A Shock-Fitting Primer

ISBN-10: 1439807582

ISBN-13: 9781439807583

A defining feature of nonlinear hyperbolic equations is the occurrence of shock waves. While the popular shock-capturing methods are easy to implement, shock-fitting techniques provide the most accurate results. A Shock-Fitting Primer presents the proper numerical treatment of shock waves and other discontinuities.The book begins by recounting the events that lead to our understanding of the theory of shock waves and the early developments related to their computation. After presenting the main shock-fitting ideas in the context of a simple scalar equation, the author applies Colombeau’s theory of generalized functions to the Euler equations to demonstrate how the theory recovers well-known results and to provide an in-depth understanding of the nature of jump conditions. He then extends the shock-fitting concepts previously discussed to the one-dimensional and quasi-one-dimensional Euler equations as well as two-dimensional flows. The final chapter explores existing and future developments in shock-fitting methods within the framework of unstructured grid methods.Throughout the text, the techniques developed are illustrated with numerous examples of varying complexity. On the accompanying CD-ROM, MATLAB® codes serve as concrete examples of how to implement the ideas discussed in the book.

Preface xiii1 Introduction 11.1 Prelude 11.2 The Curious Events Leading to the Theory of Shock Waves 51.3 Early Attempts at Computing Flows with Shocks 232 Shock-Fitting Principles 332.1 The Inviscid Burgers' Equation 362.1.1 Continuum Hypothesis 362.1.2 Material Derivative 362.1.3 Divergence Form 372.1.4 Characteristics 382.1.5 Characteristics for a System of Equations 392.1.6 Breakdown of the Classical Solution 412.1.7 Weak Solutions 412.1.7.1 Functions and Distributions 422.1.7.2 Generalized Derivative 452.1.7.3 Other Generalized Operations 482.1.8 Problem with Products: Colombeau's Generalized Functions 502.1.8.1 Application of Colombeau's Generalized Functions 522.1.9 Returning to the Divergence Equation 532.1.10 Shock Waves: Standard Analysis 542.1.11 Entropy Conditions 562.2 The One-Saw tooth Problem602.2.1 Area Conservation Rule 622.3 Background Numerical Schemes 652.3.1 Preliminaries 652.3.2 Truncation Error and Consistency 662.3.3 Order of Accuracy 672.3.4 Stability 672.3.5 Dissipation and Dispersion 712.3.6 Lax's Equivalence Theorem 712.3.7 Lax-Wendroff and MacCormack Schemes 722.3.8 Generalized Stability Analysis 752.3.9 Interpolation 762.3.10 Total Variation Diminishing Property 792.3.11 Limiter 842.3.12 Riemann's Problem 852.3.13 Approximate Riemann Solvers 862.3.14 Time Integration: Runge-Kutta Methods 912.3.15 Verification and Validation 932.3.16 Application of Background Scheme 962.4 Mappings, Conservation Form, and Transformation Matrices 972.5 Boundary Shock-Fitting 992.6 Gaussian Pulse Problem 1022.6.1 Shock Formation andBranch-Points 1042.6.2 Gaussian Pulse Shock Path 1082.7 Boundary Shock-Fitting Revisited 1102.8 Floating Shock-Fitting 1152.9 Detection of Shock Formation 1242.10 Application of Colombeau's Generalized Functions to a Nonconservative System of Equations 1292.10.1 The Mask of Batman 138Problems 1413 Fundamental Concepts and Equations 1433.1 Physical Problem 1443.1.1 Thermodynamics of the Medium 1453.1.2 Mechanical Assumptions 1463.1.3 Conservation Laws 1473.1.4 Boundary Conditions 1473.1.5 Discontinuities 1483.2 Mathematical Formulation 1483.2.1 Nondimensionalization 1483.2.2 Conservation Laws in Integral Form 1503.2.3 Conservation Laws in Differential Form 1513.3 Explicit Form of the Equations of Motion 1533.3.1 Divergence Form 1533.3.2 Euler's Form 1543.3.3 Modified Euler's Form: Conservation of Entropy 1563.3.4 Crocco's Equation 1563.4 Orthogonal Curvilinear Coordinates 1583.5 Differential Geometry of Singular Surfaces 1603.5.1 Speed of Sound and Propagation of Small Disturbances 1643.6 Finite Discontinuities 1643.6.1 Standard Analysis 1653.6.2 Nonstandard Analysis 1653.6.3 Contact Discontinuities 1703.6.4 ShockWaves 1713.6.4.1 Vorticity Jump across a Shock Wave 1733.7 Shock Wave Structure 1743.7.1 Exact Solution to the Shock Layer 1743.7.2 Description of the Shock Layer 1773.7.3 Shock Wave Thickness1783.7.4 Shock Relations in the Limit of Vanishing Viscosity 1803.7.5 Entropy as a Generalized Function 184Problems 1864 Euler Equations: One-Dimensional Problems 1894.1 Piston-Driven Flows 1894.1.1 Characteristics 1904.1.2 Riemann Invariants 1914.1.3 Range of Influence and Domain of Dependence 1914.1.4 Simple Waves 1924.1.5 Mathematical Background for Piston-Driven Flows 1934.2 Numerical Analysis of a Simple Wave Region 1974.3 Shock Wave Computation 2044.3.1 Shock Interactions 2084.3.2 Piston-Driven Flows: Numerical Examples 2124.4 Quasi-One-Dimensional Flows 2144.4.1 Governing Equations 2144.4.2 Steady Laval Nozzle 2164.4.3 Interaction of a Shock with an Abrupt Area Change 2184.4.4 Quasi-One-Dimensional Flows: Numerical Examples 224Problems 2365 Euler Equations, Two-Dimensional Problems 2395.1 The Blunt Body Problem 2395.1.1 Cylindrical and Spherical Coordinates 2395.1.2 Euler Equations in Cylindrical and Spherical Coordinates 2415.1.3 Computational Coordinates 2425.1.4 Initial Conditions 2435.1.5 Outflow Boundary 2445.1.6 Shock Computation 2455.1.6.1 Characteristic Conoid 2455.1.6.2 Standard Shock Computation 2505.1.6.3 Nuovo Shock Computation 2525.1.7 Wall Computation 2545.1.8 The Lambda Scheme in Two Space Dimensions 2545.1.9 Blunt Body Flow Results 2575.1.10 Running the Blunt Body Code 2635.2 External Conical Corners 2675.2.1 Governing Equations 2685.2.2 Flow Properties in the Neighborhood of the Sonic Line 2705.2.3 Jump Conditions at a Singular Sonic Point 2745.2.4 Flow Behavior at a Non-Stagnating Corner Point 2765.2.5 Flow Properties in the Neighborhood of a Cross-Flow Stagnation Point 2785.2.6 Higher Order Singularities 2825.2.7 Bifurcation of Higher Order Singularities 2855.2.8 Flow Model near the Comer Cross Flow Stagnation Point 2885.2.9 Generalized Coordinate Transformation 2915.2.10 Initial Conditions 2945.2.11 Shock Computation 2955.2.12 Wall Computation 2965.2.13 External Comer Row Results 2975.2.14 Running the External Corner Flow Code 3025.3 Supersonic Flow over Elliptical Wings 3035.3.1 Governing Equations 3035.3.2 Conformal Mapping 3055.3.3 Computational Coordinates 3085.3.4 Body Surface Computation 3105.3.5 Bow Shock Computation 3125.3.6 Second Derivatives of the Mappings 3145.3.7 Results for Flow over Elliptical Cross Sections 3155.3.7.1 Circular Cone 3155.3.7.2 Ogive 3185.3.7.3 Elliptical Wing 3195.3.8 Running the Supersonic Marching Code 321Problems 3236 Floating Shock-Fitting with Unstructured Grids 3276.1 Introduction 3276.2 Unstructured Grids: Preliminaries 3286.2.1 Delaunay Triangulation 3316.2.2 Some Mathematical Relations 3326.3 Unstructured Grid Solver 3346.3.1 Fluctuation Splitting for the Advection Equation 3356.3.2 Central Differences 3386.3.3 Linearity Preservation Property 3386.3.4 Positivity Property 3396.3.5 Low Diffusion Schemes3396.3.6 The N Scheme 3406.3.7 The Positive Streamwise Invariance Scheme 3416.4 Application to Euler Equations 3426.4.1 Change of Variables 3426.4.2 Linearization 3456.4.3 Fluctuation Distribution Matrix 3466.5 Floating Shock-Fitting Implementation 3486.5.1 Cell Removal in Neighborhood of Shock Front 3486.5.2 Local Re-Meshing near a Shock Front 3506.5.3 Computation of Shock Tangent and Normal Vectors 3506.5.4 Solution Update 3526.5.5 Interpolation at Phantom Nodes 3546.5.6 Shock Front Location Update 3556.5.7 Interpolation at Shock-Crossed Nodes 3566.5.8 Shock-Node Redistribution 3576.5.9 Shock Detection 3576.5.10 Shock Interactions 3596.6 Unstructured Grids Shock-Fitting Results 3636.6.1 Mach Reflection 3636.6.2 Edney Type IV Shock Interaction 364Problems 358Appendix 371References 373Index 389