Numerical Marine Hydrodynamics
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
- 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
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.