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Figure 1: The (y,z) coordinate system, and the water entry of a body with flow separation on the right side and no separation on the left side.

tion. Each section has knuckles. The pressure and the total slamming forces are measured for the whole period of water entry. Systematic comparisons between the numerical and experimental results have been carried out. The three-dimensional flow effects in the drop tests have been theoretically estimated. The maximum estimated three-dimensional effect represents a 20% reduction of the two-dimensional results. This method of estimating three-dimensional flow effects can in a qualitative way be used to analyze an error source in strip theory predictions of bow flare slamming. It is shown that the common method of estimating bow flare impact forces based on fluid momentum conservation and infinite frequency added mass coefficients as a function of submergence relative to undisturbed free surface, gives too low maximum force and wrong time history of the force.

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FULLY NONLINEAR SOLUTION

A general two-dimensional asymmetric body with time dependent vertical downwards velocity, is forced through initially calm water. The problem is solved as an initial value problem. Since viscous effects are neglected, the problem can be solved by potential theory. The effects of compressibility of the water and air cushions between the water and the structure are neglected. Gravity is not included in the analysis. This has a negligible effect in the initial phase of water entry of a blunt body, but will be more important at a later stage after flow separation has occurred. Hydroelastic effects are disregarded. This can be significant when the slamming loads are large, like for wetdeck slamming(Faltinsen(1996), Kvålsvold et al.(1995)). Cavitation and ventilation can also occur in the latter case.

The reference coordinate system yz is fixed in space. The origin of the coordinate system is in the plane of the undisturbed water surface. The z-axis is positive upwards(see figure 1). The velocity potential satisfies the Laplace

Figure 2: The free-surface elevation and pressure(p) distribution on the body surface during water entry of a symmetric wedge with constant velocity V. Deadrise angle α=30°. pa is atmospheric pressure. The results are based on a similarity solution(Zhao and Faltinsen(1993)).

equation

(1)

in the fluid domain. The dynamic free-surface condition on the exact free surface can be written as

(2)

D/Dt means the substantial derivative and t is the time variable. The kinematic free-surface condition is that a fluid particle remains on the free surface. Hence the free surface can be found by convecting particles on the free surface with the local fluid velocity. The body boundary condition on the wetted body surface is satisfied on the instantaneous body surface. It can be written as

(3)

where Vn is the body velocity in the normal direction on the body surface. Positive direction of is into the fluid domain. The initial conditions are zero velocity potential and free-surface elevation.

The problem without flow separation will be studied first. A jet flow is created at the intersection between the free surface and the body surface. A similarity solution can be found for a symmetrical wedge with constant vertical impact velocity(Dobrovol'skaya(1969), Zhao and Faltinsen(1993)). Figure 2 shows the similarity solution for the free-surface elevation and pressure distribution during water entry of a symmetrical wedge with deadrise angle 30°. Figure 2 illustrates that the thin jet will not contribute much to the total force on the body. The part of the jet, where the pressure is close to atmospheric pressure, can therefore be neglected. This makes it unnecessary to find the intersection point between the body



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