Numerical wave tank based on a sigma-transformed finite element inviscid flow solver

Inviscid two-dimensional free surface wave motions are simulated using a sigma-transformed finite-element model based on potential theory for irrotational, incompressible fluid flow over a flat fixed bed. The free surface boundary condition is fully non-linear, and so non-linear effects up to very high order can be modelled. The sigma-transformation involves linear stretching of the mesh between the bed and free surface. This has two major advantages. First, remeshing due to the moving free surface is avoided. Second, the mesh nodes are aligned vertically, allowing a high order calculation of the free surface vertical velocity component to be implemented without smoothing, except for very steep waves. The model however is further restricted to non-overturning, non-breaking waves because of the uniqueness of the sigma-transformation. Excellent agreement is obtained with analytical and alternative numerical data for small amplitude free sloshing in a rectangular tank and forced sloshing in a horizontally base-excited rectangular tank. At higher amplitudes, non-linear effects are evident in the simulations by the present numerical model. The model is also able to reproduce steep progressive waves due to a wavemaker in agreement with Stokes 5th theory, second-order shallow water waves in agreement with cnoidal theory, and focused wave groups that match the experimental measurements acquired by Baldock et al. [A laboratory study of non-linear surface waves on water.phil. trans. R. soc. lond. A 1996;354:649-676].