Computational Gasdynamics

by Culbert B. Laney

• ISBN13: 9780521570695
• ISBN10: 0521570697
• Format: Hardcover
• Copyright: 1998-06-28
• Publisher: Cambridge University Press

Summary

Numerical methods are indispensable tools in the analysis of complex fluid flows. This book focuses on computational techniques for high-speed gas flows, especially gas flows containing shocks and other steep gradients. The book decomposes complicated numerical methods into simple modular parts, showing how each part fits and how each method relates to or differs from others. The text begins with a review of gasdynamics and computational techniques. Next come basic principles of computational gasdynamics. The last two parts cover basic techniques and advanced techniques. Senior- and graduate-level students, especially in aerospace engineering, as well as researchers and practicing engineers, will find a wealth of invaluable information on high-speed gas flows in this text.

Table of Contents

Preface	xiii
Chapter 1 Introduction	1	(2)
Part I: Gasdynamics Review	3	(100)
Chapter 2 Governing Equations of Gasdynamics	5	(16)
2.0 Introduction	5	(1)
2.1 The Integral Form of the Euler Equations	5	(8)
2.1.1 Conservation of Mass	5	(1)
2.1.2 Conservation of Momentum	6	(1)
2.1.3 Conservation of Energy	7	(1)
2.1.4 Equations of State for a Perfect Gas	8	(2)
2.1.5 Entropy and the Second Law of Thermodynamics	10	(2)
2.1.6 Vector Notation	12	(1)
2.2 The Conservation Form of the Euler Equations	13	(2)
2.2.1 Vector and Vector-Matrix Notation	14	(1)
2.2.2 Rankine-Hugoniot Relations	15	(1)
2.3 The Primitive Variable Form of the Euler Equations	15	(2)
2.3.1 Vector-Matrix Notation	16	(1)
2.4 Other Forms of the Euler Equations	17	(4)
Chapter 3 Waves	21	(27)
3.0 Introduction	21	(1)
3.1 Waves for a Scalar Model Problem	22	(4)
3.2 Waves for a Vector Model Problem	26	(2)
3.3 The Characteristic Form of the Euler Equations	28	(9)
3.3.1 Examples	33	(3)
3.3.2 Physical Interpretation	36	(1)
3.4 Simple Waves	37	(2)
3.5 Expansion Waves	39	(3)
3.6 Compression Waves and Shock Waves	42	(2)
3.7 Contact Discontinuities	44	(4)
Chapter 4 Scalar Conservation Laws	48	(23)
4.0 Introduction	48	(1)
4.1 Integral Form	49	(1)
4.2 Conservation Form	49	(1)
4.3 Characteristic Form	50	(1)
4.4 Expansion Waves	51	(1)
4.5 Compression Waves and Shock Waves	51	(1)
4.6 Contact Discontinuities	52	(1)
4.7 Linear Advection Equation	52	(2)
4.8 Burgers' Equation	54	(4)
4.9 Nonconvex Scalar Conservation Laws	58	(3)
4.10 Entropy Conditions	61	(3)
4.11 Waveform Preservation, Destruction, and Creation	64	(7)
Chapter 5 The Riemann Problem	71	(32)
5.0 Introduction	71	(1)
5.1 The Riemann Problem for the Euler Equations	72	(3)
5.2 The Riemann Problem for Linear Systems of Equations	75	(7)
5.3 Three-Wave Linear Approximations -- Roe's Approximate Riemann Solver for the Euler Equations	82	(12)
5.3.1 Secant Line and Secant Plane Approximations	82	(2)
5.3.2 Roe Averages	84	(4)
5.3.3 Algorithm	88	(2)
5.3.4 Performance	90	(4)
5.4 One-Wave Linear Approximations	94	(2)
5.5 Other Approximate Riemann Solvers	96	(1)
5.6 The Riemann Problem for Scalar Conservation Laws	96	(7)
Part II: Computational Review	103	(82)
Chapter 6 Numerical Error	105	(12)
6.0 Introduction	105	(1)
6.1 Norms and Inner Products	105	(3)
6.2 Round-Off Error	108	(2)
6.3 Discretization Error	110	(7)
Chapter 7 Orthogonal Functions	117	(15)
7.0 Introduction	117	(1)
7.1 Functions as Vectors	118	(2)
7.2 Legendre Polynomial Series	120	(3)
7.3 Chebyshev Polynomial Series	123	(3)
7.4 Fourier Series	126	(6)
Chapter 8 Interpolation	132	(18)
8.0 Introduction	132	(1)
8.1 Polynomial Interpolation	133	(12)
8.1.1 Lagrange Form	134	(1)
8.1.2 Newton Form	135	(3)
8.1.3 Taylor Series Form	138	(4)
8.1.4 Accuracy of Polynomial Interpolation	142	(3)
8.1.5 Summary of Polynomial Interpolation	145	(1)
8.2 Trigonometric Interpolation and the Nyquist Sampling Theorem	145	(5)
Chapter 9 Piecewise-Polynomial Reconstruction	150	(22)
9.0 Introduction	150	(2)
9.1 Piecewise Interpolation-Polynomial Reconstructions	152	(6)
9.2 Averaged Interpolation-Polynomial Reconstructions	158	(3)
9.3 Reconstruction via the Primitive Function	161	(6)
9.4 Reconstructions with Subcell Resolution	167	(5)
Chapter 10 Numerical Calculus	172	(13)
10.0 Introduction	172	(1)
10.1 Numerical Differentiation	172	(3)
10.1.1 Linear Approximations	172	(1)
10.1.2 Quadratic Approximations	173	(2)
10.2 Numerical Integration	175	(3)
10.2.1 Constant Approximations	176	(1)
10.2.2 Linear Approximations	177	(1)
10.3 Runge-Kutta Methods for Solving Ordinary Differential Equations	178	(7)
Part III: Basic Principles of Computational Gasdynamics	185	(122)
Chapter 11 Conservation and Other Basic Principles	187	(27)
11.0 Introduction	187	(1)
11.1 Conservative Finite-Volume Methods	187	(16)
11.1.1 Forward-Time Methods	193	(3)
11.1.2 Backward-Time Methods	196	(5)
11.1.3 Central-Time Methods	201	(2)
11.2 Conservative Finite-Difference Methods	203	(5)
11.2.1 The Method of Lines	205	(1)
11.2.2 Formal, Local, and Global Order of Accuracy	206	(2)
11.3 Transformation to Conservation Form	208	(6)
Chapter 12 The CFL Condition	214	(8)
12.0 Introduction	214	(1)
12.1 Scalar Conservation Laws	215	(4)
12.2 The Euler Equations	219	(3)
Chapter 13 Upwind and Adaptive Stencils	222	(27)
13.0 Introduction	222	(1)
13.1 Scalar Conservation Laws	223	(5)
13.2 The Euler Equations	228	(1)
13.3 Introduction to Flux Averaging	228	(3)
13.4 Introduction to Flux Splitting	231	(6)
13.4.1 Flux Split Form	234	(3)
13.4.2 Introduction to Flux Reconstruction	237	(1)
13.5 Introduction to Wave Speed Splitting	237	(6)
13.5.1 Wave Speed Split Form	239	(4)
13.6 Introduction to Reconstruction-Evoluation Methods	243	(6)
Chapter 14 Artificial Viscosity	249	(6)
14.0 Introduction	249	(1)
14.1 Physical Viscosity	249	(1)
14.2 Artificial Viscosity Form	250	(5)
Chapter 15 Linear Stability	255	(17)
15.0 Introduction	255	(2)
15.1 von Neumann Analysis	257	(7)
15.2 Alternatives to von Neumann Analysis	264	(1)
15.3 Modified Equations	265	(3)
15.4 Convergence and Linear Stability	268	(4)
Chapter 16 Nonlinear Stability	272	(35)
16.0 Introduction	272	(4)
16.1 Monotonicity Preservation	276	(1)
16.2 Total Variation Diminishing (TVD)	277	(5)
16.3 Range Diminishing	282	(3)
16.4 Positivity	285	(4)
16.5 Upwind Range Condition	289	(3)
16.6 Total Variation Bounded (TVB)	292	(1)
16.7 Essentially Nonoscillatory (ENO)	293	(1)
16.8 Contraction	294	(1)
16.9 Monotone Methods	294	(1)
16.10 A Summary of Nonlinear Stability Conditions	295	(2)
16.11 Stability and Convergence	297	(1)
16.12 The Euler Equations	298	(1)
16.13 Proofs	299	(8)
Part IV: Basic Methods of Computational Gasdynamics	307	(148)
Chapter 17 Basic Numerical Methods for Scalar Conservation Laws	309	(42)
17.0 Introduction	309	(3)
17.1 Lax-Friedrichs Method	312	(4)
17.2 Lax-Wendroff Method	316	(7)
17.3 First-Order Upwind Methods	323	(13)
17.3.1 Godunov's First-Order Upwind Method	325	(4)
17.3.2 Roe's First-Order Upwind Method	329	(3)
17.3.3 Harten's First-Order Upwind Method	332	(4)
17.4 Beam-Warming Second-Order Upwind Method	336	(7)
17.5 Fromm's Method	343	(8)
Chapter 18 Basic Numerical Methods for the Euler Equations	351	(79)
18.0 Introduction	351	(4)
18.1 Flux Approach	355	(7)
18.1.1 Lax-Friedrichs Method	355	(1)
18.1.2 Lax-Wendroff Methods	355	(7)
18.2 Wave Approach I: Flux Vector Splitting	362	(44)
18.2.1 Steger-Warming Flux Vector Splitting	373	(1)
18.2.2 Van Leer Flux Vector Splitting	374	(4)
18.2.3 Liou-Steffen Flux Vector Splitting	378	(1)
18.2.4 Zha-Bilgen Flux Vector Splitting	379	(1)
18.2.5 First-Order Upwind Methods	379	(2)
18.2.6 Beam-Warming Second-Order Upwind Method	381	(25)
18.3 Wave Approach II: Reconstruction-Evolution	406	(24)
18.3.1 Godunov's First-Order Upwind Method	410	(1)
18.3.2 Roe's First-Order Upwind Method	410	(7)
18.3.3 Harten's First-Order Upwind Method	417	(1)
18.3.4 First-Order Upwind Method Based on One-Wave Solver	417	(3)
18.3.5 Second-and Higher-Order Accurate Methods	420	(10)
Chapter 19 Boundary Treatments	430	(25)
19.0 Introduction	430	(3)
19.1 Stability	433	(1)
19.2 Solid Boundaries	434	(7)
19.3 Far-Field Boundaries	441	(14)
Part V: Advanced Methods of Computational Gasdynamics	455	(150)
Chapter 20 Flux Averaging I: Flux-Limited Methods	459	(45)
20.0 Introduction	459	(4)
20.1 Van Leer's Flux-Limited Method	463	(3)
20.2 Sweby's Flux-Limited Method (TVD)	466	(16)
20.2.1 The Linear Advection Equation with a (is greater than) 0	466	(6)
20.2.2 The Linear Advection Equation with a (is less than) 0	472	(1)
20.2.3 Nonlinear Scalar Conservation Laws with a(u) (is greater than) 0	473	(1)
20.2.4 Nonlinear Scalar Conservation Laws with a(u) (is less than) 0	474	(1)
20.2.5 Nonlinear Scalar Conservation Laws at Sonic Points	475	(7)
20.2.6 The Euler Equations	482	(1)
20.3 Chakravarthy-Osher Flux-Limited Methods (TVD)	482	(9)
20.3.1 A Second-Order Accurate Method: A Semidiscrete Version of Sweby's Flux-Limited Method	485	(2)
20.3.2 Another Second-Order Accurate Method	487	(1)
20.3.3 Second- and Third-Order Accurate Methods	488	(2)
20.3.4 Higher-Order Accurate Methods	490	(1)
20.4 Davis-Roe Flux-Limited Method (TVD)	491	(4)
20.4.1 Scalar Conservation Laws	491	(3)
20.4.2 The Euler Equations	494	(1)
20.5 Yee-Roe Flux-Limited Method (TVD)	495	(9)
20.5.1 Scalar Conservation Laws	495	(3)
20.5.2 The Euler Equations	498	(6)
Chapter 21 Flux Averaging II: Flux-Corrected Methods	504	(37)
21.0 Introduction	504	(2)
21.1 Boris-Book Flux-Corrected Method (FCT)	506	(9)
21.2 Zalesak's Flux-Corrected Methods (FCT)	515	(2)
21.3 Harten's Flux-Corrected Method (TVD)	517	(6)
21.3.1 Scalar Conservation Laws	517	(4)
21.3.2 The Euler Equations	521	(2)
21.4 Shu-Osher Methods (ENO)	523	(18)
Chapter 22 Flux Averaging III: Self-Adjusting Hybrid Methods	541	(24)
22.0 Introduction	541	(1)
22.1 Harten-Zwas Self-Adjusting Hybrid Method	542	(2)
22.2 Harten's Self-Adjusting Hybrid Method	544	(6)
22.3 Jameson's Self-Adjusting Hybrid Method	550	(15)
22.3.1 Scalar Conservation Laws	550	(8)
22.3.2 The Euler Equations	558	(7)
Chapter 23 Solution Averaging: Reconstruction-Evolution Methods	565	(32)
23.0 Introduction	565	(1)
23.1 Van Leer's Reconstruction-Evolution Method (MUSCL)	565	(9)
23.1.1 Linear Advection Equation	565	(7)
23.1.2 The Lagrange Equations	572	(2)
23.2 Colella-Woodward Reconstruction-Evolution Method (PPM)	574	(3)
23.3 Anderson-Thomas-Van Leer Reconstruction-Evolution Methods	577	(5)
(TVD/MUSCL): Finite-Volume Versions of the Chakravarthy-Osher Flux-Corrected Methods	577	(1)
23.3.1 A Second-Order Accurate Method	578	(1)
23.3.2 Second- and Third-Order Accurate Methods	579	(3)
23.4 Harten-Osher Reconstruction-Evolution Method (UNO)	582	(3)
23.4.1 The Linear Advection Equation	582	(1)
23.4.2 Nonlinear Scalar Conservation Laws	583	(2)
23.4.3 The Euler Equations	585	(1)
23.5 Harten-Engquist-Osher-Chakravarthy Reconstruction-Evolution Methods (ENO)	585	(12)
23.5.1 Second-Order Accurate Temporal Evolution for Scalar Conservation Laws	587	(2)
23.5.2 Third-Order Accurate Temporal Evolution for Scalar Conservation Laws	589	(2)
23.5.3 Second-Order Accurate Temporal Evolution for the Euler Equations	591	(6)
Chapter 24 A Brief Introduction to Multidimensions	597	(8)
24.0 Introduction	597	(1)
24.1 Governing Equations	597	(2)
24.2 Waves	599	(3)
24.3 Conservation and Other Numerical Principles	602	(3)
Index	605