Stability of convective motion in a vertical layer in the presence of longitudinal vibrations

The influence of vibrations of a cavity containing a fluid on the convective stability of the equilibrium has been investigated on a number of occasions [1]. The stability of convective flows in a modulated gravity field has not hitherto been studied systematically. There is only the paper of Baxi, Arpaci, and Vest [2], which contains fragmentary data corresponding to various values of the determining parameters of the problem. The present paper investigates the linear stability of convective flow in a vertical plane layer with walls at different temperatures in the presence of longitudinal harmonic vibrations of the cavity containing the fluid. It is assumed that the frequency of the vibrations is fairly high; the motion is described by the equations of the averaged convective motion. The stability boundaries of the flow with respect to monotonic perturbations in the region of Prandtl numbers 0 ? P ? 10 are determined. It is found that high-frequency vibrations have a destabilizing influence on the convective motion. At sufficiently large values of the vibration parameter, the flow becomes unstable at arbitrarily small values of the Grashof number, this being due to the mechanism of vibrational convection, which leads to instability even under conditions of weightlessness, when the main flow is absent [3, 4].     

Convective stability of a nonisothermal fluid in a rotating horizontal coaxial gap

The thermal fluid convection in a coaxial horizontal gap uniformly rotating about its axis is investigated. The threshold above which convective flows are excited and the structure of these flows are studied. It is found that convection ensues irrespective of whether the inner or outer boundary temperature is higher. Convection manifests itself in the threshold development of rolls elongated in the direction of the rotation axis and is determined by two different mechanisms. If the layer is heated from outside, the centrifugal convection mechanism plays a leading part and the diameter of the convective rolls is comparable with the layer thickness. If the higher is the temperature of the inner boundary of the layer, the centrifugal inertia force has a stabilizing effect and convection development is related with the action of thermal vibrational mechanism. The latter is determined by gravity-generated oscillations of the nonisothermal fluid relative to the cavity. The wave number of the vibrational convective structures is several times smaller than under centrifugal convection. The results obtained broaden our understanding of thermal convection in systems rotating in external static force fields.     

Convection of a heat-generating fluid in a rotating horizontal cylinder

Thermal convection of a fluid in a horizontal cylinder rotating about its own axis with uniformly volume-distributed internal heat sources is experimentally investigated. The enclosure boundary temperature was kept constant. The threshold of the excitation of convective flows and their structure are studied as functions of the heat-release intensity and the rotation velocity. The experiments are performed with water and water-glycerin solutions. It is shown that rapidly rotating fluid is in a stable quasiequilibrium state, namely, the temperature distribution is axisymmetric and has a maximum at the center of the enclosure. It is found that with decrease in the rotation velocity a convective flow arises thresholdwise, in the form of vortex cells periodically arranged along the axis. The thermal convection in the rotating enclosure is shown to be determined by the effects of two different mechanisms. One of these is due to the centrifugal force of inertia and plays the stabilizing role, while the other, thermovibrational mechanism is connected with nonisothermal fluid oscillations under the action of gravity in the enclosure-fitted reference frame and is responsible for the occurrence of mean thermal convection. The boundaries of the convection generation are plotted in the plane of the governing dimensionless parameters and the heat transfer in the supercritical region is studied.     

Dynamics of a Fluid in a Rotating Horizontal Cylinder

The behavior of a low-viscosity fluid in a rotating horizontal circular cylinder is investigated experimentally. The stability of the centrifuged layer, the motion of the fluid with respect to the cavity, the excitation of inertial waves on the fluid surface, and the effect of the waves on the stability and flow structure are studied over a wide region of relative occupancy of the cavity. The results are analyzed from the viewpoint of vibrational mechanics in which the motion is generated by the oscillations of the fluid with respect to the cavity and the gravity force plays the role of the force oscillating in the cavity reference system.     

Vibrational Convection in the Hele-Shaw Cell. Theory and Experiment

The influence of high-frequency horizontal vibrations on convection in the Hele-Shaw cell located in a uniform gravity field is considered experimentally and theoretically. Nonlinear regimes of vibrational convection in the supercritical region are examined. It is shown that horizontal vibrations (directed toward the wide sides of the cell) decrease the threshold of quasi-equilibrium stability. Regions of existence of one- and two-vortex steady flows are found, and unsteady regular and random regimes of thermal vibrational convection are considered. New random regimes in the Hele-Shaw cell are found, which result from nonlinear interaction of the “lower” modes responsible for the formation of regular supercritical convective regimes. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 40–48, March–April, 2006.     

Convection in rotating spherical layers with allowance for centrifugal forces

The investigation of convection in rotating spherical layers with a central gravitational field g(r) is very important for the study of the global motions in the atmospheres of large planets and the convective zones of stars. In recent years, many studies of these questions have been made (they have been reviewed, for example, by Yavorskaya and Belyaev [1]), but the centrifugal convective force has been ignored in all the numerical and analytic investigations. In some cases, for example, for large planets, the centrifugal force may reach an appreciable value, O.1g, and have a strong influence on the convective motion. The present paper studies the occurrence of convection in slowly rotating spherical layers with allowance for centrifugal forces. It is shown that the centrifugal force leads to the appearance in a layer of an axisymmetric flow, at the stability limit of which convective cells of banana or toroidal shape can develop. The latter are possible only in layers with undeformable boundaries at sufficiently large values of the Froude number. Irrespective of the form of the layer and the magnitude of the centrifugal force, the banana-shaped cells propagate in a wavelike manner in the direction opposite to the rotation. In the case of undeformable boundaries, the centrifugal force stabilizes the motion of the fluid as compared with the case of a layer at rest. Deformation of one or both of the boundaries under the influence of the centrifugal force leads to destabilization of the basic flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 14–21, March–April, 1984.     

Thermal Vibrational Convection in Rotating Cavities

The average vibrational motion of a nonisothermal fluid in a uniformly rotating cavity is described theoretically. Equations are obtained using the averaging technique in the high-frequency vibration approximation. It is found that the rotation significantly affects both the intensity of the average flows and the structure of the pulsatory velocity field generating resonance amplification of the fluid vibrations ar certain ratios of the rotation frequency and the force field oscillation frequency. This makes rotation an important controlling factor ensuring a strong averaged effect under relatively weak vibrational action. The problem of excitation of vibrational convection in a plane rotating layer is considered on the basis of the equations obtained when the vibration frequency substantially exceeds the rotation frequency.     

Development and instability of steady convective flows in a square cavity heated from below and a field of vertically directed vibrational forces

The present paper is devoted to numerical investigation of the spatial structure and stability of secondary vibrational convective flows resulting from instability of the equilibrium of a fluid heated from below. Vibrations parallel to the vector of the gravitational force (vertical vibrations) are considered. As in earlier work [7–9], a region of finite size is used — a square cavity heated from below. It is shown that enhancement of the vibrational disturbance of the natural convective flow may either stabilize or destabilize flows with different spatial structures; it may also stabilize certain solutions of the system of convection equations that are unstable in the absence of vibrational forces. In addition, increase of the vibrational Rayleigh number can lead to a change of the mechanisms responsible for equilibrium instability and oscillatory instability of the secondary steady flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–18, March–April, 1991.I thank G. Z. Gershuni for assistance and extremely fruitful discussions of the results of the paper.     

Development of a steady relief at the interface of fluids in a vibrational field

The vibrations of a vessel strongly influence the behavior of the interface of the fluids in it. Thus, vertical vibrations can lead both to the parametric excitation of waves (Faraday ripples) and to the suppression of the Rayleigh-Taylor instability [1–2]. At the present time, the influence of vertical vibrations on the behavior of a fluid surface have been studied in sufficient detail (see, for example, review [3]). The behavior of an interface of fluids in the case of horizontal vibrations has been studied less. An interesting phenomenon has been revealed in the experimental papers [4, 5]: in the case of fairly strong horizontal vibrations of a vessel containing a fluid with a free surface, the fluid collects near one of the vertical vessel walls, the free surface being practically plane and stationary with respect to the vessel, while its angle of inclination to the horizon depends on the vibration rate. But if there is a system of immiscible fluids with comparable but different densities in the vessel, horizontal vibrations lead to the formation of a steady wave relief at the interface. An explanation of the behavior of a fluid with a free boundary was given in [6] on the basis of averaged equations of fluid motion in a vibrational field. The present paper is devoted to an analysis of the behavior of the interface of fluids with comparable densities in a high-frequency vibrational field. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 8–13, November–December, 1986.     

Effect of high-frequency vibrations on the stability of advective flow

The stability of plane convective flow in a horizontal layer with a longitudinal temperature gradient under the action of longitudinal vibrations is considered. The behavior of small normal plane and spiral perturbations is investigated. It is shown that the vibrations enhance the stability with respect to almost all types of perturbations. The sole exception is plane thermal waves whose existence domain extends toward low Prandtl numbers. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–22, January–February, 1998. The work was supported by the Russian Foundation for Fundamental Research (project No. 94-01-01730).     

Secondary vibrational convective motions in a flat vertical layer of liquid

Investigations of the stability of steady-state plane-parallel convective motion between vertical planes heated to different temperatures [1–5] have shown that this motion, depending on the value of the Prandtl number P, exhibits instability of two types. With small and moderate Prandtl numbers, the instability is of a hydrodynamic nature. It is brought about by monotonic perturbations which, in the supercritical region, develop into a periodic, with respect to the vertical, system of steady-state vortices at the interface between the opposing convective flows. Articles [6, 7] are devoted to the numerical investigation of nonlinear secondary steady-state flows. If P>11.4, there appears a new mode of instability, i.e., running thermal waves increasing in the flow; with P>12, this mode becomes more dangerous [4]. This instability is connected with the development of vibrational perturbations, and it can be considered that in the supercritical region the perturbations lead to the establishment of steady-state vibrations. Linear theory has made it possible to determine the boundaries of the regions of stability. In the present article a numerical investigation is made of nonlinear supercritical conditions developing as a result of a loss of stability of the steady-state flow with respect to vibrational perturbations.     

Vibrational thermal convection in a cavity executing high-frequency rocking motions

The vibrational thermal convection in a cavity executing high-frequency rocking motions is investigated. The equations of vibrational convection are obtained by the method of averaging. It is shown that rocking motions lead to some new and distinctive effects. The convective stability in a plane layer in the presence of such vibrations is investigated on the basis of the obtained equations. A comparison with known experimental data is made. The results of experiments confirm the theoretical conclusions drawn on the basis of the averaged equations of vibrational convection.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 138–144, May–June, 1988.I thank E. M. Zhukhovitskii for helpful discussion.     

A light cylinder under horizontal vibration in a cavity filled with a fluid

The behavior of a light cylindrical body of circular cross-section under horizontal vibration in a rectangular cavity filled with a fluid is experimentally investigated. At critical vibration intensity the body is repelled from the upper side of the cavity and takes up a stable suspended position, in which the gravity field is balanced by the vibrational repulsive force, executing longitudinal oscillations. As the vibrations are intensified, the gap between the cylinder and the wall widens. A new form of instability, namely, the excitation of the tangential motion of the body along the vibration axis, is found to exist on the supercritical range. The cylinder is at a finite distance from the upper side of the cavity and the tangential motion is due to the loss of symmetry of the oscillating motion. The transition of the cylinder to the suspended state and its return to the wall, as well as the excitation of the average longitudinal motion and its cessation, occur thresholdwise and have a hysteresis. The body dynamics are studied as a function of the dimensionless vibration frequency.     

On the problem of the stability of convective two-phase medium flows under high-frequency vibration

The principles and methods of constructing a model of vibrational convection in a medium consisting of a liquid (gas) and a solid admixture are discussed. A closed system of averaged equations is first obtained. The system admits passage to the limits of the equations of both vibrational convection in a homogeneous fluid and convection in a dusty medium in the static case. As a measure of the difference with respect to the homogeneous fluid, in addition to the sedimentation parameter, which also manifests itself in the absence of vibrational accelerations, it is possible to take the inhomogeneity parameter introduced in this study and responsible for the pulsatory transport of the average fields. The problem of the stability of plane parallel flow in a vertical layer of a two-phase medium under horizontal longitudinal vibration with respect to infinitesimal perturbations is considered. It is shown that the introduction of particles into the flow leads to qualitatively novel effects which cannot be predicted within the framework of the homogeneous fluid model.     

Vibrational hydrodynamic interaction between a sphere and the boundaries of a cavity

A new type of vibrational lift force [1] acting on a spherical body oscillating in a viscous fluid near a rigid boundary is experimentally investigated. The interaction between the body and the cavity boundary creates a repulsion force which is capable of holding a heavy body in the gravity field at a certain distance from the floor and a light body at a certain distance from the ceiling. The repulsion force appears at a distance comparable with the Stokesian boundary layer thickness and increases as the surface is approached. Outside the viscous interaction range, the repulsion force is replaced by an attraction force which decays with distance. Dimensionless parameters governing the vibrational interaction are found and threshold curves, corresponding to the transition of bodies of different densities to the “suspended” state, are plotted as functions of a dimensionless frequency. The dependence of the repulsion and attraction forces on the distance between the body and the wall is studied.     

On the criteria of the absolute convective stability for compressible fluids

The conditions under which natural convection is absent from compressible fluids are investigated. It is shown that in the parameter “Rayleigh number-given temperature difference” plane there is a domain in which convection occurs for neither Rayleigh numbers. It is proposed to refer to this domain as the absolute convective stability region and to name the criterion determining the boundary of this region the absolute convective stability criterion. The necessary, sufficient, and necessary and sufficient conditions of the absolute convective stability for a viscous compressible fluid are derived. It is shown that in the particular case in which the thermal properties of the fluid and the adiabatic gradient are constant, these conditions coincide with the Schwarzschild criterion.     

Streamwise vibrations of a plate in a viscous fluid flow in a channel,induced by forced transverse vibrations of the plate

Aeroelastic vibrations of a plate aligned at a zero angle of attack in a viscous incompressible fluid flow in a channel with parallel walls are considered within the framework of a plane model. Forced vibrations of the plate in the transverse direction give rise an unsteady component of the flow friction force, induced by the perturbation of the fluid flow velocity by the vibrating plate. Under the assumption of the laminar character of the fluid flow, it is demonstrated that this force can excite streamwise vibrations of the plate if the channel width is small as compared with the plate length; these streamwise vibrations have the same order as the transverse vibrations of the plate excited by external forces.     

Disturbance spectrum of a plane fluid-fluid interface in the field of tangential high-frequency vibrations underweightlessness

The spectra of time-dependent disturbances of the equilibrium state and the stability of a plane interface between two fluid layers having different but comparable densities and being in the field of tangential high-frequency vibrations under weightlessness are investigated. Plane, spiral, and three-dimensional disturbances are considered. The cases of the same and different layer thicknesses are analyzed (in the latter case one of the layers is thicker than the other by a factor of ten). It is established that it is monotonic plane disturbances that are most hazardous. It is found that at high values of a vibration parameter growing spiral oscillatory disturbances (traveling waves) appear. With intensification of the vibrations the oscillatory disturbances vanish from the spectrum. In the case of the layers of different thicknesses it is established that the wavelength of the most hazardous disturbances is of the order of the thinner layer thickness. The experimentally observed generation of alternating strata in two-layer systems in the high-frequency vibration filed under weightlessness is attributed to the growth of disturbances having the greatest growth rate. The results obtained are quantitatively compared with the experimental data.     

Magnetovibrational flows in a magnetic fluid

A new type of flow in a magnetic fluid has been experimentally detected and investigated. The interaction between an alternating nonuniform magnetic field and a magnetic fluid leads to the parametric excitation of traveling capillary-gravitational waves which are the direct cause of the average flow. A necessary condition of formation of hydrodynamic flows in an alternating field is also the presence of time-dependent harmonics with multiple frequencies synchronized with the first harmonic, so that the time dependence of the ponderomotive force is generally pulsed. It is shown that for plane vibrational flows the classical theory cannot explain the high intensity of the average flows observed. It is suggested that the high intensity of magnetovibrational flows is related to the violation of the cylindrical symmetry of the traveling capillary-gravitational waves and the transition to a three-dimensional motion. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 124–133, January–February, 1998.     

Dynamic Study of a Structure with Flexion-Torsion Coupling in the Presence of Dry Friction

In aero engines, blade vibrations are frequently reduced by centrifugal flyweights, which exert a dry friction force per unit length under blade platforms. The response of this system to a periodic load has been analysed experimentally and theoretically. From a model having mode shapes similar to those of a blade, and a dry friction link per unit length, we show that the presence of the dry friction link is very effective in reducing vibrations for a range of excitation loads. The theoretical analysis is based on the Craig and Bampton mode synthesis, the rigid movement of the platform in its plane and the replacing of the linear dry friction link by several discrete parallel systems. Direct integration of the equations of motion is carried out by using the Newmark method. The comparison with experimental results is good. This method can easily be extended to more complex structures and shows that the dry friction link is effective when stick-slip occurs in the contact zone by limiting the energy provided to the system.