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序 1

自序 1

分形几何介绍 19

博士论文 19

The Use of Fractional Brownian Motion in the Modelling of the Dispersion of Contaminants in Fluids 19

List of Symbols 19

Chapter 1 Outline of Project 23

1.Introduction 23

2.Fractals and Fractional Brownian Motion 23

3.Aim and Objectives 24

4.Structure of Thesis 25

Chapter 2 Diffusion and Dispersion in Fluids 26

1.Introduction 26

2.Molecular Diffusion： Fick’s Law and the Diffusion Equation 27

3.Statistical Theory of Diffusion： Brownian Motion 29

4.Turbulent Diffusion 32

4.1 Introduction 32

4.2 Eddies 33

4.3 Taylor’s Theorem 33

4.4 The Relationship Between Lagrangian and Eulerian Measurement 36

4.5 Relative Diffusion and Richardson’s Law 37

4.6 Okubo’s Oceanic Diffusion Diagrams 38

5.Shear Dispersion 41

5.1 Introduction 41

5.2 Taylor and Elder’s Shear Dispersion Results 42

5.3 Dispersion in Rivers 43

5.4 Dispersion in the Sea 48

6.Numerical Model of Dispersion 55

6.1 Solution of the Advection-Diffusion Equation 55

6.2 The Disadvantage of Solving the Advection-Diffusion Equation 57

7.Traditional Particle Tracking Methods 58

8.Summary 62

Chapter 3 Brownian Motion， Fractional Brownian Motion and Fractal Geometry 64

1.Brownian Motion 64

1.1 The Definition of Brownian Motion 64

1.2 Two Simple Random Walks 65

1.3 Brownian Motion Generation 67

1.4 The Properties of a One-Dimensional Brownian Motion Time Trace 70

1.5 The Skewness and Kurtosis of Random Walks 73

1.6 Random Walks in Two Dimensions 75

1.7 The Last Steps of the Random Walks in Two Dimensions 77

2.Fractional Brownian Motion 77

2.1 Introduction 77

2.2 FBM Model 85

2.3 FBMINC Model 90

2.4 The Comparison of the FBM and FBMINC Models 93

2.5 FBM Plots in One Dimension 98

2.6 The Relationship Between Memory， M， Number of Steps， NSTEP， and Number of Particles， P 104

2.7 Fractional Brownian Motion in Two Dimensions 107

2.8 Projection of Two-Dimensional Fractional Brownian Motion 109

2.9 The Use of Simpler Probability Distributions to Reduce CPU Time 111

2.10 Long Term Fickian Behaviour 115

3.fBm as a Random Fractal Function 117

3.1Fractal Geometry and Fractal Curves 117

3.2 Fractal Dimension 119

3.3 Fractal Properties of fBm 120

3.4 Methods for Determining H from Real Data 125

4.Summary 129

Chapter 4 Coastal Bay Modelling 130

1.Introduction 130

2.New Particle Tracking Method Using in the Bay 130

2.1 Advection 132

2.2 Diffusion 133

2.3 Choosing a Time Interval 136

2.4 Choosing a Diffusion Coefficient 140

2.5 Boundary Reflection 142

2.6 The Particle Tracking Model 144

2.7 Particle Clouds 155

2.8 Concentration Calculation and Plots 162

2.9 Further Reported Results 163

3.Shear Dispersion 169

3.1 Simple Shear Dispersion （Brownian Motion） 169

3.2 Shear Dispersion with Fractional Brownian Motion 174

3.3 Shear Dispersion in the Coastal Bay Model Recirculation Zone 178

4.Summary 186

Chapter 5 Simulation of Observed Coastal Dispersion 189

1.Introduction 189

2.Northumbrian Coastal Water Data Sets 190

3.Three Methods for Calculating the Standard Deviation of the Dye Patch Concentrations 197

3.1 The SQ-Method 210

3.2 The R-Method 211

3.3 The SR-Method 213

3.4 Estimation of the Direction of the Mean Advective Velocity Vector for Each Patch 213

4.Comparison of the Three Methods 215

4.1 The Reason for Introducing the SR-Method 215

4.2 Comparison of the Results Using the Three Methods 216

5.Accuracy of the Results 218

5.1 Sensitivity of the Centre 218

5.2 The Concentration Function Calculation 220

6.Simulation of the Observed Dye Patches Using an fBm Based Particle Tracking Model 220

6.1 The Accelerated Fractional Brownian Motion （AFBM） Model 223

6.2 Simulation Using the FBMINC and AFBM Models 227

6.3 Concentration Calculations 236

6.4 Contour Plots 237

7.Summary 239

Chapter 6 Conclusions， Discussion and Recommendations 241

1.Introduction 241

2.Achievement of Objectives 241

3.Discussion 244

4.Recommendations for Future Work 246

Appendix 249

brown.for 249

fbm.for 252

fbminc.for 256

bayflu.for 260

baycloud.for 266

concent.for 271

srb1.for 273

srbb1.for 282

sruvb1.for 285

afbm.for 287

References 292

分形应用论文选 317

分数布朗运动的简化和应用 317

从分形维数到海洋表面漂浮物轨迹的模拟 328

流体中污染物扩散的分形模拟 335

用分数型布朗运动模拟海湾的剪切湍流分散 343

Development of FBMINC model for particle diffusion in fluids 354

加速分数型布朗运动粒子追踪模型在水面污染扩散中的应用 377

后记 387