Page 9 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

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5.4 An Integro-Differential Formulation 157

Similarity of the Viscous Incompressible Fluid Flows 159
6.1 The Steady Flows Case 162

Flows With Low Reynolds Number. Stokes Theory 164
7.1 The Oseen Model in the Case of the Flows Past a
Thin Profile 167

Flows With High (Large) Reynolds Number 172
8.1 Mathematical Model 173
8.2 The Boundary Layer Equations 174

8.3 Probabilistic Algorithm for the Prandtl Equations
8.4 Example 187
8.5 Dynamic Boundary Layer with Sliding on a Plane
Plaque 191

4. INTRODUCTION TO NUMERICAL SOLUTIONS FOR
ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS 197
1 Introduction 197

2 Discretization of a Simple Equation 203
2.1 Using the Finite Difference Method 203
2.2 Using the Finite Element Method 203

2.3 Using the Finite Volume Method 205
2.4 Comparison of the Discretization Techniques 206

The Cauchy Problem for Ordinary Differential Equations 207
3.1 Examples 216

Partial Differential Equations 226
4.1 Classification of Partial Differential Equations 226

4.2 The Behaviour of Different Types of PDE 228
4.3 Burgers’ Equation 231
4.4 Stokes’ Problem 236
4.5 The Navier—Stokes System 239

5. FINITE-DIFFERENCE METHODS 247
1 Boundary Value Problems for Ordinary
Differential Equations 247

1.1 Supersonic Flow Past a Circular Cylindrical
Airfoil 249

Discretization of the Partial Differential Equations 253

The Linear Advection Equation 257
3.1 Discretization of the Linear Advection Equation 257