A numerical method for free surface problem by using unsteady incompressible Navier-Stocks equations is proposed in present paper, based upon density-function method. With the present method, two free surface problems are considered. The unsteady incompressible viscous flows around a square cylinder in uniform stream and a two-dimensional submerged rectangular cylinder oscillating under a free surface are computed. The numerical solutions successfully captured for time-domain simulation of free surface, vortex generation, separation and evolution induced by oscillating body.


Several important phenomena occur free surface, such as wave breaking, flow separation and vortex shedding induced by oscillating body. Study of near-surface wave-body interactions is of significance in many ocean engineering and naval architecture applications. Determination of wave loads on offshore structures, prediction of motion response of ships on free surface, design of buoy systems for oil drill in sea are examples of such applications. In order to accurately predict the hydrodynamic forces on marine structure while afloat or fully submerged of free surface, a CFD method for numerical simulation of incompressible flow around oscillating body with free surface is presented in this paper. In this method, the air and water flows are simultaneously solved in the time-marching solution procedure for the Navier-Stokes equations. The marker-density function technique is applied for the treatment of free-surface condition, like Miyata et al (1988). The great advantage of the density-function method for free-surface movements is the ability to compute strongly interacting interface like wave breaking and the overturning, unlike the moving grid method which requires only average slopes to ensure metric calculation. In this paper, a new algorithm introduced by Rudman (1997) is used for transport equation of density which is based on the concept of flux-corrected transport. This scheme may have smaller errors than original VOF method in the advection term.

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