A semi-numerical computation for the added mass coefficients of an oscillating hemi-sphere at very low and very high frequencies

TL;DR

The paper develops a semi-numerical method based on spherical harmonics to compute added-mass coefficients for a floating hemisphere undergoing forced harmonic oscillations in an infinite-depth fluid. By recasting the problem as a Laplace exterior problem and exploiting symmetry via domain extension, it derives asymptotic limits in the very-low and very-high frequency regimes and computes intermediate coefficients through a spherical-harmonic expansion. The results align with literature for several modes and deliver detailed, benchmark-worthy values (e.g., in the limits $A'_{11}$ and $A'_{33}$) that can validate numerical wave-body solvers. Overall, the work provides a physics-informed, near-exact reference method for seakeeping-flow analyses and code verification across frequency regimes.

Abstract

A floating hemisphere under forced harmonic oscillation at very high and very low frequencies is considered. The problem is reduced to an elliptic one, that is, the Laplace operator in the exterior domain with standard Dirichlet and Neumann boundary conditions, so the flow problem is simplified to standard ones, with well known analytic solutions in some cases. The general procedure is based in the use of spherical harmonics and its derivation is based on a physics insight. The results can be used to test the accuracy achieved by numerical codes as, for example, by finite elements or boundary elements.

Figures (4)

• Figure 1: Geometrical description of a seakeeping-like flow problem.
• Figure 2: Extension of the load $h$ for the heave-mode at very-low and very-high frequencies.
• Figure 3: Extension of the load $h=\cos\theta$ for the surge-mode at very-low frequencies.
• Figure 4: Extension of the load $h=\hbox{sign}~(\cos\varphi)\cos\theta$ for surge-mode at very-high frequencies.