Free Fluid Convection in a Closed Contour in a Heat-Conducting Half-Space

The steady-state convection of a fluid in a thin porous vertical ring located in a heat-conducting half-plane is considered. For this problem approximate equations are derived. For a circular ring an analytic solution is obtained. For an elliptic ring a numerical-analytic solution is found. The Nusselt number and the fluid flow rate as functions of the Rayleigh number, the aspect ratio, and the contour depth are investigated.Many studies have been devoted to fluid convection in a porous ring [1–3]. In [1] two-dimensional convection with an isothermal internal boundary was considered when a temperature stratification is given on the outer boundary. A feature of this problem is the fact that the ring is located inside an impermeable heat-conducting medium in which a thermal gradient directed vertically downward is specified at a large distance from the ring. In [2, 3] two-dimensional convection in an annular ring occupied by a porous medium was investigated. From the results obtained in these studies it follows that in the formulation considered the hydraulic approximation can be used with satisfactory accuracy. In the present study this question is discussed more concretely and the necessary estimates are found. The results obtained could be useful for investigating hydrothermal convection in the Earth's crust, which has important geophysical applications [4–6].     

Plasma oscillations of an electron beam in an external electric field

Many papers have been devoted to the problem of the interaction of beams of charged particles with a plasma (a detailed bibliography is given, for example, in [1]). Analysis of the dispersion equation shows that in the case of a sufficiently slow monoenergetic electron beam of low density, growing longitudinal waves are not excited in a system consisting of such a beam and a plasma [2–4].The problem of the penetration of an external longitudinal electric field into a semiconfined plasma with an electron beam in the absence of instabilities in the system is studied (the boundary-value problem for growing waves was examined in [5]). This problem is, in a certain sense, an extension of the second part of L. D. Landau's well-known work [6] to include the case of a plasma with a beam. On the other hand, in the absence of an external electric field, this problem may be considered a boundary-value problem of the interaction of a weakly modulated electron beam with a plasma.The authors thank M. L. Levin for his useful comments.     

Suppression of forced and convective flows of a stably stratified fluid in vertical channels

In order to improve certain technological processes, for example, single-crystal growing, it is desirable to be able to control the flow rate in order to influence the heat and mass transfer processes. For this purpose it is usual to employ rotation, an electromagnetic field or reduced gravity [1]. Here, with reference to simple solutions of the system of equations of free convection in infinite vertical channels, it is shown that the problem of reducing the intensity of the flow can be solved given a suitable relation between the degree of stable stratification (with respect to density) and the factors responsible for the flow. The possibility of using temperature stratification is considered, but all the conclusions are also fully applicable to concentration stratification.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 172–174, March–April, 1987.The authors wish to thank B. Ya. Martuzan for his useful comments and interest in the work.     

Method of generalized similitude in free convection problems with arbitrary distribution of the temperature or heat flux on a vertical wall

The self-similar problem of free convection near a heated vertical plate was solved for the first time in [1] for the simplest case of a constant wall temperature. In [2], Yang proved the existence of a self-similar solution to the problem of free convection for vertical plates and cylinders on the surfaces of which the temperature has a power-law distribution. In [3], Yang's solution was generalized to the case of free convection near a slender figure of revolution, but also only in the self-similar case of a power-law distribution of the temperature on the wall. In [4], this problem was solved in an extended nonsimilar formulation but by an artificial and not general method similar to Gertler's, the convergence of the approximations being slow. The present paper contains the solution to the problem of free convection near a vertical plate with arbitrary distribution of the temperature or heat flux on its surface. Rigorous application of the method of generalized similitude [5] leads in this case to universal equations that present insuperable computational difficulties, which forces one to use a simplified but fairly general method to solve this class of problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 167–170, May–June, 1980.I thank L. G. Loitsyanskii and E. M. Smirnov for discussing the results and for valuable comments.     

Convection of a viscous incompressible gas in rectangular regions with inlet and outlet channels

A study is made of two-dimensional problems of thermal convection of a viscous incompressible gas in rectangular regions that have gas inlet and outlet channels in the presence of a temperature difference between the bottom and the top (the bottom is heated). In contrast to the well-studied case of natural convection, when no-slip conditions are specified on all boundaries of the region and motion in the region occurs only through the temperature difference [1–4], the heat transfer in the investigated flows is complicated by the additional influence of the forced convection of the gas due to the motion of gas through the inlet and outlet channels. Flows of such type simulate well the processes that take place in many heat transfer devices and in ventilated and air-conditioned industrial premises. Two formulations of the problem are considered. In the first, the gas flow through the inlet and outlet channels is assumed given, and the solution of the problem is determined by the dimensionless Prandtl, Grashof, and Reynolds numbers. In the second case, this flow rate is not given but determined during the solution of the problem. The motion in the region arises from the difference between the temperatures of the bottom and the top of the region, and the motion, in its turn, causes a flow of gas through the inlet and outlet channels. As in the case of natural convection, the solution of the problem in this case is determined by only two dimensionless numbers — the Grashof and Prandtl numbers. By numerical solution of the boundary-value problems for the equations of heat transfer a study is made of the influence of the characteristic dimensionless numbers on the hydrodynamic and temperature fields and the heat fluxes through the boundaries of the region. The solutions of the problems in the two formulations are compared for different positions of the outlet channels.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 126–131, September–October, 1979.We thank G. I. Petrov for discussing the results.     

Convection in a rotating medium above a thermally inhomogeneous horizontal surface

The linear stationary problem of convection in a medium rotating about a vertical axis above a thermally inhomogeneous horizontal surface is theoretically investigated. Attention is mainly focused on the case of a homogeneous medium, but certain stratification effects and especially the convection characteristics in binary mixtures (for example, in saline sea water) are also considered. When the rotation is rapid (large Taylor numbers) the convective cells are strongly elongated in the vertical direction, though they also contain a thin Ekman boundary layer. The importance of the boundary conditions on the horizontal surface (in parallel with the no-slip conditions, more general conditions that may follow from the quadratic turbulent friction model are considered) is shown. In the case of binary mixtures, the differential diffusion and rotation effects may together result in the appearance of “induced salt fingers”, the deep penetration of convection into an arbitrarily stably stratified medium. The convective motions may then have a considerable effect on the background vertical temperature and admixture distributions. Attention is drawn to an original manifestation of the analogy between the rotation and stratification effects: in a non-rotating, stably stratified medium, near a thermally inhomogeneous vertical surface, the convection also penetrates deep into the medium, but in the horizontal direction, so that, when the coordinate system is rotated through 90°, the solution coincides with the case of a rotating non-stratified fluid considered here.     

Hydrodynamics in weak force fields onset of steady thermocapillary convection in a spherical fluid layer under zero-g conditions

In the study of cellular convection in an infinite plane fluid layer with a free surface, both the Archimedes and thermocapillary forces [1–3] have been cited as reasons for the onset of convection. This has also been confirmed experimentally [4], When mass forces are absent or negligibly small it is natural to pose the question of the onset of pure thermocapillary convection or convection caused only by the surface tension gradients (see [2–3]). In the present paper, this problem is examined for a spherical fluid layer under zero-g conditions.     

Average stationary binary-mixture convection in connected channels in the presence of high-frequency vibration

The effect of vertical high-frequency vibration on steady-state binary-mixture flows in connected channels is studied theoretically. Mixtures with both positive and negative thermal diffusion are considered. It is shown that the convection excitation mode changes with the sign of the thermal diffusion. The dependence of the flow amplitude on the supercriticality is analytically obtained for various vibrational Rayleigh numbers.     

Convection in an Obliquely Heated Thin Porous Elliptic Ring

The problem of convection in a thin porous elliptic ring located in an impermeable rock mass is considered. The geothermal gradient is assumed to deviate by a certain angle from the vertical axis of the ellipse. An analytic solution of the problem is obtained in the hydraulic approximation. The stability of the solutions obtained is investigated for a circular ring. The profile of the thermal anomaly in the outer mass due to convection in the contour is given.     

A Nonlinear Stability Analysis of Convection in a Porous Vertical Channel Including Local Thermal Nonequilibrium

The problem is considered of thermal convection in a saturated porous medium contained in an infinite vertical channel with differentially heated sidewalls. The theory employed allows for different solid and fluid temperatures in the matrix. Nonlinear energy stability theory is used to derive a Rayleigh number threshold below which convection will not occur no matter how large the initial data. A generalized nonlinear analysis is also given which shows convection cannot occur for any Rayleigh number provided the initial data is suitably restricted.     

On the evolution of rotation of a solid under inelastic collisions with a plane

The motion of a solid in a homogeneous gravity field under inelastic collisions with an immovable absolutely smooth horizontal plane is considered. The body is a homogeneous ellipsoid of revolution. There exists a motion in which the ellipsoid symmetry axis is directed along a fixed vertical, the ellipsoid itself rotates about this axis at a constant angular velocity, and the ellipsoid bounce height over the plane decreases from impact to impact because of the collisions. We study the motion of the ellipsoid in a small neighborhood of the motion corresponding to this infinite-impact process. The main goal is to compute the angle between the ellipsoid symmetry axis and the vertical at the discrete time instants corresponding to the collisions. The problem is solved in the first (linear) approximation. The analysis is based on the canonical transformation method used earlier in [1] to solve problems with absolutely elastic collisions. There are quite a few studies dealing with infinite-impact processes (e.g., see the monographs [2, 3]). A method for continuous representation of systems with inelastic collisions was proposed in [4] and efficiently used in [3–5] when analyzing specific mechanical systems.     

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.     

Vibrational thermal convection about a uniformly heated cylinder

Vibrational convection under conditions of weightlessness has now been investigated for closed cavities of various geometries (see, for example, [1–3]). However, the question of vibrational convection developing around a heated body in an unconfined fluid remains open. Here, the convection developing under conditions of weightlessness about a uniformly heated infinite cylinder vibrating at high frequencies together with the fluid in a direction perpendicular to the cylinder axis is considered. The nonlinear equations of averaged motion are solved numerically by a finite-difference method. It is shown that at high values of the vibrational Grashof number boundary-layer type flow with a structure consisting of two symmetrical jets perpendicular to the direction of vibration is formed. The flow and heat transfer characteristics are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 23–26, May–June, 1989.The author wishes to thank E. M. Zhukovitskii for supervising the work.     

Numerical simulation of two-dimensional convection: Role of boundary conditions

The problem of two-dimensional, periodic in the horizontal coordinate, convection of an incompressible fluid heated from below between two horizontal planes is considered. The problem is solved in two formulations: with (stress-)free and hard (no-slip) boundary conditions on the horizontal planes. It is shown that at small supercriticalities the two-dimensional convection calculation leads to more correct results with hard than with free boundary conditions. It is established that the difference between the free and hard conditions is most strongly manifested in the pulsations of the vertical velocity component, whereas the dependence of the Nusselt number and the pulsations of the horizontal velocity component on the boundary conditions is more weakly expressed.     

Convection and heat transfer in a vertical layer with allowance for radiation from nonisothermic walls

Many studies (for example, [1–5]) consider motion and heat transfer in closed vertical cavities with given different temperatures of the lateral boundaries. The majority of studies cover the case of convection, but of late studies have appeared (for example, [4]) in which joint radiative—convective heat transfer is taken into account. In the present study we consider motion and heat transfer in a rectangular cavity separating two media with given different temperatures. In contrast to [4], the temperature of the lateral boundaries is determined from the condition for interaction with the surrounding medium, and the air in the cavity is assumed to be transparent for the heat radiation of the walls. The problem considered is a mathematical model of the heat transfer through windows, and is necessary for the analysis of methods of improving the heat proofing of buildings.Translated-from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 25–30, 1987.     

Developing mixed convection in vertical eccentric annuli

This paper deals with the problem of combined (forced–free) convection in vertical eccentric annuli with simultaneously developing hydrodynamic and thermal boundary layers. A bipolar model has been developed and a numerical algorithm for solving this model is outlined. Results, not available in the literature, are presented for the developing velocity profiles, axial variation of pressure, full development length, and heat transfer parameters under thermal boundary conditions of having one of the annulus boundaries at a constant temperature while the other boundary is insulated. Both aiding and opposing free convection have been considered and possibilities of flow reversal occurrence have also been checked. After a distance from the channel entrance and provided that the value of Gr/Re is sufficiently large, aiding free convection can develop to overcome the fluid friction and the eccentric annular channel eventually works as a diffuser. The value of Gr/Re for which a vertical eccentric annular channel can work as a diffuser decreases as the eccentricity increases. The axial distance from the entrance at which the channel starts to work as a diffuser decreases as Gr/Re increases.     

Equations of convective instability of a two-phase medium with modulation of the gravity force

Thermal convection in a heterogeneous medium consisting of a fluid and solid particles is studied under conditions of finite-frequency vibrations. Equations of convection are derived within the framework of the generalized Boussinesq approximation, and the problem of stability of a horizontal layer to infinitesimal perturbations under the condition of vertical vibrations is considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 21–28, March–April, 2008.     

Parametric studies and correlations for combined conduction-mixed convection–radiation from a non-identically and discretely heated vertical plate

The present paper reports the parametric studies and correlations for the problem of combined conduction-mixed convection–radiation from a non-identically and discretely heated vertical plate. Three discrete heat sources of non-identical heights but with identical volumetric rate of heat generation are assumed to be flush-mounted in a thin vertical plate. The longest and the shortest heaters are provided at the leading and trailing edges of the plate, while the remaining heater is located centrally. The governing fluid flow and heat transfer equations are considered in their full strength without the boundary layer approximations and are solved using the finite volume method. A computer code is written to solve the problem and various parametric studies have been performed. The relative roles of free convection, forced convection and radiation in various fluid flow and heat transfer results have been elucidated. In conclusion, based on a large set of data generated from the code, correlations for maximum non-dimensional plate temperature, average non-dimensional plate temperature and mean friction coefficient have been evolved.     

Exact solutions of linearized equations of convection of a weakly compressible fluid

A mathematical model of fluid convection under microgravity conditions is considered. The equation of state is used in a form that allows considering the fluid as a weakly compressible medium. Based on the previously proposed mathematical model of convection of a weakly compressible fluid, unsteady convective motion in a vertical band, with a heat flux periodic in time set on the solid boundaries of this band, is considered. This model of convection allows one to study the problem with the boundary thermal model oscillating in an antiphase rather than in-phase mode, while the latter was required for the model of microconvection of an isothermally incompressible fluid. Exact solutions for velocity components and temperature are derived, and the trajectories of fluid particles are constructed. For comparison, the trajectories predicted by the classical Oberbeck-Boussinesq model of convection and by the microconvection model are presented.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 52–63, March–April, 2005.     

Calculation of nonstationary finiteamplitude wind waves

Reviews of the present state of wind-wave investigation can be found in many papers (for instance, [1, 2]). A method for solving the problem of motion of finite-amplitude internal waves was proposed in [3]. However, the algorithm used there did not prove useful in the case of wind waves. In connection with this, we propose here a new algorithm. The calculation results are in agreement with data obtained in observing actual wind waves.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 82–86, November–December, 1977.