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COMPUTATIONAL FLUID DYNAMICS: PRINCIPLES AND APPLICATIONS
J. Blazek
Alstom Power Ltd.,
Baden-Daettwil, Switzerland
Contents
Acknowledgements xi
List of Symbols xiii
Abbreviations xix
1. Introduction
2. Governing Equations
2.1 The Flow and its Mathematical Description
2.2 Conservation Laws
2.2.1 The Continuity Equation
2.2.2 The Momentum Equation
2.2.3 The Energy Equation
2.3 Viscous Stresses
2.4 Complete System of the Navier-Stokes Equations
2.4.1 Formulation for a Perfect Gas
2.4.2 Formulation for a Real Gas
2.4.3 Simplifications to the Navier-Stokes Equations
Bibliography
3. Principles of Solution of the Governing Equations
3.1 Spatial Discretisation
3.1.1 Finite Difference Method
3.1.2 Finite Volume Method
3.1.3 Finite Element Method
3.1.4 Other Discretisation Methods
3.1.5 Central versus Upwind Schemes
3.2 Temporal Discretisation
3.2.1 Explicit Schemes
3.2.2 Implicit Schemes
3.3 Turbulence Modelling
3.4 Initial and Boundary Conditions
Bibliography
4. Spatial Discretisation: Structured Finite Volume Schemes
4.1 Geometrical Quantities of a Control Volume
4.1.1 Two-Dimensional Case
4.1.2 Three-Dimensional Case
4.2 General Discretisation Methodologies
4.2.1 Cell-Centred Scheme
4.2.2 Cell-Vertex Scheme: Overlapping Control Volumes
4.2.3 Cell-Vertex Scheme: Dual Control Volumes
4.2.4 Cell-Centred versus Cell-Vertex Schemes
4.3 Discretisation of Convective Fluxes
4.3.1 Central Scheme with Artificial Dissipation
4.3.2 Flux-Vector Splitting Schemes
4.3.3 Flux-Difference Splitting Schemes
4.3.4 Total Variation Diminishing Schemes
4.3.5 Limiter Functions
4.4 Discretisation of Viscous Fluxes
4.4.1 Cell-Centred Scheme
4.4.2 Cell-Vertex Scheme
Bibliography
5. Spatial Discretisation: Unstructured Finite Volume Schemes Geometrical Quantities of a Control Volume
5.1.1 Two-Dimensional Case
5.1.2 Three-Dimensional Case General Discretisation Methodologies
5.2.1 Cell-Centred Scheme
5.2.2 Median-Dual Cell-Vertex Scheme
5.2.3 Cell-Centred versus Median-Dual Scheme
5.3 Discretisation of Convective Fluxes
5.3.1 Central Schemes with Artificial Dissipation
5.3.2 Upwind Schemes
5.3.3 Solution Reconstruction
5.3.4 Evaluation of Gradients
5.3.5 Limiter Functions
5.4 Discretisation of Viscous Fluxes
5.4.1 Element-Based Gradients
5.4.2 Average of Gradients
Bibliography
6. Temporal Discretisation
6.1 Explicit Time-Stepping Schemes
6.1.1 Multistage Schemes (Runge-Kutta)
6.1.3 Treatment of the Source Term
6.1.4 Determination of the Maximum Time Step
6.1.2 Hybrid Multistage Schemes
6.2 Implicit Time-Stepping Schemes
6.2.1 Matrix Form of Implicit Operator
6.2.2 Evaluation of the Flux Jacobian
6.2.3 AD1 Scheme
6.2.4 LU-SGS Scheme
6.2.5 Newton-Krylov Method
6.3 Methodologies for Unsteady Flows
6.3.1 Dual Time-Stepping for Explicit Multistage Schemes
6.3.2 Dual Time-Stepping for Implicit Schemes
Bibliography
7 Turbulence Modelling
7.1 Basic Equations of Turbulence
7.1.1 Reynolds Averaging
7.1.2 Favre (Mass) Averaging
7.1.3
7.1.4 Favre- and Reynolds-Averaged Navier-Stokes Equations
7.1.5 Eddy-Viscosity Hypothesis
7.1.6 Non-Linear Eddy Viscosity
7.2 First-Order Closures
7.2.1
7.2.2 K-a Two-Equation Model
7.2.3 Reynolds-Averaged Navier-Stokes Equations
7.1.7 Reynolds-Stress Transport Equation Spalart-Allmaras One-Equation Model SST Two-Equation Model of Menter
7.3 Large-Eddy Simulation
7.3.1 Spatial Filtering
7.3.2 Filtered Governing Equations
7.3.3 Subgrid-Scale ModelliIig
7.3.4 Wall Models
Bibliography
8 Boundary Conditions
8.1 Concept of Dummy Cells
8.2 Solid Wall
8.2.1 Inviscid Flow
8.2.2 Viscous Flow
8.3 Fafield
8.3.1 Concept of Characteristic Variables
8.3.2 Modifications for Lifting Bodies
8.4 Inlet/Outlet Boundary
8.5 Symmetry Plane
8.6 Coordinate Cut
8.7 Periodic Boundaries
8.8 Interface Between Grid Blocks
8.9 Flow Gradients at Boundaries of Unstructured Grids
Bibliography
9 Acceleration Techniques
9.1 Local Time-Stepping
9.2 Enthalpy Damping
9.3 Residual Smoothing
9.3.1 Central IRS on Structured Grids
9.3.2 Central IRS on Unstructured Grids
9.3.3 Upwind IRS on Structured Grids
9.4 Multigrid
9.4.1 Basic Multigrid Cycle
9.4.2 Multigrid Strategies
9.4.3 Implementation on Structured Grids
9.4.4 Implementation on Unstructured Grids
9.5 Preconditioning for Low Mach Numbers
Bibliography
10 Consistency. Accuracy and Stability
10.1 Consistency Requirements
10.2 Accuracy of Discretisation
10.3 Von Neumann Stability Analysis
10.3.1 Fourier Symbol and Amplification Factor
10.3.2 Convection Model Equation
10.3.3 Convection-Diffusion Model Equation
10.3.4 Explicit Time-Stepping
10.3.5 Implicit Time-Stepping
10.3.6 Derivation of the CFL Condition
Bibliography
11.1 Structured Grids
11.1.1 C-, H-, and 0-Grid Topology
11.1.2 Algebraic Grid Generation
11.1.3 Elliptic Grid Generation
11.1.4 Hyperbolic Grid Generation
11.2 Unstructured Grids
11.2.1 Delaunay Triangulation
11.2.2 Advancing-Front Method
11.2.3 Generation of Anisotropic Grids
11.2.4 Mixed-Element/Hybrid Grids
11.2.5 Assessment and Improvement of Grid Quality
Bibliography
12.1 Programs for Stability Analysis
12.2 Structured 1-D Grid Generator
12.3 Structured 2-D Grid Generators
12.4 Structured to Unstructured Grid Converter
11 Principles of Grid Generation
12 Description of the Source Codes
12.5 Quasi 1-D Euler Solver
12.6 Structured 2-D Euler Solver
12.7 Unstructured 2-D Euler Solver
Bibliography
A.1 Governing Equations in Differential Form
A.2.2 Parabolic Equations
A.2.3 Elliptic Equations
A.3 Navier-Stokes Equations in Rotating Frame of Reference
A.4 Navier-Stokes Equations Formulated for Moving Grids
A.5 Thin Shear Layer Approximation
A.6 Parabolised Navier-Stokes Equations
A.7 Convective Flux Jacobian
A.8 Viscous Flux Jacobian
A.9 Transformation from Conservative to Characteristic Variables
A.10 GMRES Algorithm
A . l l Tensor Notation
Bibliography
Index
A Appendix
A.2 Mathematical Character of the Governing Equations
A.2.1 Hyperbolic Equations
