A new model of shoaling and breaking waves: one-dimensional solitary wave on a mild sloping beach

Abstract

We present a new approach to model coastal waves in the shoaling and surf zones. The model can be described as a depth-averaged large-eddy simulation model with a cutoff in the inertial subrange. The large-scale turbulence is explicitly resolved through an extra variable called enstrophy while the small-scale turbulence is modelled with a turbulent-viscosity hypothesis. The equations are derived by averaging the mass, momentum and kinetic energy equations assuming a shallow-water flow, a negligible bottom shear stress and a weakly turbulent flow assumption which is not restrictive in practice. The model is fully nonlinear and has the same dispersive properties as the Green–Naghdi equations. It is validated by numerical tests and by comparison with experimental results of the literature on the propagation of a one-dimensional solitary wave over a mild sloping beach. The wave breaking is characterized by a sudden increase of the enstrophy which allows us to propose a breaking criterion based on the new concept of virtual enstrophy. The model features three empirical parameters. The first one governs the turbulent dissipation and was found to be a constant. The eddy viscosity is determined by a turbulent Reynolds number depending only on the bottom slope. The third parameter defines the breaking criterion and depends only on the wave initial nonlinearity. These dependences give a predictive character to the model which is suitable for further developments.

Modeling shallow water waves

“…The derivation of (90) relies on quite sound physical arguments but a good amount of physical modeling is still required to propose expressions for the eddy viscosity ν T and the dissipation term D. There is still no consensus regarding what these terms should be. For instance, ν T = C ν h √ gh is proposed in [154] while [107] suggests expressions based on the enstrophy, ν T = C p h 2 √ ϕ and D = 1 2 C r h 2 ϕ 3/2 , with C p and C r dimensionless numerical coefficients. A drawback of this last choice is that the enstrophy (or turbulent energy) stays equal to zero if it is initially zero, but good matching with experimental data are observed in many cases [107,162].…”

Hyperbolic model for free surface shallow water flows with effects of dispersion, vorticity and topography

“…In this section, dispersion relations of the two-layer dispersive system (5) and its hyperbolic approximation (19) are obtained and studied. This analysis allows one to follow the influence of the relaxation parameter α on the accuracy of the hyperbolic approximation.…”

“…Therefore, for flows over a flat bottom, solutions of Eqs. (12) in the class of travelling waves are described by equations (19). The normal form of Eqs.…”

“…This leads to a jump of the variables of the upper turbulent layer. System (19) can be used to describe the structure of turbulent bore if the dispersion effects are negligible (α = 0). Let the flow depth and velocity be the following h = H 0 , u = U 0 and η = 0 for x < x 0 .…”

MRT-Lattice Boltzmann Model for Multilayer Shallow Water Flow

“…The shallow water equations can also be coupled to transport equations to model pollutant transport [8], salinity and temperature [9], and sediment transport [6,10], which are important subjects in many industrial and environmental projects. More advanced depth-averaged models were developed for shear shallow water flows [11,12] and for coastal waves in the shoaling and surf zones [13,14].…”