Boundary conditions for the simulation of wave breaking

In this paper we propose a new numerical model for the simulation of the wave breaking. The three-dimensional equations of motion are expressed in integral contravariant form and are solved on a curvilinear boundary conforming grid that is able to represent the complex geometry of coastal regions. A time-dependent transformation of the vertical coordinate that is a function of the oscillation of the turbulent wave boundary layer is proposed. A new numerical scheme for the simulation of the resulting equations is proposed. New boundary conditions at the free surface and bottom for the equations of motion expressed in contravariant form are proposed. We present an analysis of the importance of the correct positioning, inside the oscillating turbulent boundary layer, of the centre of the calculation grid cell closest to the bottom, in order to correctly simulate the height of the breaking waves

Wave Fields and Nearshore Currents in the Coastal Region Opposite San Mauro Cilento (Italy)

In this paper in order to simulate nearshore currents in computational domains representing the complex morphology of real coastal regions we use a model based on a contravariant integral form of the fully nonlinear Boussinesq equations (FNBE). The contravariant integral form, in which Christoffel symbols are absent, of the continuity equation does not contain the dispersive term. The Boussinesq equation system is numerically solved by a hybrid finite volume-finite difference scheme. The wave breaking is represented by discontinuities of the weak solution of the integral form of the nonlinear shallow water equations (NSWE). The capacity of the proposed model to correctly simulate the wave train propagation on a highly distorted grid is verified against test case present in the literature. The simulation of wave fields and nearshore currents in the coastal region, opposite San Mauro Cilento (Italy) in presence of a system of T-head groins, is numerically reproduced by using the proposed model

A Three-Dimensional Numerical Study of Wave Induced Currents in the Cetraro Harbour Coastal Area (Italy)

In this paper we propose a three-dimensional numerical study of the coastal currents produced by the wave motion in the area opposite the Cetraro harbour (Italy), during the most significant wave event for the coastal sediment transport. The aim of the present study is the characterization of the current patterns responsible for the siltation that affects the harbour entrance area and the assessment of a project solution designed to limit this phenomenon. The numerical simulations are carried out by a three-dimensional non-hydrostatic model that is based on the Navier–Stokes equations expressed in integral and contravariant form on a time-dependent curvilinear coordinate system, in which the vertical coordinate moves in order to follow the free surface variations. The numerical simulations are carried out in two different geometric configurations: a present configuration, that reproduces the geometry of the coastal defence structures currently present in the harbour area and a project configuration, which reproduces the presence of a breakwater designed to modify the coastal currents in the area opposite the harbour entrance

A New Turbulence Model for Breaking Wave Simulations

In this paper, the hydrodynamic and free surface elevation fields in breaking waves are simulated by solving the integral and contravariant forms of the three-dimensional Navier–Stokes equations that are expressed in a generalized time-dependent curvilinear coordinate system, in which the vertical coordinate moves by following the free surface. A new k−l turbulence model in contravariant form is proposed; in this model, the mixing length, l, is defined as a function of the maximum water surface elevation variation. A new original numerical scheme is proposed. The main element of originality of the numerical scheme consists of the proposal of a new fifth-order reconstruction technique for the point values of the conserved variables on the cell face. This technique, named in the paper as WTENO, allows the choice procedure of the reconstruction polynomials for the point values to be modified in a dynamic way

Turbulence models under breaking waves: a new two-equation model based on turbulent kinetic energy and specific dissipation rate

In this thesis, a new model for the simulation of the breaking waves is proposed. This model is based on the solution of the three-dimensional equations of motion expressed in contravariant formulation. These equations are in integral form and are expressed in terms of the conserved variables H and Hu (H is the total water depth and u is fluid velocity vector). The three-dimensional ensemble-averaged motion equations are solved by a new high-order shock-capturing numerical scheme. The elements of novelty in this new numerical scheme are two. The first element of novelty consists in the proposal of a new reconstruction technique of the point values of the conserved variables on the cell faces of the computational grid (starting from the cell-averaged values of the same variables). This reconstruction technique is named WTENO and it is specifically designed for the three-dimensional simulation of breaking waves. The second element of novelty consists in the use of an exact solution for the Riemann problem to advancing in time the point values of the conserved variables at the cell faces. In this thesis, two turbulence models, which belong to the context of the URANS models, k-l and k-w (k is the turbulent kinetic energy, l is the mixing length and w is the specific dissipation rate) are proposed. In the new k-l turbulence model, the k-equation is expressed in a new integral contravariant form on a generalized time-dependent curvilinear coordinate system. In this model, the equations of motion are solved also in the buffer layer, while the k-equation is solved starting from the buffer layer in the proximity of the viscous sublayer. Outside the oscillating wave boundary layer, a new formula for the mixing length is proposed as a function of the first and second spatial derivatives of the local maximum water surface elevation. In the oscillating wave boundary layer, the mixing length is calculated by the hypothesis of the balance between production and dissipation of turbulent kinetic energy. In the new k-w turbulence model, the k and w equations are written in a new integral contravariant form on a generalized time-dependent curvilinear coordinate system. The new k-w turbulence model admits the possibility to assign the boundary condition for the specific dissipation rate directly at the bottom. In this model, the equations of motion are solved starting from the buffer layer and the k and w-equations are solved in the buffer layer at the border with the viscous sublayer. The production of turbulent kinetic energy in the zone between the breaking wave fronts and the oscillating wave boundary layer is reduced by introducing a dynamic coefficient for the dissipation of w and a limiter in the eddy viscosity. In this thesis, the new k-w turbulence model is used to directly simulate the unsteady quasi-periodic vortex structures due to the interaction between breaking waves and coastal works

Integral contravariant form of the Navier-Stokes equations

An original integral formulation of the three-dimensional contravariant Navier-Stokes equations, devoid of the Christoffel symbols, in general time-dependent curvilinear coordinates is presented. The proposed integral form is obtained from the time derivative of the momentum of a material fluid volume and from the Leibniz rule of integration applied to a control volume that moves with a velocity which is different from the fluid velocity. The proposed integral formulation has general validity and makes it possible to obtain, with simple passages, the complete differential form of the contravariant Navier-Stokes equations in a time dependent curvilinear coordinate system. The integral form, devoid of the Christoffel symbols, proposed in this work is used in order to realise a three-dimensional non-hydrostatic numerical model for free surface flows, which is able to simulate the discontinuities in the solution related to the wave breaking on domains that reproduce the complex geometries of the coastal regions. The proposed model is validated by reproducing experimental test cases on time dependent curvilinear grids

A three-dimensional high-order numerical model for the simulation of the interaction between waves and an emerged barrier

We present a new three-dimensional numerical model for the simulation of breaking waves. In the proposed model, the integral contravariant form of the Navier-Stokes equations is expressed in a curvilinear moving coordinate system and are integrated by a predictor-corrector method. In the predictor step of the method, the equations of motion are discretized by a shock-capturing scheme that is based on an original high-order scheme for the reconstruction of the point values of the conserved variables on the faces of the computational grid. On the cell faces, the updating of the point values of the conserved variables is carried out by an exact Riemann solver. The final flow velocity field is obtained by a corrector step which is based exclusively on conserved variables, without the need of calculating an intermediate field of primitive variables. The new three-dimensional model significantly reduces the kinetic energy numerical dissipation introduced by the scheme. The proposed model is validated against experimental tests of breaking waves and is applied to the three-dimensional simulation of the local vortices produced by the interaction between the wave motion and an emerged barrier

A Wave-Targeted Essentially Non-Oscillatory 3D Shock-Capturing Scheme for Breaking Wave Simulation

A new three-dimensional high-order shock-capturing model for the numerical simulation of breaking waves is proposed. The proposed model is based on an integral contravariant form of the Navier–Stokes equations in a time-dependent generalized curvilinear coordinate system. Such an integral contravariant form of the equations of motion is numerically integrated by a new conservative numerical scheme that is based on three elements of originality: the time evolution of the state of the system is carried out using a predictor–corrector method in which exclusively the conserved variables are used; the point values of the conserved variables on the cell face of the computational grid are obtained using an original high-order reconstruction procedure called a wave-targeted essentially non-oscillatory scheme; the time evolution of the discontinuity on the cell faces is calculated using an exact Riemann solver. The proposed model is validated by numerically reproducing several experimental tests of breaking waves on computational grids that are significantly coarser than those used in the literature to validate the existing 3D shock-capturing models. The results obtained with the proposed model are also compared with those obtained with a previously published model, which is based on second-order total variation diminishing reconstructions and an approximate Riemann solver usually adopted in the existing 3D shock-capturing models. Through the above comparison, the main drawbacks of the existing 3D shock-capturing models and the ability of the proposed model to simulate breaking waves and wave-induced currents are shown. The proposed 3D model is able to correctly simulate the wave height increase in the shoaling zone and to effectively predict the location of the wave breaking point, the maximum wave height, and the wave height decay in the surf zone. The validated model is applied to the simulation of the interaction between breaking waves and an emerged breakwater. The numerical results show that the proposed model is able to simulate both the large-scale circulation patterns downstream of the barrier and the onset of quasi-periodic vortex structures close to the edge of the barrier

Simulation of wave motion and wave breaking induced energy dissipation

We propose a one-equation turbulence model based on a modified closure relation for the length scale of turbulence. The proposed model is able to adequately represent the energy dissipation due to the wave breaking and does not need any criterion to a priori locate the wave breaking point and the region in which the turbulence model has to be activated. The numerical simulation of wave transformation is carried out by solving the Navier-Stokes equations expressed in an integral formulation on a time dependent curvilinear coordinate system where the vertical coordinate varies in time and vectors and tensors are represented in a Cartesian base system. The model performance is assessed by numerically reproducing a laboratory test which consists in producing the breaking of a spilling wave on a sloping beach