Numerical Methods in Fluid Dynamics

1. General Introduction.- 1.1 Introduction.- 1.2 Boundary Value Problems and Initial Problems.- 1.3 One-Dimensional Unsteady Flow Characteristics.- 1.4 Steady Supersonic Plane or Axi-Symmetric Flow. Equations of Motion in Characteristic Form.- 1.5 Basic Concepts Used in Finite Difference Methods.- References.- 2. The Godunov Schemes.- 2.1 The Origins of Godunov's First Scheme.- 2.2 Godunov's First Scheme. One-Dimensional Eulerian Equations.- 2.3 Godunov's First Scheme in Two and More Dimensions.- 2.4 Godunov's Second Scheme.- 2.5 The Double Sweep Method.- 2.6 Execution of the Second Scheme on the Intermediate Layer.- 2.7 Boundary Conditions on the Intermediate Layer.- 2.8 Procedure on the Final Layer.- 2.9 Applications of the Second Godunov Scheme.- 2.10 Glimm's Method.- 2.11 Outline of Solution for Gas Dynamic Equations.- 2.12 The Glimm Scheme for Simple Acoustic Waves.- 2.13 Random Choice for the Gas Dynamic Equations.- 2.14 Solution of the Riemann Problem.- 2.15 Extension to Unsteady Flow with Cylindrical or Spherical Symmetry.- 2.16 Remarks on Multi-Dimensional Problems.- References.- 3. The BVLR Method.- 3.1 Description of Method for Supersonic Flow.- 3.2 Extensions to Mixed Subsonic-Supersonic Flow. The Blunt Body Problem.- 3.3 The Double Sweep Method for Unsteady Three-Dimensional Flow.- 3.4 Worked Problem. Application to Circular Arc Airfoil.- 3.5 Results and Discussion.- Appendix-Shock Expansion Theory.- References.- 4. The Method of Characteristics for Three-Dimensional Problems in Gas Dynamics.- 4.1 Introduction.- 4.2 Bicharacteristics Method (Butler).- 4.3 Optimal Characteristics Methods (Bruhn and Haack, Schaetz).- 4.4 Near Characteristics Method (Sauer).- References.- 5. The Method of Integral Relations.- 5.1 Introduction.- 5.2 General Formulation. Model Problem.- 5.3 Flow Past Ellipses.- 5.4 The Supersonic Blunt Body Problem.- 5.5 Transonic Flow.- 5.6 Incompressible Laminar Boundary Layer Equations. Basic Formulation.- 5.7 The Method in the Compressible Case.- 5.8 Laminar Boundary Layers with Suction or Injection.- 5.9 Extension to Separated Flows.- 5.10 Application to Supersonic Wakes and Base Flows.- 5.11 Application to Three-Dimensional Laminar Boundary Layers.- 5.12 A Modified Form of the Method of Integral Relations.- 5.13 Application to Viscous Supersonic Conical Flows.- 5.14 Extension to Unsteady Laminar Boundary Layers.- 5.15 Application to Internal Flow Problems.- Model Problem (Chu and Gong).- References.- 6. Telenin's Method and the Method of Lines.- 6.1 Introduction.- 6.2 Solution of Laplace's Equation by Telenin's Method.- 6.3 Solution of a Model Mixed Type Equation by Telenin's Method.- 6.4 Application of Telenin's Method to the Symmetrical Blunt Body Problem.- 6.5 Extension to Unsymmetrical Blunt Body Flows.- 6.6 Application of Telenin's Method to the Supersonic Yawed Cone Problem.- 6.7 The Method of Lines. General Description.- 6.8 Applications of the Method of Lines.- 6.9 Powell's Method Applied to Two Point Boundary Value Problems.- Telenin's Method. Model Problems (Klopfer).- References.