A semi-analytical high-order translating-pulsating source method for forward-speed ship motions

Abstract

This paper presents a semi-analytical high-order translating and pulsating Green's function method for the hydrodynamic calculations of ships advancing in waves. In this method, the boundary integral equation (BIE) for the radiation and diffraction problems is discretized by using 9-node bi-quadratic curvilinear elements. The translating-pulsating Green's function used in the BIE is decomposed into Rankine source GS and Froude-dependent component GF. For the panel integrals related to Rankine source, the element geometry and velocity potential are described by the same interpolation functions, and the traditional Gauss quadrature is employed to calculate the influence coefficients. But when it comes to the Froude-dependent component (which is highly oscillatory in the horizontal direction), we take a different integral strategy: the 9-node element is divided into four quadrilateral elements, and a semi-analytical scheme is established to calculate the integral over each quadrilateral element. In this scheme, an analytical expression is derived for the integral along the horizontal direction, while Gauss quadrature is used for the integral computation along the vertical direction. Based on the semi-analytical method, the integrals of GF and its x-derivative over a unit square are calculated. Numerical results show that present method is of good convergence and accuracy as compared with other existing results.Then this semi-analytical high-order method is applied to solve the forward-speed ship motions in waves. The hydrodynamic coefficients, wave-exciting forces and motion responses of a mathematical ship model (modified Wigley hull) are first calculated. The computed results are compared with the experimental data and other numerical results of constant panel method (CPM) and traditional high-order boundary method (HOBEM) based on Gauss quadrature. Numerical results show that the present semi-analytical high-order method appears to be more stable than the traditional HOBEM and more accurate than the CPM. Further investigation is conducted on the wave-induced motions of a Navy surface combatant model DTMB 5512 and a full formed ship modified KVLCC2. Comparisons between the numerical results and experimental data indicate that the present semi-analytical high-order method can produce satisfactory predictions for the practical ships.