CONTENT: Preface 11 Acknowledgements 13 1Introduction 15 1.1 What is CFD? 15 1.2 How does a CFD code work? 16 1.3 Problem solving with CFD 18 1.4 Scope of this book 20 2Conservation laws of fluid motion and boundaryconditions 23 2.1 Governing cquation of fluid flow and heat transfc. 2.1.1Mass conservation in three dimensions 24 2.1.2Rates of change following a fluid particle and for a fluid element. 26 2.1.3 Momentum equation in three dimensions 28 2.1.4 Energy equation in dimcnsions. 30 2.3 Navier-Stokes equations for a Newtonian fluid 35 2.4 Conservative form of the governn,t equations of fluid flow. 36 2.5 Differential and integral forms of the general transgort equations 38 2.6 Classification of physical behaviours 40 2.7 The role of characteristics in hyperbolic equations 43 2.8 Classification method for simple PDEs 46 2.9Classification of fluid flow equations 47 2.10 Auxiliary conditions for viscous fluid flow equations 49 2.11 Problems in transonic and supersonic compressible flows 50 2.12 Summary 52 3 Turbulence and its modelling 54 3.1What is turbulence? 54 3.2 Transition from laminar to turbulent flow 58 3.3 Descriptors of turbulent flow 63 3.4 Characteristics of simple turbulent flows 66 3.4.1Free turbulent67 3.4.2Flat plate boundary layer and pipe flow71 3.4.3Summary 75 3.5The effect of turbulent fluctuations on properties of the mean flow 75 3.6Turbulent flow calculations79 3.7 Reynolds-averaged Navier-Stokes equations and classicalturbulence models 80 3.7.1Mixing length model 83 3.7.2The *- model 86 3.7.3Reynolds stress equation models 94 3.7.4Advanced turbulence models 99 3.7.5Closing remarks - RANS turbulence model 111 3.8Large eddy simulatio112 3.8.1Spacial filtering of unsteady Navier-Stokes equations 112 3.8.2Smagorinksy-Lilly SGS model116 3.8.3Higher-order SGS models 118 3.8.4Advanced SGS models 119 3.8.5Initial and boundary conditions for LES120 3.8.6LES applications in flows with complex geometry 122 3.8.7General comments on performance of LES 123 3.9 Direct numerical simulation 124 3.9.1Numerical issues in DNS 125 3.9.2Some achievements of DNS 127 3.10 Summary 127 4 The finite volume method for diffusion problems 129 4.1 Introduction129 4.2Finite volume method for one-dimensional steady state diffusion 129 4.3Worked examples: one-dimensional steady state diffusion 132 4.4 Finite volume method for two-dimensional diffusion problem 143 4.5 Finite volume method for three-dimensional diffusion probelems 145 4.6 Summary 146 5The finite volume method for convection- diffusion problems 148 5.1 Introduction148 5.2 Steady one-dimensional convection and diffusion 149 5.3 The central differencing scheme 150 5.4Properties of discretisation schemes 155 5.4.1 Conservativeness 155 5.4.2 Boundedness157 5.4.3 Transportiveness 157 5.5 Assessment of the central differencing scheme for convection-diffusion problems159 5.6 The upwind differencing scheme160 5.6.1 Assessment of the upwind differencing scheme 163 5.7 The hybrid differencing scheme 165 5.7.1Assessment of the hybrid differencing scheme 168 5.7.2 Hybrid differencing scheme for multi-dimensionalconvection-diffusion 168 5.8The power-law scheme 169 5.9Higher-order differencing schemes for convection-diffusion problems 170 5.9.1 Quadratic upwind differencing scheme: the QUICK scheme 170 5.9.2 Assessment of the QUICK scheme 176 5.9.3 Stability problems of the QUICK scheme and remedies 177 5.9.4 General comments on the QUICK differencing scheme 178 5.10TVD schemes . 178 5.10.1 Generalisation of upwind-biased discretisation schemes 179 5.10.2 Total variation and TVD schemes181 5.10.3 Criteria for TVD schemes 182 5.10.4 Flux limiter functions 184 5.10.5Implementation of TVD schemes 185 5.10.6 Evaluation of TVD schemes189 5.11 Summary 190 6 Solution algorithms for pressure-velocity couplinin steady flows 193 6.1 Introduction 193 6.2 The staggered grid 194 6.3The momentum equations 197 6.4The SIMPLE algorithm 200 6.5 Assembly of a complete method204 6.6 The SIMPLER algorithm . 205 6.7 The SIMPLEC algorithm 207 6.8 The PISO algorithm 207 6.9 General comments on SIMPLE, SIMPLER, SIMPLEC and PISO 210 6.10 Worked examples of the SIMPLE algorithm 211 6.11 Summary 225 7Solution of discretised equations 226 7.1 Introduction 226 7.2 The TDMA 227 7.3Application of the TDMA to two-dimensional problems229 7.4Application of the TDMA to three-dimensional problems 229 7.5Examples 230 7.5.1 Closing remarks236 7.6Point-iterative methods237 7.6.1 Jacobi iteration method 238 7.6.2 Gauss-Seidel iteration method 239 7.6.3 Relaxation methods240 7.7 Multigrid techniques243 7.7.1 An outline of a multigrid procedure 245 7.7.2 An illustrative example 246 7.7.3 Multigrid cycles 253 7.7.4Grid generation for the multigrid method 255 7.8Summary 256 8The finite volume method for unsteady flows 257 8.1 Introduction 257 8.2 One-dimensional unsteady heat conduction 257 8.2.1 Explicit scheme 260 8.2.2 Crank-Nicolson scheme 261 8.2.3The fully implicit scheme 262 8.3Illustrative examples 263 8.4Implicit method for two- and three-dimensional problems 270 8.5Discretisation of transient convection-diffusion equation 271 8.6Worked example of transient convection-diffusion using QUICKdifferencing 272 8.7 Solution procedures for unsteady flow calculations 276 8.7.1 Transient SIMPLE 276 8.7.2 The transient PISO algorithm 277 8.8 Steady state calculations using the pseudo-transient approach279 8.9 A brief note on other transient schemes 279 8.10 Summary 280 9 Implementation of boundary conditions 281 9.1 Introduction 281 9.2 Inlet boundary conditions282 9.3 Outlet boundary conditions285 9.4Wall boundary conditions 287 9.5 The constant pressure boundary condition 293 9.6 Symmetry boundary condition 294 9.7 Periodic or cyclic boundary condition 295 9.8 Potential pitfalls and final remarks 295 10 Errors and uncertainty in CFD modelling 299 10.1 Errors and uncertainty in CFD 299 10.2 Numerical errors 300 10.3 Input uncertainty 303 10.4 Physical model uncertainty305 10.5 Verification and validation 307 10.6Guidelines for best practice in CFD 312 10.7 Reporting/documentation of CFD simulation inputs and results 314 10.8 Summary316 11 Methods for dealing with complex geometries 318 11.1 Introduction 318 11.2 Body-fitted co-ordinate grids for complex geometries319 11.3 Catesian vs. curvilinear grids - an example 320 11.4 Curvilinear grids - difficulties 322 11.5 Block-structured grids 3 24 11.6Unstructured grids325 11.7 Discretisation in unstructured grids 326 11.8Discretisation of the diffusion term 330 11.9Discretisation of the convective term 334 11.10 Treatment of source terms 338 11.11 Assembly of discretised equations 339 11.12 Example calculations with unstructured grids 343 11.13 Pressure-velocity coupling inunstructured meshes 350 11.14Staggered vs. co-located grid arrangements 351 11.15Extension of the face velocity interpolation method tounstructured meshes 354 11.16 Summary 356 12 CFD modelling of combustion 357 12.1 Introduction357 12.2 Application of the first law of thermodynamics to a combustion system 358 12.3Enthalpy of formation 359 12.4Some important relationships and properties of gaseous mixtures 360 12.5Stoichiometry362 12.6 Equivalence ratio 362 12.7 Adiabatic flame temperature 363 12.8 Equilibrium and dissociation 365 12.9Mechanisms of combustion and chemical kinetics 369 12.10 Overall reactions and intermediate reactions 369 12.11 Reaction rate 370 12.12Detailed mechanisms375 12.13 Reduced mechanisms 375 12.14Governing equations for combusting flows 377 12.15 The simple chemical reacting system (SCRS) 381 12.16Modelling of a laminar diffusion flame - an example 384 12.17 CFD calculation of turbulent non-premixed combustion 390 12.18 SCRS model for turbulent combustion 394 12.19 Probability density function approach 394 12.20Beta pdf 396 12.21The chemical equilibrium model 398 12.22 Eddy break-up model of combustion 399 12.23 Eddy dissipation concept 402 12.24 Laminar flamelet model 402 12.25 Generation of laminar flamelet libraries 404 12.26Statistics of the non-equilibrium parameter 413 12.27 Pollutant formation in combustion 414 12.28 Modelling of thermal NO formation in combustion415 12.29 Flamelet-based NO modelling 416 12.30An example to illustrate laminar flamelet modelling and NOmodelling of a turbulent flame417 12.31 Other models for non-premixed combustion 429 12.32 Modelling of premixed combustion 429 12.33Summary 430 13 Numerical calculation of radiative heat transfer 431 13.1 Introduction 431 13.2 Governing equations of radiative heat transfer 438 13.3 Solution methods 440 13.4 Four popular radiation calculation techniques suitable for CFD 441 13.4.1 The Monte Carlo method441 13.4.2 The discrete transfer method443 13.4.3 Ray tracing447 13.4.4 The discrete ordinates method 447 13.4.5 The finite volume method 451 13.5 Illustrative examples 451 13.6Calculation of radiative properties in gaseous mixtures 456 13.7Summary 457 Appendix A Accuracy of a flow simulation 459 Appendix B Non-uniform grids 462 Appendix C Calculation of source terms 464 Appendix D Limiter functions used in Chapter 5 466 Appendix E Derivation of one-dimensional governing equations forsteady, incompressible flow through a planar nozzle 470 Appendix F Alternative derivation for the term (n . grad <f>A,) inChapter 11 473 Appendix G Some examples 476 Bibliography486 Index 509