�uyPxzPyy + 6�uyPxyPyz + uxuzQyyy + 3u2yQxyz+ 3uxuyQyyz + 3uyuzQxyy + 1�PxzQyyy + 3�PyyQxyz + 3�PxyQyyz+ 3�PyzQxyy + 3uyKxyyz + uxKyyyz + uzKxyyy ; (B.229)F 3533 = �uxu2yu2z + 2uyu2zPxy + 2u2yuzPxz + uxu2zPyy + uxu2yPzz + 4uxuyuzPyz+ 1�uxPyyPzz + 2�uxP 2yz + 2�uyPxyPzz + 4�uzPxyPyz + 2�uzPxzPyy+ 4�uyPxzPyz + 2uxuyQyzz + 2uxuzQyyz + u2yQxzz + u2zQxyy + 4uyuzQxyz+ 2�PxyQyzz + 2�PxzQyyz + 1�PyyQxzz + 1�PzzQxyy + 4�PyzQxyz+ uxKyyzz + 2uyKxyzz + 2uzKxyyz ; (B.230)F 3534 = �uxuyu3z + u3zPxy + 3uxuyuzPzz + 3uyu2zPxz + 3uxu2zPyz + 3�uxPyzPzz+ 3�uyPxzPzz + 3�uzPxyPzz + 6�uzPxzPyz + uxuyQzzz + 3u2zQxyz+ 3uxuzQyzz + 3uyuzQxzz + 1�PxyQzzz + 3�PzzQxyz + 3�PxzQyzz+ 3�PyzQxzz + 3uzKxyzz + uxKyzzz + uyKxzzz ; (B.231)

289F 3535 = �uxu4z + 6uxu2zPzz + 4u3zPxz + 3�uxP 2zz + 12� uzPxzPzz + 6u2zQxzz+ 4uxuzQzzz + 6�PzzQxzz + 4�PxzQzzz + uxKzzzz + 4uzKxzzz : (B.232)B.3 Evaluation of the Source TermsUsing the two-time-scale BGK approximation the source terms become���t = 0; (B.233)�ui�t = 0; (B.234)�Pij�t = �1� �Pij � 13P���ij� ; (B.235)�Qijk�t = �1� Pr Qijk; (B.236)�Kijkl�t = �1� "PrKijkl � 1� �Pij � 13P���ij��Pkl � 13P���kl�� 1� �Pik � 13P���ik��Pjl � 13P���jl��1� �Pil � 13P���il��Pjk � 13P���jk�# : (B.237)Therefore the components of the primitive source term vector S35p becomeS35p1 = 0; (B.238)S35p2 = 0; (B.239)S35p3 = 0; (B.240)S35p4 = 0; (B.241)S35p5 = � 13� �2Pxx � Pyy � Pzz�; (B.242)

290S35p6 = �1� Pxy; (B.243)S35p7 = �1� Pxz; (B.244)S35p8 = � 13� �2Pyy � Pxx � Pzz�; (B.245)S35p9 = �1� Pyz ; (B.246)S35p10 = � 13� �2Pzz � Pxx � Pyy�; (B.247)S35p11 = �1� Pr Qxxx; (B.248)S35p12 = �1� Pr Qxxy; (B.249)S35p13 = �1� Pr Qxxz; (B.250)S35p14 = �1� Pr Qxyy; (B.251)S35p15 = �1� Pr Qxyz; (B.252)S35p16 = �1� Pr Qxzz; (B.253)S35p17 = �1� Pr Qyyy ; (B.254)S35p18 = �1� Pr Qyyz ; (B.255)S35p19 = �1� Pr Qyzz ; (B.256)

291S35p20 = �1� Pr Qzzz ; (B.257)S35p21 = �1� hPr Kxxxx � 13��2Pxx � Pyy � Pzz�2i; (B.258)S35p22 = �1� hPrKxxxy � 1��2Pxx � Pyy � Pzz�Pxyi; (B.259)S35p23 = �1� hPrKxxxz � 1��2Pxx � Pyy � Pzz�Pxzi; (B.260)S35p24 = �1� hPrKxxyy � 19��2Pxx � Pyy � Pzz��2Pyy � Pxx � Pzz�� 2�P 2xyi;(B.261)S35p25 = �1� hPr Kxxyz � 13��2Pxx � Pyy � Pzz�Pyz � 2�PxyPxzi; (B.262)S35p26 = �1� hPr Kxxzz � 19��2Pxx � Pyy � Pzz��2Pzz � Pxx � Pyy�� 2�P 2xzi;(B.263)S35p27 = �1� hPr Kxyyy � 1��2Pyy � Pxx � Pzz�Pxyi; (B.264)S35p28 = �1� hPrKxyyz � 13��2Pyy � Pxx � Pzz�Pxz � 2�PxyPyzi; (B.265)S35p29 = �1� hPr Kxyzz � 13��2Pzz � Pxx � Pyy�Pxy � 2�PxzPyzi; (B.266)S35p30 = �1� hPrKxzzz � 1��2Pzz � Pxx � Pyy�Pxzi; (B.267)S35p31 = �1� hPrKyyyy � 13��2Pyy � Pxx � Pzz�2i; (B.268)

292S35p32 = �1� hPr Kyyyz � 1��2Pyy � Pxx � Pzz�Pyzi; (B.269)S35p33 = �1� hPrKyyzz � 19��2Pyy � Pxx � Pzz��2Pzz � Pxx � Pyy�� 2�P 2yzi;(B.270)S35p34 = �1� hPrKyzzz � 1��2Pzz � Pxx � Pyy�Pyzi; (B.271)S35p35 = �1� hPr Kzzzz � 13��2Pzz � Pxx � Pyy�2i: (B.272)Likewise, the components of the conservative source term vector S35c areS35c1 = 0; (B.273)S35c2 = 0; (B.274)S35c3 = 0; (B.275)S35c4 = 0; (B.276)S35c5 = � 13� �2Pxx � Pyy � Pzz�; (B.277)S35c6 = �1� Pxy; (B.278)S35c7 = �1� Pxz; (B.279)S35c8 = � 13� �2Pyy � Pxx � Pzz�; (B.280)S35c9 = �1� Pyz ; (B.281)

293S35c10 = � 13� �2Pzz � Pxx � Pyy�; (B.282)S35c11 = �1� hPr Qxxx + ux�2Pxx � Pyy � Pzz�i; (B.283)S35c12 = �1� hPr Qxxy + 2uxPxy + 13uy�2Pxx � Pyy � Pzz�i; (B.284)S35c13 = �1� hPr Qxxz + 2uxPxz + 13uz�2Pxx � Pyy � Pzz�i; (B.285)S35c14 = �1� hPr Qxyy + 13ux�2Pyy � Pxx � Pzz�+ 2uyPxyi; (B.286)S35c15 = �1� hPr Qxyz + uxPyz + uyPxz + uzPxyi; (B.287)S35c16 = �1� hPr Qxzz + 13ux�2Pzz � Pxx � Pyy�+ 2uzPxzi; (B.288)S35c17 = �1� hPr Qyyy + uy�2Pyy � Pxx � Pzz�i; (B.289)S35c18 = �1� hPr Qyyz + 2uyPyz + 13uz�2Pyy � Pxx � Pzz�i; (B.290)S35c19 = �1� hPr Qyzz + 13uy�2Pzz � Pxx � Pyy�+ 2uzPyzi; (B.291)S35c20 = �1� hPr Qzzz + uz�2Pzz � Pxx � Pyy�i; (B.292)

294S35c21 = �1� hPr Kxxxx + 4Pr uxQxxx + 3�P 2xx � 13��Pxx + Pyy + Pzz�2+ 2u2x�2Pxx � Pyy � Pzz�i; (B.293)S35c22 = �1� hPr Kxxxy + 3Pr uxQxxy + Pr uyQxxx + 3�PxxPxy + 3u2xPxy+ uxuy�2Pxx � Pyy � Pzz�i; (B.294)S35c23 = �1� hPr Kxxxz + 3Pr uxQxxz + Pr uzQxxx + 3�PxxPxz + 3u2xPxz+ uxuz�2Pxx � Pyy � Pzz�i; (B.295)S35c24 = �1� hPr Kxxyy + 2Pr uxQxyy + 2Pr uyQxxy + 1�PxxPyy + 2�P 2xy� 19��Pxx + Pyy + Pzz�2 + 13u2x�2Pyy � Pxx � Pzz�+ 4uxuyPxy+ 13u2y�2Pxx � Pyy � Pzz�i; (B.296)S35c25 = �1� hPrKxxyz + 2Pr uxQxyz + Pr uyQxxz + Pr uzQxxy + 1�PxxPyz+ 2�PxyPxz + u2xPyz + 2uxuyPxz + 2uxuzPxy + 13uyuz�2Pxx � Pyy � Pzz�i;(B.297)S35c26 = �1� hPrKxxzz + 2Pr uxQxzz + 2Pr uzQxxz + 1�PxxPzz + 2�P 2xz� 19��Pxx + Pyy + Pzz�2 + 13u2x�2Pzz � Pxx � Pyy�+ 4uxuzPxz+ 13u2y�2Pxx � Pyy � Pzz�i; (B.298)S35c27 = �1� hPr Kxyyy + Pr uxQyyy + 3Pr uyQxyy + 3�PxyPyy+ uxuy�2Pyy � Pxx � Pzz�+ 3u2yPxyi; (B.299)S35c28 = �1� hPrKxyyz + Pr uxQyyz + 2Pr uyQxyz + Pr uzQxyy + 1�PxzPyy+ 2�PxyPyz + 2uxuyPyz + 13uxuz�2Pyy � Pxx � Pzz�+ u2yPxz + 2uyuzPxyi;(B.300)

295S35c29 = �1� hPrKxyzz + Pr uxQyzz + Pr uyQxzz + 2Pr uzQxyz + 1�PxyPzz+ 2�PxzPyz + 13uxuy�2Pzz � Pxx � Pyy�+ 2uxuzPyz + 2uyuzPxz + u2zPxyi;(B.301)S35c30 = �1� hPr Kxzzz + Pr uxQzzz + 3Pr uzQxzz + 3�PxzPzz+ uxuz�2Pzz � Pxx � Pyy�+ 3u2zPxzi; (B.302)S35c31 = �1� hPr Kyyyy + 4Pr uyQyyy + 3�P 2yy � 13��Pxx + Pyy + Pzz�2+ 2u2y�2Pyy � Pxx � Pzz�i; (B.303)S35c32 = �1� hPr Kyyyz + Pr 3uyQyyz + Pr uzQyyy + 3�PyyPyz + 3u2yPyz+ uyuz�2Pyy � Pxx � Pzz�i; (B.304)S35c33 = �1� hPr Kyyzz + 2Pr uyQyzz + 2Pr uzQyyz + 1�PyyPzz + 2�P 2yz� 19��Pxx + Pyy + Pzz�2 + 13u2y�2Pzz � Pxx � Pyy�+ 4uyuzPyz+ 13u2z�2Pyy � Pxx � Pzz�i; (B.305)S35c34 = �1� hPrKyzzz + Pr uyQzzz + 3Pr uzQyzz + 3�PyzPzz+ uyuz�2Pzz � Pxx � Pyy�+ 3u2zPyzi; (B.306)S35c35 = �1� hPr Kzzzz + 4Pr uzQzzz + 3�P 2zz � 13��Pxx + Pyy + Pzz�2+ 2u2z�2Pzz � Pxx � Pyy�i: (B.307)

BIBLIOGRAPHY

296

297BIBLIOGRAPHY[1] H. Alsmeyer. Density pro�les in argon and nitrogen shock waves measured bythe absorption of an electron beam. Journal of Fluid Mechanics, 74:497{513,April 1976.[2] K. E. Atkinson. Elementary Numerical Analysis. John Wiley & Sons, 1993.[3] P. L. Bhatnagar. A model for collisional processes in gases. I. Small ampli-tude processes in charged and neutral one-component systems. Physical Review,94(3):511{525, 1954.[4] G. A. Bird. Approach to transitional equilibrium in a rigid sphere gas. Physicsof Fluids, 6:1518{1519, 1963.[5] G. A. Bird. Shock wave structure in a rigid sphere gas. Rare�ed Gas Dynamics,4:216{221, 1965.[6] G. A. Bird. Perception of numerical methods in rare�ed gas dynamics. Progressin Aeronautics and Astronautics, 118:211{226, 1989.[7] G. A. Bird. Molecular Gas Dynamics and the Direct Simulation of Gas Flows.Oxford University Press, 1994.[8] A. V. Bobylev. The Chapman-Enskog and Grad methods for solving the Boltz-mann equation. Sov. Phys. Dokl., 27(1):29, 1982.[9] L. Boltzmann. Weitere studien �uber das w�armegleichgewicht unter gas-molek�ulen. Sitz. Math.-Naturwiss. Cl. Akad. Wiss. Wien, 66:275{370, 1872.[10] S. L. Brown, P. L. Roe, and C. P. T. Groth. Numerical solution of a 10-momentmodel for nonequilibrium gasdynamics. AIAA-95-1677-CP, 1995.[11] D. Burnett. The distribution of molecular velocities and the mean motion in anon-uniform gas. Proceedings of the London Mathematics Society, 40:382{435,1935.[12] D. Burnett. The distribution of velocities and the mean motion in a slightlynon-uniform gas. Proceedings of the London Mathematics Society, 39:385{430,1935.

298[13] S. Chapman. On the kinetic theory of a gas. part II - a composite monatomicgas: Di�usion, viscosity, and thermal conduction. Phil. Trans. Royal Soc. Lon-don, 217A:115, 1916.[14] S. Chapman and T. G. Cowling. The Mathematical Theory of Non-UniformGases. Cambridge University Press, 1970.[15] H. K. Cheng, C. J. Lee, E. Y. Wong, and H. T. Yang. Hypersonic slip ows andissues on extending continuum model beyond the Navier-Stokes level. AIAA-89-1663, 1989.[16] H. K. Cheng and E. Y. Wong. A shock-layer theory based on thirteen-momentequations and DSMC calculations of rare�ed hypersonic ows. AIAA-91-0783,1991.[17] P. Colella and P. R. Woodward. The piecewise parabolic method (PPM) for gasdynamical simulations. JCP, 54:174{210, 1984.[18] K. A. Comeaux, D. R. Chapman, and R. W. MacCormack. An analysis of theBurnett equations based on the second law of thermodynamics. AIAA-95-0415,1995.[19] D. Enskog. Kinetishe Theorie der Vorg�aange in Massing Verdumten Gases.PhD thesis, University of Upsala, 1917.[20] K. A. Fiscko and D. R. Chapman. Comparison of Burnett, super-Burnett andMonte Carlo solutions for hypersonic shock structure. Progress in Aeronauticsand Astronautics, 118:374{395, 1989.[21] W. Fiszdon, R. Herczynski, and A. Walenta. The structure of a plane shockwave of a monatomic gas: Theory and experiment. Rare�ed Gas Dynamics:Proceedings of the Ninth International Symposium, 2:Ax.B. 23{1 { Ax.B. 23{57,1974.[22] D. Gilbarg and D. Paolucci. The structure of shock waves in the continuumtheory of uids. Journal of Rational Mechanical Analysis, 2:617{642, 1953.[23] J. Glimm. Solutions in the large for nonlinear hyperbolic systems of equations.CPAM, 18:695{715, 1965.[24] S. K. Godunov. Finite di�erence method for numerical computation of discon-tinuous solutions of the equations of uid dynamics. Matematicheskii Sbornik,47(3):271{306, 1959.[25] S. K. Godunov, A. W. Zabrodyn, and G. P. Prokopov. A di�erence scheme fortwo-dimensional unsteady problems of gas dynamics and computation of owwith a detached shock wave. Zhurnal Vychyslitelnoi Matematiki i Matematich-eskoi Fiziki, 1(6):1020, 1961.

299[26] T. I. Gombosi. Gaskinetic Theory. Cambridge University Press, 1994.[27] T. I. Gombosi, C. P. T. Groth, P. L. Roe, and S. L. Brown. 35-moment clo-sure for rare�ed gases: Derivation, transport equations, and wave structure. inpreparation.[28] J. J. Gottlieb and C. P. T. Groth. Assessment of Riemann solvers for unsteadyone-dimensional inviscid ows of perfect gases. JCP, 78(2):437{458, October1988.[29] H. Grad. On the kinetic theory of gases. Communications on Pure and AppliedMathematics, 2:331{407, 1949.[30] H. Grad. The pro�le of a steady plane shock wave. CPAM, 5:257{300, 1952.[31] H. Grad. Theory of rare�ed gases. In F. M. Devienne, editor, Rare�ed GasDynamics, volume 3 of International Series on Aeronautical Sciences and SpaceFlight, pages 100{138. Pergamon Press, 1960.[32] C. P. T. Groth. Private communication. University of Michigan, 1995.[33] C. P. T. Groth, P. L. Roe, T. I. Gombosi, and S. L. Brown. On the nonstationarywave structure of a 35-moment closure for rare�ed gas dynamics. AIAA Paper95-2312, 1995.[34] A. Harten. High-resolution schemes for hyperbolic conservation laws. Journalof Computational Physics, 49:357{393, 1983.[35] F. Hertwick. Allgemeine 13-momenten-n�aherung zur Fokker-Planck-gleichungeines plasmas. Zeitschrift f�ur Naturforschung, 20a:1243{1255, 1965.[36] C. Hirsch. Numerical Computation of Internal and External Flows, Volume 1,Fundamentals of Numerical Discretization. John Wiley & Sons, 1989.[37] C. Hirsch. Numerical Computation of Internal and External Flows, Volume 2,Computational Methods for Inviscid and Viscous Flows. John Wiley & Sons,1990.[38] B. L. Holian, C. W. Patterson, M. Mareschal, and E. Salomons. Modelingshock waves in an ideal gas: Going beyond the Navier-Stokes level. Phys. Rev.E, 47(1):R24{R27, 1993.[39] T. Holtz and E. P. Muntz. Molecular velocity distribution functions in an argonnormal shock wave at mach number 7. Physics of Fluids, 26(9):2425{2436, 1983.[40] L. H. Holway. Approximation Procedures for Kinetic Theory. PhD thesis, Har-vard University, 1963.

300[41] L. H. Holway. Kinetic theory of shock structure using an ellipsoidal distributionfunction. In J. H. de Leeuw, editor, Rare�ed Gas Dynamics, volume 1, pages193{215. Academic Press, 1966.[42] L. H. Holway. The e�ect of collisional models upon shock structure. In C. L.Brundin, editor, Rare�ed Gas Dynamics, volume 1, pages 759{784. AcademicPress, 1967.[43] S. T. Imlay. Solution of the Burnett equations for hypersonic ows near thecontinuum limit. AIAA 92-2922, 1992.[44] G. Koppenwallner. Low Reynolds number in uence on aerodynamic perfor-mance of hypersonic lifting vehicles. In Aerodynamics of Hypersonic LiftingVehicles, CP-428, pages 11{1{11{14. AGARD, 1987.[45] P. D. Lax. Hyperbolic Systems of Conservation Laws and the MathematicalTheory of Shock Waves. Society for Industrial and Applied Mathematics, 1975.[46] B. Van Leer, W.-T. Lee, and K. G. Powell. Sonic-point capturing. AIAA-89-1945-CP, 1989.[47] R. J. Leveque. Numerical Methods for Conservation Laws. Birkh�auser, 1992.[48] C. D. Levermore. Moment closure heirarchies for kinetic theory. Journal ofStatistical Physics, May 1995. submitted.[49] C. D. Levermore, W. J. Moroko�, and B. T. Nadiga. Moment realizibilityand the validity of the Navier-Stokes approximation for rare�ed gas dynamics.Physics of Fluids, 1995. submitted.[50] T.-P. Liu. Hyperbolic conservation laws with relaxation. Communications inMathematical Physics, 108:153{175, 1987.[51] F. E. Lumpkin and D. R. Chapman. Accuracy of the Burnett equations forhypersonic real gas ows. AIAA Paper 91-0771, 1991.[52] J. C. Maxwell. On the dynamical theory of gases. Philosophical Transactionsof the Royal Society of London, 157:49{88, 1867.[53] I. M�uller and T. Ruggeri. Extended Thermodynamics. Springer-Verlag, 1993.[54] V. Oraevskii, R. Chodura, and W. Feneberg. Hydrodynamic equations for plas-mas in strong magnetic �elds - I collisionless approximation. Plasma Physics,10:819{828, 1968.[55] G. C. Pham-Van-Diep and D. A. Erwin. Validation of MCDS by comparison ofpredicted with experimental velocity distribution functions in rare�ed normalshocks. Progress in Aeronautics and Astronautics, 118:271, 1989.

301[56] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. NumericalRecipes. Cambridge University Press, 1988.[57] P. L. Roe. The use of the Riemann problem in �nite-di�erence schemes. LectureNotes in Physics, 141:354{359, 1980.[58] P. L. Roe. Approximate Riemann solvers, parameter vectors, and di�erenceschemes. Journal of Computational Physics, 43:357{372, 1981.[59] P. L. Roe. Some contributions to the modelling of discontinuous ows. In Proc.1983 AMS-SIAM Summer Seminar on Large Scale Computing in Fluid Dynam-ics, volume 22, pages 163{193. Society for Industrial and Applied Mathematics,1985.[60] P. L. Roe and M. Arora. Characteristic-based schemes for dispersive waves, I.The method of characteristics for smooth solutions. Numer. Meth. Part. Di�.Eq., 9:459{505, 1993.[61] P. L. Roe and J. Pike. E�cient construction and utilisation of approximateRiemann solutions. In R. Glowinski and J. L. Lions, editors, Computing Methodsin Applied Sciences and Engineering. North Holland, 1984.[62] F. S. Sherman and L. Talbot. Experiment versus kinetic theory for rare�edgases. In F. M. Devienne, editor, Rare�ed Gas Dynamics, volume 1, pages161{191. Pergamon Press, 1960.[63] J. L. Steger and R. F. Warming. Flux vector splitting of the inviscid gas dy-namics equations with application to �nite-di�erence methods. JCP, 40:263,1981.[64] G. Strang. On the construction and comparison of di�erence schemes. SIAMJour. Num. Anal., 5:506, 1968.[65] L. Talbot and F. S. Sherman. Structure of weak shock waves in a monatomicgas. Technical report, NASA Memorandum 12-14-58W, 1959.[66] G. D. van Albada, B. van Leer, and Jr. W. W. Roberts. A comparative study ofcomputational methods in cosmic gas dynamics. Astronomy and Astrophysics,108:76{84, 1982.[67] B. van Leer. Towards the ultimate conservative di�erence scheme. V. A second-order sequel to Godunov's method. JCP, 32:101{136, 1979.[68] B. van Leer. Flux-vector splitting for the Euler equations. Lecture Notes inPhysics, 170:507, 1982.[69] J. P. Vila. Construction of Roe-type matrices, a new approach. application toreal gas dynamics. In Engquist and Gustafsson, editors, Proceedings of the 3rdInternational Conference on Hyperbolic Problems. Chartwell-Bratt, 1991.

302[70] W. G. Vincenti and Jr. C. H. Kruger. Introduction to Physical Gas Dynamics.Krieger Publishing Company, 1965.[71] M. Vinokur. Flux Jacobian matrices and generalized Roe average for an equi-librium real gas. NASA CR 177512, 1988.[72] R. von Mises. On the thickness of steady shock waves. Journal of AeronauticalSciences, 17:551{554, 1950.[73] G. B. Whitham. Linear and Nonlinear Waves. John Wiley & Son, 1974.[74] L. C. Woods. Frame-indi�erent kinetic theory. Journal of Fluid Mechanics,136:423{433, 1983.[75] X. Zhong, R. W. MacCormack, and D. R. Chapman. Stabilization of the Bur-nett equations and application to high-altitude hypersonic ows. AIAA Paper91-0770, 1991.

ABSTRACTAPPROXIMATE RIEMANN SOLVERS FOR MOMENT MODELS OF DILUTEGASESbyShawn Lee BrownChairperson: Philip L. RoeTwo sets of transport equations based on the 10- and 35-moment models ofkinetic theory are presented as possible models for the prediction of transitional ows. The 10-moment model is based on the Gaussian distribution function and isde�cient in heat transfer while the 35-moment distribution function is an expansionof the Gaussian and is capable of modelling heat transfer. Detailed eigensystemand linearized dispersion analyses on these systems revealed hyperbolicity of thetransport equations with elegant wave structures. This hyperbolicity allows for theuse of powerful numerical schemes which take advantage of the wave-like nature ofthe physics. Roe-type approximate Riemann solvers are developed for both of themoment models with three Riemann solvers given for the 35-moment model. Two ofthe 35-moment solvers use eigenstructures obtained at near-equilibrium conditionsalong with a correction so that the correct ux jumps are obtained. The third solver

1uses the non-equilibrium eigenstructure with the eigenvectors expressed as functionsof the eigenvalues. A relationship exists among terms which make up the non-equilibrium characteristic polynomial of the 35-moment model which is exploitedto provide e�cient numerical solution of the non-equilibrium eigenvalues. Thesesolvers were signi�cantly less expensive in terms of computational costs than a 35-moment solver which obtained the eigenvectors numerically. An additional conditionis presented for Property U which leads to unique Roe-averages for moment models.This condition must be satis�ed if the left eigenvectors obtained from the primitiveform of the transport equations are used in the solver.The �rst known solutions of these models are obtained for the problem of one-dimensional shock structure for a monatomic gas with comparison made to theNavier-Stokes and direct simulation Monte Carlo methods. The solutions are ex-tended to second-order accuracy using Hancock's Predictor/Corrector technique.The results reveal that for the 10-moment model there is no upper limit on thein ow Mach number while solutions are unobtainable above an in ow Mach numberof approximately two for the 35-moment model. At low in ow Mach numbers the 35-moment model provides solutions in excellent agreement with the DSMC solutionsat a computational cost several orders of magnitude less expensive. At higher in owMach numbers the agreement, while not excellent, remains good. An e�ciency studyreveals that even though the cell cost of the moment models is greater, with respectto the Navier-Stokes cell cost, there is signi�cant saving in iteration count at thehigher Mach numbers.