1. Computational Fluid Dynamics: An Introduction.- 1.1 Advantages of Computational Fluid Dynamics.- 1.2 Typical Practical Problems.- 1.2.1 Complex Geometry, Simple Physics.- 1.2.2 Simpler Geometry, More Complex Physics.- 1.2.3 Simple Geometry, Complex Physics.- 1.3 Equation Structure.- 1.4 Overview of Computational Fluid Dynamics.- 1.5 Further Reading.- 2. Partial Differential Equations.- 2.1 Background.- 2.1.1 Nature of a Well-Posed Problem.- 2.1.2 Boundary and Initial Conditions.- 2.1.3 Classification by Characteristics.- 2.1.4 Systems of Equations.- 2.1.5 Classification by Fourier Analysis.- 2.2 Hyperbolic Partial Differential Equations.- 2.2.1 Interpretation by Characteristics.- 2.2.2 Interpretation on a Physical Basis.- 2.2.3 Appropriate Boundary (and Initial) Conditions.- 2.3 Parabolic Partial Differential Equations.- 2.3.1 Interpretation by Characteristics.- 2.3.2 Interpretation on a Physical Basis.- 2.3.3 Appropriate Boundary (and Initial) Conditions.- 2.4 Elliptic Partial Differential Equations.- 2.4.1 Interpretation by Characteristics.- 2.4.2 Interpretation on a Physical Basis.- 2.4.3 Appropriate Boundary Conditions.- 2.5 Traditional Solution Methods.- 2.5.1 The Method of Characteristics.- 2.5.2 Separation of Variables.- 2.5.3 Green's Function Method.- 2.6 Closure.- 2.7 Problems.- 3. Preliminary Computational Techniques.- 3.1 Discretisation.- 3.1.1 Converting Derivatives to Discrete Algebraic Expressions.- 3.1.2 Spatial Derivatives.- 3.1.3 Time Derivatives.- 3.2 Approximation to Derivatives.- 3.2.1 Taylor Series Expansion.- 3.2.2 General Technique.- 3.2.3 Three-point Asymmetric Formula for
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$$.- 3.3 Accuracy of the Discretisation Process.- 3.3.1 Higher-Order vs Low-Order Formulae.- 3.4 Wave Representation.- 3.4.1 Significance of Grid Coarseness.- 3.4.2 Accuracy of Representing Waves.- 3.4.3 Accuracy of Higher-Order Formulae.- 3.5 Finite Difference Method.- 3.5.1 Conceptual Implementation.- 3.5.2 DIFF: Transient Heat Conduction (Diffusion) Problem.- 3.6 Closure.- 3.7 Problems.- 4. Theoretical Background.- 4.1 Convergence.- 4.1.1 Lax Equivalence Theorem.- 4.1.2 Numerical Convergence.- 4.2 Consistency.- 4.2.1 FTCS scheme.- 4.2.2 Fully Implicit Scheme.- 4.3 Stability.- 4.3.1 Matrix Method: FTCS Scheme.- 4.3.2 Matrix Method: General Two-Level Scheme.- 4.3.3 Matrix Method: Derivative Boundary Conditions.- 4.3.4 Von Neumann Method: FTCS Scheme.- 4.3.5 Von Neumann Method: General Two-Level Scheme.- 4.4 Solution Accuracy.- 4.4.1 Richardson Extrapolation.- 4.5 Computational Efficiency.- 4.5.1 Operation Count Estimates.- 4.6 Closure.- 4.7 Problems.- 5. Weighted Residual Methods.- 5.1 General Formulation.- 5.1.1 Application to an Ordinary Differential Equation.- 5.2 Finite Volume Method.- 5.2.1 Equations with First Derivatives Only.- 5.2.2 Equations with Second Derivatives.- 5.2.3 FIVOL: Finite Volume Method Applied to Laplace's Equation.- 5.3 Finite Element Method and Interpolation.- 5.3.1 Linear Interpolation.- 5.3.2 Quadratic Interpolation.- 5.3.3 Two-Dimensional Interpolation.- 5.4 Finite Element Method and the Sturm-Liouville Equation.- 5.4.1 Detailed Formulation.- 5.4.2 STURM: Computation of the Sturm Liouville Equation.- 5.5 Further Applications of the Finite Element Method.- 5.5.1 Diffusion Equation.- 5.5.2 DUCT: Viscous Flow in a Rectangular Duct.- 5.5.3 Distorted Computational Domains: Isoparametric Formulation.- 5.6 Spectral Method.- 5.6.1 Diffusion Equation.- 5.6.2 Neumann Boundary Conditions.- 5.6.3 Pseudospectral Method.- 5.7 Closure.- 5.8 Problems.- 6. Steady Problems.- 6.1 Nonlinear Steady Problems.- 6.1.1 Newton's Method.- 6.1.2 NEWTON: Flat-Plate Collector Temperature Analysis.- 6.1.3 NEWTBU: Two-Dimensional Steady Burgers' Equations.- 6.1.4 Quasi-Newton Method.- 6.2 Direct Methods for Linear Systems.- 6.2.1 FACT/SOLVE: Solution of Dense Systems.- 6.2.2 Tridiagonal Systems: Thomas Algorithm.- 6.2.3 BANFAC/BANSOL: Narrowly-Banded Gauss Elimination.- 6.2.4 Generalised Thomas Algorithm.- 6.2.5 Block Tridiagonal Systems.- 6.2.6 Direct Poisson Solvers.- 6.3 Iterative Methods.- 6.3.1 General Structure.- 6.3.2 Duct Flow by Iterative Methods.- 6.3.3 Strongly Implicit Procedure.- 6.3.4 Acceleration Techniques.- 6.3.5 Multigrid Methods.- 6.4 Pseudotransient Method.- 6.4.1 Two-Dimensional, Steady Burgers' Equations.- 6.5 Strategies for Steady Problems.- 6.6 Closure.- 6.7 Problems.- 7. One-Dimensional Diffusion Equation.- 7.1 Explicit Methods.- 7.1.1 FTCS Scheme.- 7.1.2 Richardson and DuFort-Frankel Schemes.- 7.1.3 Three-level Scheme.- 7.1.4 DIFEX: Numerical Results for Explicit Schemes.- 7.2 Implicit Methods.- 7.2.1 Fully Implicit Scheme.- 7.2.2 Crank-Nicolson Scheme.- 7.2.3 Generalised Three-Level Scheme.- 7.2.4 Higher-Order Schemes.- 7.2.5 DIFIM: Numerical Results for Implicit Schemes.- 7.3 Boundary and Initial Conditions.- 7.3.1 Neumann Boundary Conditions.- 7.3.2 Accuracy of Neumann Boundary Condition Implementation.- 7.3.3 Initial Conditions.- 7.4 Method of Lines.- 7.5 Closure.- 7.6 Problems.- 8. Multidimensional Diffusion Equation.- 8.1 Two-Dimensional Diffusion Equation.- 8.1.1 Explicit Methods.- 8.1.2 Implicit Method.- 8.2 Multidimensional Splitting Methods.- 8.2.1 ADI Method.- 8.2.2 Generalised Two-Level Scheme.- 8.2.3 Generalised Three-Level Scheme.- 8.3 Splitting Schemes and the Finite Element Method.- 8.3.1 Finite Element Splitting Constructions.- 8.3.2 TWDIF: Generalised Finite Difference/Finite Element Implementation.- 8.4 Neumann Boundary Conditions.- 8.4.1 Finite Difference Implementation.- 8.4.2 Finite Element Implementation.- 8.5 Method of Fractional Steps.- 8.6 Closure.- 8.7 Problems.- 9. Linear Convection-Dominated Problems.- 9.1 One-Dimensional Linear Convection Equation.- 9.1.1 FTCS Scheme.- 9.1.2 Upwind Differencing and the CFL Condition.- 9.1.3 Leapfrog and Lax-Wendroff Schemes.- 9.1.4 Crank-Nicolson Schemes.- 9.1.5 Linear Convection of a Truncated Sine Wave.- 9.2 Numerical Dissipation and Dispersion.- 9.2.1 Fourier Analysis.- 9.2.2 Modified Equation Approach.- 9.2.3 Further Discussion.- 9.3 Steady Convection-Diffusion Equation.- 9.3.1 Cell Reynolds Number Effects.- 9.3.2 Higher-Order Upwind Scheme.- 9.4 One-Dimensional Transport Equation.- 9.4.1 Explicit Schemes.- 9.4.2 Implicit Schemes.- 9.4.3 TRAN: Convection of a Temperature Front.- 9.5 Two-Dimensional Transport Equation.- 9.5.1 Split Formulations.- 9.5.2 THERM: Thermal Entry Problem.- 9.5.3 Cross-Stream Diffusion.- 9.6 Closure.- 9.7 Problems.- 10. Nonlinear Convection-Dominated Problems.- 10.1 One-Dimensional Burgers' Equation.- 10.1.1 Physical Behaviour.- 10.1.2 Explicit Schemes.- 10.1.3 Implicit Schemes.- 10.1.4 BURG: Numerical Comparison.- 10.1.5 Nonuniform Grid.- 10.2 Systems of Equations.- 10.3 Group Finite Element Method.- 10.3.1 One-Dimensional Group Formulation.- 10.3.2 Multidimensional Group Formulation.- 10.4 Two-Dimensional Burgers' Equation.- 10.4.1 Exact Solution.- 10.4.2 Split Schemes.- 10.4.3 TWBURG: Numerical Solution.- 10.5 Closure.- 10.6 Problems.- A.1 Empirical Determination of the Execution Time of Basic Operations.- A.2 Mass and Difference Operators.- References.

The purpose of this two-volume textbook is to provide students of engineer­ ing, science and applied mathematics with the specific techniques, and the framework to develop skill in using them, that have proven effective in the various branches of computational fluid dynamics (CFD). Volume 1 de­ scribes both fundamental and general techniques that are relevant to all branches of fluid flow. Volume 2 provides specific techniques, applicable to the different categories of engineering flow behaviour, many of which are also appropriate to convective heat transfer. An underlying theme of the text ist that the competing formulations which are suitable for computational fluid dynamics, e.g. the finite differ­ ence, finite element, finite volume and spectral methods, are closely related and can be interpreted as part of a unified structure. Classroom experience indicates that this approach assists, considerably, the student in acquiring a deeper understanding of the strengths and weaknesses of the alternative computational methods. Through the provision of 24 computer programs and associated exam­ ples and problems, the present text is also suitable for established research workers and practitioners who wish to acquire computational skills without the benefit of formal instruction. The text includes the most up-to-date techniques and is supported by more than 300 figures and 500 references.