Effect of a permeable partition on convective instability in a rectangular cavity

The equilibrium stability of a fluid, heated from below, in a rectangular cavity with a vertical permeable partition is investigated. The small perturbation problem is solved by the Galerkin-Kantorovich method. The relations obtained for the dependence of the critical Rayleigh numbers on the partition parameters and the cavity dimensions make it possible to identify regions in which either even or odd perturbations, sensitive to only the normal or only the tangential resistance of the partition, respectively, are responsible for equilibrium crisis. The effect of a permeable partition on the convective instability of a horizontal layer of fluid under various heating conditions was considered in [1–3], where a significant dependence of the critical Rayleigh numbers on the properties of the partition was established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 6–10, May–June, 1989.     

The nonlinear evolution of two-dimensional and three-dimensional waves in mixing layers

The authors consider problems connected with stability [1–3] and the nonlinear development of perturbations in a plane mixing layer [4–7]. Attention is principally given to the problem of the nonlinear interaction of two-dimensional and three-dimensional perturbations [6, 7], and also to developing the corresponding method of numerical analysis based on the application to problems in the theory of hydrodynamic stability of the Bubnov—Galerkin method [8–14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza, No. 1, pp. 10–18, January–February, 1985.     

The uniqueness of the solution of boundary-value problems of a thermal model of discharge

The paper is devoted to a nonlinear analysis of superheating [1, 2] instability of an electric discharge stabilized by electrodes [3] in the framework of a thermal model [4] where the stability of the discharge relative to the long-wave and short-wave perturbations is proved in a linear approximation. Similar boundary-value problems arise in the theories of chemically and biologically reacting mixtures [5–7], thermal breakdown of dielectrics [8], thermal explosion [9], in the investigation of nonlinear waves in semiconductors and superconductors [10, 11], and in the investigation of Couette flow with variable viscosity [12]. The uniqueness of the one-dimensional steady solutions of the thermal model of discharge and the stability relative to the small spatial perturbations, respectively, for the exponential and step dependence of the electrical conductivity on the temperature are proved in [3, 13]. The uniqueness of the solutions in the one-dimensional case for the same electrode temperature and arbitrary dependences of the electrical and thermal conductivity on the temperature is established in paper [14]. In the present paper, the existence and uniqueness of steady solutions of the thermal model of discharge in a three-dimensional formulation for arbitrary fairly smooth electrical and thermal conductivity functions of the temperature in the case of isothermal isopotential electrodes are proved analytically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 140–145, January–February, 1986.The author expresses his gratitude to A. G. Kulikovskii and A. A. Barmin for the formulation of the problem and their discussions.     

Connective instability of a two-layer system with thermally insulated boundaries

The investigation of the convective stability of mechanical equilibrium of two horizontal layers of immiscible fluids has revealed the characteristics of such systems [1–3]. In particular, it has been found that, as distinct from a homogeneous horizontal layer, under certain conditions two-layer systems experience convective instability when uniformly heated from above and, moreover, oscillatory instability when heated from below. In [1–3] the problem was solved for a system with isothermal outer boundaries. In this paper the stability of equilibrium of two-layer systems is investigated for thermally insulated outer boundaries. Special attention is given to the study of the long wave instability mode.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 22–28, March–April, 1986.The authors wish to thank O. V. Kustova for assisting with the computations.     

Cavitation flow around a plate in a channel by a stream of heavy liquid

The nonlinear problem of cavitation flow around a plate by a stream of heavy liquid is investigated in precise formulation; the plate is located on the horizontal floor of a channel when the gravity vector is directed perpendicular to the wall of the channel. Two flow systems are considered-Ryabushinskii's and Kuznetsov's system [1]. This problem was investigated in linear formulation in [2], Similar problems were considered earlier in [3–7] for unrestricted flow. Below, on the basis of a method proposed by Birkhoff [8, 9], all the principal hydrodynamic and geometric characteristics are calculated for the problem being considered.Translated from Ivestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–9, May–June, 1973.     

Viscous flow in a partially-filled horizontal cylinder rotating at constant speed

The problem under consideration is that of the stationary shape of the free surface of a viscous fluid in a steadily rotating horizontal cylinder. In the majority of investigations of this problem the thickness of the fluid layer coating the inner surface of the cylinder is assumed to be small [1–3]. The case of a near-horizontal free surface, with the bulk of the fluid at the cylinder bottom, was considered in [4], where, after considerable simplification, the governing equations were reduced to ordinary differential equations. In the present study the behavior of the free surface is investigated using a creeping flow approximation. The controlling parameters vary over a wide range. In the numerical computations a boundary element method was used. The numerical results have been confirmed experimentally.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–30, May–June, 1993.     

Boundary layer on a blunt body in a flow of dusty gas

The problem of interaction of gas-dust flows with solid surfaces arose in connection with the study of the motion of aircraft in a dusty atmosphere [1–2], the motion of a gas suspension in power generators, and in a number of other applications [3]. The presence of a disperse admixture may lead to a significant increase in the heat fluxes [4] and to erosion of the surface [5]. These phenomena are due to the joint influence of several factors — the change in the structure of the carrier-phase boundary layer due to the presence of the particles, collisions of the particles with the surface, roughness of the ablating surface, and so forth. This paper continues an investigation begun earlier [6–7] into the influence of particles on the structure of the dynamical and thermal two-phase boundary layer formed around a blunt body in a flow. The model of the dusty gas [8] has an incompressible carrier phase. The method of matched asymptotic expansions [9] is used to obtain the equations of the two-phase boundary layer. In the frame-work of the refined classification made by Stulov [6], it is shown that the form of the boundary layer equations is different in the presence and absence of inertial precipitation of the particles. The equations are solved numerically in the neighborhood of the stagnation point of the blunt body. The temperature and phase velocity distributions in the boundary layer, and also the friction coefficients and the heat transfer of the carrier phase are found for a wide range of the determining parameters. In the case of an admixture of low-inertia particles that are not precipitated on the body, it is shown that even when the mass concentration of the particles in the undisturbed flow is small their accumulation in the boundary layer can lead to a sharp increase in the thermal fluxes at the stagnation point.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–107, September–October, 1985.I thank V. P. Strulov for a discussion.     

Stability of a plane-parallel convective flow of a liquid in a horizontal layer

We consider the stationary plane-parallel convective flow, studied in [1], which appears in a two-dimensional horizontal layer of a liquid in the presence of a longitudinal temperature gradient. In the present paper we examine the stability of this flow relative to small perturbations. To solve the spectral amplitude problem and to determine the stability boundaries we apply a version of the Galerkin method, which was used earlier for studying the stability of convective flows in vertical and inclined layers in the presence of a transverse temperature difference or of internal heat sources (see [2]). A horizontal plane-parallel flow is found to be unstable relative to two critical modes of perturbations. For small Prandtl numbers the instability has a hydrodynamic character and is associated with the development of vortices on the boundary of counterflows. For moderate and for large Prandtl numbers the instability has a Rayleigh character and is due to a thermal stratification arising in the stationary flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 95–100, January–February, 1974.     

Stability of nonplane-parallel three-dimensional jet flows

The problem of the stability of nonplane-parallel flows is one of the most difficult and least studied problems in the theory of hydrodynamic stability [1]. In contrast to the Heisenberg approximation [1], the basic state whose stability is investigated depends on several variables, and the stability problem reduces to the solution of an eigenvalue problem for partial differential equations in which the coefficients depend on several variables [2–7]. In the case of a periodic dependence of these coefficients on the time [2] or the spatial coordinates [3, 4], the analog of Floquet theory for the partial differential equations is constructed. With rare exceptions, the case of a nonperiodic dependence has usually been considered under the assumption of weak nonplane-parallelism, i.e., a fairly small deviation from the plane-parallel case has been assumed and the corresponding asymptotic expansions in the linear [6] and nonlinear [7] stability analyses considered. The present paper considers the case of an arbitrary dependence of the velocity profile of the basic flow on two spatial variables. The deviation from the plane-parallel case is not assumed to be small, and the corresponding eigenvalue problem for the partial differential equations is solved by means of the direct methods of [5], which were introduced for the first time and justified in the theory of hydrodynamic stability by Petrov [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 21–28, May–June, 1987.     

Radiative and convective heat transfer during blowing into a compressed hypersonic layer of two-phase products from the ablation of a covering

The study examines the screening of the radiative heat flux in conditions of hypersonic flow around blunt bodies with ablated carbon-based coverings. In contrast to the studies already known [1–3], allowance is made for the presence of condensed microscopic particles in the products of ablation. In [4] the problem of radiative transfer is considered in a layer of two-phase ablation products with parametrically prescribed dimensions, particle temperature, and layer thickness. The present study uses a closed system of equations which describes the processes of heat and mass transfer. This gives rise to considerable differences in the numerical results, according to the degree of screening.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 161–166, November–December 1985.deceased     

Existence and uniqueness of a solution to a problem of steady composite waves on the surface of a liquid flowing over a wavy bottom

Composite waves on the surface of the stationary flow of a heavy ideal incompressible liquid are steady forced waves of finite amplitude which do not disappear when the pressure on the free surface becomes constant but rather are transformed into free nonlinear waves [1]. It will be shown that such waves correspond to the case of nonlinear resonance, and mathematically to the bifurcation of the solution of the fundamental integral equation describing these waves. In [2], a study is made of the problem of composite waves in a flow of finite depth generated by a variable pressure with periodic distribution along the surface of the flow. In [3], such waves are considered for a flow with a wavy bottom. In this case, composite waves are defined as steady forced waves of finite amplitude that, when the pressure becomes constant and the bottom is straightened, do not disappear but are transformed into free nonlinear waves over a flat horizontal bottom. However, an existence and uniqueness theorem was not proved for this case. The aim of the present paper is to fill this gap and investigate the conditions under which such waves can arise.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–98, July–August, 1980.     

On the convective instability of a liquid in an inclined layer of a porous medium

In this paper we study the stability of the equilibrium of a liquid heated from below, wherein the liquid saturates a planar layer of a porous medium arbitrarily inclined to the direction of gravity. We consider the cases for which the boundaries of the layer are heat-conducting and also thermally insulated. In a horizontal layer with heat-conducting boundaries equilibrium is destroyed by perturbations of cellular structure [1], In a vertical layer the minimum critical temperature gradient corresponds to perturbations of plane-parallel structure. The transition to cellular perturbations in the case of heat-conducting boundaries takes place at an arbitrarily small angle of inclination of the layer to the vertical. For the thermally insulated layer the crisis of equilibrium is connected with plane-parallel perturbations at all angles of inclination.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 127–131, May–June, 1973.The author thanks G. Z. Gershuni for stating the problem and his interest in the work.     

Propagation of an ion jet near a dielectric surface

Difficulties in determining experimentally the local electrical parameters of unipolar-charged jets are arousing interest in the theoretical investigation of electrogasdynamic (EGD) flows. Free EGD jets were examined, for example, in [1–3]. In order to control the charge on the dielectric parts of aircraft surfaces, which results from their static electrification and may have certain negative consequences [4], and, moreover, to influence the flow in the boundary layer use is being made of unipolar-charged jets propagating near the dielectric [5, 6]. In [6] the case of an ion jet near a dielectric surface possessing surface conductivity was investigated. In these circumstances it is possible to neglect charge diffusion, which considerably simplifies the problem. Space charge diffusion was taken into account in [7], but subject to certain very important simplifications. The author has calculated the electrical parameters of a unipolar-charged jet propagating in a viscous incompressible gas near an ideal dielectric plate, with allowance for surface and polarization charges and, moreover, the diffusion processes near the surface. An asymptotic solution is obtained for the equations of the ionic diffusion layer as the ratio of the thickness of the diffusion layer to the thickness of the hydrodynamic boundary layer tends to zero.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 174–180, September–October, 1984.The author is grateful to V. V. Mikhailov and A. V. Kazakov for valuable advice and comments.     

Method of factorization in unsteady problems of gravity-elastic wave excitation in a fluid with a partially free boundary

An approach to the solution of unsteady problems with mixed boundary conditions for a layer of heavy fluid is developed. The plane problem of wave excitation by displacements given in a certain region of the lower boundary of the layer when the upper boundary is partially covered by an elastic plate is examined by way of illustration. As distinct from [1, 2], the proposed approach makes it possible to construct a solution in the form of a sum of harmonics and to carry out an analytic investigation into the nature of the propagation and stabilization of the wave fields. The space-time regions of the forming and formed wave packets are identified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 100–106, November–December, 1985.     

Measurement of the velocity profile in liquid streams of large volumes using a laser Doppler velocimeter

The use of the laser Doppler velocimeter (LDVM) in fluid mechanics already has a solid past history and work on its development is proceeding rather intensively [1, 2], This method has been used successfully to measure various parameters of flows in channels or chutes of small size [3–6]. Fundamental difficulties due to possible local variations in the index of refraction along the path of the laser beam [7], in addition to technical difficulties, have not been eliminated for its use in other cases of practical importance, particularly in basins or reservoirs of large size. The aim of the present paper is the investigation of the velocity structure of streams in large volumes using an LDVM on the example of a turbulent submerged water jet. In order to estimate the effect of the thickness of the water layer on the applicability of the method we tested three schemes: based on direct [8] and back scattering [9] and a scheme with reflection of forward-scattered light from a mirror [10]. The results of the investigation of a submerged turbulent jet using an LDVM are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 170–173, September–October, 1977.     

Flow past a plate under the free surface of a weightless liquid

A study is made of the nonlinear problem of the flow without separation of a perfect weightless liquid past a plate near the free surface. This problem was first posed by Gurevich [1]. At present, there are only a general solution to the problem [2–4] and some numerical calculations [5], which have been made under definite restrictions and are inadequate for detailed information about the interaction between the free surface and the plate. In the present paper, a complete investigation of the problem is given. Convenient computational formulas are obtained together with asymptotic expansions of them, and detailed calculations are made for all depths of the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 158–162, January–February, 1980.     

Mass transfer between bubbles and the continuous phase in a pseudofluidized layer

The problem of mass transfer between an isolated bubble and the continuous phase in a pseudofluidized layer is considered, when the rising velocity of the bubble exceeds the pseudofluidization rate. In this case the bubble with the surrounding region, a so-called two-phase system, is surrounded by a surface current impermeable to the liquid [1–3], and the problem reduces to determining the concentration field and the total flow on the material surface. The problem is solved for large and small Peclet numbers by a boundary layer diffusion method and by asymptotic expansion matching.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–49, July–August, 1973.     

Numerical investigation of convective motion in a closed cavity

The investigation of thermal convection in a closed cavity is of considerable interest in connection with the problem of heat transfer. The problem may be solved comparatively simply in the case of small characteristic temperature difference with heating from the side, when equilibrium is not possible and when slow movement is initiated for an arbitrarily small horizontal temperature gradient. In this case the motion may be studied using the small parameter method, based on expanding the velocity, temperature, and pressure in series in powers of the Grashof number—the dimensionless parameter which characterizes the intensity of the convection [1–4]. In the problems considered it has been possible to find only two or three terms of these series. The solutions obtained in this approximation describe only weak nonlinear effects and the region of their applicability is limited, naturally, to small values of the Grashof number (no larger than 10³).With increase of the temperature difference the nature of the motion gradually changes—at the boundaries of the cavity a convective boundary layer is formed, in which the primary temperature and velocity gradients are concentrated; the remaining portion of the liquid forms the flow core. On the basis of an analysis of the equations of motion for the plane case, Batchelor [4] suggested that the core is isothermal and rotates with constant and uniform vorticity. The value of the vorticity in the core must be determined as the eigenvalue of the problem of a closed boundary layer. A closed convective boundary layer in a horizontal cylinder and in a plane vertical stratum was considered in [5, 6] using the Batchelor scheme. The boundary layer parameters and the vorticity in the core were determined with the aid of an integral method. An attempt to solve the boundary layer equations analytically for a horizontal cylinder using the Oseen linearization method was made in [7].However, the results of experiments in which a study was made of the structure of the convective motion of various liquids and gases in closed cavities of different shapes [8–13] definitely contradict the Batchelor hypothesis. The measurements show that the core is not isothermal; on the contrary, there is a constant vertical temperature gradient directed upward in the core. Further, the core is practically motionless. In the core there are found retrograde motions with velocities much smaller than the velocities in the boundary layer.The use of numerical methods may be of assistance in clarifying the laws governing the convective motion in a closed cavity with large temperature differences. In [14] the two-dimensional problem of steady air convection in a square cavity was solved by expansion in orthogonal polynomials. The author was able to progress in the calculation only to a value of the Grashof numberG=10⁴. At these values of the Grashof numberG the formation of the boundary layer and the core has really only started, therefore the author's conclusion on the agreement of the numerical results with the Batchelor hypothesis is not justified. In addition, the bifurcation of the central isotherm (Fig. 3 of [14]), on the basis of which the conclusion was drawn concerning the formation of the isothermal core, is apparently the result of a misunderstanding, since an isotherm of this form obviously contradicts the symmetry of the solution.In [5] the method of finite differences is used to obtain the solution of the problem of strong convection of a gas in a horizontal cylinder whose lateral sides have different temperatures. According to the results of the calculation and in accordance with the experimental data [9], in the cavity there is a practically stationary core. However, since the authors started from the convection equations in the boundary layer approximation they did not obtain any detailed information on the core structure, in particular on the distribution of the temperature in the core.In the following we present the results of a finite difference solution of the complete nonlinear problem of plane convective motion in a square cavity. The vertical boundaries of the cavity are held at constant temperatures; the temperature varies linearly on the horizontal boundaries. The velocity and temperature distributions are obtained for values of the Grashof number in the range 0<G4·10⁵ and for a value of the Prandtl number P=1. The results of the calculation permit following the formation of the closed boundary layer and the very slowly moving core with a constant vertical temperature gradient. The heat flux through the cavity is found as a function of the Grashof number.     

Nonsmall liquid oscillations in moving vessels

The system of approximate nonlinear equations describing liquid oscillations in axisymmetric vessels is constructed. The equations are obtained for the case in which two coordinates belonging to the family of generalized coordinates characterizing the liquid motion are not small. This family is selected so that from the resulting nonlinear equations we can obtain as a particular case the nonlinear equations of [1–3], which are valid for the class of cylindrical vessels, and the requirements are satisfied that the resulting nonlinear equations correspond to the widely adopted linearized equations of liquid oscillations [4–6], Nonlinear equations are obtained which describe liquid oscillations in arbitrary vessels of rotation with radial baffles.     

Numerical simulation of instabilities and secondary regimes in spherical couette flow

In all previous numerical investigations of spherical Couette flow only axisymmetric regimes were considered. At the same time, in experiments [1–4] it was found that when both spheres rotate and the layer is thin centrifugal instability of the main flow leads to the appearance of nonaxisymmetric secondary flows of the azimuthal traveling wave type. The results of an initial numerical investigation of these flows are presented below. Solving the linear problem of the stability of the main flow and simulating the secondary flows on the basis of the complete nonlinear Navier-Stokes equations has made it possible to supplement and explain many of the results obtained experimentally. The type of bifurcation and the structure of the disturbances whose growth leads to the appearance of three-dimensional nonstationary flows are determined, and the transitions between different secondary regimes in the region of weak supercriticality are described.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–15, January–February, 1995.