Havelock form translating-pulsating panel source Green's function and its numerical calculation

Abstract

The numerical approximation of the three-dimension translating-pulsating (3DTP) source Green's function is the basis for solving the diffraction-radiation problem of marine vessels with forward speed. However, the highly oscillation of the 3DTP source near the free surface makes the hydrodynamic results unstable. The panel source Green's function, which means the analytical integral of the Green's function over panels, can ease the numerical instability problem. In the present study, an analytical expression of Havelock form panel source Green's function is proposed and the corresponding numerical integral method is presented. A coordinate transformation is performed and the triangular panel is mapped into a unit rectangle. The space integral is carried out beforehand by an analytical method. And the determination of the control factor is elaborated. Thus, the panel source Green's function can be expressed as a θ-type single integral. LOBATTO rule is adopted to eliminate singularities caused by infinite discontinuity and a fraction polynomial is used to evaluate complex exponential integral. Therefore, the integral respect to θ can be performed by an adaptive iterative integration. Finally, the numerical results of present method is compared with other existing results. It shows that the present analytical integral expression is accurate and reliable.