This course deals with analytical and numerical methods dedicated to free surface unsteady flows especially under non viscous potential flow theory.

Various approaches of modeling of nonlinear free surface boundary conditions and time marching are presented. Other aspects like mass ans energy conservation and wave absorption are studied.

Two methods are then detailed :

• the Boundary Element Method (BEM)
• the spectral method

At the end of the course a method for nonlinear diffraction based on a spectral approach for modeling incident wavetrains is presented. This approach using a computation of the diffracted flow by a potential flow solver or a viscous flow solver is very efficient and leads to various applications of practical interest (long time numerical simulation of wave-body fully nonlinear interactions).

Specification of a boundary value problem

• Nonlinear free surface boundary conditions
• Development using Stokes hypothesis
• Other boundary conditions

Time domain simulation

• Mixed Euler-Lagrange approach
• Variants

Time marching

• Runge-Kutta schemes
• Predictor corrector schemes
• Implicit Euler scheme

Controle of accuracy

• Mass conservation
• Energy conservation

Wave absorption

• Condition of Orlanski