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

Abstract

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.

Numerical Investigation into the Performance of an OWC Device under Regular and Irregular Waves

“…where ∆ = D∆ξ 1 ∆ξ 2 is the characteristic dimension of the calculation grid and C s is the Smagorinsky coefficient (ranging between 0.1 and 0.2). The equations of motion ( 5) and ( 6) are numerically solved by a predictor-corrector time integration method, which is based on an innovative, recently published shockcapturing scheme [24] and an iterative procedure for a Poisson-like equation. In the predictor stage, Equation (5) without the dynamic pressure term is discretized by a finite volume shock-capturing scheme, which is based on high-order reconstructions carried out by a so-called Wave Targeted Essentially Nonoscillatory (WTENO) scheme and an exact Riemann solver.…”

“…The final velocity field, which is divergence-free, is introduced in Equation ( 6) to update the free-surface elevation. A detailed description of the numerical model can be found in [24].…”

“…On this boundary, the pressure boundary condition and kinematic boundary condition at the free surface can be exactly and easily assigned. As demonstrated by several papers [20,21,24], this approach allows researchers to accurately simulate the wave motion by also using a few grid cells (less than 15) along the vertical direction.…”

“…In this paper, we propose a numerical study on the hydrodynamic efficiency of an OWC device under regular and irregular waves and its application to a real case. The numerical simulations were carried out by two-dimensional versions of a recently published three-dimensional numerical model [24], which is based on an integral form of the equations of motion expressed in contravariant formulations using a moving co-ordinate system that adapts to the variations of the free surface and, consequently, does not require large numbers of grid cells along the vertical direction. The above equations of motion are discretized by an innovative finite-volume-shock-capturing scheme and are numerically solved by a predictor-corrector method that allows us to take into account the nonhydrostatic pressure component.…”

Numerical Study on the Performance of an OWC under Breaking and Non-Breaking Waves

“…We propose a pneumatic model that overcomes the simplifying hypothesis of air incompressibility, and we evaluate the performances of an OWC in two different hydrodynamic conditions, one characterized by non-breaking waves and one by breaking waves. The proposed numerical model consists of two sub-models: (a) a hydrodynamic model in which the Navier-Stokes equations are numerically integrated by a recently proposed shockcapturing numerical scheme that is specifically designed to simulate breaking waves [17]; and (b) a pneumatic model based on the thermodynamic equations proposed in [16], which takes into account the effect of air compressibility on the pressure variations in the chamber.…”