Numerical modelling of flows involving submerged bodies and free surfaces

Abstract

Kinetic energy extraction devices for ocean and river flows are often located in the vicinity of the fluid free surface. This differs from wind turbines where the atmosphere may be considered to extend to infinity for the purposes of numerical modelling. As most kinetic energy extraction devices are based on lifting surfaces, a numerical model is sought which can model both lifting and free surface flows. One such model is the boundary element method which has been successfully applied to free surface problems and to lifting flows as well as the combined problem. This study seeks to develop a high order boundary element method that is capable of modelling unsteady lifting and free surface flows in three dimensions. Although high order formulations of boundary element methods are common for free surface problems, providing improved accuracy and computational time, their usage for lifting flows is less frequent. This may be due to the hypersingular boundary integral equation (HBIE) which must be solved in order to find the velocity of the vortex wakes behind lifting surfaces. In previous lifting flow studies using high order boundary element methods the wake velocities have been determined at the element centres and then interpolated to the collocation points. Not until the paper of Gray et al. (2004b) has a method been available for the direct solution of the HBIEs at the edges of three dimensional high order elements with C0 continuous interfaces. The solution employs a technique known as the Galerkin boundary element method. This study shows, for the first time, that the Galerkin boundary element method is applicable to the solution of the HBIE on the vortex wake of a lifting body. The application of the technique is then demonstrated as part of the numerical model developed herein. The model is based on the high order boundary element method developed by Xu (1992) for non-linear free surface flows. This formulation is extended to include steady uniform flow throughout the computational domain as well as the presence of lifting and non-lifting bodies. Several verification cases are implemented to test the accuracy of the model.