# 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)