Angular vibrations of an ellipsoid of revolution in a confined viscous fluid

Vibrations of bodies in confined viscous fluids have been studied on a number of occasions, transverse vibrations of rods being the main subject of investigation [1–3]. The present authors [4] have considered the general problem of translational vibrations of an axisymmetric body in an axisymmetric region containing a low-viscosity fluid. The present paper follows the same approach and considers the problem of small angular vibrations of an ellipsoid of revolution in a circular cylinder with flat ends. In the general case, the hydrodynamic coefficients in the equation of motion of the ellipsoid are determined numerically for different values of the dimensionless geometrical parameters using Ritz's method. In the case of an unconfined fluid, analytic dependences in terms of elementary functions are obtained for the hydrodynamic coefficients. The theoretical results agree well with experimental investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 34–39, March–April, 1984.     

Formulation of the problem of the motion of a small body in a perturbed flow

The classical problem of the hydrodynamic reactions on a body of arbitrary shape moving in a fluid at rest [1] was generalized by Sedov to the case of an accelerated translational flow [2]. In the expressions for the hydrodynamic reactions, the shape of the body is represented only by the coefficients λ_ij of the added masses and the volume Ω of the body. In the general case of motion of a body in a nontranslational flow the shape of the body cannot be represented by a finite set of coefficients in the determination of the hydrodynamic reactions. An important simplification occurs in the small-body formulation, which again leads to expressions for the force and torque similar to the classical expressions. The problem of the motion of a small body in a perturbed nontranslational flow was posed by Grigoryan and Yakimov [3], and with allowance for deformation of the body by Yakimov [4]. Later studies containing this formulation have been reviewed by Vil'khovchenko and Yakimov [5]. The aim of the present paper is to formulate the small-body problem more precisely. The order of smallness of the terms in the earlier studies was estimated solely as a function of the power of a small parameter — the size of the body. In the present paper it is shown that if it is additionally required that the final expression for the reactions should contain only principal terms containing the components ν_i and ω_i of the translational and angular velocities, and also terms describing the flow structure, then the expression found by Grigoryan and Yakimov [3] for the hydrodynamic reaction is valid. The terms are estimated on the basis of dimensional analysis. Such arguments have already been used by the author for special examples [6, 7].     

Small vibrations of a sphere in a spherical volume of viscous fluid

Problems of the vibration of bodies in confined viscous fluids have been solved to determine the added masses and damping coefficients of rods [1–3] and floats [4–5]. The solutions of these problems, based on the use of simplifications of the boundary-layer method [4–6], are obtained analytically in general form and are in good agreement with the experimental data. However, in each specific case the possibility of using such solutions for given values of the fluid viscosity and vibration frequency must be justified either experimentally [2, 4, 5] or theoretically as, for example, in [1], where an analytic solution was obtained for concentric cylinders. The present paper offers a general solution of the problem of the small vibrations of a sphere in a spherical volume of fluid valid over a broad range of variation of the dimensionless kinematic viscosity. The limiting cases of this solution for both high and low viscosity are considered. The asymptotic expressions obtained are compared with calculations based on the analytic solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 29–34, March–April, 1986.     

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.     

On modelling and linear vibrations of arbitrarily sagged inclined cables in a quiescent viscous fluid

A theory of free linear vibrations of arbitrarily sagged inclined cables in a viscous fluid is presented in the framework of the heavy fluid loading concept. The static equilibrium shape of the cable is found by using the model of inextensible catenary and the validity ranges of this approximation are assessed. The dynamics of the viscous fluid is described by the linearised Navier–Stokes equations and their solution is pursued analytically by formulating the fluid field variables via potential functions. The vibration problem of a submerged cable is solved by Galerkin's method and the modal added mass and modal viscous damping coefficients are calculated. As a prerequisite for this analysis, the free vibrations of a cable in vacuum are addressed and a very good agreement with known results is observed. The physical interpretation of the dependence of modal added mass and modal damping coefficients on the ‘design variables’ for a fluid-loaded cable is given and the possible extensions of the suggested theory to capture weakly nonlinear effects are highlighted.     

Structure of Averaged Flow Driven by a Vibrating Body with a Large-Curvature Edge

The averaged viscous incompressible fluid flow driven by a vibrating body with a large-curvature edge is investigated experimentally and numerically. The case of an axisymmetric body immersed in fluid and performing translational vibrations along its axis is considered. Experiments carried out on fluids of various viscosity over a wide vibration frequency and amplitude range and direct numerical calculations based on the complete time-dependent equations of viscous fluid dynamics show that the global structure of the averaged flow significantly depends on the relation between the curvature radius of the body edge and the viscous skin-layer thickness. Different averaged flow regimes are detected and the flow restructuring process is investigated as a function of the vibration amplitude and frequency.     

Symmetric shapes of contact of a jet with the surface of a heavy liquid

The plane linearized problem of oblique impingement of a weightless jet of an ideal incompressible fluid on the surface of a heavy fluid is considered. Flows are sought with symmetric forms of the contact region. Mathematically we arrive at the problem of the eigenvalues and eigenfunctions of an integral equation; solving this problem we obtain various contact forms. The fundamental result for the infinitesimally thin jet of finite intensity is derived by passing to the limit, yielding a result analagous with the forms of free vibrations of a string. Some results are presented for the problem under consideration in the nonlinear formulation.The two-dimensional problem on (vertical) impingement of a jet on a liquid was solved by Olmstead and Raynor [1]. Some results for oblique impingement of a sufficiently thin, slightly curved jet are presented by Frolov [2], Information on other studies, primarily experimental, is presented in [3].This problem is related to the model of a jet curtain of an air-cushion vehicle; in this regard we note the study of Stepanov [4] in which, in particular, a result is obtained for an infinitesimally thin jet curtain.     

Linearized equations of motion of a viscous fluid in a porous medium of periodic structure

The investigation of flow in essentially inhomogeneous porous systems through the analysis of model periodic structures [1] is considered. In the acoustic approximation, an integrodifferential equation is obtained that describes the motion of a viscous fluid in a rigid porous medium of periodic structure. The velocity vector and pressure are represented in the form of asymptotic series with respect to a small parameter that characterizes the size of the periodicity cell, and the well-known procedure for averaging linearized hydrodynamic equations with small coefficients of viscosity [2, 3] is also used. A solution is presented to the local problem in the periodicity cell for a structure consisting of a doubly periodic system of infinitely long rods of circular section and a compressible viscous fluid that fills the space between them, and also for a structure formed by a system of orthogonal rectilinear channels, filled with viscous fluid, in a solid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 123–130, March–April, 1988.     

Small oscillations of a liquid in a partitioned cylindrical vessel

The equations of motion of a rigid body whose cavity is partially filled with an ideal fluid have been obtained in works of Moiseev [1, 2, 3], Okhotsimskii [4], Narimanov [5], and Rabinovich [6]. All the equation coefficients have been calculated for a cavity in the form of a circular cylinder or two concentric cylinders.The problem of fluid motion in a partitioned cylindrical cavity was considered by Rabinovich [7]. It was also considered by Bauer [8], who analyzed the particular case of vessel motion in the plane of one of the partitions.In the following we consider the two-dimensional motion of a cylinder with radial and annular baffles, and a definition is given of the velocity potential in the case of arbitrary positioning of the radial baffles with respect to the motion plane. Formulas are obtained for determining the parameters of a mechanical analog of the wave oscillations, which consists of two mathematical pendulum subsystems.     

Unsteady three-dimensional viscous shock layer in nonuniform gas flow in the neighborhood of a stagnation point

The combined influence of unsteady effects and free-stream nonuniformity on the variation of the flow structure near the stagnation line and the mechanical and thermal surface loads is investigated within the framework of the thin viscous shock layer model with reference to the example of the motion of blunt bodies at constant velocity through a plane temperature inhomogeneity. The dependence of the friction and heat transfer coefficients on the Reynolds number, the shape of the body and the parameters of the temperature inhomogeneity is analyzed. A number of properties of the flow are established on the basis of numerical solutions obtained over a broad range of variation of the governing parameters. By comparing the solutions obtained in the exact formulation with the calculations made in the quasisteady approximation the region of applicability of the latter is determined. In a number of cases of the motion of a body at supersonic speed in nonuniform media it is necessary to take into account the effect of the nonstationarity of the problem on the flow parameters. In particular, as the results of experiments [1] show, at Strouhal numbers of the order of unity the unsteady effects are important in the problem of the motion of a body through a temperature inhomogeneity. In a number of studies the nonstationary effect associated with supersonic motion in nonuniform media has already been investigated theoretically. In [2] the Euler equations were used, while in [3–5] the equations of a viscous shock layer were used; moreover, whereas in [3–4] the solution was limited to the neighborhood of the stagnation line, in [5] it was obtained for the entire forward surface of a sphere. The effect of free-stream nonuniformity on the structure of the viscous shock layer in steady flow past axisymmetric bodies was studied in [6, 7] and for certain particular cases of three-dimensional flow in [8–11].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 175–180, May–June, 1990.     

Jet Outflow from a Small Hole in a Plane

On the basis of the method of matched asymptotic expansions, the problem of the outflow of a nonswirling axisymmetric laminar jet from a hole in a plane is solved for large Reynolds numbers. Since directly matching the leading terms of the asymptotic expansions for the axial boundary layer and the main flow region is impossible, the problem is solved by introducing an intermediate region. In the axial region the solution is the Schlichting solution [1] for an axisymmetric jet in the boundary-layer approximation, in the intermediate region the solution is found analytically, and in the main flow region the problem is reduced to that of viscous flow induced by a sink line in the presence of a transverse wall [2].     

Asymptotic law of expansion of a cavity in the presence of hydrodynamic singularities in the flow

The asymptotic law of expansion of an axisymmetric half-body of finite resistance exposed to a flow of an ideal incompressible fluid and its relation with the force acting on the half-body were obtained in [1]. It is obvious that the deformation of the nose of the body will not change the form of the expressions found. It will be shown below how the asymptotic law and magnitude of hydrodynamic reactions change for half-bodies of finite resistance in the presence of sources (sinks) in the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 153–155, May–June, 1977.The author thanks G. Yu. Stepanov for attention to the work.     

Subsonic axisymmetric wake in a viscous gas

The problem of the subsonic axisymmetric flow of a compressible viscous perfect gas in the wake behind a cylindrical body with a flat base section is considered under the condition that the stream parameters are given at infinity and at some distance x_w upstream of the base section. (Let us note that the possibility of the existence of an axisymmetric wake at moderate Reynolds numbers has been shown experimentally [1].) The problem is solved by the numerical build-up method in a cylindrical x, y coordinate system on the basis of the Navier-Stokes equations, by a method elucidated in [2, 3]. Equations obtained from the fundamental system by a passage to the limit while taking into account the symmetry conditions on the axis y=0 are used on the axis of symmetry.     

Exact Analytic Solutions of the Nonlinear Long-Wave Equations in the Case of Axisymmetric Fluid Vibrations in a Parabolic Rotating Vessel

A class of exact analytic solutions of the system of nonlinear long-wave equations is found. This class corresponds to the axisymmetric vibrations of an ideal incompressible homogeneous fluid in a rotating vessel in the shape of a paraboloid of revolution. The radial velocity of these motions is a linear function, and the azimuthal velocity and free surface displacements are polynomials in the radial coordinate with time-dependent coefficients. The nonlinear vibration frequency is equal to the frequency of the lowest mode of linear axisymmetric standing waves in the parabolic vessel.     

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.     

Dynamics of interaction between a squeezed layer of a viscous incompressible fluid and an elastic three-layer plate

We study the hydrodynamic response of a thin layer of a viscous incompressible fluid squeezed between impermeable walls. We consider the distribution of pressure and force dynamic characteristics of the fluid layer in the case of forced flows along the gap between a vibration generator (which is a rigid plane) exhibiting harmonic vibrations and a stator (which is an elastic freely supported three-layer plate). The inertial forces of the viscous fluid motion and the stator elastic properties are taken into account. The amplitude and phase frequency characteristics of the elastic three-layer plate are found from the solution of the plane problem.     

Taylor instability of a fluid film on a body

The Taylor instability develops in a parallel flows when the body force acts in the direction from the heavier fluid toward the lighter [1]. It has been suggested that an increase in flow vorticity may have a stabilizing influence on the Taylor instability [2]. In studying the hydrodynamic stability of a viscous film on a body in a flow of a low-viscosity fluid [3], the author noted some stabilization of the Taylor instability with increase in Reynolds number, and suggested that cases of complete stabilization of the flow with respect to two-dimensional disturbances are possible with some increase in Reynolds number. In the present investigation, calculations revealed cases in which with increase in Reynolds number the Taylor instability goes over into a Helmholtz instability, which increases with increase in Reynolds number, and also cases in which the Taylor instability completely disappears at some value of the Reynolds number before a Helmholtz instability has developed, i.e., cases of complete stabilization of the flow with respect to two-dimensional disturbances as a result of an increase in Reynolds number.     

Determining the form of a body resulting in minimum heat flux,with laminar flow in the boundary layer

The exact solution of the problem of determining the optimal body shape for which the total thermal flux will be minimal for high supersonic flow about the body involves both computational and theoretical difficulties. Therefore, at the present time wide use is made of the inverse method, based on comparing the thermal fluxes for bodies of various specified form [1, 2]. The results of such calculations cannot always replace the solution of the direct variational problem. Therefore it is advisable to consider the direct variational problem of determining the form of a body with minimal thermal flux by using the approximate Newton formula for finding the gasdynamic parameters at the edge of the boundary layer. This approach has been used in finding the form of the body of minimal drag in an ideal fluid [3–5] arid with account for friction [6], and also for determining the form of a thin two-dimensional profile with minimal thermal flux for given aerodynamic characteristics [7].     

Strong viscous interaction on slender three-dimensional bodies

Hypersonic flow of a viscous gas past axisymmetric power-law bodies in the regime of strong interaction of the laminar boundary layer with the outer inviscid flow was studied in [1, 2]. In this paper the results obtained in [1, 2] are extended to the case of flow past a slender three-dimentional body.