Interaction of flexural-gravity waves in ice cover with vertical walls

Diffraction of flexural-gravity waves in an ice cover by a bottom mounted structure with vertical walls is studied. The problem is solved by using the so-called vertical modes corresponding to the roots of the dispersion relation for flexural-gravity waves. These modes reduce the original three-dimensional problem to a set of two-dimensional diffraction problems with non-homogeneous boundary conditions on the rigid walls. Two unknown functions presenting in the boundary conditions for each mode are determined using the conditions at the contact line between the ice cover and the vertical walls. The clamped conditions at the contact line, where the ice cover is frozen to the wall, are considered in this study. The solution of the problem is obtained for a single vertical circular cylinder frozen in the ice cover. A general approach to the problem for vertical cylinders of any shapes is presented. The diffraction problems with vertical walls extended to infinity are discussed

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

Hydroelastic effects during the fast lifting of a disc from a water surface

Here we report the results of an experimental study where we measure the hydrodynamic force acting on a plate which is lifted from a water surface, suddenly starting to move upwards with an acceleration much larger than gravity. Our work focuses on the early stage of the plate motion, when the hydrodynamic suction forces due to the liquid inertia are the most relevant ones. Besides the force, we measure as well the acceleration at the centre of the plate and the time evolution of the wetted area. The results of this study show that, at very early stages, the hydrodynamic force can be estimated by a simple extension of the linear exit theory by Korobkin (J. Fluid Mech., vol. 737, 2013, pp. 368–386), which incorporates an added mass to the body dynamics. However, at longer times, the measured acceleration decays even though the applied external force continues to increase. Moreover, high-speed recordings of the disc displacement and the radius of the wetted area reveal that the latter does not change before the disc acceleration reaches its maximum value. We show in this paper that these phenomena are caused by the elastic deflection of the disc during the initial transient stage of water exit. We present a linearised model of water exit that accounts for the elastic behaviour of the lifted body. The results obtained with this new model agree fairly well with the experimental results

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

Water entry of an elastic conical shell

The axisymmetric problem of a conical shell impact onto an inviscid and incompressible liquid of infinite depth is studied. The shell is thin, and its deadrise angle is small. The problem is inertia dominated. Gravity, surface tension and viscous effects are not taken into account. The hydrodynamic loads acting on the shell and the shell displacements are determined at the same time. The model by Scolan (J. Sound Vib., vol. 277, issue 1–2, 2004, pp. 163–203) is used to find the flow and hydrodynamic pressure caused by the shell impact. This model is based on the Wagner theory of water impact, which was generalised to axisymmetric problems of hydroelastic slamming. Dry and wet modes of the conical shell, as well as the corresponding frequencies, are calculated. It is shown that a conical shell can be approximated by a circular plate only for a very small deadrise angle. Deflections and strains in the conical shell during the impact stage, when the wetted part of the shell increases at high rate, as well as the hydrodynamic loads, are determined and analysed

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

Blunt body impact onto viscoelastic floating ice plate with a soft layer on its upper surface

Two-dimensional vertical impact of a rigid blunt body onto a floating ice plate is studied. The problem is coupled and unsteady. The liquid is inviscid, incompressible, and of infinite depth. The ice floe is modeled as a thin viscoelastic plate of constant thickness. The plate edges are free of bending stresses and shear forces. The upper surface of the plate is covered with a viscoelastic layer of constant small thickness and negligible inertia. The reaction force of this soft layer is predicted by a nonlinear and one-dimensional Winkler-Kelvin-Voigt model, which does not permit a contact between the rigid body and the ice plate. The soft layer may describe either the presence of snow on the ice or a layer of crushed ice in the place of impact, or can be considered as a way of regularization of problems with concentrated loads. The rigid body touches the upper surface of the soft layer and then suddenly starts to move downward with constant velocity. It is shown that the strains in the ice plate caused by the impact are weakly dependent on the characteristics of the soft layer. The magnitudes and distributions of the strains are studied depending on the length of the ice plate, retardation time of the ice model, thickness of the plate, shape of the rigid body, place of impact, and the impact speed. The value of the retardation time in the soft layer model is discussed with relation to the ice crushing by impact