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

Abstract

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.

Boundary Conditions for the Simulation of Wave Breaking

“…In this paper we adopt the governing equations proposed in [9][10] in which the Navier-Stokes equations are expressed in integral contravariant form in a time-dependent curvilinear coordinate system.…”

“…Equations (1) and (2) represent the general integral form of the Navier-Stokes equations expressed in a time dependent curvilinear coordinate system. The complete derivation of these equations can be found in [10]. In [9] it has been demonstrated that, by taking the limit as the volume approaches zero, the integral Equations (1) and (2) are reduced to the complete differential form of the contravariant Navier-Stokes equations in a time dependent curvilinear coordinate system that have been proposed in the literature by Luo and Bewley [12].…”

“…In this paper, in order to simulate the fully dispersive wave processes and the wave breaking, we start from the model proposed in [9][10] and obtain the following governing equations…”

“…In [9][10], the numerical scheme is based on a fractional-step method in which the sequence of steps to update the numerical solution consists in the calculation of a predictor velocity field, followed by a corrector step (in which a Poisson-like equation is numerically solved) and, finally, by the updating of the free-surface elevation. Furthermore, in [9][10] the finite difference numerical approximation of the differential terms in the Poisson-like equation are expressed in conservative form.…”

“…In the proposed numerical model, the integral contravariant form of the continuity and momentum equations are discretized by a finite-volume shockcapturing that uses an HLL approximate Riemann solver [11] and are updated by a fractional-step method that is different from the one presented in [9][10]. In the proposed fractional step method, the calculation of the predictor velocity field is followed by updating the free surface elevation and, finally, by a corrector step based on the numerical solution of a Poisson-like equation.…”