## Separation of the free surface from a moving body during early stage

The two-dimensional problems of the liquid flows with separation of the free surface, which are generated by sudden motion of a rigid body or given loads, are considered. Initially the liquid is at rest and the body is in contact with the liquid. The body starts to move either with a non-zero velocity or non-zero acceleration. The generated flow is assumed inviscid and potential. Gravity is included into the model but the surface tension effects are neglected. We need to find the free-surface shape, the flow and pressure distribution during the early stage of the body
motion.
Four configurations of unsteady problems are studied: (a) impulsive vertical motion of a floating plate; (b) impulsive rotation of a floating plate; (c) impulsive horizontal motion of a half-submerged circular cylinder; (d) motion of a vertical wall with constant acceleration. Small-time asymptotic solutions are derived by the method of matched asymptotic expansions. Second-order outer solutions are obtained analytically and matched with the
leading-order inner solutions close to the separation points. The motions of the separation points in problems (b) - (d) are determined by using the condition that the displacement of the free surface is bounded during the early stage. The inner flows close to the separation points for all configurations are self-similar and nonlinear in the leading order with unknown in advance shape of the free surface.
The asymptotic solution for the problem (a) was compared with both experimental and numerical results in terms of the free surface shape and the hydrodynamic loads. It was shown that the asymptotic solution can be used even for moderate penetration depths. In the problem (b), the initial position of the separation point was
determined by using the Sedov theory of water impact with separation. It was shown that after impact the wetted area starts to grow at infinite velocity. The hydrodynamic pressure is of order of $O(t^{-\frac{1}{3}})$ inside the wetted part of the plate. The inner solution at the separation point predicts the jet flow.  The pressure in the contact region for the configuration (c) was found to be below atmospheric pressure if the Froude number is higher than 1.27. The configuration (d) corresponds to the coupled problem of dam-break flow. The motion of the vertical dam wall was determined together with the initial flow and the initial position of the separation point. The obtained solution approaches the classical dam-break solution for small dam mass. The position of the separation point was obtained as a function of the initial acceleration of the dam.
Static nonlinear problem of floating ice sheet with unknown positions of the contact points between the ice and the water surface was studied. The ice deflection was due to given external loading acting on the upper surface of the ice sheet. Bending stresses in the ice sheet with and without account for separation were calculated to demonstrate the effect of the free surface separation on the stresses in ice.

The work was supported by The Royal Society of London, FP7 project TULCS and Federal Programme 2011-1.5-503-002-027.

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