Numerical Methods in Fluid Dynamics

This monograph is based on a graduate course, Mechanical Engipeering 266, which was developed over a number of years at the University of California-Berkeley. Shorter versions of the course were given at the University of Paris VI in 1969, and at the University of Paris XI in 1972. The course was originally presented as the last of a three quarter sequence on Compressible Flow Theory, with emphasis on the treatment of non-linear problems by numerical techniques. This is reflected in the material of the first half of the book, covering several techniques for handling non-linear wave interaction and other problems in Gas Dynamics. The techniques have their origins in the Method of Characteristics (in both two and three dimensions). Besides reviewing the method itself the more recent techniques derived from it, firstly by Godunov and his group, and secondly by Rusanov and his co-workers, are described. Both these approaches are applicable to steady flows calculated as asymptotic states of unsteady flows and treat elliptic prob­ lems as limiting forms of unsteady hyperbolic problems. They are there­ fore applicable to low speed as well a~ to high speed flow problems. The second half of the book covers the treatment of a variety of steady flow problems, including effects of both viscosity and compressibi­ lity, by the Method of Integral Relations, Telenin's Method, and the Method of Lines.

Table of Contents:
1. General Introduction.- 1.1 Introduction.- 1.2 Boundary Value Problems and Initial Value 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.- 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.- 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 BaseFlows.- 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.- 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.- References.