分形几何与流体

内容概要

瞿波 博士

江苏南通人。

1983年华东师范大学数学学士。

1986年华东师范大学数学硕士。

1992年赴英国在爱丁堡龙比亚大学（Edingburgh, Napier University）攻读计算机硕士课程和流体力学博士学位。研究方向是分形在流体中的应用。1999年获英国博士学位(PhD degree)。

1999年英国贝尔法斯特女王大学（Queen’s University of Belfast）研究助理。

2000年香港大学(Hong Kong University)土木工程系博士后。

2003年澳大利亚格里菲里斯大学（Griffith University）研究员。

2008年回国，在南通大学任教。硕士生导师。承担国家自然科学基金（2012年度）“北极的生态系统和二甲基硫对当地气候的影响”等多项课题研究。

20年来致力于分形在流体力学中的应用研究，以及环境模型，水利模型等国际国内课题研究。热衷于数学分形的普及推广，有多项成果在《国际流体数值方法》、《极地生物学》等国际权威杂志发表。

书籍目录

序

自序

分形几何介绍（只需要初等数学的知识）

博士论文

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

of Contaminants in Fluids

Chapter 1:
Outline of Project





1

1.1
Introduction






1

1.2
Fractal and Fractional Brownian Motion 


1

1.3
Aim and Objectives





2

1.4
Structure of Thesis





3

Chapter 2:
Diffusion and Dispersion in Fluids

-- A Literature Review




4

2.1
Introduction






4

2.2
Molecular Diffusion: Fick’s Law and the Diffusion

Equation






5

2.3
Statistical Theory of Diffusion: Brownian Motion

8

2.4
Turbulent Diffusion





11

2.4.1
Introduction





11

2.4.2
Eddies






12

2.4.3
Taylor’s Theorem




13

2.4.4
The Relationship Between Lagrangian and

Eulerian Measurement



15

2.4.5
Relative Diffusion and Richardson’s Law

17

2.4.6
Okubo’s Oceanic Diffusion Diagrams

19

2.5
Shear Dispersion





22

2.5.1
Introduction





22

2.5.2
Taylor and Elder’s Shear Dispersion Results

22

2.5.3
Dispersion in Rivers




24

2.5.3.1
Dispersion in Uniform Depth Open Channel
25

2.5.3.2
The Three-Dimensional Diffusion

Coefficients in an Open Channel

28

2.5.3.3
Dispersion in a Natural Channel

30







2.5.4
Dispersion in the Sea




31

2.5.4.1
Introduction




31

2.5.4.2
Relative Diffusion on the Ocean Surface
32

2.5.4.3
Coastal Region



36

2.6
Numerical Model of Dispersion



38

2.6.1
Solution of the Advection-Diffusion Equation
38

2.6.2
The Disadvantage of Solving the

Advection-Diffusion Equation


40

2.7
Particle Tracking Methods




42

2.7.1
Traditional Particle Tracking Methods

42

2.8
Summary






46

Chapter 3
Brownian Motion, Fractional Brownian Motion

and Fractal Geometry




47

3.1
Brownian Motion





47


3.1.1
The Definition of Brownian Motion


47


3.1.2
Two Simple Random Walks



48


3.1.3
Brownian Motion Generation



51

3.1.3.1
Central Limit Theorem Method

52

3.1.3.2
The Box-Muller Method


53


3.1.4
The Properties of a One-Dimensional

Brownian Motion Time Trace


54

3.1.5
The Skewness and Kurtosis of Random Walks
57


3.1.6
Random Walks in Two Dimensions


59

3.1.6.1
Delta Random Walks in Two Dimensions
60

3.1.6.2
Constant Random Walks in Two

Dimensions




60

3.1.6.3
Brownian Motion in Two Dimensions
61


3.1.7
The Last Steps of the Random Walks in Two

Dimensions





62

3.2- Fractional Brownian Motion




63

3.2.1
Introduction





63



3.2.1.1
Fractional Brownian Motion:

A Generalisation of Brownian Motion
63

3.2.1.2
Applications of Fractal Brownian Motion 
64

3.2.1.3
The Definition of Fractional Brownian

Motion





67

3.2.1.4
Properties of Fractional Brownain Motion
68



3.2.1.5
Methods for the Generation of Fractional

Brownian Motion



70

3.2.2
FBM Model





71


3.2.3
FBMINC Model




76


3.2.4
The Comparison of the FBM and FBMINC Models
80





3.2.5
fBm Plots in One Dimension



85



3.2.5.1
Fractional Random Walk Plots for the

FBM Model




85

3.2.5.2
The Effect of the Different Random

Number Sequences



89



3.2.5.3
The Mean Absolute Separation of an

fBm Trace




90


3.2.6
The Relationship Between M, NSTEP and P

92



3.2.6.1
Relationship Between NSTEP and M

92

3.2.6.2
The Effect of the Number of Particles

in a Diffusing Cloud



94

3.2.6.3- A Check on Random Number Seeds

95


3.2.7
Fractional Brownian Motion in Two Dimensions
96

3.2.8
Projection of Two-Dimensional Fractional Brownian

Motion






98

3.2.9
The Use of Simpler Probability Distributions to

Reduce CPU Time




100

3.2.10
Long Term Fickian Behaviour


104

3.3
fBm as a Random Fractal Function



106

3.3.1
Fractal Geometry and Fractal Curves


106

3.3.2
Fractal Dimension




109


3.3.3
Fractal Properties of fBm



110

3.3.3.1
The Box Counting Dimension

111

3.3.3.2
The Dimension of an fBm Trace

111

3.3.3.3
The Dimension of fBm Trajectories

113

3.3.4
Method for Determining H from Real Data

116

3.4- Summary







121

Chapter 4
Coastal Bay Modelling




122

4.1
Introduction






122

4.2
New Particle Tracking Method Using in the Bay

122

4.2.1
Advection 





123

4.2.2
Diffusion





124

4.2.2.1
Traditional Random Walk Model

124

4.2.2.2
Diffusion Using Fractional Brownian

Motion Model




125

4.2.2.3
The New fBm Particle Tracking Model
127

4.2.3
Choosing a Time Interval



128

4.2.4
Choosing a Diffusion Coefficient


129

4.2.5
Boundary Reflection




131

4.2.5.1
important Note on FBM Reflection

133

4.2.6
The Particle Tracking Model



133

4.2.6.1
The Particle Tracking Algorithm

133







4.2.6.2
Typical Particle Trajectory Plots for the

Bay Model




136

4.2.7
Particles Clouds




137

4.2.7.1
Computational Effort



137

4.2.8
Concentration Calculation and Plots


139

4.2.8.1
Algorithm for Calculation of Pollution

Concentration




140

4.2.8.2
Contour Plots and 3D Surface Plots

141

4.2.9
Further Reported Results



142

4.3
Shear Dispersion





143

4.3.1
Simple Shear Dispersion (Brownian Motion)

144

4.3.2
Shear Dispersion with Fractional Brownian Motion
147

4.3.3
Shear Dispersion in the Coastal Bay Model

Recirculation Zone




150

4.4
Summary






153

Chapter 5
Simulation of Observed Coastal Dispersion

189

5.1
Introduction






189

5.2
Northumbrian Coastal Water Data Sets


190

5.3
Three Methods for Calculating the Standard Deviation of

the Dye Patch Concentrations 



191

5.3.1
The SQ-Method




192

5.3.2
The R-Method





193

5.3.3
The SR-Method




194

5.3.4
Estimation of the Direction of the Mean Advective

Velocity Vector for Each Patch


194

5.4
Comparison of the Three Methods



195

5.4.1
The Reason for Introducing the SR-Method

195

5.4.2
Comparison of the Results Using the Three

Methods





196

5.5
Accuracy of the Results




197

5.5.1
The Sensitivity of the Centre



197

5.5.2
The Concentration Function Calculation

198

5.6
Simulation of the Observed Dye Patches Using an fBm

Based Particle Tracking Model



198

5.6.1
The Accelerated Fractional Brownian Motion

(AFBM) Model




199

5.6.2
Simulation Using the FBMINC and AFBM

Models





202

5.6.3
Concentration Calculations



202

5.6.4
Contour Plots





203

5.7
Summary






205





Chapter 6
Conclusions, Discussion and Recommendations

243

6.1
Introduction






243

6.2
Achievement of Objectives




243

6.3
Discussion






247

6.4
Recommendations for Future Work



249

Appendix 1
FORTRAN 77 Programs




253

References








293

分形应用论文选

1.
分数布朗运动的简化和应用  




317

2.
从分形维数到海洋表面漂浮物轨迹的模拟


328

3.
流体中污染物扩散的分形模拟 




335

4.
用分数型布朗运动模拟海湾的剪切湍流分散


343

5.
 Development of FBMINC model for particle diffusion

in fluids 







354

7 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
387

作者简介

本书是国内也是国际第一本分形几何在流体中的应用的参考书。本书介绍的方法不仅可以用于流体， 还可以用于其他任何有关连续随机的运动轨迹的模拟，用于粒子云的随机散布轨迹。

本书是瞿波博士在英国的博士学位论文的核心成果。书中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（fBm）模拟流水中污染物的轨迹，包括对海湾和海洋中的污染物传播轨迹的模拟。

书中介绍的方法是基于分数布朗运动(fBm)， 这是带有记忆的著名的布朗运动（随机散步）的推广。 作者对著名的fBm作了改进，创建了分数布朗运动粒子跟踪模型，并推广到加速分数布朗运动粒子跟踪模型。基于豪斯特指数（H）的灵活性，分数布朗运动粒子追踪模型的应用非常广泛。可以用于经融、股市、脑电图、岩石的裂缝，道路的分布、海洋的浮标轨迹、粒子的散布、医学上人体肺的分布及毛细胞血管、脑电图曲线（图14）等具有分形的特征物体和现象中。

本书的第一部分介绍了分形。不同于其他书介绍分形时所用的复杂的数学工具，使人望而生畏，书中介绍的分形及分形维数的计算都是用最通俗易懂的方法。这是一本实用性强、浅显易懂的应用数学学习和研究参考用书。书中还附有程序供直接使用。相信此书对大学生、研究生、大学青年教师搞科研有一定的实用和参考价值。

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