Initial stage of the inclined impact of a smooth body on a thin fluid layer

The two-dimensional problem of the oblique impact of a free rigid body with a smooth flat bottom on a thin layer of an ideal incompressible fluid is considered. The initial stage of the impact when the body motion is accompanied by the formation of jets on the boundary of the body-fluid contact zone is investigated. The problem is solved jointly, i.e., the fluid flow initiated by the body motion and the motion of the body itself are determined simultaneously. A priori the “body-fluid” contact zone is unknown and determination of its time evolution represents a significant difficulty and the method of asymptotic matched expansions is used to overcome this difficulty. A system of integro-differential equations is obtained and the motion of the body under the action of hydrodynamic loads is investigated numerically on the basis of this system. It is shown that the hydrodynamic force exerted on the body during the impact is maximum precisely in the initial stage; therefore, the motion of the body varies fairly significantly in time considered.     

Falling of a body with a flat elastic bottom on a thin liquid layer at a small angle

Nonlinear shallow water equations and the method of matched asymptotic expansions are used to solve the problem of the impact of a box-type body with a flat bottom on a thin elastic liquid layer at a small angle in the plane formulation. It is established that, at certain values of the input parameters of the problem, the liquid pressure near the body edges becomes less than atmospheric pressure, and the liquid separates from the bottom of the box. Calculations demonstrating the influence of elastic bottom and liquid separation on the body motion are performed. It is shown that the presence of an elastic bottom significantly changes the hydrodynamic pressure distribution and can cause loads higher than in the case of a rigid body.     

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.     

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.     

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.     

Plane problem of asymmetrical wave impact on an elastic plate

The problem of wave impact on the edge of an elastic horizontal plate is studied within the framework of the Wagner approach using the normal-modes method. The plate is governed by the Euler beam equation with simply supported ends. The liquid is assumed to be ideal and incompressible. The problem is coupled: the elastic and hydrodynamic characteristics of the impact process and the dimension of the contact region should be found simulatenously. An algorithm that permits a detailed study of the impact on an elastic plate is proposed. The phenomenon of unlimited increase of hydrodynamic loads owing to the plate flexibility (blockage) is revealed for fairly long plates. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirisk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 148–158, September–October, 1998.     

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.     

Elastic wedge impact onto a liquid surface: Wagner's solution and approximate models

The problem of elastic wedge impact onto the free surface of an ideal incompressible liquid of infinite depth is considered. The liquid flow is two-dimensional, symmetric and potential. The side walls of the wedge are modelled as Euler beams, which are either simply supported or connected to the main structure by linear springs. The liquid flow, the deflection of wedge walls and the size of wetted region are determined simultaneously within the Wagner theory of water impact. We are concerned with the impact conditions of strong coupling between the hydrodynamic loads and the structural response. The coupling is well pronounced for elastic wedges with small deadrise angles. This is the case when the fully nonlinear models fail and approximate models based on the Wagner approach are used. In contrast to the existing approximate models, we do not use any further simplifications within the Wagner theory. Calculations of the velocity potential are reduced to analytical evaluation of the added-mass matrix. Hydrodynamic pressures are not evaluated in the present analysis. In order to estimate the maximum bending stresses, both stages when the wedge surface is partially and totally wetted are considered.Three approximate models of water impact, which are frequently used in practical computations, are examined and their predictions are tested against the present numerical solution obtained by the normal mode method within the Wagner theory. It is shown that the decoupled model of elastic wedge impact, which does not account for the beam inertia, provides a useful formula for estimating the maximum bending stress in thick wedge platings.     

Influence of the atmosphere on the magnitude of the hydrodynamic forces in the case of a disk in a flat encounter with the surface of a compressible liquid

If the velocities with which bodies enter liquids are small, and the bodies are not too blunt, the magnitudes of the hydrodynamic forces can be satisfactorily determined in the framework of the approximation of an incompressible liquid and depend on the density of the liquid, the velocity of entry, and the geometrical parameters (shape of the body, angles of entrance and attack). If the velocity is increased or the encounter with the surface becomes nearly flat, the compressibility of the liquid and the presence of an atmosphere begin to influence the hydrodynamic forces significantly. The influence of the compressibility on the magnitude of the impact loads has been investigated theoretically and experimentally [1–8]. The influence of the atmosphere in the case of an incompressible liquid has also been taken into account [9–11]. In the case of a flat encounter the two factors (compressibility of the liquid and presence of the atmosphere) simultaneously influence the development of the impact process. The present paper reports experimental results and computer calculations of the impact loads in the case of a flat encounter of a disk and the surface of a compressible liquid in atmospheres of helium, air, and freon.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 15–20, May–June, 1984.     

Numerical simulation of plate autorotation in a viscous fluid flow

A general formulation of the plane coupled dynamical and aerodynamical problem of the motion of a rigid body with a rotational degree of freedom in a viscous incompressible fluid flow is given. A computation technique for solving the Navier-Stokes equations based on the meshless viscous vortex domain method is used. The autorotation of a single plate and a pair of plates is investigated. The effect of the reduced moment of inertia and the Reynolds number on the angular rotation velocity is determined. The time dependences of the hydrodynamic loads are compared with the corresponding instantaneous flow patterns. The increased the autorotation velocity of two plates in tandem is detected.     

A Pressure Field in an Acoustic Medium Containing a Cylindrical Solid and a Sphere Oscillating in a Predetermined Manner

A problem on the interaction of a spherical body oscillating in a predetermined fashion and a rigid cylinder is formulated. The bodies do not intersect, are immersed into an ideal compressible liquid, and their centers are in one plane. The solution is based on the possibility of representing the partial solution of the Helmholtz equation, written in cylindrical coordinates, in terms of partial solutions in spherical coordinates, and vice versa. An infinite system of linear algebraic equations is obtained by satisfying the boundary conditions on the sphere and cylinder surfaces. The system is intended for determining the coefficients of the expansion of the velocity potential into a series in terms of spherical and trigonometric functions. The system obtained is solved by the reduction method. The appropriateness of this method is substantiated. The hydrodynamic characteristics of the liquid surrounding the spherical and cylindrical bodies are determined. A comparison is made with the problem on a sphere oscillating in an infinite incompressible liquid that contains also a cylinder and in a compressible liquid that contains nothing more. Two types of motion of the sphere — pulsation and oscillation — are considered     

Unsteady nonlinear problem of the horizontal motion of a contour under the interface between two liquids

A method of solving the initial boundary-value problem of the horizontal motion of a circular cylinder under the interface between two liquids is developed within the framework of nonlinear theory and implemented numerically. Profiles of generated waves and hydrodynamic loads are calculated for the problem of the acceleration of a circular cylinder under the free surface of a heavy liquid. The phenomenon of wave breaking is considered in detail. Omsk Department, Sobolev Institute of Mathematics, Siberian Division, Russian Academy of Sciences, Omsk 644099. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 37–43, May–June, 1999.     

Fluid–structure interaction during the impact of a cylindrical shell on a thin layer of water

A two-dimensional unsteady analysis of an elastic circular cylindrical shell that enters a thin layer of an ideal incompressible liquid is considered. The cylinder initially touches the liquid free surface at a single point and then penetrates the liquid layer at a constant vertical velocity. The problem is coupled because the liquid flow, the shape of the elastic shell and the geometry of the contact region between the body and the liquid must be determined simultaneously. The flow region is subdivided into four complementary regions that exhibit different properties: the region beneath the entering body surface, the jet root, the spray jet, and the outer region. A complete solution is obtained by matching the solutions within these four subdomains. The structural analysis is based on the normal-mode method. Strain-time histories of the inner surface of the cylinder are of particular interest. In the case of a very flexible shell three distinct regimes of the impact process were found. For a high impact velocity the lower part of the shell flattens and the shell does not enter the water. For a moderate impact velocity the shell reaches the bottom and an effect of “fluid capture” may occur. For a low impact velocity the shell penetrates the liquid, but the size of the contact region decreases before the shell reaches the bottom. This behaviour corresponds to exit or “reflection” of the shell from the water layer.     

Braking of a Body by a Soft Inflatable Shell on Impact on a Surface

The results of mathematical simulation of a solid velocity damping by a soft skeleton fabric shell filled with air on impact on a hard surface are given. The equations of motion of a falling body and of the loading dynamics of membrane shells and the reinforcement rings in the fabric shell are considered together. Themathematical model and the numerical algorithm for solving the spatial problem of the dynamics of inflation of a shell with reinforcement rings are explicitly realized by the finite difference method. The boundary conditions are posed with regard to the contact of the shell elements in compression near the ring belts. The results of numerical experiments considering the interaction of the falling body with the deformable skeleton shell are discussed. The parameters influencing the process of the body braking on impact on a surface are determined.     

Dynamics of a rigid cylinder near a plane boundary in the radiation field of an acoustic wave

An approach is described for investigation of the interaction between a rigid body and a viscous fluid boundary under acoustic wave propagation. The influence of the liquid on the rigid body is determined as a mean force, which is a constant in the time component of the hydrodynamic force. This enables the use of a previously developed technique for calculation of pressure in a compressible viscous liquid. The technique takes into account the second-order terms with respect to the wave field parameters and is based on investigation of a system of initially nonlinear hydromechanics equations that can be simplified with respect to the wave motion parameters of the liquid. It has proven possible to retain the second-order terms for determination of stresses in the liquid without having to solve the system of nonlinear equations. The stresses can be expressed in terms of parameters found in the solution of the linearized equations of the compressible viscous liquid. In this way, the solution of linearized equations is expressed in terms of a scalar and vector potentials. The problem statement is derived for a rigid cylinder located near a rigid flat wall under the effects of a wave propagating perpendicular to the wall. The solution for this particular example is obtained.     

Impact of a body with a plane bottom on a thin liquid layer at a small angle

The problem of the impact of a body with a plane bottom (of the type of a box) on a thin liquid layer at a small angle is solved in the two-dimensional formulation. The nonlinear shallow water equations are used, together with the method of matched asymptotic expansions. It is found that at certain values of the input parameters of the problem the liquid pressure diminishes near the lower end of the body and becomes smaller than the atmospheric pressure, which results in liquid separation from the box bottom. The numerical results show that all input parameters of the problem have a considerable effect on the nature of body motion. The liquid separation effect on body motion is analyzed.     

HYDRODYNAMIC RESISTANCE EFFECT OF FLUID LAYER BETWEEN TWO IMMERSED APPROACHING PARTICLES

1. Introduction The mechanisms of impact and rebound of solid parti- cles in particulate flow systems are of interest over a wide range of application areas such as fluidized beds, pneu- matic transport, filtration processes, erosion and pollution control of suspended particles. In many cases, the colli- sions of particles against themselves and against walls may affect the properties of the mixture. Efforts have been made to describe the fundamental mechanics of particle collisions. The conta…     

A contact force model in the dynamic analysis of contactable cracks

In this paper a finite element method in the time domain is proposed to analyze the dynamic response of contactable cracks. By this method, the contact normal stress caused by impact or contact of the crack faces can be directly obtained and then the motion of the model containing the contactable cracks can be calculated, and the processes of contact, slip and separation of the crack faces can also be correctly determined and simulated. The method proposed can be used to study the scattering problems of an elastic wave caused by the crack and the dynamic extension problems of the crack arising from impact or cyclic loading. Numerical examples indicate that the method is efficient and accurate enough. It is also shown that the effect of dynamic contact of crack faces on the motion of itself and the area near the crack is significant by comparing the results with the contact considered and neglected.     

Horizontal impact of a sphere half immersed in a liquid of finite depth

We consider the problem of bottom influence during horizontal hydrodynamic impact of a spherical solid body of diameter 2a which is half submerged in a liquid layer of finite depth. The sphere is subjected to the action of a shock pulse, as a result of which it acquires an initial translational velocity u directed along the x-axis.The influence of the spherical bottom on the sphere impact phenomenon was first studied by Zhukovskii [1]. Vertical impact in a layer of finite thickness was considered in [2]; horizontal impact in a halfspace was examined in [3].     

Hydroelastic coupling of beam finite element model with Wagner theory of water impact

The purpose of the paper is to demonstrate the feasibility of the direct coupling of the finite element method for the structural part with a Wagner representation of the hydrodynamic loads during the impact of an elastic body onto the water surface. An efficient and very general method is developed and validated in two dimensions. Advantages of the present method are outlined for the elastic wedge impact problem; however, the method is applicable to any elastic body with small deadrise angle entering water vertically at moderate velocity. Strategy for coupling of this method with commercial finite element codes is discussed.