Numerical study on run-up heights of solitary wave with hydrodynamic pressure model

For many shallow water flows, it is sufficient to consider the depth-averaged equations, referred as the shallow water equations, which are two-dimensional in the horizontal plane, since the length scale of the vertical direction is much smaller than that of the horizontal directions. Assuming that the pressure distribution is hydrostatic, the mathematical formulation and its numerical implementation are considerably simplified. In this study, a numerical model is newly developed to investigate various free surface fl ow problems. The governing equations are the Navier-Stokes equations with the pressure decomposed into the sum of a hydrostatic and a hydrodynamic components. The equation for the free surface movement is a depth-averaged continuity equation which is a free surface equation. These governing equations are simultaneously solved by using a finite difference method with a semi-implicit method and fractional step method. At the first step, the vertical momentum equations are discretized by using an implicit method over the vertical direction. In the second step, the discrete horizontal momentum equations are projected on to the free surface equation. Finally, the hydrodynamic pressure and final velocity field are calculated. To verify the accuracy and stability, the present numerical model is applied to move practical problems such as the run-up process of solitary waves attacking a circular island. The numerically obtained maximum run-up heights around a circular island are compared with available laboratory measurements. A very reasonable agreement is observed.