Lecture Notes

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)