Computer Modelling of Heat and Fluid Flow in Materials Processing

Chun-Pyo Hong Yonsei University, Korea

Preface, 1 Mechanisms of transport phenomena, 1.1 Heat transfer, 1.1.1 Conduction—Fourier’s Law of Conduction, 1.1.2 Convection, 1.1.3 Radiation, 1.2 Mass transfer, 1.2.1 Diffusion—Fick’s Law of Diffusion, 1.2.2 Convective mass transfer, 1.3 Momentum transfer, 1.3.1 Viscous momentum transfer—Newton’s Law of Viscosity, 1.3.2 Convective momentum transfer, Reference, 2 Governing equations for transport phenomena, 2.1 Governing equations for mass transfer, 2.1.1 Integral form of mass balance equation, 2.1.2 Differential form of mass balance equation—equation of continuity, 2.2 Governing equations for momentum transfer, 2.2.1 Integral form of momentum balance equation, 2.2.2 Differential form of momentum balance equation—equation of motion, 2.2.3 Boundary conditions, 2.3 Governing equations for energy transfer, 2.3.1 Integral form of energy balance equation, 2.3.2 Differential form of energy balance equation, 2.3.3 Initial and boundary conditions, 2.4 Governing equations for species transfer, 2.4.1 Integral form of mass balance equation for species A, 2.4.2 Differential form of mass balance equation for species A, 2.4.3 Initial and boundary conditions, References, 3 Similarities among three types of transport phenomena, 3.1 Basic flux laws, 3.1.1 Heat transfer (Fourier’s law of conduction), 3.1.2 Mass transfer (Fick’s law of diffusion), 3.1.3 Momentum transfer (Newton’s law of viscosity), 3.2 Convective transfer, 3.3 Governing equations, Further readings for chapters 1 through 3, 4 Basics of finite difference methods, 4.1 Introduction, 4.2 Finite difference methods, 4.2.1 Taylor-series formulation, 4.2.2 Integral method, 4.2.3 Finite volume method—control volume approach, References, 5 Steady state heat conduction, 5.1 Mathematical formulation, 5.1.1 Governing equation, 5.1.2 Boundary conditions, 5.2 Finite volume approach for steady state problems, 5.2.1 Computational grids, 5.2.2 Derivation of finite difference equations, 5.2.3 Solution of linear algebraic equations, 5.3 One-dimensional cylindrical and spherical coordinates, 5.3.1 Control volumes inside a domain, 5.3.2 Control volumes on the outer boundary of a domain, 5.4 Multi-dimensional problems, 5.4.1 Two-dimensional problems, 5.4.2 Three-dimensional problems, 5.5 Worked examples, 5.5.1 Example 5.1, 5.5.2 Example 5.2, 5.5.3 Example 5.3, 5.6 Case study: one-dimensional steady state heat conduction problems, 5.6.1 Description of the problem, 5.6.2 Glossary of FORTRAN notation, 5.6.3 Simulations, 5.6.4 Program list, 6 Transient heat conduction, 6.1 Mathematical formulation, 6.1.1 Governing equation, 6.1.2 Initial and boundary conditions, 6.2 Finite volume approach for transient problems, 6.2.1 Computational grids, 6.2.2 Derivation of finite difference equations, 6.3 Solving schemes, 6.3.1 Fully explicit method, 6.3.2 Fully implicit method, 6.3.3 Crank-Nicolson method, 6.4 Stability analysis—von Neumann stability analysis, 6.5 Multi-dimensional problems, 6.6 Worked examples, 6.6.1 Example 6.1, 6.6.2 Example 6.2, 6.6.3 Example 6.3, 6.7 Case study: one-dimensional transient heat conduction problems, 6.7.1 Description of the problem, 6.7.2 Glossary of FORTRAN notation, 6.7.3 Simulations, 6.7.4 Program list, References, 7 Phase change problems, 7.1 Introduction, 7.2 Methods of solution for phase change, 7.2.1 Numerical methods, 7.2.2 Alloy solidification, 7.3 Case study: one-dimensional phase change problems, 7.3.1 Description of the problem, 7.3.2. Glossary of FORTRAN notation, 7.3.3 Simulation, 7.3.4 Program List, References, 8 Discretization schemes for convection and diffusion terms, 8.1 Introduction, 8.2 Steady one-dimensional convection and diffusion, 8.2.1 Governing equations, 8.2.2 The analytical solution, 8.2.3 A control volume approach, 8.2.4 The central difference scheme, 8.2.5 The upwind difference scheme, 8.2.6 The hybrid difference scheme, 8.2.7 The power-law scheme, 8.3 Comparison among difference schemes, References, 9 Solution algorithms for fluid flow analysis, 9.1 Governing equations, 9.2 Solving schemes, 9.2.1 Vorticity-stream function approach, 9.2.2 Primitive variable approaches, 9.3 Summary, References, 10 Fluid flow analysis using the SIMPLE method based on the Cartesian coordinate system, 10.1 Governing equations, 10.2 Staggered and non-staggered grids, 10.3 Discretization method, 10.3.1 Discretization of the integral form of transport equation, 10.3.2 Discretization of momentum equations, 10.3.3 The SIMPLE algorithm, 10.4 Treatment of free surfaces, 10.4.1 The MAC method, 10.4.2 The VOF (volume of fluid) method, 10.5 Boundary conditions, 10.6 Turbulent flow, 10.7 Case studies, 10.7.1 Flow over a semi-circular core between two plates, 10.7.2 Internal flow in a U-tube, References, 11 Fluid flow analysis using the SIMPLE method based on the body-fitted coordinate system, 11.1 Introduction, 11.2 Transformation of coordinate systems, 11.3 Transformation of basic equations, 11.4 Discretization method, 11.4.1 Discretization of the integral form of transport equation, 11.4.2 Discretization of momentum equations, 11.4.3 The SIMPLE algorithm, 11.5 The VOF method in the body-fitted coordinate system, 11.6 Treatment of a surface cell, 11.6.1 Momentum equations, 11.6.2. Pressure at a surface cell, 11.6.3 Contravariant velocity, 11.6.4 Determination of the direction normal to a free surface, 11.7 Case studies, 11.7.1 Flow over a semi-circular core between two plates, 11.7.2 Internal flow in a U-tube, References, 12 Modelling of mould filling, 12.1 Introduction, 12.2 Numerical analysis of filling process, 12.2.1 Governing equations, 12.2.2 Free surface tracking in mould filling, 12.2.3 Algorithms for free surface tracking in the SIMPLE method, 12.3 Examples of mould filling simulation, 12.3.1 Filling in a mould cavity with a straight and tapered gating system using the SIMPLE-VOF method, 12.3.2 Filling in a mould cavity with a curved gating system using the SIMPLE-BFC-VOF method, 12.3.3 Comparison of the standard SIMPLE-VOF and the SIMPLE-BFC-VOF methods, 12.4 Case studies, 12.4.1 Filling in a rectangular cavity with a semicircular core on the bottom plate, References, 13 Modelling of microstructure evolution, 13.1 Introduction, 13.2 Nucleation and growth kinetics, 13.2.1 Nucleation, 13.2.2 Growth kinetics, 13.3 Classical cellular automaton models, 13.3.1 Model description, 13.3.2 Nucleation and growth algorithm implemented into CA, 13.3.3 Coupling the macroscopic heat flow calculation with CA models, 13.3.4 Examples of classical CA simulation, 13.4 Modified cellular automaton models, 13.4.1 Model description, 13.4.2 Growth, 13.4.3 Coupling the continuum model with a modified CA model, 13.4.4 Examples of modified CA simulation, 13.5 Case studies, 13.5.1 Simulation of solidification grain structures by classical cellular automaton models, 13.5.2 Simulation of dendritic growth by modified cellular automaton models, References, Index