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Petr A. Nikrityuk: Computational Thermo-Fluid Dynamics — 2011/6/8 — page 351 — le-tex

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351

Index

aadiabatic wall 250advection-diffusion problem 14Al-Cu alloy 6Al-Si alloy 6Ampere-Maxwell equation 24Ampere’s law 25, 26angular velocity 202, 221, 228, 233, 237anisotropic interface tracking 320anisotropy 153, 158, 159ANS 257

– simulation 258arc

– plasma 27– welding 27

aspect ratio 221-226, 229axismetry 170axisymmetry 197, 242, 252

– acceleration 213– inertial mode 211– numerical simulation 237– oscillatory instability 192, 200, 201

azimuthal– flow 260– Lorentz force 31, 191, 222– velocity 175, 177, 199, 203, 207, 208, 214,

221, 225, 235, 238, 245, 250, 275, 307

bbackward difference scheme (BDS) 42, 49Batchelor length scale 152, 256Bessel function 33, 170, 195, 253binary alloy 120, 124, 153

– phase diagram 122– solidification

– mathematical model 137– standard mixture model 137

binary metal alloy 294, 310– solidification 296, 311

– continuum model 286– electromagnetic control 277– use if alternating current fields 277

Biot-Savart law 25, 27, 301, 312, 320, 322blending

– coefficient 43, 58– scheme 58

Bödewadt layer 5, 192, 196, 202, 206, 207, 212,218, 223, 227, 233, 236, 244, 253, 260, 272

Boltzmann– constant 21– equation 19

boundary– condition 67, 169– layer thickness 230

Boussinesq approximation 15, 109, 167Bridgman

– technique 162, 165, 221, 231-233, 252– porous medium 100, 169

buoyancy– dynamic suppression 292– effect 197, 223, 237, 242, 289– force 275

– impact of spin-up dynamics 212buoyancy-induced

– flow 293– global velocity 175

cCarman-Konzeny equation 134, 141, 142, 297Cartesian

– coordinate system 8, 10, 11, 16, 331– grid 41, 46, 52, 53, 87, 99, 102, 159

– complex geometry problems 98– sharp-interface model 157

cell-face area 54cellular automaton method 117, 157central difference scheme (CDS) 4, 42, 54, 63,

171, 224, 256– fourth order 55

checkerboard oscillation 92Clausius-Clapeyron equation 21, 120columnar solidification 141, 279, 280columnar-to-equiaxed transition (CET) 116,

191, 277combustion modeling 67concentration-conservation equation 159conduction-dominated solidification (CDS)

rule 280, 298

Computational Thermo-Fluid Dynamics, First Edition. Petr A. Nikrityuk.© 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA.

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Petr A. Nikrityuk: Computational Thermo-Fluid Dynamics — 2011/6/8 — page 352 — le-tex

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352 Index

conservation equation 4, 39– boundary condition 19– conservative form 44– for continuum media 7– in electromagnetics 24– initial condition 19– nonconservative form 44– nondimensional form 17– unified form 16

conservation of chemical species 14constitutional undercooling 125, 126continuous forcing 107, 169continuous-forcing approach 100convection

– diffusion– equation 72– phase-change problem 115

– effect 115convective driven solidification 151convergence criterion 79cooling

– curve 148, 303– rate 146, 286, 324

Coriolis force 9correction pressure 88Courant number 48Crank-Nicholson scheme 49, 50, 52critical Taylor number 235cross-diffusion phenomenon 15, 252cross-transport phenomenon 15crystal growing 1, 164

– Bridgman technique 221– modeling 162, 180– RMF control 165– simulation 163– VGF-RMF 173

curvature– correction term 158– undercooling 127

curved interface surface energy 127cut-cell method 102Czochralski method 162, 165

ddamping function 150Darcy’s

– drag 100, 169– law 139-142, 279– model 134

Davidson’s– scaling 242– theoretical analysis 236

deferred correction– scheme 55, 66– source term 65

dentrite– arm spacing (DAS) 130, 152– growth model 320, 329

– diffusion-controlled 324direct closure 146direct numerical simulation (DNS) 3, 149, 151Dirichlet boundary condition 20, 22, 68, 70,

74, 81, 156, 321, 331– thermal 109

discrete particle model (DPM) 143discretization 39, 46, 100

– equation 40, 110– explicit method 47– implicit method 49– scheme 5

– of unsteady equations 46discretized conservation equation, linearized

form 63dopant

– segregation 166– transport 165, 166

Dufour effect 15, 252

eEkman

– layer 205– number 196, 197, 202, 203, 206, 210-212,

223, 226, 232, 254, 260– pumping effect 165, 174, 206, 260, 277,

278, 283, 293elastic remelting 148electric

– discharge 318– field intensity 301

electroconductivity 37electromagnetic

– field 30, 303, 329– force 25– Lorentz force 318– processing of materials (EMP) 1, 24– stirring (EMS) 6, 249, 268, 277

electroslag remelting (ESR) 310electrovortex flow 295, 304, 313-315, 328

– scaling 321energy-conservation equation 11, 135, 160, 297enthalpy 119enthalpy-porosity model 134, 181entropy 118, 119equiaxed solidification 279, 280equilibrium

– condition 125

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Petr A. Nikrityuk: Computational Thermo-Fluid Dynamics — 2011/6/8 — page 353 — le-tex

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Index 353

– liquidus temperature 126– solidification 144, 145

equivalent– average specific heat 136– boundary layer 126

Ergun equation 144Euler scheme 49, 50, 52

– explicit method 51Eulerian method 9, 130, 154eutectic

– alloy 123, 124– fraction 7

extrapolation 94, 101

fFaraday’s law 24-26fictitious boundary method (FBM) 103finite difference method (FDM) 39, 40

– conservative property 44finite element method (FEM) 39finite volume method (FVM) 39, 52fixed-grid

– enthalpy 132, 133– method 129, 130– virtual front-tracking model 164

floating dentrites 284Fluent 96, 111fluid

– dynamics 22– flow 1, 251, 320

forcing– angular velocity 196– cell 102

FORTRAN– code 77– program 75, 80

forward difference scheme (FDS) 42Fourier number 48, 51Fourier’s law of heat conduction 11fractional-step method 86freckles 130free surface 22Froude number 17full multigrid (FMG) method 80

ggallium melting 182, 183, 188Garside and Al Dibouni correlation 144Gau and Viskanta experiment 187Gauss

– law 24– theorem 8

Gauss-Seidel algorithm 75, 76

Gaussian elimination 73, 74, 78– back substitution 73– forward elimination 73

generic source term linearization 66geostrophic mode 201, 210ghost cell 101, 102Gibbs free energy 119, 121Gibbs-Thomson

– coefficient 321– curvature term 332– effect 334– equation 127, 128– relation 157, 159

Gidaspow drag function 144global velocitiy 185

– buoyancy-induced 175– of solidification 304– short-term oscillations 186

Grashof number 18, 170, 213, 215, 220, 252,274, 275, 304– solutal 18

grid-convergence test 198, 224, 256, 289, 312guessed pressure 88Gulliver-Scheil equation 144, 146

hH-based model 134Hartmann number 28, 29, 170, 253, 304heat conduction equation 46heat transfer 1, 4, 20, 94, 103, 162, 180, 218,

233, 274– calculation of flows 85– in phase-change systems 130– simulation 103

heat-conduction-controlled solidification 131homogenous Neumann condition 21, 68hybrid

– Cartesian method 101– model 142– scheme 58

hypereutectic metal alloy 124, 295, 317

ii-sweep 77immersed boundary method (IBM) 99, 101,

152– continuous forcing 107

immersed surface reconstruction 104impermeable surface 21induction equation 25, 26interaction parameter 195interdentritic arm spacing 7interface

– cell 102, 104

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Petr A. Nikrityuk: Computational Thermo-Fluid Dynamics — 2011/6/8 — page 354 — le-tex

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354 Index

– curvature 160– point 160– surface energy 127

interfacial– nonequilibrium 127– Stefan condition 166

interpolation scheme 54, 99isoconcentrational surface 21isothermal

– flow 212, 220, 249, 274– axisymmetric code 281

– spin-up 214– surface 20

jj-sweep 77Jacobi algorithm 75, 76Joule

– dissipation 197, 223, 242– heating 13, 167, 237, 251, 254, 255, 297,

312, 317, 329

kk-epsilon 148K-model 279, 280, 283Kelvin-Helmholtz instability 240kinematic viscosity 18, 29, 109, 192kinetic

– energy 11, 205– turbulence 243-245

– undercooling 128kinetics of atomic attachment 127Kolmogorov scale 152, 256Krieger-Dougherty equation 140Kronecker delta function 10

lLagrangian method 130laminar flow 221

– RMF-driven 221Langrangian

– formulation 131– method 9, 154– type of model 143

Laplace equation 44, 77, 79, 81, 313large vortex regime 187latent heat 115

– of fusion 120lattice Boltzmann method 39level-set

– equation 157– method 117, 160– model 155

lever rule 144-146, 297, 299

lid-driven cavity (LDC) 94Lin-Lin TVD scheme 62linear

– algebraic equation 66– equation

– direct methods 73– iterative methods 73– system 72

– extrapolation 94– upwind difference scheme (LUDS) 4, 57,

65, 94Lipton-Glicksman-Kurtz (LGK) model 323liquid metal

– contactless mixing 249– spin-up in cylindrical cavities 191

liquidus line 123local composition equilibrium approach 181local thermodynamic equilibrium 125Lorentz force 4, 6, 25, 26, 29, 115, 139, 148,

170, 191, 193, 195, 202, 206, 213, 221, 224,278– azimuthal 31, 191, 194– in electroconducting containers 37– influence of the os/ol ratio 34– modeling 300, 312– oscillation 283– RMF-driven 288– RMF-induced 36– three-dimensional 250– TMF-driven 288

low-Peclet number 181low-voltage direct electrical current (LVDC)

310, 317Lykoudis number 304

mMach number 9, 85macroconservation equation 148macroscale 285

– phase-change phenomenon 129macrosegregation 7, 115, 116, 129, 180, 285,

292, 295, 307, 315magnetic

– damping time 28– Reynolds number 26, 28, 195, 254– Taylor number 29, 222, 226, 237, 253,

254, 274, 294magnetohydrodynamics (MHD) 1, 3, 24

– flow 243Marangoni convection 23, 115Marker-And-Cell (MAC) method 86mass

– conservation 8

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Petr A. Nikrityuk: Computational Thermo-Fluid Dynamics — 2011/6/8 — page 355 — le-tex

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Index 355

– transfer 1, 4, 233– calculation of flows 85– equation 48– simulation 103

mathematical model 2Maxwell equations 24, 30melt-crystal interface 164, 174, 180melt-solid interface 163melting

– of pure gallium 182, 183– velocity 185

meridional velocity 227, 230mesocale 117Metzner relation 142microscale 117

– modeling 163– phase-change phenomenon 152

microsegregation 115, 116midpoint rule 53Millere’s numerical experiment 314mixing

– different scenarios 268– impact of gravitiy 273– of binary and multiphase viscous fluids

249– RMF-driven 273– under zero-gravity conditions 249– using RMF 258– using TMF 267

mixture– electrical conductivity model 32– enthalpy 133– viscosity 139, 140, 142

model development 2modified cellular automation model 5, 157molar free energy 121molecular viscosity 97, 250, 264, 297molecular-dynamic model 117momentum conservation equation 9, 134, 300Mooney equation 140moving-grid method 130, 164multigrid method 80multivortex structure 185, 306, 307, 309mushy zone 121, 123, 129, 130, 137, 283, 285,

294, 306, 311

nnanoscale 117Navier-Stokes equation 4, 10, 11, 59, 80, 85, 99,

134, 151, 155, 169, 197, 213, 224, 251, 320,330– discretization 86– incompressible 85

Neumann boundary condition 20, 75– homogenous 20, 21

Newton’s law 295– of motion 9, 143

Newtonian fluid 10Newtonian imcompressible fluid 10, 197, 223,

242no-slip condition 22non-Newtonian fluid 139nonhomogeneous Neumann boundary

condition 68normalized

– residual 79– variable diagram (NVD) 61

numerical diffusion 56– estimation 263

numerical method 37

oOhm’s law 25-27, 300, 319overrelaxation 82, 83

ppartition coefficient 122Pb-Sn alloy 6Peclet number 17, 56, 58-60, 72

– global 70, 81– local 70, 95

permeability 142– approach 106, 109– K-model 279

phase diagram 122phase-field model 117, 154, 317piecewise linear interface construction (PLIC)

106pinch force 314, 319Poisson equation 25, 76, 77, 80, 82, 86, 301,

312, 319,331porous-medium approach 100power law discretization scheme (PDS) 5, 58,

59, 65Prandtl number 17, 183, 252prescribed heat flux 68pressure

– correction 86– coupling 86

pressure-correction equation 89– boundary condition 90– for collocated grids 94

pressure-velocity coupling 171, 224, 289, 312Primitive-variable Implicit Split Operator

(PISO) method 86pseudo-steady state (PSS) approach 163

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Petr A. Nikrityuk: Computational Thermo-Fluid Dynamics — 2011/6/8 — page 356 — le-tex

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356 Index

pseudovelocity 90, 91pulsating electric discharge (PED) 317

qquadratic upwind interpolation for convective

kinematics (QUICK) 5, 57, 62, 65, 94

rRANS turbulence model 242Rayleigh number 18, 183, 213

– dimensionless 18Rayleigh-Bernard

– configuration 212, 213– convection 149

rectangular test cell 181Reynolds number 17, 28, 29, 108, 143, 149,

152, 170, 253, 256– magnetic 26, 28, 195

Rhie-Chow stabilization 93, 110, 171, 187, 256,289, 303, 312, 322

Robin boundary condition 20, 22, 69, 75Robinson’s experiment 240Rosby initial value problem 201rotating magnetic field (RMF) 5, 28, 164, 191,

193, 249, 253, 277– frequency 170– low-frequency, low-induction 221– pulse sequence

– of alternating direction (RMF-PSAD)270

– of constant direction (RMF-PSCD) 270– stirring 272– time-modulated 270

sScheil rule 297, 299, 300Schmidt number 17, 250, 252, 264segregation coefficient 123Semi-Implicit Method for Pressure Linked

Equation (SIMPLE) method 86, 89, 92– algorithm 89, 90-92, 110, 171, 183, 187,

198, 224, 289, 303, 312– Revised (SIMPLER) 86, 90– with collocated variables arrangement 91

shape function 40sharp-interface model 32, 117, 157, 320Sheil rule 148side-wall cooling system 285, 294side-wall layer 227silicon Czochralski growth 164simple line interface calculation (SLIC) 105

– approximation of interface 108single-domain mixture model 5

solid-liquid– interface 128, 163, 164, 172, 233

– evolution 181– phase transition 180

solidification 154, 172– alloy 278– columnar 279, 280– conduction-dominated rule 280– convective-driven 151– equiaxed 279, 280– heat-conduction-controlled 131– hybrid model 149– microscale morphology 155– thermodynamics 118, 129

– conduction-dominated 138– unsteady RANS modeling 148– velocity 21, 173, 233

solidus line 123solutal transfer 21solute trapping 127solute-conservation equation 160Soret effect 15, 252source term 66

– bounded linearization 67species mass fraction conservation equation

14species-mass-conservation equation 297spin-up

– dynamics 220– flow structure 206– of isothermal flow driven 193– RMF-driven 202– time 254

steady– electric discharge (SED) 310– electromagnetic field 294– viscous regime 212

steady-state heat conduction equation 45Stefan boundary condition 21, 22Stefan condition 130, 132, 154, 156, 159, 164,

179– interfacial 166

Stefan number 183Stefan problem 131Stewartson layer 263Stokes

– flow 191, 221, 226– regime 206, 212

Stone’s strongly implicit procedure (SIP) 33,75, 78, 79, 171, 256, 303

striatious 180Strouhal number 17Stuart number interaction parameter 28

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Petr A. Nikrityuk: Computational Thermo-Fluid Dynamics — 2011/6/8 — page 357 — le-tex

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Index 357

successive overrelaxation (SOR) 76, 77surface energy of a curved interface 127Sutherland-Hodgman clipping algorithm 110Syamlal-O’Brien model 144symmetry boundary 68SZ model 181

tt-based equation 135

– numerical stabilization 136tangent rule construction 121Taylor number 36, 43, 191, 193, 194, 198, 203,

213, 221, 224, 229, 234-236, 260, 263, 275,279– critical 235– magnetic 29, 222, 226

Taylor-Görtler vortex 5, 176, 178, 192, 200, 212,218, 220, 236, 278, 283– dissipation 244

Taylor’s formula 41temperature formulation 135terrestrial gravity 274thermal

– buoyancy 289, 303, 306– conductivity 45, 138, 297– Dirichlet boundary condition 68– energy 115– undercooling 125, 330, 331

thermal-buoyancy-driven convection 176, 179thermodynamic equilibrium 167, 286thermodynamics

– first law 118– second law 119

thermoelectrical effect 167, 279, 296, 318thermofluid dynamics

– physical phenomena 7thermosolutal

– buoyancy 294– convection 129, 285, 294, 295

Thomas algorithm 77three-time-level scheme 50, 52time-step-convergence test 289, 312toroidal

– fluid flow circulation 306– meridional vortex 294– microvortex 318– vortex 174, 176, 309, 315, 327

total variation diminishing differencingscheme (TVD) 5, 61, 65

transport equation 135traveling magnetic field (TMF) 149, 255, 277

triangle geometry,stair-step approximation 107tridiagonal matrix algorithm (TDMA) 46, 74

– line-by line 75, 77truncation error 42turbulence kinetic energy 150, 243-245, 257turbulence model 149, 242turbulent

– flow 236– solidification 148– viscosity 321

uundercooling 125

– temperature 324unidirectional solidification (UDS) 314unsteady diffusion equation 50upstream difference scheme 55upstream weighted differencing scheme

(UWDS) 4, 58, 64upwind difference scheme (UDS) 4, 43, 55, 64,

224, 256upwind first-order scheme (UDS) 171upwind third-order scheme (QUICK) 5

vV-segregation channel 307, 309vacuum arc remelting (VAR) 310validation test 263, 301velocity

– coupling 86– field 85– of floating dentrites 143– of solidification 233– vector 8

velocity-correction equation 88, 89vertical

– Bridgman method 166– gradient freeze (VGF) method 162

virtual interface tracking algorithm 161viscous

– flow 221– steady-state regime 205

volume force 192von Neumann

– rule 158– stability analysis 48, 50

wwall boundary 68weight function 40Wen-Yu drag function 144