Study on Havelock form translating–pulsating source Green's function distributing on horizontal line segments and its applications

Abstract

Calculation of free surface Green's function is the fundamental element in numerically solving and analyzing hydrodynamic performance of ships or floating structures. Three dimensional translating and pulsating (3DTP) frequency domain Green's function in Havelock form is of explicit physical meaning, while it is difficult to be evaluated for its integrand has infinity multipliers k and high-frequency oscillatory behavior. To extend it to ship hydrodynamic problems, integration of 3DTP along line segment is outlined and analytically performed in advance. A θ-type integral of Havelock form Green's function distributing on a horizontal line segment is obtained. The integrals of both 3DTP source Green's function distributing on a horizontal line segment and partial derivatives are derived, as a result, both the order of infinity multiplier and the amplitude of oscillation function decrease, and the term with infinity number vanishes as well. It avoids the difficulties and also provides a new robust approach to obtain integral of Green's function over a panel accurately and stably.To calculate the 3DTP Green's function distributing on horizontal line segment and its partial derivatives. Fraction polynomial is used to evaluate complex exponential integral. Lobatto rule is applied to eliminate the effect of singularity at end points. The horizontal line segment is divided by critical points to make sure the original integrand is continuous on each line subsection. For the integration of 3DTP Green's function over a panel, the profiles along vertical direction can be obtained by accumulating a series of 3DTP source distributing on various horizontal line segments. The corresponding computing codes are developed and validated. 3DTP Green's function distributing on a horizontal line segments and its integral over a panel are performed by present method and Gaussian integral method. Numerical results show that present method is more efficient and accurate than normal Gaussian integral with point sources. By use of it, hydrodynamic of Wigley III and S175 container ship advancing in waves are calculated and analyzed. The calculated motions of ship are acquired and agree well with experimental data.