Ill-posedness of waterline integral of time domain free surface Green function for surface piercing body advancing at dynamic speed

TL;DR

This paper demonstrates that the waterline integral of the time domain free surface Green function in hydrodynamic computations is ill-posed, complicating the determination of source strength for floating bodies at dynamic speeds.

Contribution

It reveals the ill-posedness of the waterline integral in time domain Green function formulations, highlighting a fundamental challenge in hydrodynamic modeling of moving bodies.

Findings

Waterline integral is ill-posed in time domain Green function formulation.

Source strength for velocity potential cannot be obtained due to ill-posedness.

Implications for computational methods in hydrodynamics of floating bodies.

Abstract

In the linear time domain computation of a floating body advancing at a dynamic speed, the source formulation for the velocity potential of the hydrodynamic problem is commonly used so that the velocity potential is expressed as the integral of time domain free surface sources distributed on the two-dimensional wetted body surface and the one-dimensional waterline, which is the intersection of the wetted body surface and the mean free water surface. A time domain free surface source is corresponding to the time domain free surface Green function associated with a suitable source strength, which is to be solved from body boundary condition and normal velocity boundary integral equation of the source formulation. The normal velocity boundary integral equation contains an integral of the normal derivative of the time domain free surface Green function on the waterline. It is shown that the waterline integral is ill-posed. Thus the source strength of velocity potential is not obtainable.