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E-raamat: High Accuracy Computing Methods: Fluid Flows and Wave Phenomena

(Indian Institute of Technology, Kanpur)
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  • Ilmumisaeg: 16-May-2013
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781107069657
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High Accuracy Computing Methods: Fluid Flows and Wave PhenomenaSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 16-May-2013
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781107069657
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This book presents topics in a single source format using unified spectral theory of computing. With developments of DNS and LES, practitioners are rediscovering waves as important in fluid flows, and capturing these numerically is central to high accuracy computing. Analysis of waves and its use in numerical methods in propagating energy at the right velocity (dispersion effects) and with right amplitude (dissipation) are essential. Most industrial codes using Reynolds-averaged NavierStokes equations with turbulence models cannot conceive of capturing waves. The new themes covered in this book are: Correct error propagation analysis Practical compact schemes and global analysis tool Aliasing error and its alleviation Spurious upstream propagating q-waves Explanation of the Gibbs phenomenon New 1D and 2D filters for LES/DNS without SGS modelling Anisotropic skewed wave propagation Development and analysis of dispersion relation preservation (DRP) schemes Flow instabilities and wave propagation phenomena

Muu info

This book presents topics in a single source format using unified spectral theory of computing.
Foreword xiii
Preface xvii
Chapter 1 Basic Ideas of Scientific Computing
1(7)
1.1 Overview on Scientific Computing
1(1)
1.2 Major Milestones in Electronic Computing
2(1)
1.3 Supercomputing and High Performance Computing
3(2)
1.3.1 Parallel and cluster computing
4(1)
1.3.2 Algorithmic issues of HPC
5(1)
1.4 Computational Fluid Mechanics
5(1)
1.5 Role of Computational Fluid Mechanics
6(2)
Chapter 2 Governing Equations in Fluid Mechanics
8(23)
2.1 Introduction
8(1)
2.2 Basic Equations of Fluid Mechanics
8(3)
2.2.1 Finite control volume
9(1)
2.2.2 Infinitesimal fluid element
9(1)
2.2.3 Substantive derivative
10(1)
2.3 Equation of Continuity
11(1)
2.4 Momentum Conservation Equation
11(3)
2.5 Energy Conservation Equation
14(2)
2.6 Alternate Forms of Energy Equation
16(1)
2.7 The Energy Equation in Conservation Form
17(1)
2.8 Notes on Governing Equations
17(1)
2.9 Strong Conservation and Weak Conservation Forms
18(1)
2.10 Boundary and Initial Conditions (Auxiliary Conditions)
19(1)
2.11 Equations of Motion in Non-Inertial Frame
19(4)
2.12 Equations of Motion in Terms of Derived Variables
23(2)
2.13 Vorticity-Vector Potential Formulation
25(1)
2.14 Pressure Poisson Equation
26(2)
2.15 Comparison of Different Formulations
28(1)
2.16 Other Forms of Navier-Stokes Equation
28(3)
Chapter 3 Classification of Quasi-Linear Partial Differential Equations
31(7)
3.1 Introduction
31(1)
3.2 Classification of Partial Differential Equations
31(4)
3.3 Relationship of Numerical Solution Procedure and Equation Type
35(1)
3.4 Nature of Well-Posed Problems
36(1)
3.5 Non-Dimensional Form of Equations
37(1)
Chapter 4 Waves and Space-Time Dependence in Computing
38(33)
4.1 Introduction
38(1)
4.2 The Wave Equation
39(8)
4.2.1 Plane waves
42(1)
4.2.2 Three-dimensional axisymmetric wave
43(1)
4.2.3 Doppler shift
43(1)
4.2.4 Surface gravity waves
43(4)
4.3 Deep and Shallow Water Waves
47(1)
4.4 Group Velocity and Energy Flux
47(7)
4.4.1 Physical and computational implications of group velocity
49(1)
4.4.2 Wave-packets and their propagation
50(1)
4.4.3 Waves over layer of constant depth
50(2)
4.4.4 Waves over layer of variable depth H(x)
52(1)
4.4.5 Wave refraction in shallow waters
52(1)
4.4.6 Finite amplitude waves of unchanging form in dispersive medium
53(1)
4.5 Internal Waves at Fluid Interface: Rayleigh-Taylor Problem
54(5)
4.5.1 Internal and surface waves in finite over an infinite deep layer of fluid
55(2)
4.5.2 Barotropic or surface mode
57(1)
4.5.3 Baroclinic or internal mode
57(1)
4.5.4 Rotating shallow water equation and wave dynamics
57(2)
4.6 Shallow Water Equation (SWE)
59(4)
4.6.1 Various frequency regimes of SWE
61(2)
4.7 Additional Issues of Computing: Space-Time Resolution of Flows
63(2)
4.7.1 Spatial scales in turbulent flows
63(2)
4.8 Two- and Three-Dimensional DNS
65(2)
4.9 Temporal Scales in Turbulent Flows
67(1)
4.10 Computing Time-Averaged and Unsteady Flows
68(3)
Chapter 5 Spatial and Temporal Discretizations of Partial Differential Equations
71(21)
5.1 Introduction
71(1)
5.2 Discretization of Differential Operators
72(2)
5.2.1 Functional representation by the Taylor series
72(1)
5.2.2 Polynomial representation of function
73(1)
5.3 Discretization in Non-Uniform Grids
74(1)
5.4 Higher Order Representation of Derivatives Using Operators
75(2)
5.5 Higher Order Upwind Differences
77(2)
5.5.1 Symmetric stencil for higher derivatives
78(1)
5.6 Numerical Errors
79(1)
5.7 Time Integration
79(13)
5.7.1 Single-step methods
80(2)
5.7.2 Single-step multi-stage methods
82(1)
5.7.3 Runge-Kutta methods
83(7)
5.7.4 Multi-step time integration schemes
90(2)
Chapter 6 Solution Methods for Parabolic Partial Differential Equations
92(14)
6.1 Introduction
92(1)
6.2 Theoretical Analysis of the Heat Equation
92(2)
6.3 A Classical Algorithm for Solution of the Heat Equation
94(1)
6.4 Spectral Analysis of Numerical Methods
94(1)
6.4.1 A higher order method or Milne's method
95(1)
6.5 Treating Derivative Boundary Condition
95(1)
6.6 Stability, Accuracy and Consistency of Numerical Methods
96(5)
6.6.1 Richardson's method
97(2)
6.6.2 Du Fort--Frankel method
99(2)
6.7 Implicit Methods
101(1)
6.8 Spectral Stability Analysis of Implicit Methods
102(4)
Appendix I
104(2)
Chapter 7 Solution Methods for Elliptic Partial Differential Equations
106(24)
7.1 Introduction
106(2)
7.2 Jacobi or Richardson Iteration
108(1)
7.3 Interpretation of Classical Iterations
109(2)
7.4 Different Point and Line Iterative Methods
111(2)
7.4.1 Gauss-Seidel point iterative method
111(1)
7.4.2 Line Jacobi method
112(1)
7.4.3 Explanation of line iteration methods
112(1)
7.5 Analysis of Iterative Methods
113(1)
7.6 Convergence Theorem for Stationary Linear Iteration
114(1)
7.7 Relaxation Methods
115(1)
7.8 Efficiency of Iterative Methods and Rate of Convergence
116(1)
7.8.1 Method of acceleration due to Lyusternik
117(1)
7.9 Alternate Direction Implicit (ADI) Method
117(6)
7.9.1 Analysis of ADI method
119(2)
7.9.2 Choice of acceleration parameters
121(1)
7.9.3 Estimates of maximum and minimum eigenvalues
122(1)
7.9.4 Explanatory notes on ADI and other variant methods
122(1)
7.10 Method of Fractional Steps
123(1)
7.11 Multi-Grid Methods
124(6)
7.11.1 Two-Grid method
126(2)
7.11.2 Multi-Grid method
128(1)
7.11.3 Other classifications of multi-grid method
129(1)
Chapter 8 Solution of Hyperbolic PDEs: Signal and Error Propagation
130(20)
8.1 Introduction
130(1)
8.2 Classical Methods of Solving Hyperbolic Equations
130(3)
8.2.1 Explicit methods
131(2)
8.3 Implicit Methods
133(1)
8.4 General Characteristics of Various Methods for Linear Problems
134(1)
8.5 Non-linear Hyperbolic Problems
134(1)
8.6 Error Dynamics: Beyond von Neumann Analysis
135(7)
8.6.1 Dispersion error and its quantification
138(4)
8.7 Role of Group Velocity and Focussing
142(8)
8.7.1 Focussing phenomenon
144(6)
Chapter 9 Curvilinear Coordinate and Grid Generation
150(46)
9.1 Introduction
150(1)
9.2 Generalized Curvilinear Scheme
151(1)
9.3 Reciprocal or Dual Base Vectors
152(1)
9.4 Geometric Interpretation of Metrics
152(1)
9.5 Orthogonal Grid System
153(1)
9.6 Generalized Coordinate Transformation
154(1)
9.7 Equations for the Metrics
154(2)
9.8 Navier-Stokes Equation in the Transformed Plane
156(3)
9.9 Linearization of Fluxes
159(2)
9.10 Thin Layer Navier-Stokes Equation
161(1)
9.11 Grid Generation
162(1)
9.12 Types of Grid
162(1)
9.13 Grid Generation Methods
163(1)
9.14 Algebraic Grid Generation Method
164(1)
9.14.1 One-dimensional stretching functions
164(1)
9.15 Grid Generation by Solving Partial Differential Equations
165(1)
9.16 Elliptic Grid Generators
165(1)
9.17 Hyperbolic Grid Generation Method
166(1)
9.18 Orthogonal Grid Generation for Navier-Stokes Computations
166(3)
9.19 Coordinate Transformations and Governing Equations in Orthogonal System
169(3)
9.19.1 Gradient operator
170(1)
9.19.2 Divergence operator
170(1)
9.19.3 The Laplacian operator
171(1)
9.19.4 The curl operator
171(1)
9.19.5 The line integral
172(1)
9.19.6 The surface integral
172(1)
9.19.7 The volume integral
172(1)
9.20 The Gradient and Laplacian of Scalar Function
172(1)
9.21 Vector Operators of a Vector Function
173(1)
9.22 Plane Polar Coordinates
173(1)
9.23 Navier-Stokes Equation in Orthogonal Formulation
174(2)
9.24 Improved Orthogonal Grid Generation Method for Cambered Airfoils
176(8)
9.24.1 Orthogonal grid generation for GA(W)-1 airfoil
176(5)
9.24.2 Orthogonal grid generation for an airfoil with roughness element
181(1)
9.24.3 Solutions of Navier-Stokes equation for flow past SHM-1 airfoil
182(2)
9.24.4 Compressible flow past NACA 0012 airfoil
184(1)
9.25 Governing Euler Equation, Auxiliary Conditions, Numerical Methods and Results
184(3)
9.26 Flow Field Calculation Using Overset or Chimera Grid Technique
187(9)
Chapter 10 Spectral Analysis of Numerical Schemes and Aliasing Error
196(60)
10.1 Introduction
196(2)
10.2 Spatial Discretization of First Derivatives
198(1)
10.2.1 Second order central differencing (CD2) scheme
198(1)
10.3 Discrete Computing and Nyquist Criterion
198(1)
10.4 Spectral Accuracy of Differentiation
198(2)
10.5 Spectral Analysis of Fourth Order Central Difference Scheme
200(1)
10.6 Role of Upwinding
200(3)
10.6.1 First order upwind scheme (UD1)
200(2)
10.6.2 Third order upwind scheme (UD3)
202(1)
10.7 Numerical Stability and Concept of Feedback
203(1)
10.8 Spectral Stability Analysis
204(1)
10.9 High Accuracy Schemes for Spatial Derivatives
204(3)
10.10 Temporal Discretization Schemes
207(5)
10.10.1 Euler time integration scheme
207(5)
10.10.2 Four-stage Runge-Kutta (RK4) method
212(1)
10.11 Multi-Time Level Discretization Schemes
212(20)
10.11.1 Mid-point leapfrog scheme
214(9)
10.11.2 Second order Adams-Bashforth scheme
223(9)
10.12 Aliasing Error
232(11)
10.12.1 Why aliasing error is important?
233(7)
10.12.2 Estimation of aliased component
240(3)
10.13 Numerical Estimates of Aliasing Error
243(5)
10.14 Controlling Aliasing Error
248(8)
10.14.1 Aliasing removal by zero padding
253(1)
10.14.2 Aliasing removal by phase shifts and grid-staggering
254(2)
Chapter 11 Higher Accuracy Methods
256(85)
11.1 Introduction
256(1)
11.2 The General Compact Schemes
257(1)
11.2.1 Approximating first derivatives by central scheme
257(1)
11.3 Method for Solving Periodic Tridiagonal Matrix Equation
258(2)
11.4 An Example of a Sixth Order Scheme
260(2)
11.5 Order of Approximation versus Resolution
262(5)
11.6 Optimization Problem Associated with Discrete Evaluation of First Derivatives
267(3)
11.7 An Optimized Compact Scheme For First Derivative by Grid Search Method
270(1)
11.8 Upwind Compact Schemes
271(3)
11.9 Compact Schemes with Improved Numerical Properties
274(4)
11.9.1 OUCS1 scheme
274(1)
11.9.2 OUCS2 scheme
274(1)
11.9.3 OUCS3 scheme
275(2)
11.9.4 OUCS4 scheme
277(1)
11.10 Approximating Second Derivatives
278(2)
11.11 Optimization Problem for Evaluation of the Second Derivatives
280(1)
11.12 Solution of One-Dimensional Convection Equation
281(5)
11.13 Symmetrized Compact Difference Schemes
286(22)
11.13.1 High accuracy symmetrized compact scheme
291(4)
11.13.2 Solving bidirectional wave equation
295(5)
11.13.3 Transitional channel flow
300(2)
11.13.3.1 Establishment of equilibrium flow
302(1)
11.13.3.2 Receptivity of channel flow to convecting single viscous vortex
303(2)
11.13.4 Transitional channel flow created by vortex street
305(3)
11.14 Combined Compact Difference (CCD) Schemes
308(18)
11.14.1 A new combined compact difference (NCCD) scheme
314(5)
11.14.2 Solving the Stommel Ocean Model problem
319(2)
11.14.3 Operational aspects of the CCD schemes
321(1)
11.14.4 Calibrating NCCD method to solve Navier-Stokes equation for 2D lid-driven cavity problem
322(4)
11.15 Diffusion Discretization and Dealiasing Properties of Compact Schemes
326(15)
11.15.1 Dynamics and aliasing in square LDC problem
331(3)
11.15.2 Receptivity calculation of an adverse pressure gradient flow
334(7)
Chapter 12 Introduction to Finite Volume and Finite Element Methods
341(64)
12.1 Introduction
341(1)
12.2 Finite Volume Method
342(1)
12.3 Finite Volume Discretization for Two-Dimensional Flows
343(1)
12.4 Geometric Constraints of FVM
343(1)
12.5 FVM for Three-Dimensional Flows
343(2)
12.6 Evaluating Viscous Terms
345(2)
12.7 High Resolution Finite Volume Upwind Schemes
347(2)
12.8 Properties of Reconstruction Schemes
349(1)
12.9 Spectral Resolution of Flux-Vector Splitting (FVS) Scheme
350(7)
12.10 Dispersion Relation Preservation Property
357(1)
12.11 Solution of 1D Convection Equation
357(1)
12.12 Explaining the Gibbs' Phenomenon for FVS-FVM
358(3)
12.13 Derivation of the FV2S Scheme
361(13)
12.13.1 Spectral properties of the FV2S scheme
362(6)
12.13.2 Test cases for the FV2S scheme
368(6)
12.14 Introduction to Finite Element Methods
374(1)
12.15 Weighted Residual Methods
375(3)
12.15.1 The collocation method
376(1)
12.15.2 The subdomain method
376(1)
12.15.3 The least square method
376(1)
12.15.4 The Bubnov-Galerkin method
376(1)
12.15.5 General weighted residual method or the Petrov-Galerkin method
377(1)
12.16 Finite Element Approximation and Discretization
378(5)
12.16.1 FEM basis functions
378(5)
12.17 Dispersion Properties of the Galerkin FEM
383(3)
12.18 The Petrov--Galerkin Finite Elements Method
386(3)
12.18.1 Further notes on the SUPG method
388(1)
12.19 Higher Order Basis Function for the Bubnov--Galerkin FEM
389(10)
12.19.1 Solution of 1D convection equation by the quadratic basis function Galerkin method (G2FEM)
390(9)
12.20 A Comparative Study of FEM, FVM and FDM
399(6)
Chapter 13 Solution of Navier-Stokes Equation
405(37)
13.1 Introduction
405(1)
13.2 Stream Function-Vorticity Formulation for 2D Flows
406(2)
13.3 Start-up and Initial Condition
408(2)
13.4 Solution of Stream Function Equation (SFE)
410(2)
13.5 Wall-Vorticity Estimation
412(1)
13.6 Solution of Vorticity Transport Equation (VTE)
413(1)
13.6.1 Explicit upwind methods
413(1)
13.7 Implicit Upwind Methods
414(1)
13.8 Solution of Navier-Stokes Equation using Pressure-Velocity Formulation
414(2)
13.8.1 The MAC method of Harlow and Welch
414(1)
13.8.2 Operator splitting projection methods or fractional step methods
415(1)
13.9 Solution of Navier-Stokes Equation: Comparative Study of Different Formulations
416(3)
13.10 Vorticity-Velocity Formulation: Detailed Study
419(8)
13.11 Solution of Flow in Lid Driven Cavity by (V, ω)-Formulation
427(4)
13.12 Role of Grid Staggering for LDC Flow using (V, ω)-Formulation
431(2)
13.13 Receptivity of Flow Past a Flat Plate using (V, ω)-Formulation
433(3)
13.14 Effects of Initial Acceleration: Solution of Navier-Stokes Equation
436(6)
Chapter 14 Recent Developments in Discrete Finite Difference Computing
442(93)
14.1 Introduction
442(1)
14.2 One-Dimensional Filters for DES, LES and DNS
443(2)
14.3 Design, Order and Transfer Function of Explicit Filters
445(4)
14.4 Transfer Functions of Filters for Non-Periodic Problems
449(2)
14.5 Numerical Amplification and Dispersion Properties of Filters
451(1)
14.6 Upwind One-Dimensional Filter: A New Approach
452(3)
14.7 Application of One-Dimensional Filters
455(10)
14.7.1 Accelerated flow past a NACA 0015 aerofoil
456(2)
14.7.2 Flow past a cylinder executing rotary oscillation
458(7)
14.8 Two-Dimensional Higher Order Filters
465(6)
14.8.1 Implementation procedure for two-dimensional filters
469(1)
14.8.2 Performance comparison between 1D and 2D filters
470(1)
14.9 Solutions of Navier-Stokes Equation
471(8)
14.9.1 Flow past a cylinder executing rotary oscillation
471(4)
14.9.2 Accelerated flow past a NACA 0015 aerofoil
475(1)
14.9.3 Equilibrium flow past SHM-1 aerofoil
476(2)
14.9.4 Algorithmic cost estimate for 2D filters
478(1)
14.10 Optimal Time Advancing DRP Schemes
479(11)
14.10.1 Formulating an optimization problem for time integration
482(4)
14.10.2 Optimized RK2 scheme coupled to CD2, CD4 and CD6 schemes
486(1)
14.10.3 Optimized RK3 scheme coupled with CD2, CD4 and CD6 schemes
487(1)
14.10.4 Optimized RK4 scheme coupled with CD2, CD4 and CD6 schemes
488(2)
14.11 Numerical Properties of Coupled Temporal and Spatial Schemes
490(16)
14.11.1 Solving 1D convection problem
492(6)
14.11.2 Solving 2D lid-driven cavity (LDC) problem
498(5)
14.11.3 Benchmark Problem for DRP schemes for application in computational aeroacoustics (CAA)
503(3)
14.12 Optimized Explicit Runge-Kutta DRP Schemes for Compact Spatial Discretization Schemes
506(6)
14.13 Anisotropy of Numerical Wave Solutions by Finite Difference Methods
512(1)
14.14 Analysis of Numerical Methods for One-Dimensional Skewed Wave Propagation
513(1)
14.15 Finite Difference Spatial Semi-Discretization in 2D
514(3)
14.15.1 Fourier analysis of finite difference semi-discretization
515(2)
14.16 Analysis of Full Discretization in 2D
517(9)
14.16.1 An example of Skewed wave propagation at θ = 30°
524(2)
14.17 Analysis of Numerical Methods for Linearized Rotating Shallow Water Wave Equation
526(9)
Exercises 535(11)
References 546(17)
Index 563
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