Oblique impact of an elongated three-dimensional body on a thin liquid layer

The problem of the impact of an elongated solid body with a blant bottom on a thin layer of an ideal incompressible liquid is considered in the case where the horizontal component of the body velocity is much greater than its vertical component. The initial stage of the impact, during which the contact area between the body and the liquid is rapidly expanding, is studied. The loads on the body are determined by strip theory. The method of matched asymptotic expansions is used to determine the position and size of the contact area in each section. The considered problem is coupled: the liquid flow due to the motion of the body and the body motion itself are determined simultaneously. A system of integrodifferential equations was derived and used for both numerical investigation of the body motion under the action of hydrodynamic loads and for determination of the hydrodynamic pressure distribution over the contact area.     

Asymptotic modeling of the impact of a spherical indenter on an elastic half-space

The impact of a rigid sphere on a homogeneous, isotropic elastic half-space in the absence of friction and adhesion is considered. The influence of the superseismic stage immediately following the moment of first contact upon the impact process is investigated in the frame of the Hertzian impact theory. The first order asymptotic approximation for the contact force in a three-dimensional dynamic contact problem with the slowly moving contact zone boundary is obtained and the corresponding asymptotic model of impact is developed. The motion of the indenter as it indents and rebounds from the elastic medium is analytically described. Explicit formulas are derived for the peak indentation depth, contact time, and rebound velocity as functions of the initial impact velocity, indenter mass, and characteristics of the elastic half-space.     

Wave front in the blunt body immersion problem

The immersion of a three-dimensional blunt convex body in a compressible fluid with nonpositive acceleration is considered in the linear formulation. It is shown that at every instant the perturbation zone will be convex. The fluid particle velocity and pressure are calculated at each point on the wave front. At every instant the wave front is wholly determined by the initial supersonic stage of propagation of the boundary of the body-fluid interaction zone.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 5–11, July–August, 1992.The author wishes to thank A. G. Khovanskii for his constant interest.     

Variational methods of solving dynamic problems for fluid-containing bodies

A variational approach to solving linear and nonlinear problems for a body with cavities partially filled with a perfect incompressible fluid is enunciated. The approach applies a nonclassical variational principle to describe the spatial motion of a finite fluid with a free surface and the classical variational principle, which is widely used in rigid body dynamics. These principles are used to formulate variational problems that are the basis of direct methods of solving nonlinear and linear dynamic problems for body-fluid systems. The approach allows us to derive an infinite system of nonlinear ordinary differential equations describing the joint motion of the rigid body and fluid and to develop an algorithm for determining the hydrodynamic coefficients. Linearized differential equations of motion of the mechanical system are presented and approximate methods are given to solve linear boundary-value problems and to determine the hydrodynamic coefficients.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 37–77, October 2004.The study was partially sponsored by the German Research Fund (der Deutsche Forschungsgemeinschaft), Grant 436 UKR113/33/0-3.     

Sliding a body on snow

Snow is considered as an ideal nonlinear elastoplastic medium. A body performs planeparallel motion on snow. The area of its contact with snow is a part of a rectangular plate. The contact zone changes during the motion of the body. Steady motions are found from the derived equations of motion in the case when the constant external forces and the moment exerted on the body are given. The inverse problem of determining the forces and moments is solved for a given steady motion of a vehicle.     

Methods of Solving Nonlinear Problems of Hydrodynamic Impact in Bounded Regions

The plane problem of separation impact on a plate floating on the surface of an ideal incompressible fluid in a bounded vessel is considered. In this problem the zone of contact between the body and the fluid is not known in advance and must be determined together with the fluid flow. As a result, the problem formulated is nonlinear and belongs to the class of free-boundary problems. The effect of vessel walls of different shapes on the fluid particle separation zone formed on the plate surface is studied. As examples, the problems for a layer or a truncated circular meniscus are considered.     

Separation impact of a circular disk floating on the surface of an ideal incompressible fluid of infinite depth

The three-dimensional mixed problem of the separation impact of a circular disk floating on the surface of an ideal incompressible unlimited fluid is considered. The position and shape of the contact area between the body and the fluid (and the separation zone) are not known and depend on the relation between the translational and angular velocities acquired by the disk upon impact. Because of this, the problem in question is nonlinear and belongs to the class of free-boundary problems. The problem is solved using the method of Hammerstein-type nonlinear boundary integral equations. This approach allows the fluid flow after impact and the unknown zone of separation of fluid particles to be determined simultaneously. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 76–86, July–August, 2009.     

Initial stage of the circular cylindermotion in a fluid after an impact with the formation of a cavity

The joint motion of an ideal fluid and a submerged circular cylinder is considered in the initial stage after an impact. The dynamics of separation points on the inner free boundary (cavity boundary) and the shapes of the inner and outer free boundaries of the fluid are determined. An asymptotic analysis of the inner free boundary near the separation points is made. The effects of the Froude number, the pressure difference, and the cylinder immersion depth are investigated.     

Vibrations of a body in a bounded volume of viscous fluid

The problem of the vibrations of a body in a bounded volume of viscous fluid has been studied on a number of occasions [1–4]. The main attention has been devoted to determining the hydrodynamic characteristics of elements in the form of rods. Analytic solution of the problem is possible only in the simplest cases [2]. In the present paper, in which large Reynolds numbers are considered, the asymptotic method of Vishik and Lyusternik [5] and Chernous' ko [6] is used to consider the general problem of translational vibrations of an axisymmetric body in an axisymmetric volume of fluid. Equations of motion of the body and expressions for the coefficients due to the viscosity of the fluid are obtained. It is shown that in the first approximation these coefficients differ only by a constant factor and are completely determined if the solution to the problem for an ideal fluid is known. Examples are given of the determination of the “viscous” added mass and the damping coefficient for some bodies and cavities. In the case of an ideal fluid, general estimates are obtained for the added mass and also for the influence of nonlinearity. Ritz's method is used to solve the problem of longitudinal vibrations of an ellipsoid of revolution in a circular cylinder. The hydrodynamic coefficients have been determined numerically on a computer. The theoretical results agree well with the results of experimental investigations.     

Numerical simulation of water entry of twin wedges

The hydrodynamic problem of twin wedges entering water vertically at constant speed is analysed based on the velocity potential theory. The gravity effect on the flow is ignored based on the assumption that the ratio of the entry speed to the acceleration due to gravity is much larger than the time scale of interest. The problem is solved using the complex velocity potential together with the boundary element method through three stages. When the body touches water, the similarity solution is obtained for each wedge in isolation. This is used as the initial solution at the second stage for the time stepping technique for each wedge in a stretched system defined through the ratio of the Cartesian system to the distance the wedge travelled into water. When the disturbed zone of each wedge begins to affect the flow generated by the other wedge, the stretched system is abandoned and the original system is used. At the third stage the full interactions between the two wedges are included. Various results are provided for the wave elevation, pressure distribution and force at different deadrise angles. They are compared with those obtained from a single wedge and the interaction effect is investigated.     

Transient motion of a floating body having a rectangular shape

The twodimensional transient problem of a floating body having a rectangular shape in a fluid layer of finite depth is considered. Vertical displacements of the body are specified. The problem is studied within the framework of the linear theory of potential ideal incompressible flow. The fluid flow equations reduce to an infinite system of Volterra integral equations of the second kind by the method of decomposition of the flow region. The system obtained is studied and solved numerically by the reduction method. A method of solving the problem for the flow velocity potential is proposed. The distribution of the hydrodynamic pressure and force acting on the body is determined.     

Prediction of the Compressible Stage Slamming Force on Rigid and Elastic Systems Impacting on the Water Surface

In this paper some models focusing on hydrodynamic and elasticforces arising during the impact of rigid and elastic systems on thewater surface are investigated. In particular, the supersoniccompressible stage of the impact is considered by modelling the slammingphenomenon through the Skalk–Feit acoustic approximation. The dynamicequations of the dropping system are coupled to those of the fluid and anonlinear fluid-solid interaction problem is stated. Generalrelationships between the body's shape, slamming force and body motionare determined. These equations are applied to the wedge water entrycases, and a closed-form expression for the maximum hydrodynamic forceis found. Moreover the theoretical correlation between the hydrodynamicforce and the body geometry allows us to control the inverse problem andthe shape associated to a constant slamming force is determined.Due to some simplifications allowed in the supersonic compressibleimpact, the results of the hydrodynamic analysis hold in closed form.This permits us to focus on the basic result of the paper addressed to asystematic correlation between hydrodynamic and elastic maximum forcesin terms of some characteristic dimensionless quantities involved influid-solid interaction.In particular, critical conditionscorresponding to those hydorelastic parameters combinations areinvestigated, leading to severe elastic response of the impactingsystem.     

Numerical Simulation of the Fall of a Body with a Corrugated Bottom on Water

A numerical model is proposed for the potential flow of an ideal incompressible fluid produced by impact of a body with concave bottom on water. Compression of the entrapped air is taken into account. The algorithm is based on joint solution of the equations of motion for the body and the fluid by the finite difference method with approximation in time. At each time, the boundaryvalue problem for the Laplace equation is solved by the boundaryelement method. Calculation results are given. The effects of the air layer, dimensions and shape of the corrugations, initial velocity, and other parameters on the impact process are shown.     

Oblique water entry of a wedge into waves with gravity effect

The hydrodynamic problem of a two dimensional wedge entering waves with gravity effect is analysed based on the incompressible velocity potential theory. The problem is solved through the boundary element method in the time domain. The stretched coordinate system in the spatial domain, which is based on the ratio of the Cartesian system in the physic space to the vertical distance the wedge has travelled into the water, is adopted based on the consideration that the decay of the effect of the impact away from the body is proportional to this ratio. The solution is sought for the total potential which includes both the incident and disturbed potentials, and decays towards the incident potential away from the body. A separate treatment at initial stage is used, in which the solution for the disturbed potential is sought to avoid the very large incident potential amplified by dividing the small travelled vertical distance of the wedge. The auxiliary function method is used to calculate the pressure on the body surface. Detailed results through the free surface elevation and the pressure distribution are provided to show the effect of the gravity and the wave, and their physical implications are discussed.     

Two-dimensional problem of the impact of a vertical wall on a layer of a partially aerated liquid

The two-dimensional unsteady problem of the impact of a vertical wall on a layer of a liquid which is mixed with air near the wall and does not contain air bubbles away from the wall is solved in a linear approximation. The gas-liquid mixture is modeled by a homogeneous, ideal, and weakly compressible medium with a reduced sound velocity dependent on the air concentration in the gas-liquid mixture. Outside the gas-liquid layer, the liquid is considered ideal and incompressible. During the initial stage of the impact, the liquid flow and the hydrodynamic pressure are determined using the linear theory of the potential motion of an inhomogeneous liquid. The dependence of the amplitude of the impact pressure along the wall on the air concentration in the gas-liquid layer and on the thickness of this layer is investigated. For a small relative thickness of the layer, the thin-layer approximation is used. It is shown that the solution of the original problem tends to the approximate solution as the thickness of the layer decreases. It is shown that the presence of the gas-liquid layer leads to wall pressure oscillations. Estimates are obtained for the pressure amplitude and the oscillation period. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 34–46, September–October, 2006.     

On the impulsive motion of a flat plate with suction

Summary The impulsive motion of a flat plate through a viscous incompressible fluid with time-dependent suction applied normally at the plate is considered. An approximate solution is obtained which describes the motion of the fluid during the initial stage of the motion of the plate, and also one which describes the motion at later stages when the suction velocity is large. The particular case when the suction velocity depends linearly on the time is considered in detail.     

Wave impact on the center of an Euler beam

The problem of a symmetric wave impact on the Euler beam is solved by the normal modes method. The liquid is supposed to be ideal and incompressible. The initial stage of impact when hydrodynamic loads are very high and the beam is wetted only partially is considered. The flow of a liquid and the size of the wetted part of the body are determined by the Wagner approach with a simultaneous calculation of the beam deflection. The specific features of the developed numerical algorithm are demonstrated and the criterion of its stability is specified. In addition to a direct solution of the problem, two approximate approaches within the framework of which the dimension of the contact region is found ignoring the deformations of the plate are considered. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhnaika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 134–147, September–October, 1998.     

Coupled hydrodynamic and structural analysis of compressible jet impact onto elastic panels

This paper is concerned with the initial stage of a compressible liquid jet impact onto an elastic plate. The fluid flow is governed by the linear wave equation, while the response of the plate is governed by the classical linear dynamical plate equation. The coupling between the fluid flow and the plate deflection is taken into account through the dynamic and kinematic conditions imposed on the wetted part of the plate. The problem is solved numerically by the normal mode method. The principal coordinates of the hydrodynamic pressure and plate deflections satisfy a system of ordinary differential and integral equations. A time stepping method based on the Runge–Kutta scheme is used for the numerical integration of the system. Calculations are performed for two-dimensional, axisymmetric and three-dimensional jet impacts onto an elastic plate. The effects of the impact conditions and the elastic properties of the plate on the magnitudes of the elastic deflections and bending stresses are analysed.     

Experimental study of disk impact onto shallow water

A problem of the impact of circular disks with a flat, convex, or concave onto a finite-depth fluid layer is experimentally studied. The influence of the small curvature of the lower surface of the body on the added mass, characteristic time of the impact on the free surface, parameters of the air cavity trapped during the impact, and type of the body impact on the bottom of the hydrodynamic test tank is examined     

Motion of a slender body in a fluid under a floating plate

This paper studies the three-dimensional unsteady problem of the hydroelastic behavior of a floating infinite plate under the impact of waves generated by horizontal rectilinear motion of a slender solid in a fluid of infinite depth. An analytic solution of the problem is found based on the known solutions for the unsteady motion of a point source of mass in a fluid of infinite depth under a floating plate. Asymptotic formulas are obtained which model the motion of a solid slender body in a fluid by replacing the body with a source-sink system. These formulas are used to numerically analyze the effect of plate thickness, depth of the body, its dimensions and the velocity of rectilinear motion on the amplitude of deflection of the floating plate. The motion of a submarine under a nonbreakable plate was modeled experimentally. Theoretical and experimental data are in good agreement.