Boussinesq modeling of breaking waves: Description of turbulence

Abstract

[1] Improved turbulent closures for use in fully nonlinear Boussinesq-type models are described here. The approach extends previous works in order to give a more flexible and accurate description of the turbulence due to a breaking wave. Turbulent stresses are handled by means of the Boussinesq hypothesis, and the eddy viscosity is assumed to vary over the water depth according to different laws. The model is described in detail, and its performances are evaluated both against available analytical solutions and against experimental data of regular waves breaking over a slope. The influence of the vertical structure of turbulence under a breaking wave is analyzed by means of four different vertical profiles of eddy viscosity; the differences in terms of hydrodynamic features are also discussed. Among the four selected profiles, two of them (the uniform one and that with uniform eddy viscosity over the top half of the water column which linearly decreases to zero over the lower half ) give better overall performances when compared with experimental data concerning velocity profiles.

On wave breaking for Boussinesq-type models

“…A highly non-linear Boussinesq model with the same turbulence wave breaking model of Nwogu, has been used by Elnaggar and Watanabe [23]. More recent related works are those of Briganti et al [12] in which a Boussinesq model is coupled with a vorticity transport equation, as well as the work of Zang et al [95] in which they use a Green-Naghdi system with an eddy viscosity PDE derived from the TKE model. Here we propose a version of the model proposed by Nwogu modified according to some of the definitions proposed in [95], and embedding the detection criteria discussed in the beginning of this section.…”

“…in (1) and (5). There exist improved variants of this idea, either also embedding some notion of vertical (along the depth) kinematics or vorticity transport [12], or attempting at improving the behavior of the total energy dissipation by also including a water elevation dissipation [18], up to more complex approaches including partial differential equations for an average turbulent kinetic energy [29,95], or even multilayer approaches embedding PDEs for a turbulent layer flowing on top and interacting with the bulk of the wave, well representative of spilling flows [14,53,59,64,27]. In this work, we have chosen to focus our attention on an intermediate complexity approach: a wave breaking closure based on a one equation PDE for the Turbulent Kinetic Energy (TKE).…”

“…To provide some quantitative evaluation of the modelling closures proposed, we use here a simpler strategy: we compare the dissipation of the shallow water energy introduced by the upwinding terms of the finite volume scheme (12), with those of the eddy viscosity term. To evaluate these quantities we have used the standard relation between the total shallow water energy E and the shallow water fluxes F, namely (see e.g.…”

“…While based on a better physical background, these models still rely on an algebraic ad-hoc definition of a momentum dissipation, and require an appropriate calibration. A more advanced version of these roller models has been proposed in [12], and more recently extended in [83]. These models attempt at accounting for depth variations of some of the physical quantities (eddy viscosity, horizontal velocity), thus going beyond the irrotational hypothesis when computing the vorticity and/or dissipation generated in breaking regions.…”

A nonlinear rotational, quasi-2DH, numerical model for spilling wave propagation

“…As discussed in [16], it can be demonstrated that the two mentioned approaches are equivalent as regards to their effect on the momentum equation. Regarding the breaking model, two main strategies were adopted: (i) artificial eddy viscosity models [17,14] and (ii) surface roller models, which in turn can be subdivided into (a) non-explicit rotational models [15,18], and (b) rotational models [19][20][21][22].…”