- Email: [email protected]
Subject index for Volume III
Subject index
Acoustic wave velocity, 14 Adaptive mesh generation, incompressible Newtonian laminar flows, for transient problems, 131,133 Adaptive mesh refinement, incompressible Newtonian laminar flows: about adaptive mesh refinement, 123 choice of variables, 130-1 element elongation, 128-30 estimation of second derivatives at nodes, 128 first derivative- (gradient) based refinement, 130 lid-driven cavity example, 131-2 local patch interpolation: superconvergent values, 127 second gradient- (curvature) based refinement, 123-7 interpolation errors, 125-6 principal values and directions, 127 see also Compressible high-speed gas flow, adaptive refinement and shock capture in Euler problems Aerofoil, potential flow solution example, 21-2 Anisotropic shock capturing, with high-speed gas flow, 205 Arbitrary-Lagrangian-Eulerian (ALE) methods, free surface flows, 172, 185-9 Artificial compressibility: and the character-based split (CBS) algorithm, 95-6 and dual time stepping, 96 Artificial diffusion concept, 40 Astley's shape function, with conjugated infinite elements, 331 Babu~ka-Brezzi restriction (BB), 81 circumvention of, 97-8 Balancing diffusion, 46 streamline balancing diffusion, 47 Bore, shallow water example, 300-1 Boundary conditions: with the CBS algorithm, 100-3
convection-diffusion-reaction equation, 72-3 Dirichlet type, 15 governing equations, 9 Neumann type, 15 radiation, 62-4 Boundary layer-inviscid flow coupling, 410-12 Boussinesq approximation, porous medium flow, 279, 285 Boussinesq assumption, with turbulent flows, 250 Brinkman extensions, porous medium flow, 284 Bristol channel, shallow water example, 301-5 Buoyancy driven incompressible flows: about buoyancy driven flows, 189-91 flow in an enclosure example, 191-3 Grashoff number, 190 Prandtl number, 191 Rayleigh number, 191 Burger equation, 68-70 Cauchy-Poisson free surface condition, waves, 339 Character-based split (CBS) algorithm: about the CBS algorithm and compressible and incompressible flow, 79-81,104-5 about the split, 82-3 artificial compressibility, 95-6 in transient problems (dual time stepping), 96 boundary conditions: application of real boundary conditions, 101-3 fictitious boundaries, 100-1 prescribed traction boundary conditions, 101 solid boundaries in inviscid flow (slip conditions), 101 solid boundaries with no slip, 101 Chorin split, 82-3 circumvention of the Babu~ka-Brezzi (BB) restrictions, 97-8 forms/schemes: about forms, 92 evaluation of time limits, 93-5 fully explicit form, 92 quasi- (nearly) implicit form, 93
428
Subjectindex Character-based split (CBS) algorithm- cont. semi-implicit form, 93 governing equations for, 79-82 with high-speed gas flow, 202-3 inviscid problem, performance of two- and single-step algorithms, 103-5 mass diagonalization (lumping), 91-2 with metal forming, transient, 164 with Porous medium flow, 280 with shallow water problems, 297-8 single step version, 98-9 spatial discretization and solution procedure, 86-91 split A, 86-91 split B, 91 temporal discretization, 83-5 split A, 84-5 split B, 85 with viscoelastic flows, 163 see a l s o Computer implementation of the CBS algorithm; Incompressible Newtonian laminar flow Chrzy bed friction, long and medium waves, 320 Chrzy coefficient, 294 Chorin split, 82-3 see also Character-based split (CBS) algorithm Cnoidal and solitary waves, 340-2 Coefficients of pressure and friction, postprocessing, 393 Compressible high-speed gas flow basics: about high-speed gas flow, 197-8 boundary conditions, subsonic and supersonic: Euler equation, 200-1 Navier-Stokes equations, 201-2 boundary layer-inviscid Euler solution coupling, 241 Euler equation examples: inviscid flow past an RAE2822 airfoil, 208-10, 211 isothermal flow through a nozzle in one dimension example, 207, 209 Riemann shock tube - transient problem in one dimension example, 207, 208 two-dimensional transient supersonic flow over a step example, 207-8, 210 governing equations: ideal gas law, 198 internal energy, 198 Navier-Stokes, 198 total specific energy, 198-9 numerical approximations and the CBS algorithm, 202-3 shock capture methods: about shock capture, 203-4 anisotropic viscosity shock capturing, 205 residual-based methods, 205
second derivative-based methods, 204-5 structured meshes, 197 variable smoothing, 205-6 subsonic inviscid flow past an NACA0012 airfoil, 206 see also Turbulent flows, compressible material Compressible high-speed gas flow, adaptive refinement and shock capture in Euler problems: about adaptive refinement, 212 h-refinement process and mesh enrichment, 212-13 h-refinement and remeshing in steady-state two-dimensional problems, 213-17 remeshing in steady-state two-dimensional problems examples: hypersonic inviscid flow past a blunt body example, 214-16 inviscid flow with shock reflection from a solid wall example, 214, 215 inviscid shock interaction example, 217, 220 supersonic inviscid flow past a full circular cylinder example, 217-19 Compressible high-speed gas flow, steady state three-dimensional inviscid examples: complete aircraft flow patterns: inviscid engine intake example, 221-2, 224 inviscid flow past full aircraft example, 221,223 multigrid approaches, 220-1 parallel computation, 221 recasting element formulations in an edge form, 217 THRUST- the supersonic car, 222-6 Compressible high-speed gas flow, transient twoand three-dimensional problems: exploding pressure vessel example, 226, 228 shuttle launch, 227, 229, 230 Compressible high-speed gas flow, viscous problems in three dimensions, 240 hypersonic viscous flow past a double ellipsoid, 240-1 Compressible high-speed gas flow, viscous problems in two dimensions: about viscous flow, 227-8, 231 adaptive refinement in both shock and boundary layer, 230, 234-5 special adaptive refinement for boundary layers and shocks, 230-3, 236--8 transonic viscous flow past an NACA0012 aerofoil, 235-9 viscous flow past a plate example, 229-30, 232 Compressible and incompressible flow see Character-based split (CBS) algorithm Computer implementation of the CBS algorithm: about computer implementation, 382-3 data input module, 383--4 boundary data, 383
Subject index 429 mesh data- nodal coordinates and connectivity, 383 necessary data and flags, 383-4 preliminary subroutines and checks, 384 output module, 387 solution module, 384-7 boundary conditions, 386-7 CBS algorithm; steps, 386 convergence to steady state, 387 different forms of energy equation, 387 shock capture, 385-6 solution of simultaneous equations semi-implicit form, 387 time step, 384-5 see also Character-based split (CBS) algorithm Conservation laws, 28-30 Conservation of mass: about conservation of mass, 1-2 governing equations, 6-7, 81 incompressible Newtonian laminar flow, 110 Conservation of momentum: for CBS algorithm, 80-1 dynamic equilibrium, 7 incompressible Newtonian laminar flow, 111 Constitutive equation, viscoelastic flows, 158 Continuity equation: with free surface flows, 173 viscoelastic flow, 157 Convection-diffusion equations: vector-valued variables: multiple wave speeds, 402-4 Swansea two step operation, 402 Taylor-Galerkin method, 397-9 two-step predictor-corrector methods, 399-400 two-step Taylor-Galerkin operation, 400-2 Convection-diffusion-reaction equation: about convection-diffusion equations, 28-30, 73 boundary conditions, 72-3 Galerkin process/method/procedure, 72-3 conservation laws, 28-30 convective flux quantities, 28 diffusive flux quantities, 28 Galerkin weighting, 30 pure convection treatment, 70-2 transport equation, 28 velocity field, 29 see also Steady state convection-diffusion equation in one dimension; Steady state convection-diffusion equation in two (or three) dimensions; Transient convection-diffusion equation; Waves and shocks, non-linear Convective acceleration effects, 1 Convective flux quantities, 28 Coriolis accelerations/parameters, with shallow water problems, 293-5,303
Dam break, shallow water example, 299-300 Darcy-Rayleigh number, porous medium flow, 279 Darcy's law/flow regime, porous medium flow, 274-5, 286-8 Deborah number, with viscoelastic flows, 158, 160, 162 Detached Eddy Simulation (DES), 270 Deviatoric stresses/deviatoric strain rates, 1, 5 Differential equations, self-adjoint, 391 Diffusion: artificial diffusion concept, 40 balancing diffusion, 46 cross-wind diffusion, 47 negative diffusion concept, 40 streamline balancing diffusion, 48-9 see also Convection-diffusion-reaction equation Diffusive flux quantities, 28 Direct Numerical Simulation (DNS), 270-1 Dirichlet type boundary conditions, 15 Discontinuous enrichment method, short waves, 357-9 Discontinuous Galerkin Finite Elements (DGFE), 374-7 Discretization procedure, porous medium flow, 279-82 Drag force calculation, postprocessing, 392-3 Dual time stepping, and artificial compressibility, 96 Dynamic viscosity, 1 Eddies, large eddy simulation, 267-9 eddy viscosity, 269 Kolmogorov cascade/constant, 269 Smagorinsky's model/constant, 269 standard SGS model, 269-70 Edge-based finite element formulation, 405--6 Eigenvalues, problems with waves in closed domains, 319 Elastic bulk modulus, 1, 14 Elastic springback, in non-Newtonian flows, 152-4 Electromagnetic scattering, 352-3 Energy conservation: for CBS algorithm, 80 and equation of state, 7-9 Enrichment functions, short waves, 358 Enthalphy, 8, 13 Equation of state and energy conservation, 7-9 Equations of fluid dynamics, 1-4 see also Governing equations of fluid dynamics; Inviscid, incompressible flow Ergun correlation, porous medium flow, 276-7 Euler equations, 9-11 compressible high speed gas flow examples, 206-11 with high-speed gas flow, 200-1
430 Subjectindex Euler problems s e e Compressible high-speed gas flow, adaptive refinement and shock capture in Euler problems Euler solutions, boundary layer-inviscid Euler solution coupling, 241 Eulerian form, 6 Eulerian methods, free surface flows, 176-84 Explicit characteristic-Galerkin procedures, 56-62 Extrusion: transient extrusion problem, 150-4 s e e a l s o Metal forming, transient, direct displacement approach; Non-Newtonian flows - metal and polymer forming; Viscoelastic flows Fekete points, 374 Finite element methods/approximation s e e Weighted residual and finite element methods Finite increment calculus (FIC): in multidimensional problems, 48-9 stabilization of the convection-diffusion equation, 43-4 Finite volume approximation/technique/methodology, 2, 23-4 Poisson equation in two dimensions example, 24-6 Forcheimmer extensions, porous medium flow, 284 Forming, steady-state problems, 144-7 Free surface incompressible flows: about free surface flows, 170-2 Arbitrary-Lagrangian-Eulerian (ALE) methods: about the ALE method, 172, 185-6 implementation of the ALE method, 186-7 solitary wave propagation example, 187-9 Eulerian methods: about Eulerian methods, 172, 176 hydrostatic adjustment, 178-9 mesh updating or regeneration methods, 176-84 sailing boat example, 184-5 ship motion problem, 177 submarine example, 181-3 submerged hydrofoil example, 179-82 Lagrangian methods, 171-5 continuity equation, 173 Kroneker delta, 172 model broken dam problem example, 173-5 momentum equation, 173 Frequency domain solutions, short waves, 350 Friction coefficients, postprocessing, 393
with transients, 51-3 Galerkin procedure: and boundary conditions for convection-diffusion, 72-3 simple explicit characteristic, 56-62 Galerkin scheme/equations, short waves, 367 Galerkin spatial approximation, 56, 59 and CBS algorithm, 80 Galerkin weighting, convection-diffusion-reaction equation, 30, 32 Gas flow s e e Compressible high-speed gas flow Gauss-Chebyshev-Lobatto scheme, 373-4 Gauss-Legendre integration, 355, 373 Gauss-Lobatto scheme, 373-4 Givoli's procedure, NRBCs with waves, 326 GLS s e e Galerkin least squares (GLS) approximation Governing equations of fluid dynamics: Babu~ka-Brezzi restriction, 81 balance of energy, 9 boundary conditions, 9 for character-based split (CBS) algorithm, 79-82 compressible flow, 11 conservation of energy and equation of state, 7-9, 80 non-dimensional form, 82 conservation of mass, 6-7 non-dimensional form, 81 conservation of momentum, 7, 80 non-dimensional form, 81 deviatoric stresses/deviatoric strain rates, 5 enthalphy, 8 Euler equations, 9-11 Eulerian form, 6 gradient operator, 6 indicial notation, 4 intrinsic energy, 8 inviscid flow, 11 Kroneker delta, 5, 80 Lam6 notation, 6 Navier-Stokes equations, 9-11 non-dimensional form (for CBS algorithm), 81-2 rates of strain, 5 stress-strain rate relations, 5 stresses in fluids, 4-6 turbulence/turbulent instability, 11 volumetric viscosity, 5-6 Gradient operator, 6 Grashoff number, with buoyancy driven flows, 190 Green's function, 43
Galerkin, finite element, method, 17-18 Galerkin formulation with two triangular elements example, 19-21 Galerkin least squares (GLS) approximation/method: in multidimensional problems, 48-9 in one dimension, 41-2
h-refinement: and mesh enrichment, 212-13 and remeshing in steady-state two-dimensional problems, 213-17 Hankel functions, 338 and infinite elements, 328
Subject index 431 with Trefftz type infinite elements, 333 Higdon boundary condition, short waves, 365 Hydrofoil, submerged, example, free surface flows, 179-82 Hydrostatic adjustment, free surface flows, 178-9 Ideal gas law, with high-speed gas flow, 198 Incompressibility constraint difficulties, 1-2 Incompressible flows, 13-14 about free surface and buoyancy driven flows, 170 about incompressible flows, 3 acoustic wave velocity, 14 elastic bulk modulus, 14 s e e a l s o Buoyancy driven incompressible flows; Character-based split (CBS) algorithm; Free surface incompressible flows Incompressible Newtonian laminar flow: about laminar flow, 110, 136 basic equations: conservation of energy, 111 conservation of mass, 110 conservation of momentum, 111 with CBS algorithm: fully explicit artificial compressibility form, 112 incompressible flow in a lid-driven cavity example, 113-20, 125 quasi-implicit form, 123 semi-implicit form, 112-23 steady flow past a backward facing step example, 115-21 steady flow past a sphere example, 117-22 transient flow past a circular cylinder example, 118-24 mixed and penalty discretization/formulations, 134-6 slow flows, analogy with incompressible elasticity, 131,134 s e e a l s o Adaptive mesh refinement, incompressible Newtonian laminar flows Incompressible non-Newtonian flows: about non-Newtonian effects, 141, 165 s e e a l s o Extrusion; Metal forming, transient, direct displacement approach; Non-Newtonian flows - metal and polymer forming; Viscoelastic flows Indicial notation, 4 Infinite elements: about infinite elements, 327 accuracy of, 332 Burnett and Holford ellipsoidal type infinite elements, 328-30 mapped periodic (unconjugated) infinite elements, 327-8 Hankel functions, 328 Trefftz type infinite elements, 332-3 Hankel functions, 333
wave envelope (conjugated) infinite elements, 330-2 Astley's shape function, 331 Integration formulae: linear tetrahedron, 395-6 linear triangles, 395 Interpolation errors, 125-6 Intrinsic energy, 8 Inviscid, incompressible flow, 11-13 irrotational flow, 12 stream function, 13 velocity potential solution, 11-13 Irrotational flow, 12 Kolmogorov cascade/constant, 269 Kolmogorov length scale, 248 Korteweg-de Vries wave equation, 342 Kroneker delta, 5 and CBS algorithm, 80 with free surface flows, 172 with viscoelastic flows, 157 Lagrangian methods, free surface flows, 171-5 Lam6 notation, 6 Laminar flow s e e Incompressible Newtonian laminar flow Lapidus type diffusivity, 67-8 Local Non-Reflecting Boundary Conditions (NRBCs), long and medium waves, 324-7 Long waves s e e Waves, long and medium Mass-weighted averaged turbulence transport equations, 413-15 Maxwell equation, with viscoelestic flows, 156 Medium waves s e e Waves, long and medium Mesh enrichment: and h-refinement process, 212-13 with high-speed gas flow problems, 212-13 Mesh refinement s e e Adaptive mesh refinement, incompressible Newtonian laminar flows Mesh updating, free surface flows, 176-8 Metal forming, transient, direct displacement approach, 163-5 and the CBS algorithm, 164, 165 impact of circular bar example, 165 Microlocal discretization, 354 Mixed and penalty discretization/formulations, 134-6 Modelling errors, short waves, 351 Mollifying/smoothing discontinuities, 39 Momentum equation: free surface flows, 173 viscoelastic flow, 157 Monotonically Integrated LES (MILES), 270 Multigrid method, 407-8
432
Subjectindex NACA0012 aerofoil: inviscid problems of subsonic and supersonic flow, 103-5, 206 transonic viscous flow past the aerofoil, 235-9 Navier-Stokes equations, 9-11, 26 with compressible high-speed gas flow, 198 derivation of non-conservative form, 389-90 with high-speed gas flow, 198, 201-2 with turbulent flows, 248, 249 Neumann type boundary conditions, 15 Newton-Cotes formula, 372-3 Newtonial dynamic viscosity, 157 Newtonion laminar flow s e e Adaptive mesh refinement, incompressible Newtonian laminar flows; Incompressible Newtonian laminar flow Non-Newtonian flows - metal and polymer forming: about viscosity, 141-2 elastic springback, 152-4 flow formulation, 143 Oswald de Wahle law, 142 prescribed boundary velocities, 143 steady-state problems of forming, 144-7 kinetic energy and work considerations, 147 steady state rolling example, 147-8 transient problems with changing boundaries, 147-52 punch indentation example, 149, 151-2 transient extrusion problem, 150--4 viscoelastic fluids, 152-4 viscoplastic fluids, 142 viscoplasticity and plasticity, 141-4 s e e a l s o Metal forming, transient, direct displacement approach; Viscoelastic flows Non-self-adjoint equations, 1 Oldroyd-B model, 156 ONERA-M6 wing, turbulent flow past example, 266-8 Ortiz formulation, short waves, 365 Oswald de Wahle law, 142 Partition of Unity Finite Elements (PUFEs), 357 Peclet number, 32, 33 Perfectly Matched Layers (PMLs), NRBCs with waves, 326-7 Petrov-Galerkin methods: with transients, 51-2, 60-1 for upwinding in one dimension, 34-9, 42 Poisson equation in two dimensions: finite volume formulation with triangular elements example, 24-6 Galerkin formulation with two triangular elements example, 19-21 Pollution error, waves in closed domains, 319 Polymeric liquids, 154
Porous medium flow: about flow through porous media, 274-5 Boussinesq approximation, 279 Brinkman extensions, 284 with CBS scheme, 280 continuity equation, 277, 278-9 Darcy-Rayleigh number, 279 Darcy's law, 274-5 discretization procedure, 279-82 energy equation, 277, 278, 279 Ergun correlation, 276-7 forced convection, 282-3 heat transfer in a packed channel example, 283-4 Forcheimmer extensions, 284 generalized approach, 275-9 momentum equation, 277, 278, 279 natural convection: about natural convection, 284 Boussinesq approximation, 285 buoyancy driven convection in an axisymmetric enclosure example, 288-9 buoyancy driven convection in a packed enclosure example, 285-6 buoyancy driven flow in a saturated cavity example, 286-8 constant porosity medium, 285-6 Darcy flow regime, 286-8 non-dimensional scales, 277-9 non-isothermal flows, 282 porosity definition, 276 semi- and quasi-implicit forms, 281-2 Postprocessing: coefficients of pressure and friction, 393 drag force calculation, 392-3 stream function, 393-4 Prandtl number, 82 with buoyancy driven flows, 191 Pressure coefficients, postprocessing, 393 PUFEs (Partition of Unity Finite Elements), 357 Radiation, boundary conditions, 62-4 RAE2822 airfoil, inviscid flow past example, 208-11 Rayleigh number, with buoyancy driven flows, 191 Refraction s e e Waves, short Regeneration methods, free surface flows, 176-8 Reynolds averaged Navier-Stokes equations, 251 Reynolds number: with turbulent flows, 248, 249 with viscoelastic flows, 158, 160 Reynolds stress, with turbulent flows, 250 Riemmann invariants, with shallow water transport, 313 Riemmann shock tube - high-speed gas flow example, 207 River Severn bore, shallow water example, 305-7 Robin boundary conditions, short waves, 366
Subject index 433 Sailing boat example, free surface flows, 184-5 Self-adjoint differential equations, 17, 391 SGS s e e Sub-Grid Scale (SGS) approximation/method Shallow water: about shallow water problems, 4, 292 drying areas, 310-11 equations for shallow water, basis of, 293-7 Ch6zy coefficient, 294 Coriolis accelerations/parameters, 293-5 Helrnholtz equation, 296 mass conservation with full incompressibility, 293 notation, 294 numerical approximation, 297-8 Characteristic-Based-Split (CBS) algorithm, 297-8 Taylor--Galerkin approximation, 298 Steady state solutions examples: steady state solution, 308, 310 supercritical flow, 308, 310 transient one-dimensional examples: bore, 300-1 dam break, 299-300 solitary wave, 299 transport of shallow water, 311-13 characteristic-Galerkin method/procedure, 311-12 depth-averaged transport equations, 311 Riemmann invariants, 313 tsunami wave in Severn Estuary example, 307-9 two dimensional periodic tidal motion examples: Bristol channel, 301-5 periodic wave, 300-2 River Severn bore, 305-7 Ship motion problem, free surface flow methods, 177 Shocks s e e Waves and shocks, non-linear Shuttle launch, high-speed gas problem flow example, 227, 228, 230 Slow flows, analogy with incompressible elasticity 131, 134 Smagorinsky's model, 269 Sommerfield radiation condition, 363 Spalart-Allmaras (SA) model, turbulence transport equations, 252-3, 413-14 Split, the s e e Character-based split (CBS) algorithm Sponge layers, NRBCs with waves, 326-7 Steady state convection-diffusion equation in one dimension: about the steady state problem, 31-2, 49-50 artificial diffusion concept, 40 balancing diffusion in one dimension, 39-40 continuity requirements for weighting functions, 37-9 convection diffusion example, 32-4 discretization, 31
finite increment calculus (FIC) stabilization, 43--4 Galerkin least squares approximation (GLS), 41-2 Galerkin weighting, 31-2 Green's function, 43 higher order approximations, 44-5 mollifying/smoothing discontinuities, 39 Peclet number, 32, 33 Petrov-Galerkin methods for upwinding, 34-9 sub-grid scale (SGS) approximation, 42-3 variational principle, 40-1 weight function for exact solution example, 34 Steady state convection-diffusion equation in two (or three) dimensions: about two or three dimensions, 45, 49-50 balancing diffusion, 46 finite increment calculus (FIC), 48-9 Galerkin least squares (GLS), 48-9 streamline (Upwind) Petrov-Galerkin (SUPG) weighting, 45-8 Stokes flow, 2-3 Stokes waves, 342-3 Stream function, 13 postprocessing, 393-4 Streamline balancing diffusion, 47-8 Streamline (Upwind) Petrov--Galerkin (SUPG) weighting, 45-8 Stresses in fluids, governing equations, 4-6 Strong and weak forms s e e Weighted residual and finite element methods Structures meshes, with compressible high-speed gas flow, 197 Sub-grid scale (SGS) approximation/method, 42-3 Submarine example, free surface flows, 181-3 Submerged hydrofoil example, free surface flows, 179-82 Supercritical flow, shallow water example, 308, 310 SUPG (Streamline (Upwind) Petrov-galerkin) weighting, 45-8 Swansea two step operation, convection-diffusion equations, 402 Taylor-Galerkin method/procedures, 52, 65-6 with shallow water problems, 298 used for vector-valued variables, 387-9 Tetrahedron, linear, integration formulae, 395-6 THRUST the supersonic car, Euler solution example, 222--6 Time domain solutions, short waves, 350 Transient convection-diffusion equation: about transients, 50--3 advection of a Gaussian cone in a rotating fluid, 62 boundary conditions - radiation, 62--4 characteristic directions, 51 characteristic-Galerkin method/procedures, 54-6, 61-3 discretization procedures, 51-3 -
434 Subjectindex Transient convection-diffusion equation - cont. explicit characteristic-Galerkin procedures, 56-62 Galerkin least squares (GLS) method, 51-3 mathematical background, 50-1 mesh updating and interpolation methods, 53-4 Petrov-Galerkin method, 51-2, 60-1 and the steady-state condition, 66 Taylor-Galerkin methods/procedures, 52, 65-6 see a l s o Waves and shocks, non-linear Transport equation, 28 Transport of shallow water, 311-13 Trefftz type finite elements for short waves, 362-4 Triangles, linear, integration formulae, 395 Tsunami wave in Severn Estuary example, 307-9 Turbulence transport equations, mass-weighted averaged: about turbulence models, 413 Spalart-Allmaras model, 413-44 Turbulent flows, basics: about turbulent flows, 248-9 Boussinesq assumption, 250 instability, 11 Kolmogorov length scale, 248 large eddy velocity scale, 249 Reynolds stress, 250 time averaging, 249-50 turbulent eddy/kinematic viscosity, 250-1 Turbulent flows, compressible material: detached Eddy Simulation (DES), 270 Direct Numerical Simulation (DNS), 270-1 energy conservation equation, 265 large eddy simulation, 267-9 Kolmogorov constant, 269 Smagorinsky's model, 269 standard SGS model, 269-70 mass conservation equation, 264 mass-weighted (Favre) time averaging, 265-6 momentum equation, 265 Monotonically Integrated LES (MILES), 270 turbulent flow past an ONERA-M6 wing example, 266-8 Turbulent flows, incompressible material: diffusion Prandtl number, 252, 253 governing equations, non-dimensional: i( - e model, 255 one-equation model, 255 Spalart-Allmaras model, 255 turbulent flow solution, 254-5 Reynolds averaged Navier-Stokes equations, 251 shortest distance to a solid wall, 256 solution procedure/examples: CBS scheme recommended, 256 turbulent flow past a backward facing step example, 256-9 unsteady turbulent flow past a circular cylinder example, 259-64
Spalart-Allmaras (SA) (one equation) model, 252-3 standard K - e (two equation) model, 253-4 Wolfstein i( - l (one equation) model, 251-2 Ultra weak formulation, short waves, 359-62 Ultra Weak Variational Formulation (UWVF), 361-2 Upwinding: optimal streamline upwinding, 59-60 using Petrov-Galerkin methods, 34-9 Vandermonde matrix, 374 Velocity field, 29 Viscoelastic flows: about viscoelastic flows, 154--7 and the CBS algorithm, 163 flow past a circular cylinder example, 159-63 governing equations: constitutive equation, 158 continuity, 157, 158 Deborah number, 158, 160, 162 Kroneker delta, 157 momentum, 157, 158 Newtonial dynamic viscosity, 157 Reynolds number, 158, 160 Maxwell equation, 156 Oldroyd-B model, 156 polymeric liquids, 154 Viscosity: polymers and hot metals, 141 secant velocity, 141 viscoelastic fluids, 152-4 viscoplastic fluids, 142 viscosity-strain rate dependence, 141-2 viscous flow problems, 3 volumetric viscosity, 5 see also Compressible high-speed gas flow, viscous problems .... Volumetric viscosity, 5-6 Wave envelope (conjugated) infinite elements, 330-2 Wave propagation, solitary wave example, free surface flows, 186-9 Waves, long and medium: about long and medium waves, 317-18, 344 bed friction, 320 Chrzy bed friction, 320 convection of waves, 333-5 linking finite elements to exterior solutions (DtN mapping): about linking, 336 Hankel functions, 338 to boundary integrals, 336-7 to series solutions, 337-8
Subject index local Non-Reflecting Boundary Conditions (NRBCs): about NRBCs, 324-6 Givoli's procedure, 326 Perfectly Matched Layers (PMLs), 326-7 sponge layers, 326-7 modelling difficulties, 320 refraction of waves, 333-5 three-dimensional effects in surface waves: about surface waves in deep water, 338-40 Cauchy-Poisson free surface condition, 339 cnoidal and solitary waves, 340-2 free surface condition, 338, 338-9 Korteweg--de Vries equation, 342 large amplitude waves, 340 Stokes waves, 342-3 transient problems, 335 unbounded problems, 324 waves in closed domains, finite element models, 318-19 eigenvalue problem, 319 pollution error, 319 waves in unbounded domains, 321-3 diffraction and refraction problems, 320-1 incident waves, domain integrals and nodal values, 323 radiation condition, 321-2 radiation problem, 321 scattering problem, 321 wave diffraction, 321-3 s e e a l s o Infinite elements; Shallow water Waves and shocks, non-linear, 66-70 Burger equation, 69-70 development of shock, 67-8 Lapidus type diffusivity, 67-8 propagation speeds, 67 steep wave modelling, 67-9 Waves, short: about short waves, 349-51 frequency domain solutions, 350 time domain solutions, 350 Discontinuous Galerkin finite elements (DGFE), 374-7 electromagnetic scattering, transient solution, 352-3 finite elements incorporating wave shapes: about finite elements with short waves, 352-4 discontinuous enrichment method, 357-9 enrichment functions, 358 Gauss-Legendre integration points, 355 microlocal discretization, 354 Partition of Unity Finite Elements (PUFEs), 357 shape functions using products of polynomials and waves, 354-7
shape functions using sums of polynomials and waves, 357 Sommerfield radiation condition, 363 Trefftz type finite elements for waves, 362--4 ultra weak formulation, 359--62 Ultra Weak Variational Formulation (UWVF), 361-2 modelling developments, 351 modelling errors, 351 refraction: about refraction, 364 acoustic velocity potential, 370 convected wave equation, 370 Galerkin scheme, 367 Higdon boundary condition, 365 Ortiz formulation, 365 plane scattered by stepped cylinder example, 368-9 plane wave basis finite elements, 366 refraction caused by flows, 369-72 Robin boundary conditions, 366 wave speed refraction, 364-9 weighted residual scheme, 366 spectral finite elements for waves, 372--4 Fekete points, 374 Gauss-Chebyshev-Lobatto scheme, 373-4 Gauss-Legendre integration, 373 Gauss-Lobatto scheme, 373-4 Newton-Cotes formula, 372-3 Vandermonde matrix, 374 T-complete systems, 363-4 Weak form of equations, 15 s e e a l s o Weighted residual and finite element methods Weighted residual and finite element methods: about strong and weak forms, 14-15 boundary conditions, Neumann type, 15 elements, 16 examples: free surface potential flow, 21-3 Poisson equation in two dimensions: Galerkin formulation with two triangular elements, 19-21 potential flow solution around an aerofoil, 21-2 shape functions for triangle with three nodes, 18-19 Galerkin, finite element, method, 17-18 nodal values, 16 self-adjoint differential equations, 17 test functions, 16 weak form of equations, 15 weighted residual approximation, 16-17 Wolfstein x-1 model, 251-2
435
Acoustic wave velocity, 14 Adaptive mesh generation, incompressible Newtonian laminar flows, for transient problems, 131,133 Adaptive mesh refinement, incompressible Newtonian laminar flows: about adaptive mesh refinement, 123 choice of variables, 130-1 element elongation, 128-30 estimation of second derivatives at nodes, 128 first derivative- (gradient) based refinement, 130 lid-driven cavity example, 131-2 local patch interpolation: superconvergent values, 127 second gradient- (curvature) based refinement, 123-7 interpolation errors, 125-6 principal values and directions, 127 see also Compressible high-speed gas flow, adaptive refinement and shock capture in Euler problems Aerofoil, potential flow solution example, 21-2 Anisotropic shock capturing, with high-speed gas flow, 205 Arbitrary-Lagrangian-Eulerian (ALE) methods, free surface flows, 172, 185-9 Artificial compressibility: and the character-based split (CBS) algorithm, 95-6 and dual time stepping, 96 Artificial diffusion concept, 40 Astley's shape function, with conjugated infinite elements, 331 Babu~ka-Brezzi restriction (BB), 81 circumvention of, 97-8 Balancing diffusion, 46 streamline balancing diffusion, 47 Bore, shallow water example, 300-1 Boundary conditions: with the CBS algorithm, 100-3
convection-diffusion-reaction equation, 72-3 Dirichlet type, 15 governing equations, 9 Neumann type, 15 radiation, 62-4 Boundary layer-inviscid flow coupling, 410-12 Boussinesq approximation, porous medium flow, 279, 285 Boussinesq assumption, with turbulent flows, 250 Brinkman extensions, porous medium flow, 284 Bristol channel, shallow water example, 301-5 Buoyancy driven incompressible flows: about buoyancy driven flows, 189-91 flow in an enclosure example, 191-3 Grashoff number, 190 Prandtl number, 191 Rayleigh number, 191 Burger equation, 68-70 Cauchy-Poisson free surface condition, waves, 339 Character-based split (CBS) algorithm: about the CBS algorithm and compressible and incompressible flow, 79-81,104-5 about the split, 82-3 artificial compressibility, 95-6 in transient problems (dual time stepping), 96 boundary conditions: application of real boundary conditions, 101-3 fictitious boundaries, 100-1 prescribed traction boundary conditions, 101 solid boundaries in inviscid flow (slip conditions), 101 solid boundaries with no slip, 101 Chorin split, 82-3 circumvention of the Babu~ka-Brezzi (BB) restrictions, 97-8 forms/schemes: about forms, 92 evaluation of time limits, 93-5 fully explicit form, 92 quasi- (nearly) implicit form, 93
428
Subjectindex Character-based split (CBS) algorithm- cont. semi-implicit form, 93 governing equations for, 79-82 with high-speed gas flow, 202-3 inviscid problem, performance of two- and single-step algorithms, 103-5 mass diagonalization (lumping), 91-2 with metal forming, transient, 164 with Porous medium flow, 280 with shallow water problems, 297-8 single step version, 98-9 spatial discretization and solution procedure, 86-91 split A, 86-91 split B, 91 temporal discretization, 83-5 split A, 84-5 split B, 85 with viscoelastic flows, 163 see a l s o Computer implementation of the CBS algorithm; Incompressible Newtonian laminar flow Chrzy bed friction, long and medium waves, 320 Chrzy coefficient, 294 Chorin split, 82-3 see also Character-based split (CBS) algorithm Cnoidal and solitary waves, 340-2 Coefficients of pressure and friction, postprocessing, 393 Compressible high-speed gas flow basics: about high-speed gas flow, 197-8 boundary conditions, subsonic and supersonic: Euler equation, 200-1 Navier-Stokes equations, 201-2 boundary layer-inviscid Euler solution coupling, 241 Euler equation examples: inviscid flow past an RAE2822 airfoil, 208-10, 211 isothermal flow through a nozzle in one dimension example, 207, 209 Riemann shock tube - transient problem in one dimension example, 207, 208 two-dimensional transient supersonic flow over a step example, 207-8, 210 governing equations: ideal gas law, 198 internal energy, 198 Navier-Stokes, 198 total specific energy, 198-9 numerical approximations and the CBS algorithm, 202-3 shock capture methods: about shock capture, 203-4 anisotropic viscosity shock capturing, 205 residual-based methods, 205
second derivative-based methods, 204-5 structured meshes, 197 variable smoothing, 205-6 subsonic inviscid flow past an NACA0012 airfoil, 206 see also Turbulent flows, compressible material Compressible high-speed gas flow, adaptive refinement and shock capture in Euler problems: about adaptive refinement, 212 h-refinement process and mesh enrichment, 212-13 h-refinement and remeshing in steady-state two-dimensional problems, 213-17 remeshing in steady-state two-dimensional problems examples: hypersonic inviscid flow past a blunt body example, 214-16 inviscid flow with shock reflection from a solid wall example, 214, 215 inviscid shock interaction example, 217, 220 supersonic inviscid flow past a full circular cylinder example, 217-19 Compressible high-speed gas flow, steady state three-dimensional inviscid examples: complete aircraft flow patterns: inviscid engine intake example, 221-2, 224 inviscid flow past full aircraft example, 221,223 multigrid approaches, 220-1 parallel computation, 221 recasting element formulations in an edge form, 217 THRUST- the supersonic car, 222-6 Compressible high-speed gas flow, transient twoand three-dimensional problems: exploding pressure vessel example, 226, 228 shuttle launch, 227, 229, 230 Compressible high-speed gas flow, viscous problems in three dimensions, 240 hypersonic viscous flow past a double ellipsoid, 240-1 Compressible high-speed gas flow, viscous problems in two dimensions: about viscous flow, 227-8, 231 adaptive refinement in both shock and boundary layer, 230, 234-5 special adaptive refinement for boundary layers and shocks, 230-3, 236--8 transonic viscous flow past an NACA0012 aerofoil, 235-9 viscous flow past a plate example, 229-30, 232 Compressible and incompressible flow see Character-based split (CBS) algorithm Computer implementation of the CBS algorithm: about computer implementation, 382-3 data input module, 383--4 boundary data, 383
Subject index 429 mesh data- nodal coordinates and connectivity, 383 necessary data and flags, 383-4 preliminary subroutines and checks, 384 output module, 387 solution module, 384-7 boundary conditions, 386-7 CBS algorithm; steps, 386 convergence to steady state, 387 different forms of energy equation, 387 shock capture, 385-6 solution of simultaneous equations semi-implicit form, 387 time step, 384-5 see also Character-based split (CBS) algorithm Conservation laws, 28-30 Conservation of mass: about conservation of mass, 1-2 governing equations, 6-7, 81 incompressible Newtonian laminar flow, 110 Conservation of momentum: for CBS algorithm, 80-1 dynamic equilibrium, 7 incompressible Newtonian laminar flow, 111 Constitutive equation, viscoelastic flows, 158 Continuity equation: with free surface flows, 173 viscoelastic flow, 157 Convection-diffusion equations: vector-valued variables: multiple wave speeds, 402-4 Swansea two step operation, 402 Taylor-Galerkin method, 397-9 two-step predictor-corrector methods, 399-400 two-step Taylor-Galerkin operation, 400-2 Convection-diffusion-reaction equation: about convection-diffusion equations, 28-30, 73 boundary conditions, 72-3 Galerkin process/method/procedure, 72-3 conservation laws, 28-30 convective flux quantities, 28 diffusive flux quantities, 28 Galerkin weighting, 30 pure convection treatment, 70-2 transport equation, 28 velocity field, 29 see also Steady state convection-diffusion equation in one dimension; Steady state convection-diffusion equation in two (or three) dimensions; Transient convection-diffusion equation; Waves and shocks, non-linear Convective acceleration effects, 1 Convective flux quantities, 28 Coriolis accelerations/parameters, with shallow water problems, 293-5,303
Dam break, shallow water example, 299-300 Darcy-Rayleigh number, porous medium flow, 279 Darcy's law/flow regime, porous medium flow, 274-5, 286-8 Deborah number, with viscoelastic flows, 158, 160, 162 Detached Eddy Simulation (DES), 270 Deviatoric stresses/deviatoric strain rates, 1, 5 Differential equations, self-adjoint, 391 Diffusion: artificial diffusion concept, 40 balancing diffusion, 46 cross-wind diffusion, 47 negative diffusion concept, 40 streamline balancing diffusion, 48-9 see also Convection-diffusion-reaction equation Diffusive flux quantities, 28 Direct Numerical Simulation (DNS), 270-1 Dirichlet type boundary conditions, 15 Discontinuous enrichment method, short waves, 357-9 Discontinuous Galerkin Finite Elements (DGFE), 374-7 Discretization procedure, porous medium flow, 279-82 Drag force calculation, postprocessing, 392-3 Dual time stepping, and artificial compressibility, 96 Dynamic viscosity, 1 Eddies, large eddy simulation, 267-9 eddy viscosity, 269 Kolmogorov cascade/constant, 269 Smagorinsky's model/constant, 269 standard SGS model, 269-70 Edge-based finite element formulation, 405--6 Eigenvalues, problems with waves in closed domains, 319 Elastic bulk modulus, 1, 14 Elastic springback, in non-Newtonian flows, 152-4 Electromagnetic scattering, 352-3 Energy conservation: for CBS algorithm, 80 and equation of state, 7-9 Enrichment functions, short waves, 358 Enthalphy, 8, 13 Equation of state and energy conservation, 7-9 Equations of fluid dynamics, 1-4 see also Governing equations of fluid dynamics; Inviscid, incompressible flow Ergun correlation, porous medium flow, 276-7 Euler equations, 9-11 compressible high speed gas flow examples, 206-11 with high-speed gas flow, 200-1
430 Subjectindex Euler problems s e e Compressible high-speed gas flow, adaptive refinement and shock capture in Euler problems Euler solutions, boundary layer-inviscid Euler solution coupling, 241 Eulerian form, 6 Eulerian methods, free surface flows, 176-84 Explicit characteristic-Galerkin procedures, 56-62 Extrusion: transient extrusion problem, 150-4 s e e a l s o Metal forming, transient, direct displacement approach; Non-Newtonian flows - metal and polymer forming; Viscoelastic flows Fekete points, 374 Finite element methods/approximation s e e Weighted residual and finite element methods Finite increment calculus (FIC): in multidimensional problems, 48-9 stabilization of the convection-diffusion equation, 43-4 Finite volume approximation/technique/methodology, 2, 23-4 Poisson equation in two dimensions example, 24-6 Forcheimmer extensions, porous medium flow, 284 Forming, steady-state problems, 144-7 Free surface incompressible flows: about free surface flows, 170-2 Arbitrary-Lagrangian-Eulerian (ALE) methods: about the ALE method, 172, 185-6 implementation of the ALE method, 186-7 solitary wave propagation example, 187-9 Eulerian methods: about Eulerian methods, 172, 176 hydrostatic adjustment, 178-9 mesh updating or regeneration methods, 176-84 sailing boat example, 184-5 ship motion problem, 177 submarine example, 181-3 submerged hydrofoil example, 179-82 Lagrangian methods, 171-5 continuity equation, 173 Kroneker delta, 172 model broken dam problem example, 173-5 momentum equation, 173 Frequency domain solutions, short waves, 350 Friction coefficients, postprocessing, 393
with transients, 51-3 Galerkin procedure: and boundary conditions for convection-diffusion, 72-3 simple explicit characteristic, 56-62 Galerkin scheme/equations, short waves, 367 Galerkin spatial approximation, 56, 59 and CBS algorithm, 80 Galerkin weighting, convection-diffusion-reaction equation, 30, 32 Gas flow s e e Compressible high-speed gas flow Gauss-Chebyshev-Lobatto scheme, 373-4 Gauss-Legendre integration, 355, 373 Gauss-Lobatto scheme, 373-4 Givoli's procedure, NRBCs with waves, 326 GLS s e e Galerkin least squares (GLS) approximation Governing equations of fluid dynamics: Babu~ka-Brezzi restriction, 81 balance of energy, 9 boundary conditions, 9 for character-based split (CBS) algorithm, 79-82 compressible flow, 11 conservation of energy and equation of state, 7-9, 80 non-dimensional form, 82 conservation of mass, 6-7 non-dimensional form, 81 conservation of momentum, 7, 80 non-dimensional form, 81 deviatoric stresses/deviatoric strain rates, 5 enthalphy, 8 Euler equations, 9-11 Eulerian form, 6 gradient operator, 6 indicial notation, 4 intrinsic energy, 8 inviscid flow, 11 Kroneker delta, 5, 80 Lam6 notation, 6 Navier-Stokes equations, 9-11 non-dimensional form (for CBS algorithm), 81-2 rates of strain, 5 stress-strain rate relations, 5 stresses in fluids, 4-6 turbulence/turbulent instability, 11 volumetric viscosity, 5-6 Gradient operator, 6 Grashoff number, with buoyancy driven flows, 190 Green's function, 43
Galerkin, finite element, method, 17-18 Galerkin formulation with two triangular elements example, 19-21 Galerkin least squares (GLS) approximation/method: in multidimensional problems, 48-9 in one dimension, 41-2
h-refinement: and mesh enrichment, 212-13 and remeshing in steady-state two-dimensional problems, 213-17 Hankel functions, 338 and infinite elements, 328
Subject index 431 with Trefftz type infinite elements, 333 Higdon boundary condition, short waves, 365 Hydrofoil, submerged, example, free surface flows, 179-82 Hydrostatic adjustment, free surface flows, 178-9 Ideal gas law, with high-speed gas flow, 198 Incompressibility constraint difficulties, 1-2 Incompressible flows, 13-14 about free surface and buoyancy driven flows, 170 about incompressible flows, 3 acoustic wave velocity, 14 elastic bulk modulus, 14 s e e a l s o Buoyancy driven incompressible flows; Character-based split (CBS) algorithm; Free surface incompressible flows Incompressible Newtonian laminar flow: about laminar flow, 110, 136 basic equations: conservation of energy, 111 conservation of mass, 110 conservation of momentum, 111 with CBS algorithm: fully explicit artificial compressibility form, 112 incompressible flow in a lid-driven cavity example, 113-20, 125 quasi-implicit form, 123 semi-implicit form, 112-23 steady flow past a backward facing step example, 115-21 steady flow past a sphere example, 117-22 transient flow past a circular cylinder example, 118-24 mixed and penalty discretization/formulations, 134-6 slow flows, analogy with incompressible elasticity, 131,134 s e e a l s o Adaptive mesh refinement, incompressible Newtonian laminar flows Incompressible non-Newtonian flows: about non-Newtonian effects, 141, 165 s e e a l s o Extrusion; Metal forming, transient, direct displacement approach; Non-Newtonian flows - metal and polymer forming; Viscoelastic flows Indicial notation, 4 Infinite elements: about infinite elements, 327 accuracy of, 332 Burnett and Holford ellipsoidal type infinite elements, 328-30 mapped periodic (unconjugated) infinite elements, 327-8 Hankel functions, 328 Trefftz type infinite elements, 332-3 Hankel functions, 333
wave envelope (conjugated) infinite elements, 330-2 Astley's shape function, 331 Integration formulae: linear tetrahedron, 395-6 linear triangles, 395 Interpolation errors, 125-6 Intrinsic energy, 8 Inviscid, incompressible flow, 11-13 irrotational flow, 12 stream function, 13 velocity potential solution, 11-13 Irrotational flow, 12 Kolmogorov cascade/constant, 269 Kolmogorov length scale, 248 Korteweg-de Vries wave equation, 342 Kroneker delta, 5 and CBS algorithm, 80 with free surface flows, 172 with viscoelastic flows, 157 Lagrangian methods, free surface flows, 171-5 Lam6 notation, 6 Laminar flow s e e Incompressible Newtonian laminar flow Lapidus type diffusivity, 67-8 Local Non-Reflecting Boundary Conditions (NRBCs), long and medium waves, 324-7 Long waves s e e Waves, long and medium Mass-weighted averaged turbulence transport equations, 413-15 Maxwell equation, with viscoelestic flows, 156 Medium waves s e e Waves, long and medium Mesh enrichment: and h-refinement process, 212-13 with high-speed gas flow problems, 212-13 Mesh refinement s e e Adaptive mesh refinement, incompressible Newtonian laminar flows Mesh updating, free surface flows, 176-8 Metal forming, transient, direct displacement approach, 163-5 and the CBS algorithm, 164, 165 impact of circular bar example, 165 Microlocal discretization, 354 Mixed and penalty discretization/formulations, 134-6 Modelling errors, short waves, 351 Mollifying/smoothing discontinuities, 39 Momentum equation: free surface flows, 173 viscoelastic flow, 157 Monotonically Integrated LES (MILES), 270 Multigrid method, 407-8
432
Subjectindex NACA0012 aerofoil: inviscid problems of subsonic and supersonic flow, 103-5, 206 transonic viscous flow past the aerofoil, 235-9 Navier-Stokes equations, 9-11, 26 with compressible high-speed gas flow, 198 derivation of non-conservative form, 389-90 with high-speed gas flow, 198, 201-2 with turbulent flows, 248, 249 Neumann type boundary conditions, 15 Newton-Cotes formula, 372-3 Newtonial dynamic viscosity, 157 Newtonion laminar flow s e e Adaptive mesh refinement, incompressible Newtonian laminar flows; Incompressible Newtonian laminar flow Non-Newtonian flows - metal and polymer forming: about viscosity, 141-2 elastic springback, 152-4 flow formulation, 143 Oswald de Wahle law, 142 prescribed boundary velocities, 143 steady-state problems of forming, 144-7 kinetic energy and work considerations, 147 steady state rolling example, 147-8 transient problems with changing boundaries, 147-52 punch indentation example, 149, 151-2 transient extrusion problem, 150--4 viscoelastic fluids, 152-4 viscoplastic fluids, 142 viscoplasticity and plasticity, 141-4 s e e a l s o Metal forming, transient, direct displacement approach; Viscoelastic flows Non-self-adjoint equations, 1 Oldroyd-B model, 156 ONERA-M6 wing, turbulent flow past example, 266-8 Ortiz formulation, short waves, 365 Oswald de Wahle law, 142 Partition of Unity Finite Elements (PUFEs), 357 Peclet number, 32, 33 Perfectly Matched Layers (PMLs), NRBCs with waves, 326-7 Petrov-Galerkin methods: with transients, 51-2, 60-1 for upwinding in one dimension, 34-9, 42 Poisson equation in two dimensions: finite volume formulation with triangular elements example, 24-6 Galerkin formulation with two triangular elements example, 19-21 Pollution error, waves in closed domains, 319 Polymeric liquids, 154
Porous medium flow: about flow through porous media, 274-5 Boussinesq approximation, 279 Brinkman extensions, 284 with CBS scheme, 280 continuity equation, 277, 278-9 Darcy-Rayleigh number, 279 Darcy's law, 274-5 discretization procedure, 279-82 energy equation, 277, 278, 279 Ergun correlation, 276-7 forced convection, 282-3 heat transfer in a packed channel example, 283-4 Forcheimmer extensions, 284 generalized approach, 275-9 momentum equation, 277, 278, 279 natural convection: about natural convection, 284 Boussinesq approximation, 285 buoyancy driven convection in an axisymmetric enclosure example, 288-9 buoyancy driven convection in a packed enclosure example, 285-6 buoyancy driven flow in a saturated cavity example, 286-8 constant porosity medium, 285-6 Darcy flow regime, 286-8 non-dimensional scales, 277-9 non-isothermal flows, 282 porosity definition, 276 semi- and quasi-implicit forms, 281-2 Postprocessing: coefficients of pressure and friction, 393 drag force calculation, 392-3 stream function, 393-4 Prandtl number, 82 with buoyancy driven flows, 191 Pressure coefficients, postprocessing, 393 PUFEs (Partition of Unity Finite Elements), 357 Radiation, boundary conditions, 62-4 RAE2822 airfoil, inviscid flow past example, 208-11 Rayleigh number, with buoyancy driven flows, 191 Refraction s e e Waves, short Regeneration methods, free surface flows, 176-8 Reynolds averaged Navier-Stokes equations, 251 Reynolds number: with turbulent flows, 248, 249 with viscoelastic flows, 158, 160 Reynolds stress, with turbulent flows, 250 Riemmann invariants, with shallow water transport, 313 Riemmann shock tube - high-speed gas flow example, 207 River Severn bore, shallow water example, 305-7 Robin boundary conditions, short waves, 366
Subject index 433 Sailing boat example, free surface flows, 184-5 Self-adjoint differential equations, 17, 391 SGS s e e Sub-Grid Scale (SGS) approximation/method Shallow water: about shallow water problems, 4, 292 drying areas, 310-11 equations for shallow water, basis of, 293-7 Ch6zy coefficient, 294 Coriolis accelerations/parameters, 293-5 Helrnholtz equation, 296 mass conservation with full incompressibility, 293 notation, 294 numerical approximation, 297-8 Characteristic-Based-Split (CBS) algorithm, 297-8 Taylor--Galerkin approximation, 298 Steady state solutions examples: steady state solution, 308, 310 supercritical flow, 308, 310 transient one-dimensional examples: bore, 300-1 dam break, 299-300 solitary wave, 299 transport of shallow water, 311-13 characteristic-Galerkin method/procedure, 311-12 depth-averaged transport equations, 311 Riemmann invariants, 313 tsunami wave in Severn Estuary example, 307-9 two dimensional periodic tidal motion examples: Bristol channel, 301-5 periodic wave, 300-2 River Severn bore, 305-7 Ship motion problem, free surface flow methods, 177 Shocks s e e Waves and shocks, non-linear Shuttle launch, high-speed gas problem flow example, 227, 228, 230 Slow flows, analogy with incompressible elasticity 131, 134 Smagorinsky's model, 269 Sommerfield radiation condition, 363 Spalart-Allmaras (SA) model, turbulence transport equations, 252-3, 413-14 Split, the s e e Character-based split (CBS) algorithm Sponge layers, NRBCs with waves, 326-7 Steady state convection-diffusion equation in one dimension: about the steady state problem, 31-2, 49-50 artificial diffusion concept, 40 balancing diffusion in one dimension, 39-40 continuity requirements for weighting functions, 37-9 convection diffusion example, 32-4 discretization, 31
finite increment calculus (FIC) stabilization, 43--4 Galerkin least squares approximation (GLS), 41-2 Galerkin weighting, 31-2 Green's function, 43 higher order approximations, 44-5 mollifying/smoothing discontinuities, 39 Peclet number, 32, 33 Petrov-Galerkin methods for upwinding, 34-9 sub-grid scale (SGS) approximation, 42-3 variational principle, 40-1 weight function for exact solution example, 34 Steady state convection-diffusion equation in two (or three) dimensions: about two or three dimensions, 45, 49-50 balancing diffusion, 46 finite increment calculus (FIC), 48-9 Galerkin least squares (GLS), 48-9 streamline (Upwind) Petrov-Galerkin (SUPG) weighting, 45-8 Stokes flow, 2-3 Stokes waves, 342-3 Stream function, 13 postprocessing, 393-4 Streamline balancing diffusion, 47-8 Streamline (Upwind) Petrov--Galerkin (SUPG) weighting, 45-8 Stresses in fluids, governing equations, 4-6 Strong and weak forms s e e Weighted residual and finite element methods Structures meshes, with compressible high-speed gas flow, 197 Sub-grid scale (SGS) approximation/method, 42-3 Submarine example, free surface flows, 181-3 Submerged hydrofoil example, free surface flows, 179-82 Supercritical flow, shallow water example, 308, 310 SUPG (Streamline (Upwind) Petrov-galerkin) weighting, 45-8 Swansea two step operation, convection-diffusion equations, 402 Taylor-Galerkin method/procedures, 52, 65-6 with shallow water problems, 298 used for vector-valued variables, 387-9 Tetrahedron, linear, integration formulae, 395-6 THRUST the supersonic car, Euler solution example, 222--6 Time domain solutions, short waves, 350 Transient convection-diffusion equation: about transients, 50--3 advection of a Gaussian cone in a rotating fluid, 62 boundary conditions - radiation, 62--4 characteristic directions, 51 characteristic-Galerkin method/procedures, 54-6, 61-3 discretization procedures, 51-3 -
434 Subjectindex Transient convection-diffusion equation - cont. explicit characteristic-Galerkin procedures, 56-62 Galerkin least squares (GLS) method, 51-3 mathematical background, 50-1 mesh updating and interpolation methods, 53-4 Petrov-Galerkin method, 51-2, 60-1 and the steady-state condition, 66 Taylor-Galerkin methods/procedures, 52, 65-6 see a l s o Waves and shocks, non-linear Transport equation, 28 Transport of shallow water, 311-13 Trefftz type finite elements for short waves, 362-4 Triangles, linear, integration formulae, 395 Tsunami wave in Severn Estuary example, 307-9 Turbulence transport equations, mass-weighted averaged: about turbulence models, 413 Spalart-Allmaras model, 413-44 Turbulent flows, basics: about turbulent flows, 248-9 Boussinesq assumption, 250 instability, 11 Kolmogorov length scale, 248 large eddy velocity scale, 249 Reynolds stress, 250 time averaging, 249-50 turbulent eddy/kinematic viscosity, 250-1 Turbulent flows, compressible material: detached Eddy Simulation (DES), 270 Direct Numerical Simulation (DNS), 270-1 energy conservation equation, 265 large eddy simulation, 267-9 Kolmogorov constant, 269 Smagorinsky's model, 269 standard SGS model, 269-70 mass conservation equation, 264 mass-weighted (Favre) time averaging, 265-6 momentum equation, 265 Monotonically Integrated LES (MILES), 270 turbulent flow past an ONERA-M6 wing example, 266-8 Turbulent flows, incompressible material: diffusion Prandtl number, 252, 253 governing equations, non-dimensional: i( - e model, 255 one-equation model, 255 Spalart-Allmaras model, 255 turbulent flow solution, 254-5 Reynolds averaged Navier-Stokes equations, 251 shortest distance to a solid wall, 256 solution procedure/examples: CBS scheme recommended, 256 turbulent flow past a backward facing step example, 256-9 unsteady turbulent flow past a circular cylinder example, 259-64
Spalart-Allmaras (SA) (one equation) model, 252-3 standard K - e (two equation) model, 253-4 Wolfstein i( - l (one equation) model, 251-2 Ultra weak formulation, short waves, 359-62 Ultra Weak Variational Formulation (UWVF), 361-2 Upwinding: optimal streamline upwinding, 59-60 using Petrov-Galerkin methods, 34-9 Vandermonde matrix, 374 Velocity field, 29 Viscoelastic flows: about viscoelastic flows, 154--7 and the CBS algorithm, 163 flow past a circular cylinder example, 159-63 governing equations: constitutive equation, 158 continuity, 157, 158 Deborah number, 158, 160, 162 Kroneker delta, 157 momentum, 157, 158 Newtonial dynamic viscosity, 157 Reynolds number, 158, 160 Maxwell equation, 156 Oldroyd-B model, 156 polymeric liquids, 154 Viscosity: polymers and hot metals, 141 secant velocity, 141 viscoelastic fluids, 152-4 viscoplastic fluids, 142 viscosity-strain rate dependence, 141-2 viscous flow problems, 3 volumetric viscosity, 5 see also Compressible high-speed gas flow, viscous problems .... Volumetric viscosity, 5-6 Wave envelope (conjugated) infinite elements, 330-2 Wave propagation, solitary wave example, free surface flows, 186-9 Waves, long and medium: about long and medium waves, 317-18, 344 bed friction, 320 Chrzy bed friction, 320 convection of waves, 333-5 linking finite elements to exterior solutions (DtN mapping): about linking, 336 Hankel functions, 338 to boundary integrals, 336-7 to series solutions, 337-8
Subject index local Non-Reflecting Boundary Conditions (NRBCs): about NRBCs, 324-6 Givoli's procedure, 326 Perfectly Matched Layers (PMLs), 326-7 sponge layers, 326-7 modelling difficulties, 320 refraction of waves, 333-5 three-dimensional effects in surface waves: about surface waves in deep water, 338-40 Cauchy-Poisson free surface condition, 339 cnoidal and solitary waves, 340-2 free surface condition, 338, 338-9 Korteweg--de Vries equation, 342 large amplitude waves, 340 Stokes waves, 342-3 transient problems, 335 unbounded problems, 324 waves in closed domains, finite element models, 318-19 eigenvalue problem, 319 pollution error, 319 waves in unbounded domains, 321-3 diffraction and refraction problems, 320-1 incident waves, domain integrals and nodal values, 323 radiation condition, 321-2 radiation problem, 321 scattering problem, 321 wave diffraction, 321-3 s e e a l s o Infinite elements; Shallow water Waves and shocks, non-linear, 66-70 Burger equation, 69-70 development of shock, 67-8 Lapidus type diffusivity, 67-8 propagation speeds, 67 steep wave modelling, 67-9 Waves, short: about short waves, 349-51 frequency domain solutions, 350 time domain solutions, 350 Discontinuous Galerkin finite elements (DGFE), 374-7 electromagnetic scattering, transient solution, 352-3 finite elements incorporating wave shapes: about finite elements with short waves, 352-4 discontinuous enrichment method, 357-9 enrichment functions, 358 Gauss-Legendre integration points, 355 microlocal discretization, 354 Partition of Unity Finite Elements (PUFEs), 357 shape functions using products of polynomials and waves, 354-7
shape functions using sums of polynomials and waves, 357 Sommerfield radiation condition, 363 Trefftz type finite elements for waves, 362--4 ultra weak formulation, 359--62 Ultra Weak Variational Formulation (UWVF), 361-2 modelling developments, 351 modelling errors, 351 refraction: about refraction, 364 acoustic velocity potential, 370 convected wave equation, 370 Galerkin scheme, 367 Higdon boundary condition, 365 Ortiz formulation, 365 plane scattered by stepped cylinder example, 368-9 plane wave basis finite elements, 366 refraction caused by flows, 369-72 Robin boundary conditions, 366 wave speed refraction, 364-9 weighted residual scheme, 366 spectral finite elements for waves, 372--4 Fekete points, 374 Gauss-Chebyshev-Lobatto scheme, 373-4 Gauss-Legendre integration, 373 Gauss-Lobatto scheme, 373-4 Newton-Cotes formula, 372-3 Vandermonde matrix, 374 T-complete systems, 363-4 Weak form of equations, 15 s e e a l s o Weighted residual and finite element methods Weighted residual and finite element methods: about strong and weak forms, 14-15 boundary conditions, Neumann type, 15 elements, 16 examples: free surface potential flow, 21-3 Poisson equation in two dimensions: Galerkin formulation with two triangular elements, 19-21 potential flow solution around an aerofoil, 21-2 shape functions for triangle with three nodes, 18-19 Galerkin, finite element, method, 17-18 nodal values, 16 self-adjoint differential equations, 17 test functions, 16 weak form of equations, 15 weighted residual approximation, 16-17 Wolfstein x-1 model, 251-2
435