Abstract
In the wake of theoretical, numerical and experimental advances by a large number of contributors, we revisit here some aspects of the fluid kinematics in a two-dimensional plunging breaker occurring in shallow water. In particular, we propose a simplified identification of the velocity distribution at the free surface in terms of the velocity at some characteristic points. We can then simply explain the reasons for which the velocity is maximum inside the barrel at its roof. We also show that the relative velocity field calculated in a coordinate system centered to a point where the velocity is maximum may have a possible analytic representation.
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A. Brief Description of the Numerical Method
A. Brief Description of the Numerical Method
The numerical model is based on the potential theory. The free surface is an isobar and material line. The free surface is described with a finite number of markers with cartesian coordinates \(\vec {X}=(X,Y)\) that are tracked over time. The velocity potential is transported in a Lagrangian way on those markers with the fluid velocity \(\vec {\nabla } \phi \) computed at the same markers. The free surface boundary conditions are written in a Lagrangian way as follows:
This differential system is solved using a standard Runge–Kutta of fourth order (RK4) that updates the velocity potential \(\phi _{fs}\) at the free surface. The velocity potential verifies the following boundary value problem:
The method is desingularized in the sense that the velocity potential follows from the influence of a finite number of sources (Rankine–Green function) located outside the fluid domain. The number of sources and markers being the same, there is square linear system between the velocity potential computed at the markers and the strengths of the influencing sources. Using a conformal mapping that maps the inner tank domain onto a half plane or a quarter plane, the homogeneous Neumann boundary condition on the walls can be implicitly accounted for in the expression of the Green function using images of the sources with respect to the axes. The unknowns of the problem are the velocity potential calculated at the moving markers.
The energy and mass conservations are checked at each step of the time integration. The time step is adapted to the speed of the simulated fluid motion. When the plunging breaker develops, the time step must decrease substantially. In all the computations done in the present paper, the relative errors on the mass and energy conservations are never greater than \(10^{-6}\) and \(10^{-3}\), respectively. The number of markers/sources is set to few hundreds depending on the smoothness of the free surface, it does not vary over a given simulation. No smoothing is required. Regridding can be performed when the free surface is barely distorted. A natural concentration of markers occurs where and when it is necessary. That is the case in the plunging crest where the convection velocity is the greatest. Apart from the time step and the number of Lagrangian markers, the only arbitrary parameter is the distance of desingularization. It is chosen in the range of 2 and 3 times the distance between a given a marker and its two neighbour markers as proposed by Cao et al. [5]. It is shown that this choice optimizes the conservations of mass and energy.
The elaborated software has been continuously validated since its first development in 2007. Recently, the present code has been used for the benchmark organized in the frame of the Joint Collaborative Project TANDEM (Tsunamis in the Atlantic and the English ChaNnel: Definition of the Effects through numerical Modeling). The present code yielded the reference data that describe the multiple reflections of a wave train in channel of 30 km long with a water depth 50 m (2500 markers are used). The results of Navier–Stokes solvers, Boussinesq solvers and potential solvers were benchmarked (see [1]). Another recent validation test focused on the energy distribution during the highly nonlinear sloshing in a tank of a two fluid system. Several sizes of entrapped gas pocket and flip-through configurations are treated (see [10]). The results of a Navier–Stokes solver and the present code show the limit of those approaches in terms of compressibility of the gas which should be accounted for depending on the shape of the entrapped gas pocket.
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Scolan, YM., Guilcher, PM. Wave Kinematics in a Two-Dimensional Plunging Breaker. Water Waves 2, 185–206 (2020). https://doi.org/10.1007/s42286-019-00013-w
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DOI: https://doi.org/10.1007/s42286-019-00013-w