GEM 517E - Boundary Element Methods in Ship Hydrodynamics

Course Objectives

The boundary element methods are one of very powerfull methods for the solution of complex probelms in ship hydrodynamics. It is mainly based on the discretization of boundaries of problem rather than the fluid domain itself. Some hydrodynamic singularities are distributed on the boundaries. The unknown strengths of corresponding singularities can be computed using kinematic boundary conditions. In this course, first, the basics of fluid mechanics are reviewed. Then, mathematical formulations and numerical implementations of boundary element methods are explained. The boundary element methods are applied to lots of ship hydrodynamics problems in homeworks and term projects. Students are expected to develop their own codes and algoritms as well as to use some commercial softwares.

Course Description

Introduction, What are the boundary element methods?, Where do they apply?, Some of their advantages;
Review of fundamentals of Fluid mechanics, Velocity, pressure and shear stresses of fluid flow, Conservation of momentum equations, Constitutive relations - Navier-Stokes equations, Inviscid/irrotational flow - Velocity potential - Bernoulli's equation, Kinematic boundary condition, Lifting flows - Kutta condition;
Formulation of boundary element methods for Fluids, Green's theorem, The Green's source function, Green's identity - Integral equation for the potential on the boundary, The Neumann and Dirichlet boundary conditions, Velocity vs. potential formulations, Equivalence of the dipole and vorticity distributions;
Numerical implementation, Discretization of boundary into panels, Approximation of singularity distributions on the boundary, Galerkin vs. collocation approach, Matrix of influence coeffcients, Evaluation of infuence coeffients - The self-infuence coeffcient, Low-order (constant) vs. high-order (linear, quadratic) methods, Boundary shape discontinuities (corners), Hydrofoil trailing edge – Morino-Kutta condition, Error vs. number of panels and vs. computing time;
Applications, Flow about a 2-D non-lifting body, Flow about a 2-D lifting hydrofoil, Free-surfaces – Cavities, The dynamic boundary condition, Implementation of the dynamic boundary condition, Linear vs. non-linear approach, Vortex roll-up - Numerical stability, Unsteady flow problems - Convection of vorticity;
Other topics, Viscous/inviscid flow coupling, Hybrid BEM/FEM's methods, Stokes flows.