Numerical Modeling Of Nearshore Waves And Wave-induced Currents Based On Hyperbolic Mild-Slope Equation In Curvilinear Coordinates

The nearshore waves and current induced by breaking waves are key hydrodynamic factors to coastal waters; understanding their rules of movement is the basic way to understand issues of the design of coastal projects, sediment transport, the deformation of coastlines and the diffusion of pollutants et al. However, the topography and coastline in the coastal regions are always complicated; therefore the conventional numerical modeling in rectangular grid could easily lead to the state that the boundary of computational domain does not match with the actual border, and thus reduce the accuracy of numerical results. While in practical engineering, engineers are usually more concerned about the distribution of waves and wave-induced current near irregular coastlines, such as, in the vicinity of estuaries, islands, piers and breakwaters.It is the object of this paper to establish a numerical model for coastal regions with complex terrain and variable boundaries, and to realize a more accurate hydrodynamic simulation of waves and current in these areas. Three reasons make the curvilinear coordinates transformation to be an outstanding mathematical method in dealing with complex boundary in fluid dynamics. They are being that firstly boundary-fitted grid can achieve a seamless fit to the curve border; secondly, the size of the grids can be adjusted according to changes in topography; and lastly coordinate transformation will not change the numerical method, in most cases. Based on this understanding, the present paper shows numerical models of the hyperbolic mild-slope equation and shallow water equation in orthogonal curvilinear coordinates.First of all, a comprehensive review of the development of nearshore wave modeling using the mild-slope equation was given. To facilitate numerical calculation, two different hyperbolic approximations of mild-slope equation, which are capable to describe wave transformation in large areas, were chosen to study under orthogonal curvilinear coordinates. The established wave models were discreted with a space-staggered grid and solved by the efficient ADI method (Alternating Direction Implicit).Then, wave models were applied to several physical experiments; the numerical results were compared to experimental data, analytical results, and numerical results from Cartesian coordinates, available. The numerical models were validated by their applicability to the complicated terrain and boundaries, to the distortion of orthogonal grids, and to the practical engineering application. Appropriate analysis was also given on the numerical accuracy and efficiency of the two kinds of hyperbolic mild-slope equations.Thirdly, the two-dimensional shallow water equations were transformed in orthogonal curvilinear coordinates based on the concept of the radiation stress theory. The numerical solving method of the transform current model was the same as wave model, i.e., the continuity equation being calculated by the implicit scheme and the momentum equations by alternating explicit scheme.Last but not the least, the regular wave-induced longshore currents and rip currents were calculated through the transformed current model. Here, the established hyperbolic mild-slope wave model was employed as a driving force to flow field. The good agreement of numerical results with experimental data and with published numerical results showed that the numerical model in this paper is applicable to simulate wave-induced current in areas with complicated topography and boundary conditions.