Page 9 - Introduction to Computational Fluid Dynamics

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CONTENTS
2.7.3 Boundary Conditions May 27, 2005 0:54
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2.7.4 Underrelaxation 37
2.8 Methods of Solution 38
2.8.1 Gauss–Seidel Method 38
2.8.2 Tridiagonal Matrix Algorithm 38
2.8.3 Applications 40
2.9 Problems from Related Fields 45
Exercises 47
3. 1D Conduction–Convection ........................ 55
3.1 Introduction 55
3.2 Exact Solution 55
3.3 Discretisation 57
3.4 Upwind Difference Scheme 59
3.5 Comparison of CDS, UDS, and Exact Solution 60
3.6 Numerical False Diffusion 61
3.7 Hybrid and Power-Law Schemes 63
3.8 Total Variation Diminishing Scheme 63
3.9 Stability of the Unsteady Equation 65
3.9.1 Exact Solution 65
3.9.2 Explicit Finite-Difference Form 66
3.9.3 Implicit Finite-Difference Form 67
Exercises 68

4. 2D Boundary Layers................................71
4.1 Governing Equations 71
4.2 Adaptive Grid 73
4.3 Transformation to (x,ω) Coordinates 74
4.4 Discretisation 76
4.5 Determination of ω, y, and r 79
4.6 Boundary Conditions 80
4.6.1 Symmetry 80
4.6.2 Wall 81
4.6.3 Free Stream 82
4.7 Source Terms 85
4.7.1 Pressure Gradient 85

4.7.2 Q and R k 86
4.8 Treatment of Turbulent Flows 87
4.8.1 Mixing Length Model 88
4.8.2 e− Model 89
4.8.3 Free-Shear Flows 91