A boundary integral approach to the modeling of surface waves in a wave tank

Abstract

Boundary integral equations (BIEs) are used to model surface waves in a wave tank. Assuming an ideal fluid, the velocity of the fluid can be considered as a potential flow and be modeled by the Laplace equation on the domain. The domain in this case will be a section of a wave channel with an incoming wave from the right, a rigid bottom, a reflective wall on the right and a time varying surface that represent the surface waves. A numerical solution is implemented and used to simulate a wave tank at the University of Oslo. The numerical solution to the BIEs is tested for accuracy against a known solution to the Laplace equation on the boundary and it was found that the BIEs gave a mostly accurate solution except for around the parts of the parametrized boundary that were non-smooth. Two cases of instabilities were found: 1) when the number of discrete points in time was to low and 2) when the wave amplitude got to large. Adding a diffuse term was tried in both cases and found to be ineffective. In case 1) it was ineffective since increasing the number of discrete points in time already made the solution stable and adding the diffusive term introduced another relation ship between the discrete points in time and space which had to be satisfied to remain stable. In case 2) adding the diffusive term had little to no effect, except for when the number of discrete points on time was to small, in which case the instability oc- cured earlier. It is conjectured that improving the accuracies at the non-smooth parts of the boundary will improve the stability for incoming waves.