《分形几何与流体》是在作者瞿波的博士学位论文的基础上修订而成的，具体对流体中分散和传播的背景文献做了介绍，介绍了布朗运动、分数布朗运动、和分形几何，对沿海湾水流和粒子的模拟，用加速分数布朗运动粒子追踪模型来模拟海洋上的污染物的传播。是一本很有启迪和实用性的分形入门书。

《分形几何与流体》是瞿波在英国龙比亚大学攻读博士的学位论文的核心成果，深入浅出地介绍了分形及其在流体中的应用，详细论述了如何用分形中的分数布朗运动模拟流水中污染物的轨迹，包括对海湾和海洋中污染物传播轨迹的模拟。是一本实用性强、浅显易懂的应用数学学习和研究的参考用书。

序

自序

分形几何介绍

博士论文

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

List of Symbols

Chapter 1 Outline of Project

 1. Introduction

 2. Fractals and Fractional Brownian Motion

 3. Aim and Objectives

 4. Structure of Thesis

Chapter 2 Diffusion and Dispersion in Fluids

 1. Introduction

 2. Molecular Diffusion: Fick's Law and the Diffusion Equation

 3. Statistical Theory of Diffusion: Brownian Motion

 4. Turbulent Diffusion

4.1 Introduction

4.2 Eddies

4.3 Taylor's Theorem

4.4 The Relationship Between Lagrangian and Eulerian Measurement

4.5 Relative Diffusion and Richardson's Law

4.6 Okubo's Oceanic Diffusion Diagrams

 5. Shear Dispersion

5.1 Introduction

5.2 Taylor and Elder's Shear Dispersion Results

5.3 Dispersion in Rivers

5.4 Dispersion in the Sea

 6. Numerical Model of Dispersion

6.1 Solution of the Advection-Diffusion Equation

6.2 The Disadvantage of Solving the Advection-Diffusion Equation

 7. Traditional Particle Tracking Methods

 8. Summary

Chapter 3 Brownian Motion, Fractional Brownian Motion and Fractal Geometry

 1. Brownian Motion

1.1 The Definition of Brownian Motion

1.2 Two Simple Random Walks

1.3 Brownian Motion Generation

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

1.5 The Skewness and Kurtosis of Random Walks

1.6 Random Walks in Two Dimensions

1.7 The Last Steps of the Random Walks in Two Dimensions

 2. Fractional Brownian Motion

2.1 Introduction

2.2 FBM Model

2.3 FBMINC Model

2.4 The Comparison of the FBM and FBMINC Models

2.5 FBM Plots in One Dimension

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

2.7 Fractional Brownian Motion in Two Dimensions

2.8 Projection of Two-Dimensional Fractional Brownian Motion

2.9 The Use of Simpler Probability Distributions to Reduce CPU Time

2.10 Long Term Fickian Behaviour

 3. fBm as a Random Fractal Function

3.1 Fractal Geometry and Fractal Curves

3.2 Fractal Dimension

3.3 Fractal Properties of fBm

3.4 Methods for Determining H from Real Data

 4. Summary

Chapter 4 Coastal Bay Modelling

 1. Introduction

 2. New Particle Tracking Method Using in the Bay

2.1 Advection

2.2 Diffusion

2.3 Choosing a Time Interval

2.4 Choosing a Diffusion Coefficient

2.5 Boundary Reflection

2.6 The Particle Tracking Model

2.7 Particle Clouds

2.8 Concentration Calculation and Plots

2.9 Further Reported Results

 3. Shear Dispersion

3.1 Simple Shear Dispersion (Brownian Motion)

3.2 Shear Dispersion with Fractional Brownian Motion

3.3 Shear Dispersion in the Coastal Bay Model Recirculation Zone

 4. Summary

Chapter 5 Simulation of Observed Coastal Dispersion

 1. Introduction

 2. Northumbrian Coastal Water Data Sets

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

3.1 The SQ-Method

3.2 The R-Method

3.3 The SR-Method

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

 4. Comparison of the Three Methods

4.1 The Reason for Introducing the SR-Method

4.2 Comparison of the Results Using the Three Methods

 5. Accuracy of the Results

5.1 Sensitivity of the Centre

5.2 The Concentration Function Calculation

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

6.1 The Accelerated Fractional Brownian Motion (AFBM) Model

6.2 Simulation Using the FBMINC and AFBM Models

6.3 Concentration Calculations

6.4 Contour Plots

 7. Summary

Chapter 6 Conclutions, Discussion and Recommendations

 1. Introduction

 2. Achievement of Objectives

 3. Discussion

 4. Recommendations for Future Work

Appendix

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References

分形应用论文选

分数布朗运动的简化和应用

从分形维数到海洋表面漂浮物轨迹的模拟

流体中污染物扩散的分形模拟

用分数型布朗运动模拟海湾的剪切湍流分散

Development of FBMINC model for particle diffusion in fluids

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

后记