MyOOPS開放式課程

13.024數值海洋流體動力學 ，2003 春季

13.024 Numerical Marine Hydrodynamics, Spring 2003

 週 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N－S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-Neumann的Green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法 Euler's Method, Modified Euler's Method ‧四階Runge－Kutta法 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 ‧Gerritsma與Beukelman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

m函數“rank2d”“localize”和習題集6一起提供。

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

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)
 留下您對本課程的評論 標題： 您目前為非會員，留言名稱將顯示「匿名非會員」只能進行20字留言 留言內容： 驗證碼請輸入9 + 5 = 標籤 現有標籤：1 新增標籤：

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