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課程來源:MIT
     

 

13.024數值海洋流體動力學 ,2003 春季

13.024 Numerical Marine Hydrodynamics, Spring 2003

譯者:范會渠

編輯:陳盈、洪曉慧

 

 

An ASCI three-dimensional modeling of the Rayleigh-Taylor instability.

用數值方法和美國加速戰略計算的創意計畫(ASCI)三維水動力模型展示Rayleigh-Taylor的不穩定性。(圖片由Lawrence Livermore國家實驗室Lawrence Livermore National Laboratory提供。)

 

課程重點

該課程有一整套的課堂講稿和可供下載的作業。

 

課程描述

本課程是對插值、微分、積分與線性方程系統數值方法的介紹。涵蓋用數值積分求解微分方程,以及用有限微分法、積分邊界板塊方程求解非粘性水動力偏微分方程。該課程還包括:移動介面數值方法導論,快速傅立葉轉換,確定海浪與隨機海浪的數值表示法,積分邊界層方程及其數值解。

 

技術需求

本課程網頁中的.m檔需要用MATLAB® software執行。.FIN.OUT檔是輸入、輸出資料表,可以在任意文本閱讀軟體中打開。

 

教學大綱

 

介紹

本課程向學生介紹在海洋流體動力學中碰到的問題,這些問題很難藉由手算解決,但是可以用短的電腦程式相對迅速而準確的解決。這要求學生知道或者學習使用電腦語言編寫程式。這裏選用MATLAB®,原因是它相對比較容易學習且功能強大,具有很好的繪圖功能。因此,很多問題答案的圖表可以作為學生編寫出的程式的一部分,用以解決很多問題。

由於學習課程13.024的學生編寫程式背景不同,因此課程的第一部分有必要教一下MATLAB® 。主要的教材是:《使用MATLAB®的數值方法,及其實現與應用》Numerical Methods with MATLAB®, Implementations and Applications,由erald W. Recktenwald所著,ISBN: 02010398606Prentice Hall, Inc. 2000年出版。該書的前四章講用MATLAB®語言編寫程式,每個學生都應該學習,並完成從書中選出的練習題。大約在課程的前15堂課教授MATLAB®。學生做輔助的習題集更能瞭解如何使用學過的知識,並開始在MATLAB®中編寫程式。學生還應該使用這樣一本書:《MATLAB®初級讀本》MATLAB® PRIMER, 第六版,由K. SigmonT. A. Davis所著,ISBN: 1584882948,Chapman and Hall/CRC出版。

 

 

附帶的課堂講稿包括:導論和一些針對課程主要部分的進階資料。在麻省理工學院的課堂上,老師在介紹材料時使用這些課堂講稿,每個學生有一份講稿的列印本。Recktenwald著的課本中第五章到第十二章的閱讀資料展現了書中內容的更多細節資訊,學生也要做相應的習題集。該書的例外是:缺少基於Green定理求解邊界積分方程問題的資訊。本課程包含大量海洋流體動力學的理論與練習。這些內容的基礎知識在課堂講稿和習題集6習題集8中,要求學生編寫並利用編寫的程式解決二維板塊問題。會花幾個小時講解所據理論的細節和如何編寫程式解決問題。雖然海洋流體動力學中很多板塊問題和程式都是三維的,但對只學一學期課程的學生而言,編寫程式解決這些三維問題是艱巨而耗時的。藉由在課堂上講授三維問題的理論,並讓學生編寫和使用程式來解決二維問題,學生可以做好準備,如果他們在以後的職業生涯中遇到三維邊界積分方程問題的話也能很好地處理。

通常,建議與導師互相交流來完成習題集6和習題集8。因為這些習題集比其他習題集要求學生編寫更多的 MATLAB® 源代碼。

 

 

教學時程、課堂講稿

 

所有的課堂講稿能夠作為一份檔案下載 (PDF - 5.6 MB).

 

課程單元

課堂講稿

1

不可壓縮流體力學背景知識

Incompressible Fluid Mechanics Background (PDF)

‧粒子圖像測速

Particle Image Velocimetry

‧平均化NS方程

Averaged Navier-Stokes Equations

‧不可壓縮流體壓力方程

The Pressure Equation for an Incompressible Fluid

‧渦量方程

The Vorticity Equation

‧非粘性流體力學,Euler方程

Inviscid Fluid Mechanics, Euler's Equation

‧非粘性流體Bernoulli定理

Bernoulli Theorems for Inviscid Flow

‧渦量動力學與Kelvin環量定理

Vorticity Dynamics and Kelvin's Circulation Theorem

‧勢流與近似勢流

Potential Flows and Mostly Potential Flows

Green函數,Green定理與邊界積分方程

Green Functions, Green's Theorem and Boundary Integral Equations

‧求解範例

Example of Method Solution

‧從源和偶極子層角度理解邊界積分方程

Interpretation of Boundary Integral Equation in Terms of Source and Dipole Layers

Kelvin-Neumann問題

The Kelvin-Neumann Problem

Kelvin-NeumannGreen函數

The Kelvin-Neumann Green Function

‧源分佈與偶極子分佈

Source Only and Dipole Only Distributions

‧二維Green定理

Green's Theorem in Two Dimensions

‧渦力

Force on a Vortex

‧柱狀渦升力

Lift on a Vortex in a Cylinder

‧範例:用偶極子和渦設計二維機翼的平均線

Example: Design of 2D Airfoil Mean Line Using Dipoles and Vortices

2

微積分中的有用結果Some Useful Results from Calculus (PDF)

Gauss定理推導

Derivation of Gauss' Theorem

Gauss定理應用範例:船舶的 Froude Krylov振盪力

Example of Use of Gauss Theorem: Froude Krylov Surge Force on a Ship

‧運輸定理

The Transport Theorem

‧物體的壓力與力矩

Pressure Forces and Moments on an Object

3

複數應用An Application Using Complex Numbers (PDF)

‧複數編寫程式範例:圓到機翼的共形映射

Example of Programming with Complex Numbers: Conformal Mapping of a Circle into an Airfoil

‧壓力係數計算步驟

Procedure to Compute Pressure Coefficient

4

方程求根Root Finding (PDF)

‧二分法

Bisection Method

‧方程牛頓求根法

Newton's Method for Finding Roots of y(x)

‧矩陣代數回顧

Review of Matrix Algebra

‧矩陣行列式

Determinant of a Matrix

‧矩陣轉置與求逆

Transpose of a Matrix, Calculating the Inverse of a Matrix

‧矩陣範數

Matrix Norms

‧矩陣條件數

The Condition Number of a Matrix

‧高斯消元法

Gaussian Elimination

n維方程高斯消元的運算步數

Gaussian Elimination Operation Count for n Equations

‧線性方程數值求解的誤差,比例部分旋轉法則

Errors in Numerical Solutions of Sets of Linear Equations, Scaled Partial Pivoting Rule

LU分解法求解線性方程

Solution of Linear Equations by LU Decomposition

‧矩陣分解步驟

Procedure for Factorization of A

5

曲線擬合與插值

Curve Fitting and Interpolation (PDF)

‧函數的多項式逼近

Polynomial Approximation to a Function

‧拉格朗日多項式範例

Polynomials Example

6

數值微分

Numerical Differentiation (PDF)

‧有限差分法

Finite Difference Differentiation

7

數值積分

Numerical Integration (PDF)

‧梯行法

Trapezoidal Rule

‧梯行法的誤差

Trapezoidal Rule Error

‧一般梯行法

Usual Trapezoidal Rule

‧數值積分

Numerical Integration

Simpson法則

Simpson's Rule

8

數值積分方程與數值微分方程

Numerical Integration of Differential Equations (PDF)

Euler法,修正Euler

Euler's Method, Modified Euler's Method

‧四階RungeKutta

Fourth Order Runge Kutta Method

‧預估校正法

Predictor-Corrector Methods

‧高階微分方程

Higher Order Differential Equations

‧回顧與拓展

Review and Extension

9

數值誤差範例Some Examples and Numerical Errors (PDF)

‧數值流體動力學的類型,函數計算範例

Types of Numerical Hydrodynamics Problems, Example of Function Evaluation

‧常微分方程求解範例

Example of Solution of Ordinary Differential Equation

‧偏微分方程求解範例

Example of Solution of Partial Differential Equation

‧柱面座標

Cylindrical Coordinates

‧離散積分方程範例

Example of Discretized Integral Equation

‧穩定性

Stability

10

板塊法

Panel Methods (PDF)

‧擾動勢流邊界條件,三維流

Boundary Condition of Perturbation Potential, Three Dimensional Flows

‧深入理解Green定理

Interpretation of Green's Theorem

‧積分方程的排列

Arrangement of the Integral Equation

‧積分方程的數值形式

Numerical Form of the Integral Equation

‧生成數值方程

Making the Numerical Equations

‧求解步驟

Solution Steps

‧二維板塊法

Two Dimensional Panel Methods

‧二維積分方程的數值形式

Numerical Form of the Two Dimensional Integral Equation

‧產生升力的情況

Situations with the Generation of Lift

‧壓力與力的計算

Computation of Pressures and Forces

11

邊界層

Boundary Layers (PDF - 1.3 MB)

‧二維穩定邊界層方程

Two-Dimensional Steady Boundary Layer Equations

‧邊界層參數

Boundary Layer Parameters

‧品質通量

Mass Fluxes

‧邊界層動量積分求解範例

Example of Solution of Momentum Integral BL Equation

‧壓力分佈已知的湍流邊界層積分

Calculation of Turbulent Boundary Layer When Pressure Distribution is Known

‧層流封閉關係,湍流封閉關係

Laminar Closure Relations, Turbulent Closure Relations

‧海浪

Sea Waves

‧模擬範例

Example of Simulation

‧海洋光譜

Sea Spectra

‧傅立葉轉換

Fourier Transforms

‧實數的快速傅立葉轉換與反轉快速傅立葉轉換

Computational FFT and IFFT of Real Numbers

‧隨機波模擬

Simulation of Random Waves

‧傅立葉轉換、逆傅立葉轉換、快速傅立葉轉換與反轉快速傅立葉轉換與波模擬的回顧

Review of Fourier Transforms, Inverse Fourier Transforms, FFT's IFFT's and Wave Simulation

‧高斯亂數的產生(演算法經Everett F. Carter Jr允許使用)

Generating Gaussian Random Numbers (Courtesy of Everett F. Carter Jr.)

‧波統計學

Wave Statistics

‧理論結果

Results from Theory

‧高斯隨機過程的定義

Definition of a Gaussian Random Process

1/n最高波平均波幅

Average Amplitude of the 1/n'th Highest Waves

‧極限波

Extreme Waves

‧剛性方程

Stiff Equations

‧水中水平淺吃水下垂纜線動力學

Dynamics of Horizontal Shallow Sag Cables in Water

12

剛體振動Oscillating Rigid Objects (PDF)

‧勢函數與邊界條件

Potentials and Boundary Conditions

‧細長體理論

Strip Theory

‧船體邊界條件

Boundary Conditions on Hull

‧搖擺,翻滾與偏航方程

Sway, Roll and Yaw Equations

‧船舶在隨機海況下運動模擬

Simulations of Ship Motions in Random Seas

‧附加阻力與漂流力

Added Resistance and Drift Forces

GerritsmaBeukelman的附加阻力理論

Gerritsma and Beukelman Theory for Added Resistance

‧非線性波力計算

Nonlinear Wave Force Calculations

‧垂直海洋負載

Vertical Sea Loads

   

附錄:板塊法與細長梯理論的進階資料(由Alexis Mantzaris提供)。Further Material on Panel Methods and Strip Theory  (PDF - 1.0 MB)

作業

 

本課程網頁中的.m檔需要在MATLAB® software中執行。.FIN.OUT型檔是輸入、輸出資料表,可以在任意文本閱讀軟體中打開。

 

關於習題集6和習題集8的說明:

在習題集6和習題集8中,學生在格林定理的基礎上編寫MATLAB® 程式求解二維邊界積分方程。在習題集8中,計算二維任意形狀物體的非粘性流線。解決圓柱體的問題。

在習題集8中,該方法進一步應用於升力機翼周圍的流體,該機翼的尾跡處速度勢有間斷。使用二維格林函數,G = -ln r,其中r源點場點的距離。

 

並不要求學生寫出計算圍繞面板的格林函數積分的的高效MATLAB®m檔案,相反,這個叫rank2d.m的程式是提供給學生的。該程式計算出在一個面板上格林函數的積分g,格林函數的法向導數dg/dn。該MATLAB函數在本地座標中漸移,其中源面板近似為一條x軸上中點在座標原點的線。(x,y)處的場點是在本地座標中的座標值。面板的法向量指向y軸正方向。

 

利用 rank2d.m函數,首先應該確定面板的長度和場點在本地座標中的座標值。這些工作是由m函數“localize.m”完成的,該程式也應該提供給學生,他們要編寫並利用該系列的程式剩下的部分,以完成習題集。

m函數“rank2d”“localize”和習題集6一起提供。

 

習題集

習題集1 (PDF)

習題集2 (PDF)

習題集3 (PDF)

習題集4 (PDF)

習題集5 (PDF)

習題集6 (PDF)

習題集7 (PDF)

習題集8 (PDF)

習題集9 (PDF)

習題集10 (PDF)

支援檔案

Supporting Files

64a012.fin (FIN)

LOCALIZE.M (M)

RANK2D.M (M)

wig4125.out (OUT)

wigley5.out (OUT)

wigley9.out (OUT)

 

13.024 Numerical Marine Hydrodynamics, Spring 2003

An ASCI three-dimensional modeling of the Rayleigh-Taylor instability. The Rayleigh-Taylor instability illustrated using numerical methods and ASCI three-dimensional hydrodynamic modeling. (Image courtesy of Lawrence Livermore National Laboratory.)

Highlights of this Course

The course features a complete set of lecture notes and downloadable assignments.

» Watch a video introduction featuring the course instructor.
(RM - 56K) (RM - 80K) (RM - 220K)

Course Description

This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions.

Technical Requirement

MATLAB® software is required to run the .m files found on this course site. The .FIN and .OUT are simply data offest tables. They can be viewed with any text reader. RealOne™ Player software is required to run the .rm files found on this course site.

 

Syllabus

Introduction

This subject introduces the student those problems in marine hydrodynamics that would be difficult to solve by "hand calculation", but which can be solved relatively quickly and accurately with short computer programs. It requires that the student know or learn programming in a "computer language". MATLAB® was chosen for this purpose because it is comparatively easy to learn, while still being powerful and providing a strong graphics capability so that graphs of many problem solutions can be made as part of the programs written by the student to solve many problems.

Because the students that study 13.024 have diverse programming backgrounds, it is necessary to teach MATLAB® as the first part of the subject. The principal textbook is "Numerical Methods with MATLAB®, Implementations and Applications", by Gerald W. Recktenwald, (ISBN: 02010398606), published by Prentice Hall, Inc. (2000). The first four chapters of this book are devoted to programming in MATLAB®. These should be studied and a selection of the Exercises in the text should be done by each student. About 15 lecture hours are used at the start of the course for teaching MATLAB®. Students do associated Problem Sets to better learn how to use the material taught and to do initial programming in MATLAB®. Students should also get and use the book: "MATLAB® PRIMER, Sixth Edition", by K. Sigmon and T. A. Davis, (ISBN: 1584882948), published by Chapman and Hall/CRC.

The accompanying Lecture Notes cover introductory, as well as some advanced, material for the principal part of this course. In the M.I.T. classes, these lecture notes are used by the instructor in presenting material and each student obtains a printed copy of the notes. Reading from the Recktenwald text, chapters 5 through 12, presents more detailed information on the material covered and students do Problem Sets. An exception is the lack of information in the text about solving Boundary Integral Equation Problems (Panel Methods) based on Green's Theorem. A great deal of the theory and practice of Marine Hydrodynamics involves this subject. The fundamental information about it is covered in the Lecture Notes and Problem Sets 6 and 8 require the students to write and use programs to solve two-dimensional panel method problems. Several hours of class time are used to explain some of the details of the underlying theory and how to write computer programs to solve these problems. Although many of the Panel Method problems and programs in marine hydrodynamics are three dimensional, the task of writing programs for these three dimensional problems is too difficult and time-consuming for students in a 1-term course. By teaching the theory for the three dimensional problems in the lectures and by giving students the experience of writing and using programs for two-dimensional problems, the students are well equipped to deal with three dimensional boundary integral equation problems if they subsequently encounter them in their professional careers.

Normally, student interaction with the instructor is recommended for doing Problem Sets 6 and 8. They require more MATLAB® code to be written by students than do the other Problem Sets.

Calendar

WEEK # TOPICS
 
1 Incompressible Fluid Mechanics Background
2 Some Useful Results from Calculus
3 An Application Using Complex Numbers
4 Root Finding
5 Curve Fitting and Interpolation
6 Numerical Differentiation
7 Numerical Integration
8 Numerical Intergration and Differential Equations
9 Some Examples of Numerical Errors
10 Panel Methods
11 Boundary Layers
12 Oscillating Rigid Objects

Lecture Notes

All of the lecture notes may be downloaded as a single file (PDF - 5.6 MB).

Week 1: Incompressible Fluid Mechanics Background (PDF)

Particle Image Velocimetry Averaged Navier-Stokes Equations The Pressure Equation for an Incompressible Fluid The Vorticity Equation Inviscid Fluid Mechanics, Euler's Equation Bernoulli Theorems for Inviscid Flow Vorticity Dynamics and Kelvin's Circulation Theorem Potential Flows and Mostly Potential Flows Green Functions, Green's Theorem and Boundary Integral Equations Example of Method Solution Interpretation of Boundary Integral Equation in Terms of Source and Dipole Layers The Kelvin-Neumann Problem The Kelvin-Neumann Green Function Source Only and Dipole Only Distributions Green's Theorem in Two Dimensions Force on a Vortex Lift on a Vortex in a Cylinder Example: Design of 2D Airfoil Mean Line Using Dipoles and Vortices

Week 2: Some Useful Results from Calculus (PDF)

Derivation of Gauss' Theorem Example of Use of Gauss Theorem: Froude Krylov Surge Force on a Ship The Transport Theorem Pressure Forces and Moments on an Object

Week 3: An Application Using Complex Numbers (PDF)

Example of Programming with Complex Numbers: Conformal Mapping of a Circle into an Airfoil Procedure to Compute Pressure Coefficient

Week 4: Root Finding (PDF)

Bisection Method Newton's Method for Finding Roots of y(x) Review of Matrix Algebra Determinant of a Matrix Transpose of a Matrix, Calculating the Inverse of a Matrix Matrix Norms The Condition Number of a Matrix Gaussian Elimination Gaussian Elimination Operation Count for n Equations Errors in Numerical Solutions of Sets of Linear Equations, Scaled Partial Pivoting Rule Solution of Linear Equations by LU Decomposition Procedure for Factorization of A

Week 5:Curve Fitting and Interpolation (PDF)

Polynomial Approximation to a Function Lagrange Polynomials Example

Week 6: Numerical Differentiation (PDF)

Finite Difference Differentiation

Week 7: Numerical Integration (PDF)

Trapezoidal Rule Trapezoidal Rule Error Usual Trapezoidal Rule Numerical Integration Simpson's Rule

Week 8: Numerical Integration of Differential Equations (PDF)

Euler's Method, Modified Euler's Method Fourth Order Runge Kutta Method Predictor-Corrector Methods Higher Order Differential Equations Review and Extension

Week 9: Some Examples and Numerical Errors (PDF)

Types of Numerical Hydrodynamics Problems, Example of Function Evaluation Example of Solution of Ordinary Differential Equation Example of Solution of Partial Differential Equation Cylindrical Coordinates Example of Discretized Integral Equation Stability

Week 10: Panel Methods (PDF)

Boundary Condition of Perturbation Potential, Three Dimensional Flows Interpretation of Green's Theorem Arrangement of the Integral Equation Numerical Form of the Integral Equation Making the Numerical Equations Solution Steps Two Dimensional Panel Methods Numerical Form of the Two Dimensional Integral Equation Situations with the Generation of Lift Computation of Pressures and Forces

Week 11: Boundary Layers (PDF - 1.3 MB)

Two-Dimensional Steady Boundary Layer Equations Boundary Layer Parameters Mass Fluxes Example of Solution of Momentum Integral BL Equation Calculation of Turbulent Boundary Layer When Pressure Distribution is Known Laminar Closure Relations, Turbulent Closure Relations Sea Waves Example of Simulation Sea Spectra Fourier Transforms Computational FFT and IFFT of Real Numbers Simulation of Random Waves Review of Fourier Transforms, Inverse Fourier Transforms, FFT's IFFT's and Wave Simulation Generating Gaussian Random Numbers (Courtesy of Everett F. Carter Jr.) Wave Statistics Results from Theory Definition of a Gaussian Random Process Average Amplitude of the 1/n'th Highest Waves Extreme Waves Stiff Equations Dynamics of Horizontal Shallow Sag Cables in Water

Week 12: Oscillating Rigid Objects (PDF)

Potentials and Boundary Conditions Strip Theory Boundary Conditions on Hull Sway, Roll and Yaw Equations Simulations of Ship Motions in Random Seas Added Resistance and Drift Forces Gerritsma and Beukelman Theory for Added Resistance Nonlinear Wave Force Calculations Vertical Sea Loads Appendix: Further Material on Panel Methods and Strip Theory (Courtesy of Alexis Mantzaris) (PDF - 1.0 MB)

Assignments

MATLAB® software is required to run the .m files in this section. The .FIN and .OUT are simply data offest tables. They can be viewed with any text reader.

Notes about Problem Sets 6 and 8

In problem sets 6 and 8, students write MATLAB® programs to solve two-dimensional boundary integral equations based on Green's Theorem. In problem set 8, the inviscid streaming flow about an arbitrary two-dimensional object is calculated. The solution is done for a circular cylinder.

In problem set 8, the method is extended to the flow around a lift-generating airfoil with a wake across which there is a jump in the velocity potential. The two-dimensional Green function that is used is G = -ln r, where r is the distance between a "source point" and a "field point".

The student is not expected to write an efficient MATLAB® m-file for computing the integral of the Green Function over a panel. Rather, that m-file is given to the students and it is called rank2d.m. This m-file computes the integral of the Green function, g, and of the normal derivative of the Green function, dg/dn, over a panel. This m-function works in local coordinates for which the "source panel" is approximated as a line on a local x-axis with the center of the line at the local origin. The "field point" is at (x,y) in local coordinates. The normal vector to the panel is in the positive local y-direction.

To use rank2d.m function, the panel length and the location of the field point in local coordinates must first be determined. This is done in the m-function "localize.m", which should also be provided to the student who writes and used the remainder of the set of programs needed to complete the problem sets.

The m-functions, rank2d and localize are provided with problem set 6.

Problem Sets

Problem Set 1 (PDF) Problem Set 2 (PDF) Problem Set 3 (PDF) Problem Set 4 (PDF) Problem Set 5 (PDF) Problem Set 6 (PDF) Problem Set 7 (PDF) Problem Set 8 (PDF) Problem Set 9 (PDF) Problem Set 10 (PDF)

Supporting Files

64a012.fin (FIN) LOCALIZE.M (M) RANK2D.M (M) wig4125.out (OUT) wigley5.out (OUT) wigley9.out (OUT)
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驗證碼請輸入4 + 4 =

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非常好!非常感谢!辛苦了!

Anonymous, 2011-03-03 22:54:45

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