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Apr 29, 2017

EBOOK - Computational Fluid Dynamics (T. J. Chung)


EBOOK - Động lực học tính toán - Tác giả: T. J. Chung (1022 Trang).

The development of modern computational fluid dynamics (CFD) began with the ad-vent of the digital computer in the early 1950s. Finite difference methods (FDM) and finite element methods (FEM), which are the basic tools used in the solution of par-tial differential equations in general and CFD in particular, have different origins. In 1910, at the Royal Society of London, Richardson presented a paper on the first FDM solution for the stress analysis of a masonry dam.
In contrast, the first FEM work was published in the Aeronautical Science Journal by Turner, Clough, Martin, and Topp for applications to aircraft stress analysis in 1956. Since then, both methods have been developed extensively in fluid dynamics, heat transfer, and related areas.
Earlier applications of FDM in CFD include Courant, Friedrichs, and Lewy [1928], Evans and Harlow [1957], Godunov [1959], Lax and Wendroff [1960], MacCormack [1969], Briley and McDonald [1973], van Leer [1974], Beam and Warming [1978], Harten [1978, 1983], Roe [1981, 1984], Jameson [1982], among many others. The literature on FDM in CFD is adequately documented in many text books such as Roache [1972, 1999], Patankar [1980], Peyret and Taylor [1983], Anderson, Tannehill, and Pletcher [1984, 1997], Hoffman [1989], Hirsch [1988, 1990], Fletcher [1988], Anderson [1995], and Ferziger and Peric [1999], among others.

PART ONE. PRELIMINARIES

1 Introduction 3
1.1 General 3
1.1.1 Historical Background 3
1.1.2 Organization of Text 4
1.2 One-Dimensional Computations by Finite Difference Methods 6
1.3 One-Dimensional Computations by Finite Element Methods 7
1.4 One-Dimensional Computations by Finite Volume Methods 11
1.4.1 FVM via FDM 11
1.4.2 FVM via FEM 13
1.5 Neumann Boundary Conditions 13
1.5.1 FDM 14
1.5.2 FEM 15
1.5.3 FVM via FDM 15
1.5.4 FVM via FEM 16
1.6 Example Problems 17
1.6.1 Dirichlet Boundary Conditions 17
1.6.2 Neumann Boundary Conditions 20
1.7 Summary 24
2 Governing Equations 29
2.1 Classification of Partial Differential Equations 29
2.2 Navier-Stokes System of Equations 33
2.3 Boundary Conditions 38
2.4 Summary 41


PART TWO. FINITE DIFFERENCE METHODS

3 Derivation of Finite Difference Equations 45
3.1 Simple Methods 45
3.2 General Methods 46
3.3 Higher Order Derivatives 50
3.4 Multidimensional Finite Difference Formulas 53
3.5 Mixed Derivatives 57
3.6 Nonuniform Mesh 59
3.7 Higher Order Accuracy Schemes 60
3.8 Accuracy of Finite Difference Solutions 61
3.9 Summary 62
4 Solution Methods of Finite Difference Equations 63
4.1 Elliptic Equations 63
4.1.1 Finite Difference Formulations 63
4.1.2 Iterative Solution Methods 65
4.1.3 Direct Method with Gaussian Elimination 67
4.2 Parabolic Equations 67
4.2.1 Explicit Schemes and von Neumann Stability Analysis 68
4.2.2 Implicit Schemes 71
4.2.3 Alternating Direction Implicit (ADI) Schemes 72
4.2.4 Approximate Factorization 73
4.2.5 Fractional Step Methods 75
4.2.6 Three Dimensions 75
4.2.7 Direct Method with Tridiagonal Matrix Algorithm 76
4.3 Hyperbolic Equations 77
4.3.1 Explicit Schemes and Von Neumann Stability Analysis 77
4.3.2 Implicit Schemes 81
4.3.3 Multistep (Splitting, Predictor-Corrector) Methods 81
4.3.4 Nonlinear Problems 83
4.3.5 Second Order One-Dimensional Wave Equations 87
4.4 Burgers’ Equation 87
4.4.1 Explicit and Implicit Schemes 88
4.4.2 Runge-Kutta Method 90
4.5 Algebraic Equation Solvers and Sources of Errors 91
4.5.1 Solution Methods 91
4.5.2 Evaluation of Sources of Errors 91
4.6 Coordinate Transformation for Arbitrary Geometries 94
4.6.1 Determination of Jacobians and Transformed Equations 94
4.6.2 Application of Neumann Boundary Conditions 97
4.6.3 Solution by MacCormack Method 98
4.7 Example Problems 98
4.7.1 Elliptic Equation (Heat Conduction) 98
4.7.2 Parabolic Equation (Couette Flow) 100
4.7.3 Hyperbolic Equation (First Order Wave Equation) 101
4.7.4 Hyperbolic Equation (Second Order Wave Equation) 103
4.7.5 Nonlinear Wave Equation 104
4.8 Summary 105
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