ADVANCED MECHANICS: DYNAMICS
CONTENTS PART I FOUNDATION 1 INTRODUCTION 3 1 1 WHAT IS COMPUTATIONAL FLUID DYNAMICS CFD 3 1 2 WHAT IS THE FINITE VOLUME METHOD 4 1 3 THIS BOOK 5 1 3 1 FOUNDATION 5 1 3 2 NUMERICS 6 1 3 3 ALGORITHMS 7 1 3 4 APPLICATIONS 8 1 4 CLOSURE 8 2 REVIEW OF VECTOR CALCULUS 9 2 1 INTRODUCTION 9 2 2 VECTORS AND VECTOR OPERATIONS 10 2 2 1 THE DOT PRODUCT OF TWO VECTORS 11 2 2 2 VECTOR MAGNITUDE 11 2 2 3 THE UNIT DIRECTION VECTOR 12 2 2 4 THE CROSS PRODUCT OF TWO VECTORS 12 2 2 5 THE SCALAR TRIPLE PRODUCT 14 2 2 6 GRADIENT OF A SCALAR AND DIRECTIONAL DERIVATIVES 15 2 2 7 OPERATIONS ON THE NABLA OPERATOR 17 2 2 8 ADDITIONAL VECTOR OPERATIONS 19 2 3 MATRICES AND MATRIX OPERATIONS 20 2 3 1 SQUARE MATRICES 21 2 3 2 USING MATRICES TO DESCRIBE SYSTEMS OF EQUATIONS 23 2 3 3 THE DETERMINANT OF A SQUARE MATRIX 23 2 3 4 EIGENVECTORS AND EIGENVALUES 26 2 3 5 A SYMMETRIC POSITIVE DEFINITE MATRIX 27 2 3 6 ADDITIONAL MATRIX OPERATIONS 28 2 4 TENSORS AND TENSOR OPERATIONS 29 2 5 FUNDAMENTAL THEOREMS OF VECTOR CALCULUS 32 2 5 1 GRADIENT THEOREM FOR LINE INTEGRALS 32 2 5 2 GREEN S THEOREM 33 2 5 3 STOKES THEOREM 34 2 5 4 DIVERGENCE THEOREM 35 2 5 5 LEIBNIZ INTEGRAL RULE 37 2 6 CLOSURE 38 2 7 EXERCISES 39 REFERENCES 41 3 MATHEMATICAL DESCRIPTION OF PHYSICAL PHENOMENA 43 3 1 INTRODUCTION 43 3 2 CLASSIFICATION OF FLUID FLOWS 44 3 3 EULERIAN AND LAGRANGIAN DESCRIPTION OF CONSERVATION LAWS 45 3 3 1 SUBSTANTIAL VERSUS LOCAL DERIVATIVE 46 3 3 2 REYNOLDS TRANSPORT THEOREM 47 3 4 CONSERVATION OF MASS CONTINUITY EQUATION 48 3 5 CONSERVATION OF LINEAR MOMENTUM 50 3 5 1 NON CONSERVATIVE FORM 51 3 5 2 CONSERVATIVE FORM 52 3 5 3 SURFACE FORCES 52 3 5 4 BODY FORCES 54 3 5 5 STRESS TENSOR AND THE MOMENTUM EQUATION FOR NEWTONIAN FLUIDS 55 3 6 CONSERVATION OF ENERGY 57 3 6 1 CONSERVATION OF ENERGY IN TERMS OF SPECIFIC INTERNAL ENERGY 60 3 6 2 CONSERVATION OF ENERGY IN TERMS OF SPECIFIC ENTHALPY 61 3 6 3 CONSERVATION OF ENERGY IN TERMS OF SPECIFIC TOTAL ENTHALPY 61 3 6 4 CONSERVATION OF ENERGY IN TERMS OF TEMPERATURE 62 3 7 GENERAL CONSERVATION EQUATION 65 3 8 NON DIMENSIONALIZATION PROCEDURE 67 3 9 DIMENSIONLESS NUMBERS 72 3 9 1 REYNOLDS NUMBER 72 3 9 2 GRASHOF NUMBER 73 3 9 3 PRANDTL NUMBER 73 3 9 4 P CLET NUMBER 75 3 9 5 SCHMIDT NUMBER 75 3 9 6 NUSSELT NUMBER 77 3 9 7 MACH NUMBER 77 3 9 8 ECKERT NUMBER 78 3 9 9 FROUDE NUMBER 79 3 9 10 WEBER NUMBER 79 3 10 CLOSURE 80 3 11 EXERCISES 80 REFERENCES 82 4 THE DISCRETIZATION PROCESS 85 4 1 THE DISCRETIZATION PROCESS 85 4 1 1 STEP I GEOMETRIC AND PHYSICAL MODELING 87 4 1 2 STEP II DOMAIN DISCRETIZATION 88 4 1 3 MESH TOPOLOGY 90 4 1 4 STEP III EQUATION DISCRETIZATION 93 4 1 5 STEP IV SOLUTION OF THE DISCRETIZED EQUATIONS 98 4 1 6 OTHER TYPES OF FIELDS 100 4 2 CLOSURE 101 5 THE FINITE VOLUME METHOD 103 5 1 INTRODUCTION 103 5 2 THE SEMI DISCRETIZED EQUATION 104 5 2 1 FLUX INTEGRATION OVER ELEMENT FACES 105 5 2 2 SOURCE TERM VOLUME INTEGRATION 107 5 2 3 THE DISCRETE CONSERVATION EQUATION FOR ONE INTEGRATION POINT 108 5 2 4 FLUX LINEARIZATION 109 5 3 BOUNDARY CONDITIONS 111 5 3 1 VALUE SPECIFIED DIRICHLET BOUNDARY CONDITION 111 5 3 2 FLUX SPECIFIED NEUMANN BOUNDARY CONDITION 112 5 4 ORDER OF ACCURACY 113 5 4 1 SPATIAL VARIATION APPROXIMATION 113 5 4 2 MEAN VALUE APPROXIMATION 114 5 5 TRANSIENT SEMI DISCRETIZED EQUATION 117 5 6 PROPERTIES OF THE DISCRETIZED EQUATIONS 118 5 6 1 CONSERVATION 118 5 6 2 ACCURACY 119 5 6 3 CONVERGENCE 119 5 6 4 CONSISTENCY 120 5 6 5 STABILITY 120 5 6 6 ECONOMY 120 5 6 7 TRANSPORTIVENESS 120 5 6 8 BOUNDEDNESS OF THE INTERPOLATION PROFILE 121 5 7 VARIABLE ARRANGEMENT 122 5 7 1 VERTEX CENTERED FVM 123 5 7 2 CELL CENTERED FVM 124 5 8 IMPLICIT VERSUS EXPLICIT NUMERICAL METHODS 126 5 9 THE MESH SUPPORT 127 5 10 COMPUTATIONAL POINTERS 128 5 10 1 UFVM 128 5 10 2 OPENFOAM 129 5 11 CLOSURE 133 5 12 EXERCISES 133 REFERENCES 134 6 THE FINITE VOLUME MESH 137 6 1 DOMAIN DISCRETIZATION 137 6 2 THE FINITE VOLUME MESH 138 6 2 1 MESH SUPPORT FOR GRADIENT COMPUTATION 139 6 3 STRUCTURED GRIDS 142 6 3 1 TOPOLOGICAL INFORMATION 142 6 3 2 GEOMETRIC INFORMATION 144 6 3 3 ACCESSING THE ELEMENT FIELD 145 6 4 UNSTRUCTURED GRIDS 146 6 4 1 TOPOLOGICAL INFORMATION CONNECTIVITIES 147 6 5 GEOMETRIC QUANTITIES 152 6 5 1 ELEMENT TYPES 153 6 5 2 COMPUTING SURFACE AREA AND CENTROID OF FACES 154 6 6 COMPUTATIONAL POINTERS 162 6 6 1 UFVM 162 6 6 2 OPENFOAM 164 6 7 CLOSURE 170 6 8 EXERCISES 170 REFERENCES 170 7 THE FINITE VOLUME MESH IN OPENFOAM AND UFVM 173 7 1 UFVM 173 7 1 1 AN OPENFOAM TEST CASE 173 7 1 2 THE POLYMESH FOLDER 175 7 1 3 THE UFVM MESH 178 7 1 4 UFVM GEOMETRIC FIELDS 183 7 1 5 WORKING WITH THE UFVM MESH 187 7 1 6 COMPUTING THE GAUSS GRADIENT 188 7 2 OPENFOAM 191 7 2 1 FIELDS AND MEMORY 197 7 2 2 INTERNALFIELD DATA 199 7 2 3 BOUNDARYFIELD DATA 200 7 2 4 LDUADDRESSING 200 7 2 5 COMPUTING THE GRADIENT 202 7 3 MESH CONVERSION TOOLS 204 7 4 CLOSURE 205 7 5 EXERCISES 205 REFERENCES 207 PART II DISCRETIZATION 8 SPATIAL DISCRETIZATION THE DIFFUSION TERM 211 8 1 TWO DIMENSIONAL DIFFUSION IN A RECTANGULAR DOMAIN 211 8 2 COMMENTS ON THE DISCRETIZED EQUATION 216 8 2 1 THE ZERO SUM RULE 216 8 2 2 THE OPPOSITE SIGNS RULE 217 8 3 BOUNDARY CONDITIONS 217 8 3 1 DIRICHLET BOUNDARY CONDITION 218 8 3 2 VON NEUMANN BOUNDARY CONDITION 220 8 3 3 MIXED BOUNDARY CONDITION 222 8 3 4 SYMMETRY BOUNDARY CONDITION 223 8 4 THE INTERFACE DIFFUSIVITY 224 8 5 NON CARTESIAN ORTHOGONAL GRIDS 239 8 6 NON ORTHOGONAL UNSTRUCTURED GRID 241 8 6 1 NON ORTHOGONALITY 241 8 6 2 MINIMUM CORRECTION APPROACH 242 8 6 3 ORTHOGONAL CORRECTION APPROACH 243 8 6 4 OVER RELAXED APPROACH 243 8 6 5 TREATMENT OF THE CROSS DIFFUSION TERM 244 8 6 6 GRADIENT COMPUTATION 244 8 6 7 ALGEBRAIC EQUATION FOR NON ORTHOGONAL MESHES 245 8 6 8 BOUNDARY CONDITIONS FOR NON ORTHOGONAL GRIDS 252 8 7 SKEWNESS 254 8 8 ANISOTROPIC DIFFUSION 255 8 9 UNDER RELAXATION OF THE ITERATIVE SOLUTION PROCESS 256 8 10 COMPUTATIONAL POINTERS 258 8 10 1 UFVM 258 8 10 2 OPENFOAM 260 8 11 CLOSURE 265 8 12 EXERCISES 265 REFERENCES 270