Free Surface Flow Simulation in Time-Dependent Curvilinear Coordinates

Wave motion simulation can be obtained by numerically integrating the three-dimensional Navier-Stokes equations. Some of the most recent models based on this approach use a coordinate transformation in the vertical direction, named sigma coordinate transformation, by which the Cartesian vertical coordinate is expressed as a function of a moving vertical coordinate which adjusts to the free surface motions. This coordinate transformation does not concern the horizontal coordinates, consequently does not allow to accurately represent the coastal regions complex geometries.

In order to overcome the limits imposed by Cartesian grids, the three-dimensional numerical simulation of the fluid motion on domains characterised by complex geometries can be carried out by using boundary conforming curvilinear coordinate systems and expressing the governing equations in contravariant formulation.

The differential contravariant formulation of the Navier-Stokes equations includes the covariant derivatives of contravariant vectors which imply the presence of the Christoffel symbols, that prevents the convective terms of the motion equations from being expressed in conservative form. It is known that the numerical methods for the solution of conservation laws in which the convective terms are expressed in non-conservative form do not guarantee the convergence to the weak solution, i.e. the solution that may contain discontinuities. In order to obtain a numerical model which is able to converge to the weak solution, it is necessary to express the convective terms of the differential motion equations in conservative form or express the motion equations directly in integral form.

In order to realise a three dimensional numerical model 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, we propose an integral contravariant form of the Navier-Stokes equations, devoid of the Christoffel symbols, in a time dependent curvilinear coordinate system. The resulting equations represents the general integral contravariant formulation of the momentum equation in a time dependent curvilinear coordinate system. The motion equations are numerically solved in order to realize a three dimensional non-hydrostatic shock capturing numerical model which is able to simulate wave propagation and the nonlinear hydrodynamic wave phenomena related to it.

An appropriate turbulence model is developed to estimate the eddy turbulent coefficient as a function of the kinetic energy of turbulence and the turbulent dissipation rate. The dynamic procedure presented in Derakhti et al.(2016) is introduced in the model in order to determine the destruction of turbulent dissipation rate.