2.29 | Spring 2003 | Graduate

Numerical Marine Hydrodynamics (13.024)

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)

Course Info

As Taught In
Spring 2003
Learning Resource Types
Lecture Notes
Course Introduction
Problem Sets
Programming Assignments