ABSTRACT

In the context of linear potential flow, the effect of viscous dissipation is often modeled by sudden pressure change in proportion to the local flow velocity across a surface. This model leads to a hyper-singular integral equation for the pressure jump across the surface but the jump and the normal velocity are both unknown on that surface. In this paper, we solve the problem by representing the unknowns in terms of generalized modes. The velocity potentials corresponding to the generalized modes are derived as the solutions of the hyper-singular integral equations with prescribed Neumann conditions. The coefficients of the generalized modes are then determined using the linear relationship between the pressure and the velocity. As illustrative examples, we applied the solution method to a circular tube used as a model for solar ponds, side-by-side barges and an oscillating wave surge converter, all in waves, with the consideration of the effect of dissipation.

INTRODUCTION

When the solution of a linear potential flow over-predicts the response of body motion or the motion of the fluid body, it may be necessary to take into account of the effect of viscous dissipation. For the body motion, the viscous damping forces are estimated and added to the wave damping forces in the equations of motion. As a remedy for the resonant motion of the fluid body in a moon pool or in the gap between two bodies, the boundary condition is modified on a part of the free surface to allow energy dissipation. Huijsmans ef al. (2001) placed a rigid lid and Newman (2004) placed a flexible lid on the free surface and linear damping force was applied to the vertical motion of the lid. Molin et. al. (2009) used multiple lids subject to quadratic damping force. Chen (2004) obtained the solution of the integral equation directly without introducing additional modes of motion associated with the lid.