We propose a three-dimensional non-hydrostatic shock-capturing numerical model for the simulation of wave propagation, transformation and breaking, which is based on an original integral formulation of the contravariant Navier-Stokes equations, devoid of Christoffel symbols, in general time-dependent curvilinear coordinates. A coordinate transformation maps the time-varying irregular physical domain that reproduces the complex geometries of coastal regions to a fixed uniform computational one. The advancing of the solution is performed by a second-order accurate strong stability preserving Runge-Kutta fractional-step method in which, at every stage of the method, a predictor velocity field is obtained by the shock-capturing scheme and a corrector velocity field is added to the previous one, to produce a non-hydrostatic divergence-free velocity field and update the water depth. The corrector velocity field is obtained by numerically solving a Poisson equation, expressed in integral contravariant form, by a multigrid technique which uses a four-colour Zebra Gauss-Seidel line-by-line method as smoother. Several test cases are used to verify the dispersion and shockcapturing properties of the proposed model in time-dependent curvilinear grids.
In this paper a new finite-volume non-hydrostatic and shock-capturing three-dimensional model for the simulation of wave-structure interaction and hydrodynamic phenomena (wave refraction, diffraction, shoaling and breaking) is proposed. The model is based on an integral formulation of the Navier-Stokes equations which are solved on a time dependent coordinate system: a coordinate transformation maps the varying coordinates in the physical domain to a uniform transformed space. The equations of motion are discretized by means of a finite-volume shock-capturing numerical procedure based on high order WENO reconstructions. The solution procedure for the equations of motion uses a third order accurate Runge-Kutta (SSPRK) fractional-step method and applies a pressure corrector formulation in order to obtain a divergence-free velocity field at each stage. The proposed model is validated against several benchmark test cases
In this paper we propose an integral form of the fully non-linear Boussinesq equations in contravariant formulation, in which Christoffel symbols are avoided, in order to simulate wave transformation phenomena, wave breaking and nearshore currents in computational domains representing the complex morphology of real coastal regions. Following the approach proposed by Chen (2006), the motion equations retain the term related to the approximation to the second order of the vertical vorticity. A new Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the fully non-linear Boussinesq equations on generalised curvilinear coordinate systems is proposed. The equations are rearranged in order to solve them by a high resolution hybrid finite volume-finite difference scheme. The conservative part of the above-mentioned equations, consisting of the convective terms and the terms related to the free surface elevation, is discretised by a high-order shock-capturing finite volume scheme in which an exact Riemann solver is involved; dispersive terms and the term related to the approximation to the second order of the vertical vorticity are discretised by a cell-centred finite difference scheme. The shock-capturing method makes it possible to intrinsically model the wave breaking, therefore no additional terms are needed to take into account the breaking related energy dissipation in the surf zone. The model is verified against several benchmark tests, and the results are compared with experimental, theoretical and alternative numerical solutions. (C) 2013 The Authors. Published by Elsevier B.V. All rights reserved
In this paper, we propose a model based on a new contravariant integral form of the fully nonl inearBoussinesq equations in order to simulate wave transformation phenomena, wave breaking, and nearshorecurrents in computational domains representing the complex morphology of real coastal regions. The afore-mentioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the factthat the con tinuity equation does not include any dispersive term. A procedure developed in order to correcterrors related to the difficult ies of numerically satisfying the metric identities in the numerical integration offully nonlinear Bous sinesq equation on generalized boundary-conforming grids is presented. TheBoussinesq equation system is numerically solved by a hybrid finite volume–finite difference scheme.The proposed high-order upwind weighted essentially non-oscillatory finite volume scheme involves anexact Riemann solver and is based on a genuinely two-dimensional reconstruction procedu re, which usesa convex combination of biquadratic polynomials. The wave breaking is represented by discontinuities ofthe weak solution of the integral form of the nonlin ear shallow water equations.The capacity of the proposed model to correctly represent wave propagation, wave breaking, and wave-induced currents is verified against test cases present in the literature. The results obtained are comparedwith experimental measures, analytical solutions, or alternative numerical solutions
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