Numerical Simulation Of Water Wave Motions With Free Surface Based On Unstructured Grids

The numerical models based on unstructured grids for simulating two or three-dimensional water wave flows with free surface are presented in this paper.Firstly, based upon unstructured finite volume method, a high-resolution numerical model is developed for solving two dimensional shallow water equations. The Roe's approximate Riemann solver is used in the calculation of interface flux, In order to establist balance between the bed slope source terms and the gravity gradient terms, two methods to treat the bed slope source terms with considering different components of the bed slope term separately and using a modified water depth in the interfaces are proposed. It is confirmed by an algebraic manipulation that the proposed schemes are well balance. It also can be used with a large range of existing shock-capturing schemes such as the HLL, the HLLC, the CU methods, etc. The linear reconstruction technique and multidimensional limiter are employed to achieve high-order spatial accuracy and to prevent nonphysical oscillations. In order to accomplish high-order temporal accuracy, the third-order Runge-Kutta method is used to the time discretization. The friction terms are treated fully implicitly to prevent numerical instabilities. The limiter water depth method is used for the wetting and drying interface, which improved the useful of model. Numerical tests of propagation of a smooth save over a bump, steady flow over a bump, dam break and advance over a triangular obstacle, simulation of a tidal wave over an adverse not constant slop and dam break over three mounds are employed to test the model's accuracy, conservation and capability of capturing discontinuous solutions and treating moving boundaries. At last the model is used to simulate field-scale generation of tidal currents in Bohai Sea which verify the applicability, reliability of the model.Finally, the three-dimensional non-hydrostatic models for solving water wave flows with free surface are developed by using a finite-volume formulation that solves three-dimensional Navier-Stokes equations on an unstructured, staggered, z-lever grid, with the goal of simulating non-hydrostatic processes in the free-surface flows. A semi-implicit, fractical step scheme is used in order to obtain an efficient numerical algorithm whose stability is independent from the free-surface wave speed, wind stress, vertical viscosity and bottom friction. In the first step, the provisional water velocity and surface elevation are computed by neglecting the implicit contribution of the nonhydrostatic pressure. In the second fractional step, the provisional velocities are corrected by a corrected step of the non-hystatic pressure of which its pressure Poisson equation is obtained by employing a divergence-free velocity field. The surface elevation is abtained by the free-surface equation, which is satisfied kinematic condition at the free surface. The advection terms in the momentum equation are discretized explicitly with the Euler scheme while conserves momentum in cells that do not contain the free surface. Advection terms in k-εturbulence model is discretized in a way that ensures consistency with the three-dimensional and depth-averaged continuity euqations, thereby ensuring lacal and global mass consercation using a velocity field that conserves volume on a local and global basis. To minimize the number of vertical layers and subsequently the computational cost, an integral method of the vertical momentum equation at the top-layer is applied to account for the full effects of non-hydrostatic pressure at the free-surface layer. Numerical tests including small amplitude uni-nodal standing waves, linear 3D standing waves in close basin, periodic waves propagating over a submerged bar and wave transformation over an elliptic shoal on a sloped bottom are performed. It is found that the model is capable to effectively simulate wave shoaling, non-linearity, refraction, and diffraction phenomena using a very small number of vertical layers (e.g. two-five layers). Numerical tests of the unsteady and non-uniform open channel flow, the trench channel flow and the flow around a non-submerged spur-dike are employed to test the validity of the turbulence model. Comparison between numerical results and experimental data is carried out to test the model's capability of simulating complex three-dimensional flows.