The ice response to an oscillating load moving along a frozen channel

Unsteady response of an ice cover to an oscillating load moving along a frozen rectangular channel is studied for large times. The channel is filled with ideal incompressible fluid. The ice cover is modelled by a thin elastic plate. The flow caused by the deflection of the ice cover is potential. The problem is formulated within the linear theory of hydroelasticity. External load is modelled by a smooth localized pressure distribution. The load has periodic magnitude and moves along the channel with constant speed. Joint system of equations for the ice plate and the flow potential is closed by initial and boundary conditions: the ice plate is frozen to the walls of the channel, the flow velocity potential satisfies the impermeability condition at the rigid walls of the channel and linearized kinematic and dynamic conditions at the ice-liquid interface; at the initial time the load is stationary, the fluid in the channel is at rest and the stationary ice deflection is determined from the plate equation for the initial magnitude of the load. The problem is solved with the help of the Fourier transform along the channel. The ice deflection profile across the channel is sought in the form of the series of the eigenmodes of the ice cover oscillations in a channel. The solution of the problem is obtained in quadratures and consists of three parts: (1) symmetric with respect to the load deflection corresponding to the stationary load; (2) deflection corresponding to steady waves propagating at the load speed; (3) deflection corresponding to waves propagating from the load and caused by the oscillations of the load. The number of the last waves, depending on the parameters, can not exceed four for each eigenmode. In this article the results of the analytical and numerical analysis of the considered problem is presented

Oblique elastic plate impact on thin liquid layer

The present study is concerned with possible mechanisms of air entrainment in a thin liquid layer caused by oblique impact of a deformable body on the layer. The two-dimensional unsteady problem of oblique elastic plate impact is considered within the thin-layer approximation for the first time. The plate deflection is described by the Euler beam equation. The plate edges are free of stresses and shear forces. The plate deflections are comparable with the liquid layer thickness. It is revealed in this paper that, for a stiff plate, the initial impact by the trailing edge makes the plate rotate with the leading plate edge entering water before the wetted part of the plate arrives at this edge. The air cavity trapped in such cases can be as long as 40% of the plate length. For a flexible plate, the impact does not cause the plate rotation. However, the dry part of the plate in front of the advancing wetted region is deflected toward the liquid layer also trapping the air. The numerical results are presented for elastic and rigid motions of the plate, hydrodynamic pressure in the wetted part of the plate, position of this wetted part, and the flow beneath the plate

Splashing of liquid droplet on a vibrating substrate

The unsteady axisymmetric problem of a liquid drop impacting onto a rigid vibrating substrate is studied. Initially, the drop is spherical and touches the flat substrate at a single point. Then, the substrate starts to move toward the drop and vibrate with a small amplitude and high frequency. The early stage of the impact is studied by using the potential flow theory and the Wagner approach in dimensionless variables. The effect of the substrate vibration on the drop impact is described by a single parameter. It is shown that the vibration of the substrate leads to oscillations of the pressure in the contact region. The low-pressure zone periodically appears in the wetted part of the substrate. The low-pressure zone can approach the contact line, which may lead to ventilation with separation of the liquid from the substrate. The magnitude of the low pressure grows in time. The acceleration of the contact line oscillates with time, leading to splashing of the droplet with the local increase of the thickness of the spray jet sheet at a distance from the contact line. The phase shift of the substrate vibration with respect to the impact instant is not studied. Splashing can be produced only by a forced vibration of the substrate. The impact onto an elastically supported rigid plate does not produce splashing. The obtained results and the theoretical model of the initial stage of drop impact are valid for certain ranges of parameters of the problem

Free-surface flow behind elastic plate impacting on a thin liquid layer

The problem of an inclined impact by an elastic plate on a thin liquid layer is considered. Evolution of the flow behind the plate is studied. The main input parameters are the position of the separation point of the liquid from the plate and the speed of the liquid under this point. The flow in the wake behind the plate is described by shallow water equations without gravity. Analytical formulae for the shape of the free surface behind the plate are derived. The study is focused on the possibility of the formation of jets arising from the wake perpendicular to the liquid layer. The problem is solved in two stages: before and after the formation of the first such a jet. The effects of the flow speed at the beginning of the wake, its time derivative, and the law of motion of the separation point on the formation of jets are investigated. The positions of the jets, their speeds and shapes are determined. Using the obtained results, mechanisms of the thin layer aeration behind the plate are discussed

Deflection of ice cover caused by an underwater body moving in channel

Deflections and strains in an ice cover of a frozen channel caused by an underwater body moving under the ice with a constant speed along the channel are studied. The channel is of rectangular cross section, the fluid in the channel is inviscid and incompressible. The ice cover is clamped to the channel walls. The ice cover is modeled by a thin viscoelastic plate. The underwater body is modeled by a three-dimensional dipole. The intensity of the dipole is related to the speed and size of the underwater body. The problem is considered within the linear theory of hydroelasticity. For small deflections of the ice cover the velocity potential of the dipole in the channel is obtained by the method of images in leading order without account for the deflection of the ice cover. The problem of moving dipole in the channel with rigid walls provides the hydrodynamic pressure on the upper boundary of the channel, which corresponds to the ice cover. This pressure distribution does not depend on the deflection of the ice cover in the leading approximation. The deflections of the ice and strains in the ice plate are independent of time in the coordinate system moving together with the dipole. The problem is solved numerically using the Fourier transform, method of the normal modes and the truncation method for infinite systems of algebraic equations

Generalised Wagner model of water impact by numerical conformal mapping

A numerical method to solve the problem of symmetric rigid contour entering water vertically at a given time-dependent speed is presented. The method is based upon the so-called generalised Wagner model. Within this model the body boundary condition is imposed on the actual position of the entering surface, the free-surface boundary conditions are linearised and imposed on the pile-up height, which is determined as part of the solution. The hydrodynamic pressure is given by the non-linear Bernoulli equation. The hydrodynamic pressures which are below the atmospheric value are disregarded. The conformal mapping of the flow region onto the lower half-plane is used. The velocity potential of the flow is given in analytical form once this mapping is known. The conformal mapping is calculated numerically. The obtained results are validated with respect to the known solutions for wedge and circular cylinder. The novelty and practical importance of the present approach are due to a special accurate treatment of the flows and the pressures close to the contact points between the entering body and water free surface. This special treatment is required for reliable prediction of the hydrodynamic pressure along the wetted part of the contour during its impact onto the water surface and the subsequent entry