Oscillations of a vessel containing a viscous fluid

The oscillations of a rigid body having a cavity partially filled with an ideal fluid have been studied in numerous reports, for example, [1–6]. Certain analogous problems in the case of a viscous fluid for particular shapes of the cavity were considered in [6, 7]. The general equations of motion of a rigid body having a cavity partially filled with a viscous liquid were derived in [8]. These equations were obtained for a cavity of arbitrary form under the following assumptions: 1) the body and the liquid perform small oscillations (linear approximation applicable); 2) the Reynolds number is large (viscosity is small). In the case of an ideal liquid the equations of [8] become the previously known equations of [2–6]. In the present paper, on the basis of the equations of [8], we study the free and the forced oscillations of a body with a cavity (vessel) which is partially filled with a viscous liquid. For simplicity we consider translational oscillations of a body with a liquid, since even in this case the characteristic mechanical properties of the system resulting from the viscosity of the liquid and the presence of a free surface manifest themselves.The solutions are obtained for a cavity of arbitrary shape. We then consider some specific cavity shapes.     

Added Masses of a Cylinder Intersecting the Interface of a Two-Layer Weightless Fluid of Finite Depth

The linear problem of high-frequency oscillations of a horizontal cylinder floating at the interface of a two-layer fluid was solved numerically using the boundary element method. Added masses are calculated for circular and elliptic cylinders.     

Effect of capillary wave radiation on the oscillations of a vibrator

The oscillations of a rigid body on an elastic tie (vibrator) in an ideal incompressible fluid with a free boundary, on which surface tension forces act, are considered. The linearized problem of hydrodynamics is solved approximately in the self-consistent formulation, the reaction forces exerted on the body by the fluid are calculated, and an integrodifferential equation of motion is obtained. Using asymptotic methods, the average characteristics determining the damping coefficient and the frequency shift of the oscillations of the vibrator are obtained with allowance for the effect of the capillary waves radiated by the vibrator. Qualitative effects depending on the parameters of the system are revealed. The authors' numerical simulation of the motion of the vibrator completely confirms the qualitative conclusions concerning the nature of the oscillations of a body in a fluid having surface tension.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 126–132, March–April, 1995.     

Oscillations of a Circular Cylinder in a Linearly Stratified Fluid

The plane problem of the motions of a three-layer fluid initiated by the oscillations of a circular cylinder is solved in the linear formulation in the Boussinesq approximation. The cylinder is completely immersed in the linearly stratified middle layer, and the upper and lower layers are homogeneous and bounded by rigid horizontal walls. The fluid is assumed to be ideal and incompressible. The added mass and damping coefficients are calculated as functions of the oscillation frequency of the cylinder and the layer thicknesses.     

Effect of weak compressibility of a fluid on bubble-bubble interaction in strong acoustic fields

The effect of weak compressibility of a fluid on the interaction between spherical bubbles in a strong acoustic field is considered. A small parameter ɛ which represents the ratio of the characteristic velocity of radial oscillations of the bubbles to the speed of sound in the fluid is used as a parameter characterizing the fluid compressibility. The equations governing the interaction between two bubbles are derived with an accuracy O(ɛ) in the case in which the ratio of the characteristic velocities of their translational and radial motions is of the order of ɛ. It is shown that neglecting the fluid compressibility effect due to the bubble interaction can lead to either enhancement or attenuation of their radial oscillations following the main compression stage, variation in the oscillation frequency, the bubble approach velocity, and the velocity of the spatial motion of the coupled pair, and the bubble approach and collision rather than their moving away from one another with the formation of a coupled pair.     

Oscillations of a cylinder piercing a linearly stratified fluid layer

The plane problem of the small steady-state oscillations of a horizontal cylinder arbitrarily located in a three-layer fluid whose upper and lower layers are homogeneous and whose middle layer is linearly stratified is considered in the linear formulation using the Boussinesq approximation. The fluid is assumed to be ideal and incompressible. The method of mass sources distributed along the body contour is used in the internal wave generation regime and an integral equation for the fluid pressure is derived in the non-wave regime. The hydrodynamic load acting on the body is calculated as a function of the oscillation frequency of the cylinder and its location. The results are compared with experimental data.     

Non-linear hydrodynamics of thin laminae undergoing large harmonic oscillations in a viscous fluid

Smoothed Particle Hydrodynamics is implemented to study the motion of a thin rigid lamina undergoing large harmonic oscillations in a viscous fluid. Particularly, the flow physics in the proximity of the lamina is resolved and contours of non-dimensional velocity, vorticity and pressure are presented for selected oscillation regimes. The computation of the hydrodynamic load due to the fluid–structure interaction is carried out using Fourier decomposition to express the total fluid force in terms of a non-dimensional complex-valued hydrodynamic function, whose real and imaginary parts identify added mass and damping coefficients, respectively. For small oscillations, the hydrodynamic force reflects the harmonic nature of the displacement, whereas multiple harmonics are observed as both the amplitude and frequency of oscillation increase. We propose a novel formulation of hydrodynamic function that incorporates added mass and damping coefficients for a thin rigid lamina spanning large amplitudes in viscous fluids in a broad range of the oscillation frequencies. Results of the simulations are validated against numerical and experimental works available in the literature in addition to theoretical predictions for the limit case of zero-amplitude oscillations.     

Hydromagnetic Stokes flow in a rotating fluid with suspended small particles

An initial value investigation is made of the motion of an incompressible, viscous conducting fluid with embedded small spherical particles bounded by an infinite rigid non-conducting plate. Both the plate and the fluid are in a state of solid body rotation with constant angular velocity about an axis normal to the plate. The flow is generated in the fluid-particle system due to non-torsional oscillations of a given frequency superimposed on the plate in the presence of a transverse magnetic field. The operational method is used to derive exact solutions for the fluid and the particle velocities, and the wall shear stress. The small and the large time behaviour of the solutions is discussed in some detail. The ultimate steady-state solutions and the structure of the associated boundary layers are determined with physical implications. It is shown that rotation and magnetic field affect the motion of the fluid relatively earlier than that of the particles when the time is small. The motion for large times is set up through inertial oscillations of frequency equal to twice the angular velocity of rotation. The ultimate boundary layers are established through inertial oscillations. The shear stress at the plate is calculated for all values of the frequency parameter. The small and large-time behaviour of the shear stress is discussed. The exact solutions for the velocity of fluid and the wall shear stress are evaluated numerically for the case of an impulsively moved plate. It is found that the drag and the lateral stress on the plate fluctuate during the non-equilibrium process of relaxation if the rotation is large. The present analysis is very general in the sense that many known results in various configurations are found to follow as special cases.     

Ellipsoidal Oscillations of a Gas Bubble with Periodic Variation of the Surrounding Fluid Pressure

Ellipsoidal linear and nonlinear oscillations of a gas bubble under harmonic variation of the surrounding fluid pressure are studied. The system is considered under conditions in which periodic sonoluminescence of the individual bubble in a standing acoustic wave is observable. A mathematical model of the bubble dynamics is suggested; in this model, the variation of the gas/fluid interface shape is described correct to the square of the amplitude of the deformation of the spherical shape of the bubble. The character of the air bubble oscillations in water is investigated in relation to the initial bubble radius and the fluid pressure variation amplitude. It is shown that nonspherical oscillations of limited amplitude can occur outside the range of linearly stable spherical oscillations. In this case, both oscillations with a period equal to one or two periods of the fluid pressure variation and aperiodic oscillations can be observed.     

Modeling of jet flows of a viscous fluid by the discrete vortex method

Results obtained previously by the discrete vortex method with a “viscous” correction are generalized. The boundaries of applicability of this method are determined. Previous results obtained for a flow past a flat plate are supplemented with solution convergence estimates. Exhaustion of a plane jet of a viscous incompressible fluid into the ambient space is modeled. The geometric parameters of the jet (its half-width, shapes of the streamwise velocity profiles, and intensity of oscillations) are analyzed. The calculated results are found to agree well with experimental data and with results calculated by other methods.     

Oscillations of a wedge in a free stream with particular reference to parachutes

The phenomenon of coupled breathing and longitudinal oscillations of a wedge-mass system in a free stream is examined. As a first step the unsteady pressure distribution on the surface of the oscillating wedge is calculated. For dynamic equilibrium of the wedge-mass system, the moment about the apex of the wedge must be zero. This condition establishes the amplitude and phase relation between breathing and longitudinal oscillations. As a final step the equation of motion of the store is used to calculate the frequency of the breathing oscillations. This frequency is shown to be dependent on four parameters. These parameters include the Froude number, the rigging line length to wedge breadth ratio and the rigging line stiffness and damping. Current results are compared with Hume and Stevens [1] experimental results.     

Natural oscillations of a jet membrane under hydrostatic pressure

The natural oscillations of a submerged elastic jet membrane modeling an axisymmetric hydrodynamic direct-flow radiator are analyzed. The fundamental frequency is calculated as a function of the geometric and hydrodynamic characteristics of the jet. The principle possibility of adjusting the frequency by changing the elasticity of the fluid, which is a function of hydrostatic pressure, is demonstrated. Numerical and experimental data are compared.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 92–98, December 2004.     

Wave generation by an oscillating submerged cylinder in the presence of a floating semi-infinite elastic plate

The results of solving the linear problem of steady-state oscillations of a horizontal cylinder submerged in fluid on whose upper boundary a semi-infinite elastic plate with free edge floats are given. The remaining part of the fluid surface is free. The contour-distributed mass source method is used. The corresponding Green’s function is constructed using eigenfunction expansions. The hydrodynamic load and the amplitudes of vertical displacements of the free surface and elastic plate are calculated. Equivalence relations which demonstrate both symmetry of the apparent mass and damping factors and connection of the damping factors with the wave amplitudes in the far field are derived.     

Natural Oscillations of a Heavy Fluid in an Elliptic Vessel

The problem of the natural oscillations of a heavy ideal fluid in a vertical elliptic basin (vessel) of finite depth is investigated. The eigenvalue and eigenfunction problem with Neumann boundary conditions is solved using a numerical-analytic approach developed by the authors. For the five lowest oscillation modes the natural frequencies and shapes are determined with high accuracy over a wide range of variation of the eccentricity. Graphical dependences of theoretical and applied interest are constructed and commented on.     

Dynamic behaviour of a bubble near an elastic infinite interface

This paper presents a study to describe the behaviour of a non-equilibrium bubble in a fluid (Fluid 1) that is in contact with another fluid (Fluid 2). Fluid 2 is assumed to incorporate some elastic properties, which are modelled through a pressure term at the fluid–fluid interface. The Laplace equation is assumed to be valid in both fluids and the boundary integral method is employed to simulate the dynamics of the bubble and the fluid–fluid interface. Interesting characteristic phenomena concerning bubble oscillations and the deformation of the fluid–fluid interface are studied for a range of parameters (distance from the fluid–fluid interface, density ratios of the two fluids and elastic properties of Fluid 2). Some of the phenomena observed are jet formation in the bubble, bubble splitting, a ring bubble separating from the main bubble, mushroom-shaped bubbles and the dynamic elevation of the elastic interface. Most of these phenomena are only observed when Fluid 2 possesses some elastic properties (besides the usual formation of a high speed liquid jet). Comparisons with experimental observations confirm the validity of our simulations.     

Oscillations of an ideal stratified fluid in a fixed vessel

Oscillations of a stratified fluid with a continuous distribution of the density that completely or partly fills a fixed vessel are investigated. Small oscillations of a stratified fluid with discrete distribution of the density have been investigated by Kopachevskii [1, 2]. Individual examples of the analysis of oscillations of a fluid consisting of layers of different densities have been given by Gontkevich and Ganichev et al. [3, 4].     

Shock Relations for a Detonation Regime Developing under the Action of Frequency-Modulated Laser Radiation

The oscillations of the gas dynamic parameters of plasma generated under the action of a powerful UHF-modulated laser radiation are investigated theoretically when the characteristic laser radiation intensity modulation time is significantly less than the characteristic time of variation of the gas dynamic parameters of the plasma. It is shown that in this regime the shock relations, written in the form of mass, momentum, and energy flux conservation laws, require the time derivatives of the gas dynamic quantities calculated from the corresponding differential equations to be taken into account.     

Nonlinear isochronous oscillations of a fluid in a paraboloid: theory and experiment

Within the framework of the shallow-water model, the nonlinear axisymmetric oscillations of a fluid in a paraboloid of revolution and in an unbounded parabolic channel are investigated. It is established that in the paraboloid of revolution the oscillation period does not depend on the amplitude, that is, the oscillations are isochronous. Experimental investigations of free fluid oscillations in a paraboloid confirm this theoretical result.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 131–142. Original Russian Text Copyright © 2004 by Kalashnik, Kakhiani, Lominadze, Patarashvili, Svirkunov, and Tsakadze.     

3-D inviscid self-excited vibrations of a blade row in the last stage turbine

Presented here is a three-dimensional (3-D) nonlinear time-marching method for the aeroelastic behaviour of an oscillating turbine blade row. The approach has been based in the solution of a coupled fluid–structure problem where the aerodynamic and structural dynamic equations are integrated simultaneously in time. This provides the correct formulation of a coupled problem, as the interblade phase angle (IBPA) at which stability (instability) would occur is also a part of solution. The ideal gas flow around multiple interblade passages (with periodicity in the entire annulus) is described by the unsteady Euler equations in conservative form, which are integrated by using the explicit monotonic second-order accurate Godunov–Kolgan finite-volume scheme and a moving hybrid H–O (or H–H) grid. The fluid and the structural equations are solved using the modal superposition method. An aeroelasticity prediction of a turbine blade of 0.765 m is presented. The natural frequencies and modal shapes of the blade were calculated by using 3-D finite element models. The instability regions for five mode shapes and the distribution of the aerodamping coefficient along the blade length were shown for harmonic oscillations with an assumed IBPA. The coupled fluid–structure oscillations in which the IBPA is part of the solution are shown.