Numerical wavetanks for wave generation, interaction, and dissipation: variational and computational modelling

A new computational tool has been developed for simulating water-wave motion in the context of the maritime engineering sector, with a specific focus on the formation and analysis of extreme waves generated within in-house experimental wavetanks.

A three-dimensional (3D) potential-flow-based model is built upon Luke’s variational principle (VP), using the first fully variational discretisation in space and time. The resulting numerical wavetank (NWT) is capable of emulating laboratory sea states involving complex wave-wave interactions.

After first mapping the time-dependent free surface and oscillatory wavemaker onto a fixed computational domain using a bespoke σ-coordinate transformation, two numerical models — referred to as Model 1 and Model 2 — are established and implemented within the finite-element environment Firedrake. In Model 1, weak formulations are derived manually from the space-discretised VP and explicitly formulated in code. Model 2 adopts a new methodology, whereby weak formulations are generated automatically from the encoded time-discretised VP, resulting in an implicit implementation. Both approaches employ robust time integrators to maintain stability and conservation. Verification and validation of the new tool are conducted through five test cases (TCs), including validation against experimental measurements and comparison with a newly derived analytical solution for two-soliton interactions.

To further improve physical realism and broaden applicability, a 2D model coupled with a sloping beach is revisited, removing the mild-slope approximation. The coupled model divides the domain into deep- and shallow-water regions governed by nonlinear potential-flow and nonlinear shallow-water equations, respectively. The coupling conditions are derived from a unified variational principle, and the two subdomains are solved using the finite-element-based Model 1 and a well-balanced finite-volume scheme, respectively. This extension, along with improvements in computational performance, enables the simulation of wave energy dissipation through shoaling, cresting, and breaking on the absorbing beach, resulting in a cost-effective and flexible platform for modelling the full life cycle of water waves — from generation and propagation to interaction and dissipation.