Effect of a permeable partition on convective instability of a horizontal liquid layer

We consider the effect of a thin permeable partition on the static stability of a horizontal liquid layer heated from underneath. The permeable partition is assumed to be plane and situated parallel to the boundary planes in the center of the layer. The resistance of the partition to the flow of liquid from one part of the layer to another leads to an increase in the static stability. We investigate the dependence of the minimum critical Rayleigh number-on the resistance of the partition and the form of the critical motions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 157–159, January–February, 1977.     

Convection in a closed cavity heated from below when equilibrium conditions are disrupted

The equilibrium of a fluid is possible in a closed cavity in the presence of a strictly vertical temperature gradient (heating from below) [1]. There is a distinct sequence of critical Rayleigh numbers R_i at which this equilibrium loses its stability relative to low characteristic perturbations. The presence of different finite perturbations, unavoidable in an experiment, leads to the absence of a strict equilibrium when R < R₁. The problem of the influence of the perturbation on the convection conditions near the critical points arises in this context [2, 3]. The case in which the cavity is heated not strictly from below is investigated in [2] and the case in which the perturbation of the equilibrium is due to the slow movement of the upper boundary of the region is investigated in [3]. In [2, 3] the perturbation has the structure of a first critical motion and thus the results of these papers coincide qualitatively. The perturbation of the temperature in the horizontal sections of the boundary, which creates a perturbation with a two-vortex structure corresponding to the second critical point R₂, is examined in this paper. A similar type of perturbation is characteristic for experiments in which the thermal conductivity properties of the fluid and the cavity walls are different. The nonlinear convection conditions are investigated numerically by the net-point method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 203–207, March–April, 1977.The author wishes to thank D. B. Lyubimova, V. I. Chernatynskii, and A. A, Nepomnyashchii for their helpful comments.     

Nonstationary laminar heat convection in a closed domain for a given heat flux

Characteristic modes of the time development of nonstationary heat convection in a closed planar domain upon a sudden supply of heat from the lateral surface are considered for Rayleigh numbers 10³–10⁷. Estimates of the boundaries of the beginning of the influence of convection on the temperature field and the buildup of a quasistationary convection mode in the range of Rayleigh and Fourier numbers are given. Characteristics of the circulation flow, the singularities of the temperature-field configuration and of the heat transfer from the wall to the fluid, are investigated. The mechanism for the origination and disappearance of vertical temperature differences, caused by convection, and the dependence of the vertical temperature differences on the Rayleigh and Fourier numbers, on the thermal mode of the boundary, and the domain geometry, are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 109–117, July–August, 1970.The author is grateful to T. D. Pirumov and T. V. Volokitin for assistance in performing the computations.     

Numerical study of convection of a liquid heated from below

The equilibrium of a liquid heated from below is stable only for small values of the vertical temperature gradient. With increase of the temperature gradient a critical equilibrium situation occurs, as a result of which convection develops. If the liquid fills a closed cavity, then there is a discrete sequence of critical temperature gradients (Rayleigh numbers) for which the equilibrium loses stability with respect to small characteristic disturbances. This sequence of critical gradients and motions may be found from the solution of the linear problem of equilibrium stability relative to small disturbances. If the temperature gradient exceeds the lower critical value, then (for steady-state heating conditions) there is established in the liquid a steady convective motion of a definite amplitude which depends on the magnitude of the temperature gradient. Naturally, the amplitude of the steady convective motion cannot be determined from linear stability theory; to find this amplitude we must solve the problem of convection with heating from below in the nonlinear formulation. A nonlinear study of the steady motion of a liquid in a closed cavity with heating from below was made in [1]. In that study it was shown that for Rayleigh numbers R which are less than the lower critical value R_c steady-state motions of the liquid are not possible. With R>R_c a steady convection arises, whose amplitude near the threshold is small and proportional to (R–R_c)^1/2 (the so-called soft instability)-this is in complete agreement with the results of the phenom-enological theory of Landau [2, 3].Primarily, various versions of the method of expansion in powers of the amplitude [4–8] have been used, and, consequently, the results obtained in those studies are valid only for values of R which are close to R_c, i. e., near the convection threshold.It is apparent that the study of developed convective motion far from the threshold can be carried out only numerically, with the use of digital computers. In [9, 10] the numerical methods have been successfully used for the study of developed convection in an infinite plane horizontal liquid layer.The present paper undertakes the numerical study of plane convective motions of a liquid in a closed cavity of square section. The complete nonlinear system of convection equations is solved by the method of finite differences on a digital computer for various values of the Rayleigh number, the maximal value exceeding by a factor of 40 the minimal critical value R_c. The numerical solution permits following the development of the steady motion which arises with R>R_c in the course of increase of the Rayleigh number and permits study of the oscillatory motions which occur at some value of the parameter R. The heat transfer through the cavity is studied. The corresponding linear problem on equilibrium stability is solved approximately by the Galerkin method.     

Effect of the Coriolis force on the onset of convection in a layer with a fixed heat flux on the boundaries

The effect of the Coriolis force on the onset of convection in a plane horizontal layer of viscous fluid with a fixed heat flux on the rigid lower and free upper boundaries is investigated. Expressions for the critical Rayleigh numbers and wave number are obtained analytically in the rapid rotation limit.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–46, May–June, 1994.     

A numerical study of the occurrence of convection in a liquid layer under the influence of periodic external forces

The development of convection in a horizontal liquid layer located in a periodically modulated gravitational field (or with periodically varying temperature gradient) is examined. The effect of modulation frequency on stability is studied. Modulation stabilizes equilibrium if the direction of the gravitational force remains constant at all times. In the opposite case, stabilization occurs only at sufficiently high frequencies. In [1] the dependence of the critical Rayleigh number on modulation amplitude of the external force for several fixed frequencies was examined. In all cases examined in [1], the modulation proves to have a stabilizing influence.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 81–86, May–June, 1972.     

Convective instability of a binary mixture with allowance for the Soret effect

The effect of weak mixture concentration on the threshold of convective instability of a binary mixture filling a cavity of arbitrary shape is investigated. In the case of thermally insulated boundaries in the neighborhood of the critical Rayleigh number monotonicity of perturbations is proved. This makes it possible to express the critical Rayleigh number for the mixture in terms of its analog for a single-component fluid at any values of the Soret parameter. In the general case of boundaries of arbitrary thermal conductivity an estimate of the critical Rayleigh number is obtained for small values of the Soret parameter.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 161–165, November–December, 1996.     

Parametric excitation of Rayleigh convection in the presence of a variable heat flux on the free surface

The onset of Rayleigh convection in a semi-infinite fluid layer is investigated for a heat flux harmonically modulated along the normal to the surface of the fluid. The problem of the evolution of the velocity and temperature perturbations is solved numerically by means of a finite-difference method. The stability limits and the characteristics of the critical perturbations are determined as functions of the Prandtl numbers. The behavior of the critical Rayleigh number is studied for finite layer depths.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 40–44, November–December, 1996.     

Accuracy of the Boussinesq approximation for an incompressible fluid

The corrections of first order to the eigenvalues and critical Rayleigh numbers obtained in the Boussinesq approximation are determined for convection in a fluid with zero compressibility. The ratio of the equilibrium difference of the densities to a mean density of the fluid is taken as the small parameter. The corrections are found by the methods of perturbation theory for self-adjoint operators. It is shown that in the class of problems with symmetry with respect to a horizontal plane the first-order corrections vanish. The restrictions on the system needed if the Boussinesq approximation is to be meaningful in the problem of the occurrence of convective instability are established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 2, pp. 19–26, March–April, 1981.     

Effect of rotation on the vibroconvective stability of a fluid layer with internal heat release

The equilibrium stability of a horizontal fluid layer with homogeneous internal heat release is investigated theoretically for the case in which the layer simultaneously undergoes high-frequency circular vibration in a horizontal plane and rotates about a vertical axis. The rotation frequency is assumed to be small as compared with the vibration frequency. It is found that the rotation has a stabilizing effect on the vibrational-gravitational convection. At the high-frequency limit the dependence of the critical values of the controlling parameters (gravitational and vibrational Rayleigh numbers) and the wave number on the rotation frequency is obtained.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 53–61. Original Russian Text Copyright © 2005 by Ivanova, Kozlov, and Kolesnikov.     

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.     

Numerical investigation of the shapes of convective gas motion in a cubic cavity heated from below

A numerical study has been made of the shapes of convective motion of a gas in a cubic cavity heated from below. The motion is generated by heating of the base of the cavity in accordance with a definite law. The evolution of the shapes of the motion up to the establishment of a final state has been followed. For the three basic forms of motion the regions of stability, the coefficients of heat transfer through the cavity, and the critical Rayleigh numbers have been determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–26, January–February, 1984.I am grateful to L. A. Chudov for support and valuable comments.     

Interferometric study of buoyancy-driven convection in a differentially heated circular fluid layer

The present study is concerned with buoyancy-driven convection experiments in a circular horizontal differentially heated layer of air. The radius-to-height ratio of 14, and Rayleigh numbers of 5,861 and 12,124 have been considered. A Mach–Zehnder interferometer has been used to visualize the convection patterns in the fluid layer. The fluid layer has been imaged at view angles of 0, 45 and 90°. Results obtained show that fringe patterns appropriate for a cavity square in plan are seen in the fluid layer during the early stages of the experiments. After the passage of the initial transients, steady fringes have been observed in the fluid layer for a Rayleigh number of 5,861. At Ra=12,124, a dominant pattern was detectable combined with mild unsteadiness. The steady thermal field at Ra=5,861 displayed symmetry with respect to the viewing angle. A stronger three dimensionality was seen at the higher Rayleigh number. The average steady state Nusselt numbers were found to vary with view angle from 1.91 to 2.04 at Ra=5,861 and 2.28 to 2.43 at Ra = 12,124. The cavity-averaged Nusselt numbers are in good agreement with the available correlations.     

Dynamics of a flat cavity in a low-viscosity liquid

The problem of the motion of a cavity in a plane-parallel flow of an ideal liquid, taking account of surface tension, was first discussed in [1], in which an exact equation was obtained describing the equilibrium form of the cavity. In [2] an analysis was made of this equation, and, in a particular case, the existence of an analytical solution was demonstrated. Articles [3, 4] give the results of numerical solutions. In the present article, the cavity is defined by an infinite set of generalized coordinates, and Lagrange equations determining the dynamics of the cavity are given in explicit form. The problem discussed in [1–4] is reduced to the problem of seeking a minimum of a function of an infinite number of variables. The explicit form of this function is found. In distinction from [1–4], on the basis of the Lagrauge equations, a study is also made of the unsteady-state motion of the cavity. The dynamic equations are generalized for the case of a cavity moving in a heavy viscous liquid with surface tension at large Reynolds numbers. Under these circumstances, the steady-state motion of the cavity is determined from an infinite system of algebraic equations written in explicit form. An exact solution of the dynamic equations is obtained for an elliptical cavity in the case of an ideal liquid. An approximation of the cavity by an ellipse is used to find the approximate analytical dependence of the Weber number on the deformation, and a comparison is made with numerical calculations [3, 4]. The problem of the motion of an elliptical cavity is considered in a manner analogous to the problem of an ellipsoidal cavity for an axisymmetric flow [5, 6]. In distinction from [6], the equilibrium form of a flat cavity in a heavy viscous liquid becomes unstable if the ratio of the axes of the cavity is greater than 2.06.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 15–23, September–October, 1973.The author thanks G. Yu. Stepanov for his useful observations.     

Linear analysis of convective instability of a fluid in a horizontal annular cavity occupied by a porous medium

The problem of convective instability in an infinite horizontal annular porous stratum located in an impermeable rock mass is considered. Using the Bubnov-Galerkin method, the values of the first seven critical Rayleigh numbers are found. The forms of the corresponding critical motions are established. The change in the modes of instability of the critical motions when the thickness of the porous stratum is varied is analyzed.Makhachkala. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 19–25, May–June, 1996.     

MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure

This study looks at MHD natural convection flow and heat transfer in a laterally heated enclosure with an off-centred partition. Governing equations in the form of vorticity–stream function formulation are solved using the polynomial differential quadrature (PDQ) method. Numerical results are obtained for various values of the partition location, Rayleigh, Prandtl and Hartmann numbers. The results indicate that magnetic field significantly suppresses flow, and thus heat transfer, especially for high Rayleigh number values. The results also show that the x-directional magnetic field is more effective in damping convection than the y-directional magnetic field, and the average heat transfer rate decreases with an increase in the distance of the partition from the hot wall. The average heat transfer rate decreases up to 80% if the partition is placed at the midpoint and an x-directional magnetic field is applied. The results also show that flow and heat transfer have little dependence on the Prandtl number.     

Equilibrium of a pressurized plastic fluid in a wellbore crack

This paper investigates equilibrium of a pressurized plastic fluid invading a tensile wellbore crack in a linear elastic, permeable rock. The crack is initially filled by pore fluid at ambient pressure, that is immiscibly displaced by the plastic fluid invading from the wellbore. The plastic fluid comes to rest to form a “plug” within the elastically deformed crack when the limit equilibrium between the shear stresses generated at the fracture walls and the pressure drop between the wellbore wall and the crack tip is reached. The model assumes that the leak-off of the plastic fluid into the rock is negligible, while the displaced pore fluid in the crack tip region is freely exchanged with the surrounding permeable rock to maintain the ambient pressure level. When the crack length ? is small or large compared to the wellbore radius R, the problem reduces to that of a pressurized edge or Griffith’s crack, respectively, subjected to a uniform far-field confining stress. In these two end-member cases, the normalized solution for the net pressure distribution, the plug length, and the stress intensity factor at the crack tip is obtained as a function of two numbers – the normalized net fluid pressure at the crack inlet and at the crack tip (partial plugs only) – that embody the solution’s dependence on the wellbore and the far field loading, the fluid yield strength, and the rock modulus. In the general case of an intermediate crack length (? ～ R), the normalized solution is a function of two additional parameters, the length-to-radius ratio and a normalized measure of the far field stress anisotropy, respectively, which accurate approximation is devised from an end-member solution using a rescaling argument. The equilibrium plug solutions are used to evaluate the breakdown pressure, the critical wellbore pressure at which the crack propagation condition is first met, and to analyze the stability of the ensuing crack propagation.     

Drag of porous cylinders in a viscous fluid at low reynolds numbers

The problem of flow past a permeable cylinder at low Reynolds numbers is of interest for the solution of a number of problems in chemical technology in, for example, the design of porous electrodes and porous catalysts and in the calculation of nonstationary filtration of aerosols by fibrous filters. In the present paper, we solve the problem of transverse flow of a viscous fluid past a continuous cylinder in a porous shell and, in particular, in the case of a porous cylinder under conditions of constrained flow (system of cylinders) and an isolated cylinder at arbitrary permeability. The analogous problem of Stokes flow past permeable spheres has been solved in a number of papers [1–3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 122–124, November–December, 1979.     

Experimental determination of the boundaries of the region of existence of anomalous convection flow in an tilted cube

This paper presents an experimentally study of the bifurcation of steady-state air convection in a cubic cavity heated from below under controlled deviations from equilibrium heating conditions due to a slow quasisteady-state tilt of the cavity at a predetermined angle α. It is found that in the supercritical range of Rayleigh numbers Ra at a tilt of the cavity not exceeding 7°, the existence of two stable steady-state convection regimes (normal and anomalous) with circulation in opposite directions is possible. A study is made of the transformations of the temperature distribution in the middle (with respect to the planes in which heat exchangers are located) plane during transition from the anomalous flow regime to the normal regime by instantaneous rotation of the entire mass of air in the cavity around the vertical axis by an angle of 90 to 135°. It is shown that this rotation occurs when the tilt of the cavity exceeds a critical value α _cr(Ra), which was determined experimentally for Rayleigh numbers 0 < Ra < 25Racr, where Racr is the critical Rayleigh number for stability of mechanical equilibrium for heating from below.     

Free thermoconcentration convection above a localized heat source

Shadowgraph and probe techniques are used to investigate the free convective flow pattern above various localized heat sources (electrolytic cell — point source, short vertical cylinder, sphere) in an exponentially stratified fluid. The characteristic types of structure are classified with respect to the appearance of new forms of instability. The critical values of the global Rayleigh number, at which the flow pattern is restructured, are determined. The dependence of the height of the convection zone and the height of the individual cells on the governing parameters of the problem is investigated and the results are reduced to universal form. Laboratory experiments and oceanic observations are qualitatively compared.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 27–34, March–April, 1989.