TRANSONIC, SHOCK, AND MULTIDIMENSIONAL FLOWS: ADVANCES IN SCIENTIFIC COMPUTING
CONTENTS 1 INTRODUCTION 1 1 CONSERVATION LAWS FOR A CONTINUUM MEDIUM 1 1 1 CONSERVATION OF MASS 1 1 2 CONSERVATION OF MOMENTUM 1 1 3 CONSERVATION OF ENERGY 1 1 4 CONSTITUTIVE RELATIONS 1 2 GOVERNING EQUATIONS IN TERMS OF PRIMITIVE VARIABLES 1 2 1 VECTOR FORM 1 2 2 COMPONENT FORM IN CARTESIAN COORDINATES 1 2 3 COMPONENT FORM IN CYLINDRICAL COORDINATES 1 2 4 SUMMARY 1 3 SPECIES TRANSPORT EQUATIONS 1 4 GOVERNING EQUATIONS IN TRANSLATING AND ROTATING FRAMES OF REFERENCE 1 5 BOUNDARY AND INITIAL CONDITIONS 1 5 1 GENERAL BOUNDARY CONDITIONS 1 5 2 FREE BOUNDARY CONDITIONS 1 5 3 MOVING INTERFACE CONDITIONS 1 5 4 PHASE CHANGE CONDITIONS 1 6 GOVERNING EQUATIONS FOR FLOWS THROUGH POROUS MEDIA 1 7 GOVERNING EQUATIONS IN CONSERVATION FORM EXERCISES REFERENCES 2 DISCONTINUOUS FINITE ELEMENT PROCEDURES 2 1 THE CONCEPT OF DISCONTINUOUS FINITE ELEMENTS 2 1 1 WEAKLY IMPOSED CROSS-ELEMENT CONTINUITY 2 1 2 NUMERICAL BOUNDARY FLUXES FOR DISCONTINUITY 2 1 3 BOUNDARY CONSTRAINT MINIMIZATION 2 1 4 TREATMENT OF DISCONTINUITY FOR NON-CONSERVATIVE SYSTEMS 2 1 5 TRANSIENT PROBLEMS 2 2 DISCONTINUOUS FINITE ELEMENT FORMULATION 2 2 1 INTEGRAL FORMULATION 2 2 2 TIME INTEGRATION 2 3 SOLUTION PROCEDURES 2 4 ADVANTAGES AND DISADVANTAGES OF DISCONTINUOUS FINITE ELEMENT FORMULATIONS 2 4 1 ADVANTAGES 2 4 2 DISADVANTAGES 2 5 EXAMPLES 3 SHAPE FUNCTIONS AND ELEMENTAL CALCULATIONS 3 1 SHAPE FUNCTIONS 3 1 1 1-D SHAPE FUNCTIONS 3 1 2 2-D SHAPE FUNCTIONS 3 1 2 1 TRIANGULAR ELEMENTS 3 1 2 2 QUADRILATERAL ELEMENTS 3 1 3 3-D SHAPE FUNCTIONS 3 1 3 1 TETRAHEDRAL ELEMENTS 3 1 3 2 HEXAHEDRAL ELEMENTS 3 2 CONSTRUCTION OF SPECIAL ELEMENTS 3 2 1 NON-STANDARD ELEMENTS 3 2 2 CONSTRUCTION OF ELEMENT SHAPE FUNCTIONS BY NODE COLLAPSING 3 2 3 SPECTRAL ELEMENTS 3 3 HIERARCHICAL SHAPE FUNCTIONS 3 3 1 1-D HIERARCHICAL CORRECTION 3 3 2 CANONICAL SQUARE AND CUBIC ELEMENTS 3 3 3 TRIANGULAR AND TETRAHEDRAL ELEMENTS 3 3 4 OBTAINING HIERARCHICAL ELEMENTS THROUGH COORDINATE TRANSFORMATIONS 3 3 5 ORTHOGONAL MASS MATRIX CONSTRUCTION 3 4 INTERPOLATION ERROR ANALYSIS 3 4 1 HILBERT SPACE AND VARIOUS ERROR MEASURES 3 4 2 INTERPOLATION ERROR ANALYSIS FOR 1-D ELEMENTS 3 4 3 INTERPOLATION ERROR ANALYSIS FOR 2-D/3-D ELEMENTS 3 5 NUMERICAL INTEGRATION 3 5 1 1-D NUMERICAL INTEGRATION 3 5 2 2-D AND 3-D NUMERICAL INTEGRATION 3 5 3 INTEGRATION FOR TRIANGULAR AND TETRAHEDRAL ELEMENTS 3 6 ELEMENTAL CALCULATIONS 3 6 1 DOMAIN CALCULATIONS 3 6 2 BOUNDARY CALCULATIONS 4 CONDUCTION HEAT TRANSFER AND POTENTIAL FLOWS 4 1 1-D STEADY STATE HEAT CONDUCTION 4 2 STEADY STATE HEAT CONDUCTION IN MULTIDIMENSIONS 4 3 1-D TRANSIENT HEAT CONDUCTION 4 3 1 ALTERNATING UPWINDING SCHEME 4 3 2 CENTRAL FLUXES 4 3 3 UNIFIED REPRESENTATION 4 3 4 NUMERICAL IMPLEMENTATION 4 3 5 RUNGEKUTTA TIME INTEGRATION 4 3 6 COMPUTATIONAL PROCEDURES 4 4 TRANSIENT HEAT CONDUCTION IN MULTIDIMENSIONS 4 5 POTENTIAL FLOWS AND FLOWS IN POROUS MEDIA 4 6 SELECTION OF NUMERICAL FLUXES 4 6 1 STABILITY FOR STEADY STATEPROBLEMS 4 6 1 1 STABILITY AND NUMERICAL FLUXES 4 6 1 2 DISCONTINUOUS AND MIXED FINITE ELEMENT FORMULATIONS 4 6 2 STABILITY FOR TIME DEPENDENT PROBLEMS 4 6 2 1 NUMERICAL FLUXES FOR TRANSIENT PROBLEMS 4 6 2 2 STABILITY ANALYSIS USING MATRIX 4 6 3 FOURIER ANALYSIS 5 CONVECTION-DOMINATED PROBLEMS 5 1 PURE CONVECTION PROBLEMS 5 1 1 1-D PURE CONVECTION 5 1 1 1 METHOD OF CHARACTERISTICS 5 1 1 2 DISCONTINUOUS FINITE ELEMENT FORMULATION 5 1 2 PURE CONVECTION IN MULTIDIMENSIONS 5 1 3 STABILITY ANALYSIS 5 1 3 1 L2-STABILITY – INTEGRAL ANALYSIS 5 1 3 2 L2-STABILITY – DISCRETIZED ANALYSIS 5 1 3 3 FOURIER ANALYSIS 5 2 STEADY STATE CONVECTION-DIFFUSION 5 2 1 1-D PROBLEM 5 2 2 ORIGIN OF OSCILLATORY STABILITY 5 2 3 STEADY CONVECTION-DIFFUSION IN MULTIDIMENSIONS 5 3 TRANSIENT CONVECTION-DIFFUSION 5 3 1 MULTIDIMENSIONAL PROBLEM 5 3 2 STABILITY ANALYSIS 5 3 2 1 L2-STABILITY – INTEGRAL ANALYSIS 5 3 2 2 L2-STABILITY – DISCRETIZED ANALYSIS 5 3 2 3 FOURIER ANALYSIS 5 4 NONLINEAR PROBLEMS 5 4 1 1-D INVISCID BURGERS’ EQUATION 5 4 1 1 BASIC CONSIDERATIONS 5 4 1 2 DISCONTINUOUS FINITE ELEMENT FORMULATION 5 4 2 MULTIDIMENSIONAL INVISCID BURGERS’ EQUATION 5 4 2 1 DISCONTINUOUS FINITE ELEMENT FORMULATION 5 4 2 2 CHARACTERISTIC DECOMPOSITION 5 4 3 HIGHER ORDER APPROXIMATIONS AND TVD FORMULATIONS 5 4 3 1 CONCEPT OF TOTAL VARIATION DIMINISHING 5 4 3 2 FLUX LIMITERS 5 4 3 3 SLOPE LIMITERS 5 4 3 4 TVD-RUNGEKUTTA SCHEMES 5 5 VISCOUS BURGERS’ EQUATIONS 5 5 1 1-D BURGERS’ EQUATION 5 5 2 2-D VISCOUS BURGERS’ EQUATION 6 INCOMPRESSIBLE FLOWS 6 1 PRIMITIVE VARIABLE APPROACH 6 2 FRACTIONAL STEP (PROJECTION) APPROACH 6 3 VORTICITY AND STREAM FUNCTION APPROACH 6 4 COUPLED FLOW AND HEAT TRANSFER 7 COMPRESSIBLE FLUID FLOWS 7 1 1-D COMPRESSIBLE FLOWS 7 1 1 GOVERNING EQUATIONS 7 1 2 BASIC PROPERTIES OF THE EULER EQUATIONS 7 1 3 THE RANKINE–HUGONIOT CONDITIONS 7 1 4 1-D RIEMANN SOLVER – EXACT SOLUTION 7 1 5 1-D RIEMANN SOLVER – APPROXIMATE SOLUTION 7 1 6 DISCONTINUOUS FINITE ELEMENT FORMULATION 7 1 7 LOW ORDER (FINITE VOLUME) APPROXIMATIONS 7 1 7 1 THE GODUNOV SCHEME 7 1 7 2 THE ROE SCHEME 7 1 8 HIGH ORDER TVD APPROXIMATIONS 7 1 9 NUMERICAL EXAMPLES 7 2 MULTIDIMENSIONAL INVISCID COMPRESSIBLE FLOWS 7 2 1 GOVERNING EQUATIONS 7 2 2 BASIC PROPERTIES OF THE SPLIT 3-D EULER EQUATIONS 7 2 3 DISCONTINUOUS FINITE ELEMENT FORMULATION 7 3 MULTIDIMENSIONAL COMPRESSIBLE VISCOUS FLOWS 7 4 ALE FORMULATION 7 4 1 ALE KINEMATIC DESCRIPTION 7 4 2 CONSERVATION OF MASS 7 4 3 CONSERVATION OF MOMENTUM 7 4 4 CONSERVATION OF ENERGY 7 4 5 SUMMARY OF ALE EQUATIONS 7 4 6 CONSTITUTIVE RELATIONS 7 4 7 ALE DESCRIPTION OF COMPRESSIBLE FLOWS 7 4 8 DISCONTINUOUS FINITE ELEMENT FORMULATION 8 EXTERNAL RADIATIVE HEAT TRANSFER 8 1 INTEGRAL EQUATION FOR SURFACE RADIATION EXCHANGES 8 1 1 GOVERNING EQUATION 8 1 2 KERNEL FUNCTIONS 8 2 DISCONTINUOUS GALERKIN FINITE ELEMENT FORMULATION 8 3 SHADOWING ALGORITHMS 8 3 1 SHADOWING ALGORITHM FOR 2-D GEOMETRY 8 3 2 SHADOWING ALGORITHM FOR AXISYMMETRIC CONFIGURATIONS 8 3 3 SHADOWING ALGORITHM FOR 3-D GEOMETRY 8 4 COUPLING WITH OTHER HEAT TRANSFER CALCULATIONS 8 4 1 DIRECT COUPLING 8 4 2 ITERATIVE COUPLING 8 5 NUMERICAL EXAMPLES 9 RADIATIVE TRANSFER IN PARTICIPATING MEDIA 9 1 GOVERNING EQUATION AND BOUNDARY CONDITIONS 9 1 1 RADIATIVE TRANSFER EQUATION 9 1 2 BOUNDARY CONDITIONS 9 2 APPROXIMATION METHODS 9 2 1 THE DISCRETE ORDINATE METHOD 9 2 2 THE SPHERICAL HARMONICS METHOD 9 3 DISCONTINUOUS FINITE ELEMENT FORMULATION 9 4 NUMERICAL IMPLEMENTATION 9 4 1 2-D CALCULATIONS 9 4 2 3-D CALCULATIONS 9 4 3 INTEGRATION OF THE SOURCE TERM 9 4 3 1 THE EMITTING CONTRIBUTION 9 4 3 2 THE SCATTERING CONTRIBUTION 9 5 RADIATION IN SYSTEMS OF AXISYMMETRY 9 5 1 GOVERNING EQUATION IN CYLINDRICAL COORDINATES 9 5 2 VOLUME INTEGRATION 9 5 3 SURFACE INTEGRATION OVER *P 9 5 4 INTEGRATION OVER *M 9 5 5 MAPPING 9 5 6 TREATMENT OF THE EMITTING AND SCATTERING TERM USE OF RTE FOR EXTERNAL RADIATION CALCULATIONS 9 7 COUPLING OF THE DISCONTINUOUS METHOD WITH OTHER METHODS 9 8 CONSTANT ELEMENT APPROXIMATION 10 FREE AND MOVING BOUNDARY PROBLEMS 10 1 FREE AND MOVING BOUNDARIES 10 2 BASIC RELATIONS FOR A CURVED SURFACE 10 2 1 DESCRIPTION OF A SURFACE 10 2 2 DIFFERENTIAL AND INTEGRAL RELATIONS FOR CURVED SURFACES 10 3 PHYSICAL CONSTRAINTS AT A MOVING BOUNDARY 10 3 1 KINEMATIC CONDITIONS AT A MOVING BOUNDARY 10 3 2 STRESS CONDITION AT A MOVING INTERFACE 10 3 3 THERMAL CONDITIONS AT A MOVING INTERFACE 10 4 MOVING GRIDS VS FIXED GRIDS FOR NUMERICAL SOLUTIONS 10 5 MOVING GRID METHODS 10 5 1 MOVING BOUNDARIES BETWEEN FLUIDS 10 5 2 MOVING PHASE BOUNDARIES 10 5 2 1 SOLIDLIQUID PHASE TRANSITION 10 5 2 2 LIQUIDVAPOR PHASE TRANSITION 10 6 FIXED GRID METHODS 10 6 1 VOLUME OF FLUID METHOD 10 6 1 1 STRUCTURED MESH 10 6 1 2 UNSTRUCTURED MESH 10 6 2 THE MARKER-AND-CELL METHOD 10 6 3 THE LEVEL SET METHOD 10 6 4 FIXED GRID METHODS FOR PHASE CHANGE PROBLEMS 10 6 4 1 FLOW-BASED METHODS 10 6 4 2 ENTHALPY-BASED METHODS 10 7 PHASE FIELD MODELING OF MOVING BOUNDARIES 10 7 1 BASIC IDEAS OF PHASE FIELD MODELS 10 7 2 GOVERNING EQUATIONS FOR INTERFACIAL PHENOMENA 10 7 3 DISCONTINUOUS FINITE ELEMENT FORMULATION 10 7 4 PHASE FIELD MODELING OF MICROSTRUCTURE EVOLUTION 10 7 4 1 GOVERNING EQUATIONS 10 7 4 2 DISCONTINUOUS FORMULATION 10 7 4 3 NUMERICAL EXAMPLES 10 7 5 FLOW AND ORIENTATION EFFECTS ON MICROSTRUCTURE EVOLUTION 10 7 5 1 FLOW EFFECTS ON MICROSTRUCTURE EVOLUTION 10 7 5 2 MICROSTRUCTURE EVOLUTION DURING POLYCYSTALLINE SOLIDIFICATION 11 MICRO AND NANO SCALE FLUID FLOW AND HEAT TRANSFER 11 1 MICRO SCALE HEAT CONDUCTION 11 1 1 TWO-TEMPERATURE EQUATIONS 11 1 2 PHONON SCATTERING EQUATION 11 1 3 PHONON RADIATIVE TRANSFER EQUATION 11 2 DISCONTINUOUS FINITE ELEMENT FORMULATION 11 3 MICRO AND NANO FLUID FLOW AND HEAT TRANSFER 11 4 THE BOLTZMANN TRANSPORT EQUATION AND NUMERICAL SOLUTION 11 4 1 THE BOLTZMANN INTEGRAL-DIFFERENTIAL EQUATION 11 4 2 NUMERICAL SOLUTION OF THE BOLTZMANN TRANSPORT EQUATION 11 5 THE BOLTZMANBGK EQUATION AND NUMERICAL SOLUTION 11 5 1 THE BOLTZMANNBGK EQUATION 11 5 2 DISCONTINUOUS FINITE ELEMENT FORMULATION 11 6 THE LATTICE BOLTZMAN EQUATION AND NUMERICAL SOLUTION 11 6 1 DERIVATION OF THE LATTICE BOLTZMANN EQUATION 11 6 2 BOUNDARY CONDITIONS 11 6 2 1 BOUNCE BACK BOUNDARY CONDITIONS (NO SLIP) 11 6 2 2 SYMMETRIC BOUNDARY CONDITIONS (FREE SLIP) 11 6 2 3 INFLOW AND OUTFLOW BOUNDARY CONDITIONS (NO GRADIENT) 11 6 2 4 FORCE FIELD CONDITIONS 11 6 2 5 MOVING WALL CONDITIONS 11 6 3 DISCONTINUOUS FINITE ELEMENT FORMULATION 12 FLUID FLOW AND HEAT TRANSFER IN ELECTROMAGNETIC FIELDS 12 1 MAXWELL EQUATIONS AND BOUNDARY CONDITIONS 12 1 1 MAXWELL EQUATIONS 12 1 2 BOUNDARY CONDITIONS 12 1 2 1 INTERFACE BOUNDARY CONDITION 12 1 2 2 PERFECT CONDUCTING SURFACE 12 1 2 3 IMPEDANCE CONDITION 12 1 2 4 SOMMERFIELD RADIATION CONDITION 12 1 2 5 SYMMETRY BOUNDARY CONDITION 12 2 MAXWELL STRESSES AND ENERGY SOURCES 12 3 DISCONTINUOUS FORMULATION OF THE MAXWELL EQUATIONS 12 3 1 SOLUTION IN TIME DOMAIN 12 3 2 SOLUTION IN FREQUENCY DOMAIN 12 3 3 SOLUTION IN OTHER FORMS 12 4 ELECTROOSMOTIC FLOWS 12 4 1 GOVERNING EQUATIONS 12 4 2 DISCONTINUOUS FINITE ELEMENT FORMULATION 12 5 MICROWAVE HEATING 12 6 ELECTRICALLY DEFORMED FREE SURFACES 12 7 COMPRESSIBLE FLOWS IN MAGNETIC FIELDS