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.