On wave breaking for Boussinesq-type models

Abstract

Abstract:We considers the issue of wave breaking closure for Boussinesq type models, and attempt at providing some more understanding of the sensitivity of some closure approaches to the numerical setup. In particular, we are interested in the potential for mesh refinement, and in quantifying the dissipation mechanism active. Two closure strategies are considered. The first is an eddy viscosity approach following some early work by O. Nwogu in the 90's, and some more recent developments by Zhang and co-workers. In this model, a breaking viscosity is computed starting form a turbulent kinetic energy. The latter is obtained from an ad-hoc partial differential equation, which is solved in parallel with the propagation model. The second approach considered consists in suppressing the dispersive terms in breaking regions. The dissipation of total energy obtained in a shallow water shock is used to model the energy dissipation due to breaking. Due to its simplicity and effectiveness, the second approach has gained substantial attention in the coastal engineering community. Here we propose a systematic comparison of the two approaches, to understand more of their sensitivity w.r.t. the type of propagation model used (weakly or fully nonlinear), to the type of breaking wave being simulated, as well as to understand the mesh dependence of the pointwise results obtained, and in particular the potential for achieving mesh converged simulations. Finally, we provide a quantitative analysis of the dissipation introduced by the two closures for a moving breaking bore.

Implementation and Evaluation of Breaking Detection Criteria for a Hybrid Boussinesq Model

“…In the figure, the first row reports the results obtained with the local criterion, the second those of the physical detection criterion, and the last the results obtained with the hybrid criterion. For the last two criteria our results are very similar to those of [24], and show the appearance on the finer meshes of unphysical high frequency oscillations superimposed to the trend of the coarse mesh solution. This phenomenon is a not as visible for the results obtained with the local criterion, which only flashes at the end of the upward slope.…”

“…We use this test to evaluate the mesh dependence of the closure and detection criteria. Note that a very thorough investigation of this issue has already been presented in [24]. The reference, as well as our study shows that the type of breaking closure based on the coupling of Boussinesq-type equations with the Non-linear Shallow Water model and the approximation of the rollers by bores, introduces instabilities, that lead to a blow up, as the mesh gets refined.…”

“…Note that, although the analysis is performed in the context of hybrid wave breaking closure, detection criteria are an essential element for other approaches as well. The reader might refer to the extensive references provided in [23,24] for further details. Our results will also be relevant for multi-dimensional flows, which will however require to embed some additional elements related to the direction of wave propagation, and on water level variations in the transversal direction.…”

“…While other schemes have been used in combination with a similar wave breaking closure (cf. [23,19,24]), such interaction is always necessary. This makes the understanding of the stabilisation and shock-capturing mechanisms critical.…”

“…As discussed in detail in [24], the wave breaking closure used here introduces easily oscillations at the interface of the coupling between the Bousisnesq model and the Shallow Water equations. Indeed, in [24], it is shown that this is especially important when a mesh refinement is involved, as, if uncontrolled, it would lead to the blow up of the computation, instead of the sought mesh convergence. In particular, on coarse meshes the amplitude of these perturbations remains very small.…”