AERODYNAMICS 1: FUNDAMENTALS OF THEORY. AERODYNAMICS OF AN AIRFOIL AND A WING
CONTENTS PART I FOUNDATION INTRODUCTION WHAT IS COMPUTATIONAL FLUID DYNAMICS (CFD) WHAT IS THE FINITE VOLUME METHOD THIS BOOK FOUNDATION NUMERICS ALGORITHMS APPLICATIONS CLOSURE REVIEW OF VECTOR CALCULUS VECTORS AND VECTOR OPERATIONS THE DOT PRODUCT OF TWO VECTORS VECTOR MAGNITUDE THE UNIT DIRECTION VECTOR THE CROSS PRODUCT OF TWO VECTORS THE SCALAR TRIPLE PRODUCT GRADIENT OF A SCALAR AND DIRECTIONAL DERIVATIVES OPERATIONS ON THE NABLA OPERATOR ADDITIONAL VECTOR OPERATIONS MATRICES AND MATRIX OPERATIONS SQUARE MATRICES USING MATRICES TO DESCRIBE SYSTEMS OF EQUATIONS THE DETERMINANT OF A SQUARE MATRIX EIGENVECTORS AND EIGENVALUES A SYMMETRIC POSITIVE-DEFINITE MATRIX ADDITIONAL MATRIX OPERATIONS TENSORS AND TENSOR OPERATIONS FUNDAMENTAL THEOREMS OF VECTOR CALCULUS GRADIENT THEOREM FOR LINE INTEGRALS GREEN’S THEOREM STOKES’ THEOREM DIVERGENCE THEOREM LEIBNIZ INTEGRAL RULE EXERCISES REFERENCES MATHEMATICAL DESCRIPTION OF PHYSICAL PHENOMENA CLASSIFICATION OF FLUID FLOWS EULERIAN AND LAGRANGIAN DESCRIPTION OF CONSERVATION LAWS SUBSTANTIAL VERSUS LOCAL DERIVATIVE REYNOLDS TRANSPORT THEOREM CONSERVATION OF MASS (CONTINUITY EQUATION) CONSERVATION OF LINEAR MOMENTUM NON-CONSERVATIVE FORM CONSERVATIVE FORM SURFACE FORCES BODY FORCES STRESS TENSOR AND THE MOMENTUM EQUATION FOR NEWTONIAN FLUIDS CONSERVATION OF ENERGY CONSERVATION OF ENERGY IN TERMS OF SPECIFIC INTERNAL ENERGY CONSERVATION OF ENERGY IN TERMS OF SPECIFIC ENTHALPY CONSERVATION OF ENERGY IN TERMS OF SPECIFIC TOTAL ENTHALPY CONSERVATION OF ENERGY IN TERMS OF TEMPERATURE GENERAL CONSERVATION EQUATION NON-DIMENSIONALIZATION PROCEDURE DIMENSIONLESS NUMBERS REYNOLDS NUMBER GRASHOF NUMBER PRANDTL NUMBER PÉCLET NUMBER SCHMIDT NUMBER NUSSELT NUMBER MACH NUMBER ECKERT NUMBER FROUDE NUMBER WEBER NUMBER THE DISCRETIZATION PROCESS STEP I: GEOMETRIC AND PHYSICAL MODELING STEP II: DOMAIN DISCRETIZATION MESH TOPOLOGY STEP III: EQUATION DISCRETIZATION STEP IV: SOLUTION OF THE DISCRETIZED EQUATIONS OTHER TYPES OF FIELDS THE FINITE VOLUME METHOD THE SEMI-DISCRETIZED EQUATION FLUX INTEGRATION OVER ELEMENT FACES SOURCE TERM VOLUME INTEGRATION THE DISCRETE CONSERVATION EQUATION FOR ONE INTEGRATION POINT FLUX LINEARIZATION BOUNDARY CONDITIONS VALUE SPECIFIED (DIRICHLET BOUNDARY CONDITION) FLUX SPECIFIED (NEUMANN BOUNDARY CONDITION) ORDER OF ACCURACY SPATIAL VARIATION APPROXIMATION MEAN VALUE APPROXIMATION TRANSIENT SEMI-DISCRETIZED EQUATION PROPERTIES OF THE DISCRETIZED EQUATIONS CONSERVATION ACCURACY CONVERGENCE CONSISTENCY STABILITY ECONOMY TRANSPORTIVENESS BOUNDEDNESS OF THE INTERPOLATION PROFILE VARIABLE ARRANGEMENT VERTEX-CENTERED FVM CELL-CENTERED FVM IMPLICIT VERSUS EXPLICIT NUMERICAL METHODS THE MESH SUPPORT COMPUTATIONAL POINTERS UFVM OPENFOAM® THE FINITE VOLUME MESH DOMAIN DISCRETIZATION MESH SUPPORT FOR GRADIENT COMPUTATION STRUCTURED GRIDS TOPOLOGICAL INFORMATION GEOMETRIC INFORMATION ACCESSING THE ELEMENT FIELD UNSTRUCTURED GRIDS TOPOLOGICAL INFORMATION (CONNECTIVITIES) GEOMETRIC QUANTITIES ELEMENT TYPES COMPUTING SURFACE AREA AND CENTROID OF FACES THE FINITE VOLUME MESH IN OPENFOAM® AND UFVM AN OPENFOAM® TEST CASE THE POLYMESH FOLDER THE UFVM MESH UFVM GEOMETRIC FIELDS WORKING WITH THE UFVM MESH COMPUTING THE GAUSS GRADIENT FIELDS AND MEMORY INTERNALFIELD DATA BOUNDARYFIELD DATA LDUADDRESSING COMPUTING THE GRADIENT MESH CONVERSION TOOLS PART II DISCRETIZATION SPATIAL DISCRETIZATION: THE DIFFUSION TERM TWO-DIMENSIONAL DIFFUSION IN A RECTANGULAR DOMAIN COMMENTS ON THE DISCRETIZED EQUATION THE ZERO SUM RULE THE OPPOSITE SIGNS RULE DIRICHLET BOUNDARY CONDITION VON NEUMANN BOUNDARY CONDITION MIXED BOUNDARY CONDITION SYMMETRY BOUNDARY CONDITION THE INTERFACE DIFFUSIVITY NON-CARTESIAN ORTHOGONAL GRIDS NON-ORTHOGONAL UNSTRUCTURED GRID NON-ORTHOGONALITY MINIMUM CORRECTION APPROACH ORTHOGONAL CORRECTION APPROACH OVER-RELAXED APPROACH TREATMENT OF THE CROSS-DIFFUSION TERM GRADIENT COMPUTATION ALGEBRAIC EQUATION FOR NON-ORTHOGONAL MESHES BOUNDARY CONDITIONS FOR NON-ORTHOGONAL GRIDS SKEWNESS ANISOTROPIC DIFFUSION UNDER-RELAXATION OF THE ITERATIVE SOLUTION PROCESS COMPUTING GRADIENTS IN CARTESIAN GRIDS GREEN-GAUSS GRADIENT LEAST-SQUARE GRADIENT INTERPOLATING GRADIENTS TO FACES SOLVING THE SYSTEM OF ALGEBRAIC EQUATIONS DIRECT OR GAUSS ELIMINATION METHOD GAUSS ELIMINATION FORWARD ELIMINATION FORWARD ELIMINATION ALGORITHM BACKWARD SUBSTITUTION BACK SUBSTITUTION ALGORITHM LU DECOMPOSITION THE DECOMPOSITION STEP LU DECOMPOSITION ALGORITHM THE SUBSTITUTION STEP LU DECOMPOSITION AND GAUSS ELIMINATION LU DECOMPOSITION ALGORITHM BY GAUSS ELIMINATION DIRECT METHODS FOR BANDED SPARSE MATRICES TRIDIAGONAL MATRIX ALGORITHM (TDMA) PENTADIAGONAL MATRIX ALGORITHM (PDMA) ITERATIVE METHODS JACOBI METHOD GAUSS-SEIDEL METHOD PRECONDITIONING AND ITERATIVE METHODS MATRIX DECOMPOSITION TECHNIQUES INCOMPLETE LU (ILU) DECOMPOSITION INCOMPLETE LU FACTORIZATION WITH NO FILL-IN ILU(0) ILU(0) FACTORIZATION ALGORITHM ILU FACTORIZATION PRECONDITIONERS ALGORITHM FOR THE CALCULATION OF D IN THE DILU METHOD FORWARD AND BACKWARD SOLUTION ALGORITHM WITH THE DILU METHOD GRADIENT METHODS FOR SOLVING ALGEBRAIC SYSTEMS THE METHOD OF STEEPEST DESCENT THE CONJUGATE GRADIENT METHOD THE BI-CONJUGATE GRADIENT METHOD (BICG) AND PRECONDITIONED BICG THE MULTIGRID APPROACH ELEMENT AGGLOMERATION/COARSENING THE RESTRICTION STEP AND COARSE LEVEL COEFFICIENTS THE PROLONGATION STEP AND FINE GRID LEVEL CORRECTIONS TRAVERSAL STRATEGIES AND ALGEBRAIC MULTIGRID CYCLES DISCRETIZATION OF THE CONVECTION TERM STEADY ONE DIMENSIONAL CONVECTION AND DIFFUSION ANALYTICAL SOLUTION NUMERICAL SOLUTION A PRELIMINARY DERIVATION: THE CENTRAL DIFFERENCE (CD) SCHEME THE UPWIND SCHEME THE DOWNWIND SCHEME TRUNCATION ERROR: NUMERICAL DIFFUSION AND ANTI-DIFFUSION THE CENTRAL DIFFERENCE (CD) SCHEME NUMERICAL STABILITY HIGHER ORDER UPWIND SCHEMES SECOND ORDER UPWIND SCHEME THE INTERPOLATION PROFILE THE DISCRETIZED EQUATION TRUNCATION ERROR STABILITY ANALYSIS THE QUICK SCHEME THE FROMM SCHEME COMPARISON OF THE VARIOUS SCHEMES FUNCTIONAL RELATIONSHIPS FOR UNIFORM AND NON-UNIFORM GRIDS STEADY TWO DIMENSIONAL ADVECTION ERROR SOURCES HIGH ORDER SCHEMES ON UNSTRUCTURED GRIDS REFORMULATING HO SCHEMES IN TERMS OF GRADIENTS THE DEFERRED CORRECTION APPROACH HIGH RESOLUTION SCHEMES THE NORMALIZED VARIABLE FORMULATION (NVF) THE CONVECTION BOUNDEDNESS CRITERION (CBC) HIGH RESOLUTION (HR) SCHEMES THE TVD FRAMEWORK THE NVF-TVD RELATION HR SCHEMES IN UNSTRUCTURED GRID SYSTEMS DEFERRED CORRECTION FOR HR SCHEMES THE DIFFICULTY WITH THE DIRECT USE OF NODAL VALUES THE DWF AND NWF METHODS THE DOWNWIND WEIGHING FACTOR (DWF) METHOD THE NORMALIZED WEIGHING FACTOR (NWF) METHOD INLET BOUNDARY CONDITION OUTLET BOUNDARY CONDITION WALL BOUNDARY CONDITION TEMPORAL DISCRETIZATION: THE TRANSIENT TERM THE FINITE DIFFERENCE APPROACH FORWARD EULER SCHEME STABILITY OF THE FORWARD EULER SCHEME BACKWARD EULER SCHEME CRANK-NICOLSON SCHEME IMPLEMENTATION DETAILS ADAMS-MOULTON SCHEME THE FINITE VOLUME APPROACH FIRST ORDER TRANSIENT SCHEMES FIRST ORDER IMPLICIT EULER SCHEME FIRST ORDER EXPLICIT EULER SCHEME SECOND ORDER TRANSIENT EULER SCHEMES CRANK-NICHOLSON (CENTRAL DIFFERENCE PROFILE) SECOND ORDER UPWIND EULER (SOUE) SCHEME INITIAL CONDITION FOR THE FV APPROACH NON-UNIFORM TIME STEPS NON-UNIFORM TIME STEPS WITH THE FINITE DIFFERENCE APPROACH ADAMS-MOULTON (OR SOUE) SCHEME NON-UNIFORM TIME STEPS WITH THE FINITE VOLUME APPROACH DISCRETIZATION OF THE SOURCE TERM, RELAXATION, AND OTHER DETAILS SOURCE TERM DISCRETIZATION UNDER-RELAXATION OF THE ALGEBRAIC EQUATIONS UNDER-RELAXATION METHODS EXPLICIT UNDER-RELAXATION IMPLICIT UNDER-RELAXATION METHODS RESIDUAL FORM OF THE EQUATION RESIDUAL FORM OF PATANKAR’S UNDER-RELAXATION RESIDUALS AND SOLUTION CONVERGENCE RESIDUALS ABSOLUTE RESIDUAL MAXIMUM RESIDUAL ROOT-MEAN SQUARE RESIDUAL NORMALIZATION OF THE RESIDUAL PART III ALGORITHMS FLUID FLOW COMPUTATION: INCOMPRESSIBLE FLOWS THE MAIN DIFFICULTY A PRELIMINARY DERIVATION DISCRETIZATION OF THE MOMENTUM EQUATION DISCRETIZATION OF THE CONTINUITY EQUATION THE CHECKERBOARD PROBLEM THE STAGGERED GRID THE PRESSURE CORRECTION EQUATION THE SIMPLE ALGORITHM ON STAGGERED GRID PRESSURE CORRECTION EQUATION IN TWO DIMENSIONAL STAGGERED CARTESIAN GRIDS PRESSURE CORRECTION EQUATION IN THREE DIMENSIONAL STAGGERED CARTESIAN GRID DISADVANTAGES OF THE STAGGERED GRID THE RHIE-CHOW INTERPOLATION GENERAL DERIVATION THE DISCRETIZED MOMENTUM EQUATION THE COLLOCATED PRESSURE CORRECTION EQUATION CALCULATION OF THE DF TERM THE COLLOCATED SIMPLE ALGORITHM BOUNDARY CONDITIONS FOR THE MOMENTUM EQUATION BOUNDARY CONDITIONS FOR THE PRESSURE CORRECTION EQUATION THE SIMPLE FAMILY OF ALGORITHMS THE SIMPLEC ALGORITHM THE PRIME ALGORITHM THE PISO ALGORITHM OPTIMUM UNDER-RELAXATION FACTOR VALUES FOR V AND P0 TREATMENT OF VARIOUS TERMS WITH THE RHIE-CHOW INTERPOLATION TREATMENT OF THE UNDER-RELAXATION TERM TREATMENT OF THE TRANSIENT TERM TREATMENT OF THE BODY FORCE TERM COMBINED TREATMENT OF UNDER-RELAXATION, TRANSIENT, AND BODY FORCE TERMS FLUID FLOW COMPUTATION: COMPRESSIBLE FLOWS HISTORICAL THE CONSERVATION EQUATIONS DISCRETIZATION OF THE ENERGY EQUATION DISCRETIZATION OF THE EXTRA TERMS THE ALGEBRAIC FORM OF THE ENERGY EQUATION THE COMPRESSIBLE SIMPLE ALGORITHM INLET BOUNDARY CONDITIONS OUTLET BOUNDARY CONDITIONS PART IV APPLICATIONS TURBULENCE MODELING REYNOLDS AVERAGING TIME AVERAGING SPATIAL AVERAGING ENSEMBLE AVERAGING AVERAGING RULES INCOMPRESSIBLE RANS EQUATIONS BOUSSINESQ HYPOTHESIS TURBULENCE MODELS TWO-EQUATION TURBULENCE MODELS STANDARD K ?? MODEL THE K ?? MODEL THE BASELINE (BSL) K ?? MODEL THE SHEAR STRESS TRANSPORT (SST) K ?? MODEL SUMMARY OF INCOMPRESSIBLE TURBULENT FLOW EQUATIONS DISCRETIZATION OF THE TURBULENT FLOW EQUATIONS THE DISCRETIZED FORM OF THE K EQUATION THE DISCRETIZED FORM OF THE ? EQUATION THE DISCRETIZED FORM OF THE ? EQUATION MODELING FLOW NEAR THE WALL STANDARD WALL FUNCTIONS IMPROVED WALL FUNCTIONS SCALABLE WALL FUNCTIONS WALL BOUNDARY CONDITIONS FOR LOW REYNOLDS NUMBER MODELS AUTOMATIC NEAR-WALL TREATMENT NEAR-WALL HEAT TRANSFER OTHER BOUNDARY CONDITIONS CALCULATING NORMAL DISTANCE TO THE WALL THE K ?? MODEL THE SST K – ? MODEL SIMPLEFOAMTURBULENT BOUNDARY CONDITIONS IN OPENFOAM® AND UFVM BOUNDARY CONDITIONS IN OPENFOAM® BOUNDARY CONDITION CUSTOMIZATION DEVELOPMENT OF A NEW BC: NO SLIP WALL CONDITION THE NO-SLIP BOUNDARY CONDITION IN UFVM REFERENCE AN OPENFOAM® TURBULENT FLOW APPLICATION THE AHMED BLUFF BODY INITIAL AND BOUNDARY CONDITIONS SYSTEMS FILES RUNNING THE SOLVER CONCLUSION