Numerical solution for the submerged pulsating line source in the presence of a free surface

Abstract

A modified source and dipole panel method to calculate the flow properties around an oscillating arbitrary body in the presence of a free surface is proposed. To demonstrate the feasibility of the method the problem of a pulsating line source submerged under a free surface is treated. The technique chosen is based on Green's identity whereby the boundary-value problem is expressed as a boundary integral equation which is solved numerically. The near field of the water surface is represented by singularity panels with constant strength.

The work was motivated by the reported large computing times for existing programs using Green's functions satisfying the free surface boundary condition. The present approach uses free-space Green's function. The free surface boundary condition is applied to surface singularity panels using Green's theorem. Thus free surface effects are included in the solution while panel integrations are simplified considerably by the use of simpler Green's function.

The matrix equations resulting from Green's identity were solved by using IMSL routines for Gaussian Elimination, and the behavior of the influence coefficient matrix was tested by using LINPACK routines.

The depth of the submerged-source and wave number was kept constant while the length of near field and the number of panels per wavelength was varied systematically. A minimum of 10 panels per wavelength and paneled water surface length of 2 wavelengths gives good agreement with the known exact solution. Computing times were low, indicating the feasibility of the technique for application to unsteady ship problems.