Essentials of Computational Fluid Dynamics

Covered from the vantage point of a user of a commercial flow package, Essentials of Computational Fluid Dynamics provides the information needed to competently operate a commercial flow solver. This book provides a physical description of fluid flow, outlines the strengths and weaknesses of computational fluid dynamics (CFD), presents the basics of the discretization of the equations, focuses on the understanding of how the flow physics interact with a typical finite-volume discretization, and highlights the approximate nature of CFD. It emphasizes how the physical concepts (mass conservation or momentum balance) are reflected in the CFD solutions while minimizing the required mathematical/numerical background. In addition, it uses cases studies in mechanical/aero and biomedical engineering, includes MATLAB and spreadsheet examples, codes and exercise questions. The book also provides practical demonstrations on core principles and key behaviors and incorporates a wide range of colorful examples of CFD simulations in various fields of engineering.

In addition, this author:

• Introduces basic discretizations, the linear advection equation, and forward, backward and central differences
• Proposes a prototype discretization (first-order upwind) implemented in a spreadsheet/MATLAB example that highlights the diffusive character
• Looks at consistency, truncation error, and order of accuracy
• Analyzes the truncation error of the forward, backward, central differences using simple Taylor analysis
• Demonstrates how the of upwinding produces Artificial Viscosity (AV) and its importance for stability
• Explains how to select boundary conditions based on physical considerations
• Illustrates these concepts in a number of carefully discussed case studies

Essentials of Computational Fluid Dynamics provides a solid introduction to the basic principles of practical CFD and serves as a resource for students in mechanical or aerospace engineering taking a first CFD course as well as practicing professionals needing a brief, accessible introduction to CFD.

Foreword -- 1 Introduction -- 1.1 CFD, the virtual wind tunnel -- 1.2 Examples of CFD applications -- 1.3 Prerequisites -- 1.4 Literature -- 1.5 Ingredients -- 1.6 Organisation of the chapters -- 1.7 Exercises -- 2 Physical and mathematical principles of modern CFD -- 2.1 The physical model -- 2.1.1 Continuum assumption -- 2.1.2 Lagrangian vs. Eulerian description -- 2.1.3 Conservation principles -- 2.2 The mathematical model: the equations of fluid flow -- 2.2.1 Mass conservation in 1-D -- 2.2.2 Mass conservation in 3-D -- 2.2.3 Divergence and gradient operators, total derivative -- 2.2.4 The total or material derivative -- 2.2.5 The divergence form of the total derivative -- 2.2.6 Reynolds’ transport theorem -- 2.2.7 Transport of a passive scalar -- 2.3 The momentum equations -- 2.3.1 Examples of momentum balance -- 2.3.2 The inviscid momentum equation — the Euler equation -- 2.3.3 The viscous momentum equations — Navier-Stokes -- 2.3.4 The incompressible Navier-Stokes equations -- 2.3.5 Energy balance -- 2.3.6 Summary of properties for the Navier-Stokes equations -- 2.4 Simplified model equations -- 2.4.1 Linear advection equation -- 2.4.2 Inviscid Burgers’ equation -- 2.4.3 Heat equation -- 2.5 Excercises -- 3 Discretisation of the equations -- 3.1 Discretisation of the linear advection equation -- 3.1.1 Finite difference discretisation of linear advection . -- 3.1.2 Solving the finite difference approximation -- 3.1.3 Mesh refinement -- 3.1.4 Finite volume discretisation of the 1-D advection . -- 3.1.5 Solving the finite volume approximation -- 3.1.6 Finite difference vs. finite volume formulations . . -- 3.2 Burgers’ equation: non-linear advection and conservation -- 3.3 Heat equation in 1-D -- 3.3.1 Discretising second derivatives -- 3.3.2 1-D Heat equation, differential form -- 3.3.3 Solving the 1-D heat equation -- 3.4 Advection equation in 2-D -- 3.4.1 Discretisation on a structured grid -- 3.5 Solving the Navier-Stokes equations -- 3.6 The main steps in the finite volume method -- 3.6.1 Discretisation on arbitrary grids -- 3.6.2 Transport through an arbitrary face -- 3.6.3 The concept of pseudotime-stepping -- 3.6.4 Time-stepping for compressible flows -- 3.6.5 Iterative methods for incompressible flows -- 3.6.6 The SIMPLE scheme -- 3.7 Exercises -- 4 Analysis of discretisations -- 4.1 Forward, backward and central differences -- 4.2 Taylor analysis: consistency, first- and second-order accuracy -- 4.2.1 Round-off errors -- 4.2.2 Order of accuracy and mesh refinement -- 4.3 Stability, artificial viscosity and second-order accuracy . . -- 4.3.1 Artificial viscosity -- 4.3.2 Artificial viscosity and finite volume methods -- 4.3.3 Stable second-order accurate discretisations for CFD -- 4.3.4 Monotonicity and second-order accuracy: limiters . -- 4.4 Summary of spatial discretisation approaches -- 4.5 Convergence of the time-stepping iterations -- 4.5.1 Explicit methods -- 4.5.2 Implicit methods -- 4.5.3 Increasing mesh resolution -- 4.5.4 Multigrid -- 4.6 Excercises -- 5 Boundary conditions and flow physics -- 5.1 Selection of boundary conditions -- 5.1.1 Some simple examples -- 5.1.2 Selecting boundary conditions to satisfy the equations -- 5.2 Characterisation of partial differential equations -- 5.2.1 Wave-like solutions: hyperbolic equations -- 5.2.2 Smoothing-type solutions: elliptic equations -- 5.2.3 The borderline case — parabolic equations -- 5.2.4 The domain of dependence, the domain of influence -- 5.2.5 Example of characterisation: surface waves -- 5.2.6 Compressible and incompressible flows -- 5.2.7 Characterisation of the Navier-Stokes equations . . -- 5.3 Choice of boundary conditions -- 5.3.1 Boundary conditions for incompressible flow -- 5.3.2 Boundary conditions for hyperbolic equations -- 5.4 Exercises -- 6 Turbulence modelling -- 6.1 The challenges of turbulent flow for CFD -- 6.2 Description of turbulent flow -- 6.3 Self-similar profiles through scaling -- 6.3.1 Laminar velocity profiles -- 6.3.2 Turbulent velocity profile -- 6.4 Velocity profiles of turbulent boundary layers -- 6.4.1 Outer scaling: friction velocity -- 6.4.2 Inner scaling: non-dimensional wall distance y+ -- 6.5 Levels of turbulence modelling -- 6.5.1 Direct Numerical Simulation (DNS) -- 6.5.2 Reynolds-Averaged Navier-Stokes (RANS) -- 6.5.3 Large Eddy (LES) & Detached Eddy Simulation (DES) -- 6.5.4 Summary of approaches to turbulence modelling . -- 6.6 Eddy viscosity models -- 6.6.1 Mixing length model -- 6.6.2 The Spalart-Allmaras model -- 6.6.3 The k—e model -- 6.7 Near-wall mesh requirements -- 6.7.1 Estimating the wall distance of the first point -- 6.8 Exercises -- 7 Mesh quality and grid generation -- 7.1 Influence of mesh quality on the accuracy -- 7.1.1 Maximum angle condition -- 7.1.2 Regularity -- 7.1.3 Size variation -- 7.2 Requirements for the ideal mesh generator -- 7.3 Structured grids -- 7.3.1 Algebraic grids using transfinite interpolation -- 7.4 Unstructured grids -- 7.4.1 The Advancing Front Method -- 7.4.2 Delaunay triangulation -- 7.4.3 Hierarchical grid Methods -- 7.4.4 Hexahedral unstructured mesh generation -- 7.4.5 Hybrid mesh generation for viscous flow -- 7.5 Mesh adaptation -- 7.5.1 Mesh movement: r-refinement -- 7.5.2 Mesh refinement: h-refinement -- 7.6 Exercises -- 8 Analysis of the results -- 8.1 Types of errors -- 8.1.1 Incorrect choice of boundary conditions -- 8.1.2 Insufficient convergence -- 8.1.3 Artificial viscosity -- 8.1.4 Modelling errors -- 8.2 Mesh convergence -- 8.2.1 Cost of error reduction -- 8.3 Validation -- 8.4 Summary -- 8.5 Exercises -- 9 Case studies -- 9.1 Aerofoil in 2-D, inviscid flow -- 9.1.1 Case description -- 9.1.2 Flow physics -- 9.1.3 Meshes -- 9.1.4 Simulation results for the C-mesh -- 9.1.5 Comparison of C- vs O-mesh -- 9.1.6 Analysis of lift and drag values -- 9.2 Blood vessel bifurcation in 2-D -- 9.2.1 Geometry and flow parameters -- 9.2.2 Flow physics and boundary conditions -- 9.2.3 Velocity and pressure fields -- 9.2.4 Velocity profile in the neck -- 9.2.5 Effect of outlet boundary condition -- 9.3 Aerofoil in 2-D, viscous flow -- 9.3.1 Flow physics -- 9.3.2 Turbulence modelling -- 9.3.3 Flow results -- 9.3.4 Lift and drag -- 10 Appendix -- 10.1 Finite-volume implementation of 2-D advection -- Bibliography -- Index.

Dr. Jens-Dominik Mueller is a senior lecturer in the School of Engineering and Materials Science at Queen Mary, University of London, UK. He graduated with a Dipl.-Ing in mechanical engineering in 1989 from the Technical University of Munich, obtained a VKI Diploma in aeronautics from the Von Karman Institute in Brussels in 1990, and an MSc and PhD in aerospace engineering from the University of Michigan, Ann Arbor, in 1996. He held research and academic positions at CERFACS, Toulouse, Oxford University and Queen's University Belfast. He is the author of more than 40 publications and has organized numerous international conferences.