Abstract:Further developments and applications of the 2D harmonic polynomial cell (HPC) method proposed by Shao and Faltinsen [22] are presented. First, a local potential flow solution coupled with the HPC method and based on the domain decomposition strategy is proposed to cope with singular potential flow characteristics at sharp corners fully submerged in a fluid. The results are verified by comparing them with the analytical added mass of a double-wedge in infinite fluid. The effect of the singular potential flow is not dominant for added mass and damping, but the error is non-negligible when calculating mean wave loads using direct pressure integration. Then, the double-layer nodes technique is used to simulate a thin free shear layer shed from lifting bodies, across which the velocity potential is discontinuous. The results are verified by comparing them with analytical results for steady and unsteady lifting problems of a flat plate in infinite fluid. The latter includes the Wagner problem and the Theodorsen functions. Satisfactory agreement with other numerical results is documented for steady linear flow past a foil and beneath the free surface.

The fundamental solution to steady ship waves accounting for viscous effects (the viscous-ship-wave Green function) is investigated within the framework of the weakly damped free-surface flow theory. An explicit expression of the viscous-ship-wave Green function is firstly derived, and an accurate and efficient technique is described to evaluate the Green function via decomposing the free-surface term into the local-flow component and wave component. To delve into the physical features of the viscous-ship-wave Green function, the asymptotic approximations in the far field due to Kelvin, Havelock and Peters are presented for the flow-field point located inside, at and outside the Kelvin wedge. In addition, uniform approximations to the wave component based on the Chester–Friedman–Ursell (CFU) approximation and the Kelvin–Havelock–Peters (KHP) approximation are carried out. Both numerical evaluation and asymptotic approximations show that the singular behaviour is eliminated and the divergent waves associated with large wavenumbers leading to rapid oscillations are severely damped when viscous effects are accounted for. In addition, viscous effects also alter the apparent wake angle associated with the wave pattern created by a high-speed translating source, and the apparent wake angle is dependent on both $\mathscr{U}^{-1}$ and $\mathscr{U}^{-2}$, where $\mathscr{U}$ is the translating speed of the source.

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