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Twenty-First Symposium on NAVAL HYDRODYNAMICS
FREE SURFACE CONDITION
Density-function method
The density-function method is employed for the fulfilment of the kinematic free surface condition. The density-function plays the role of conserving mass and the determination of the interface location. Assuming that both viscous stresses and the surface tension on the free surface can be ignored, the dynamic free surface condition becomes the condition of pressure continuity between the two layers. Then the kinematic and dynamic free surface conditions are approximately implemented by the following equations.
(10)
= (11)
Here, M is the density-function, u,v,w are the velocity components in the x,y,z directions which represent the longitudinal, lateral and vertical coordinates, respectively, and and are the pressures of the water and air on the interface. For the time differencing in Eq.(10) the Adams-Bashforth method is used and the third-order MUSCL-type upwind scheme for the space differencing. Eqs.(1a) and (1b) are solved with constant value of density-for respective fluid and the location of interface is defined to be the surface on which the value of the density-function takes the value of , where and are the density-function values initially set for the water and air regions, respectively. Due to the inherent numerical diffusion in the solution of Eq.(10) the sharpness of interface can be lost as calculation proceeds. However, it is demonstrated in the subsequent section that the degree of accuracy can be raised to a sufficient level in case a higher-order differencing scheme such as third-order upwind scheme used here is employed in a fine grid system.
After determining the interface location the dynamic free surface condition is implemented by the so-called “irregular star” technique[16] in the solution process of the Poisson equation. The length of leg in the irregular star technique is calculated using the density-function. The pressure on the interface is determined by extrapolating the pressure in the air region, in which the pressure is extrapolated with zero gradient in the horizontal direction while in the vertical direction the pressure difference due to the gravitation is considered. Since the velocity must be continuous between the two layers, the extrapolation of velocity from the water region to the air region is conducted by approximately satisfying zero normal gradient to the interface.
Accuracy examination
As described in the previous section the numerical diffusion takes place in the computation of the free-surface location and this should be solved to keep sufficient degree of accuracy. For the numerical test, regular periodic waves of finite-amplitude based on the linear theory is generated by the finite-difference method with density-function method and the accuracy of the generated waves is compared with the records of physically generated waves by a flap-type wave generator[ 17].
For the space derivative terms of the Eq.(10), a variety of schemes are tested. Three of them are represented as follows only for the first term for simplicity.
1st-order upwind differencing
(12)
where,
(13)
Δ is the grid spacing in the x-direction and the subscript i is used for the x location.
2nd-order centered differencing+filtering
(14)
The filtering is used only in the vertical direction as follows.
Items in cart [0]
=
(11)
, where 
and 
are the density-function values initially set for the water and air regions, respectively. Due to the inherent numerical diffusion in the solution of Eq.(10) the sharpness of interface can be lost as calculation proceeds. However, it is demonstrated in the subsequent section that the degree of accuracy can be raised to a sufficient level in case a higher-order differencing scheme such as third-order upwind scheme used here is employed in a fine grid system.
