Non-Linear Shallow Water Equations numerical integration on curvilinear boundary-conforming grids

An Upwind Weighted Essentially Non-Oscillatory scheme for the solution of the Shallow Water Equations on generalized curvilinear coordinate systems is proposed. The Shallow Water Equations are expressed in a contravariant formulation in which Christoffel symbols are avoided. The equations are solved by using a high-resolution finite-volume method incorporated with an exact Riemann Solver. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities on generalized boundary-conforming grids is presented; this procedure allows the numerical scheme to satisfy the freestream preservation property on highly-distorted grids. The capacity of the proposed model is verified against test cases present in literature. The results obtained are compared with analytical solutions and alternative numerical solutions

Bottom changes in coastal areas with complex shorelines

A model for the sea-bottom change simulations in coastal areas with complex shorelines is proposed. In deep and intermediate water depths, the hydrodynamic quantities are calculated by numerically integrating the contravariant Boussinesq equations, devoid of Christoffel symbols. In the surf zone, the propagation of the breaking waves is simulated by the nonlinear shallow water equations. The momentum equation is solved inside the turbulent boundary layer in order to calculate intrawave hydrodynamic quantities. An integral formulation for the contravariant suspended sediment advection-diffusion equation is proposed and used for the sea-bottom dynamic simulations. The proposed model is applied to the real case study of Pescara harbor (in Italy)

Numerical integration of the contravariant integral form of the Navier–Stokes equations in time-dependent curvilinear coordinate systems for three-dimensional free surface flows

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 shock-capturing properties of the proposed model in time-dependent curvilinear grids

Numerical simulation of bed evolution dynamics: the Pescara harbor

A two-dimensional phase resolving model is used for the computation of the hydrodynamic field in wave-current interaction in the sea regions opposite to the Pescara harbor. The total sediment transport is given by the contribution of the suspended sediment load, calculated by solving the advection-diffusion equation for the suspended sediment concentration, and of the spatial variation of the bed load transport. The proposed model has been used to simulate the silting phenomenon occurring in the sea region opposite to the Pescara harbor in presence of coastal defense structures

Modeling bed evolution using weakly coupled phase-resolving wave model and wave-averaged sediment transport model

In this paper, we propose a model for the simulation of the bed evolution dynamics in coastal regions characterized by articulated morphologies. An integral form of the fully nonlinear Boussinesq equations in contravariant formulation, in which Christoffel symbols are absent, is proposed in order to simulate hydrodynamic fields from deep water up to just seaward of the surf zones. Breaking wave propagation in the surf zone is simulated by integrating the nonlinear shallow water equations with a high-order shock-capturing scheme. The near-bed instantaneous flow velocity and the intra-wave hydrodynamic quantities are calculated by the momentum equation integrated over the turbulent boundary layer. The bed evolution dynamics is calculated starting from the contravariant formulation of the advection-diffusion equation for the suspended sediment concentration in which the advective sediment transport terms are formulated according to a quasi-three-dimensional approach, and taking into account the contribution given by the spatial variation of the bed load transport. The model is validated against several tests by comparing numerical results with experimental data. The ability of the proposed model to represent the sediment transport phenomena in a morphologically articulated coastal region is verified by numerically simulating the long-term bed evolution in the coastal region opposite Pescara harbor (in Italy) and comparing numerical results with the field data

Contravariant Boussinesq equations for the simulation of wave transformation, breaking and run-up

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 near shore currents in computational domains representing the complex morphology of real coastal regions. 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; 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 applied on a real case regarding the simulation of wave fields and nearshore currents in the coastal region opposite Pescara harbour (Italy)

Noll’s axioms and formulation of the closure relations for the subgrid turbulent tensor in large eddy simulation

-In this paper, the relation between the Noll formulation of the principle of material frame indifference and the principle of turbulent frame indifference in large eddy simulation, is revised. The principle of material frame indifference and the principle of turbulent frame indifference proposed by Hutter and Joenk imposes that both constitutive equations and turbulent closure relations must respect both the requirement of form invariance, and the requirement of frame independence. In this paper, a new rule for the formalization of turbulent closure relations, is proposed. The generalized SGS turbulent stress tensor is related exclusively to the generalized SGS turbulent kinetic energy, which is calculated by means of its balance equation, and the modified Leonard tensor

Simulation of wave run-up by means of the exact solution of the wet/dry Riemann problem

An innovative method for the simulation of the hydrodynamics in the swash zone, related to the wave run-up phenomenon, is presented. This method applies the exact solution of the Riemann problem over a dry bed to correctly evaluate the celerity of water waves propagating over the shore, and so to precisely track the coastline location. The simulations of velocity and wave fields outside the surf zone, inside the surf zone and in the swash zone, are carried out by means of a numerical model which solves 3D motion equations expressed in integral form, with a vertical coordinate that varies in time in order to follow the free surface evolution. Several numerical validation tests are carried out, in order to verify the capability of the method to track the coastline

Numerical simulation of the Rio Fucino dam-break flood

In this paper a dam-break flood model based on a contravariant integral form of the shallow water equations is presented. The equations of motion are numerically solved by means of a finite volume-finite difference numerical scheme that involves an exact Riemann solver and is based on a WENO reconstruction procedure. An original scheme for the simulation of the wet front progress on the dry bed is adopted. The proposed model is used to simulate the Rio Fucino dam-break and subsequent flood wave propagation, downstream of the Campotosto reservoir (Italy)

Numerical investigation of the three-dimensional velocity fields induced by wave-structure interaction

Submerged shore-parallel breakwaters for coastal defence are a good compromise between the need to mitigate the effects of waves on the coast and the ambition to ensure the preservation of the landscape and water quality. In this work we simulate, in a fully three-dimensional form, the hydrodynamic effects induced by submerged breakwaters on incident wave trains with different wave height. The proposed three-dimensional non-hydrostatic finite-volume model is based on an integral form of the Navier-Stokes equations in σ-coordinates and is able to simulate the shocks in the numerical solution related to the wave breaking. The obtained numerical results show that the hydrodynamic phenomena produced by wave-structure interaction have features of three-dimensionality (undertow), that are locally important, and emphasize the need to use a non-hydrostatic fully-three-dimensional approach