Accuracy and Efficiency of a Panel Method for Free Surface Flow Problems in Three Dimensions

Abstract

In ocean engineering many problems involve the analysis of free surface waves where the fluid flow can be described by a velocity potential ϕ which satisfies the Laplace equation

$${\nabla ^2}\phi = 0$$

(1)

throughout the fluid domain Ω. The motion of the free surface waves is described by the dynamic and the kinematic free surface boundary conditions. These boundary conditions are the nonlinear, time-dependent partial differential equations

$$\frac{{\partial \phi }}{{\partial t}} + \frac{1}{2}{\left( {\nabla \phi } \right)^2} + g\eta = 0$$

(2)

$$\frac{{\partial \eta }}{{\partial t}} + \frac{{\partial \phi }}{{\partial x}} \cdot \frac{{\partial \eta }}{{\partial x}} + \frac{{\partial \phi }}{{\partial y}} \cdot \frac{{\partial \eta }}{{\partial y}} - \frac{{\partial \phi }}{{\partial z}} = 0$$

(3)

where η is the free surface elevation and g is the gravitational acceleration.