Birfucation of solutions for convecttve flows of gas heated below in a wide range of pressure differences

Direct numerical modeling based on a net method is used to investigate the convective flow of a viscous compressible gas in a closed square region heated from below. The complete Navier-Stokes equations and some simplified models of the type of Boussinesq equations are used. The influence of the hydrostatic compressibility on the bifurcation of the solutions is investigated. A definition of the Rayleigh number that makes it possible to match results obtained using different models is proposed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 163–166, May–June, 1993.     

Theory of short waves in gas dynamics

An examination is made of the two-dimensional, almost stationary flow of an ideal gas with small but clear variations in its parameters. Such gas motion is described by a system of two quasilinear equations of mixed type for the radial and tangential velocity components [1, 2]. Partial solutions [3, 4], characterizing the variation in the gas parameters in the vicinity of the shock wave front (in the short-wave region), are known for this system of equations. The motion of the initial discontinuity of the short waves derived from the velocity components with respect to polar angle and their damping are studied in the report. A solution of the equations characterizing the arrangement of the initial discontinuity derived from the velocities is presented for one particular case of the class of exact solutions of the two parameter type [4]. Functions are obtained which express the nature of the variation in velocity of the front of the damped wave and its curvature.Translation from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 55–58, May–June, 1973.     

Nonlinear analysis of the flow initiated by the sudden motion of a wedge

Certain self-similar problems involving the sudden motion of a wedge which were treated in the linear approximation in [1–3] are studied by the method of matched asymptotic expansions. The nature of the wave boundary of the perturbed region is determined. Second-approximation solutions are constructed which describe flows behind weak shock fronts propagating in a stationary gas and behind fronts of weak discontinuity lines propagating by known uniform flows. A boundary-value problem is formulated whose solution describes, in first approximation, flows in the neighborhoods of points of interaction of the fronts. The existence of similarity rules of flows in these nieghborhoods is estimated. An approximate solution of the problems is given.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 37–47, May–June, 1976.     

Convective motions of a viscoplastic fluid in a rectangular region

Two-dimensional convective motion of a viscoplastic fluid in a long horizontal cylinder of square section heated on the side was studied numerically by the author in [1]. In the present paper, the problem of convection of a viscoplastic fluid in a rectangular region is solved numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–144, September–October, 1979.I should like to thank G. Z. Gershuni for supervising the work.     

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.     

Structure of internal waves in a channel

One method of describing wave motion in a fluid with continuous stratification is to use normal waves (modes). The propagation of internal gravity waves in closed rectangular regions whose boundaries coincide with planes through which there is no normal motion is essentially different from wave motion in an unbounded medium [1, 2]. This paper describes a theoretical and experimental investigation of the propagation of internal waves in an exponentially stratified fluid in a horizontal channel of finite height.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–110, January–February, 1935.In conclusion the authors wish to express their gratitude to A. T. Onufriev for his interest in their work.     

Motion of a shock wave through a gas of variable density

A successive approximation method is used to solve the self-similar problem of gas flow accompanying a shock wave propagated through a polytropic gas of variable density. The method is based on a special choice of independent variables and the use of Whitham's approximation [1] as the initial approximation for the motion of the discontinuity. A first approximation for the self-simulation index is calculated which is in good agreement with exact values.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 68–72, September–October 1970.The author wishes to thank S. V. Fal'kovich for suggesting this problem and for his help in the work.     

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.     

The motion of a gas around a strongly heated body

Maxwell was one of the first to study the thermal slipping and radiometry effects. In particular, he suggested [1] that the thermal stresses which occur in a gas are important in an analysis of the radiometry effect. Interest in these problems has recently increased in connection with the problem of the slow motion of a strongly heated body in a gas. The paper by Galkin et al. [2], for example, is devoted to this question. However, the paper contains certain inaccuracies, and this means that the problem needs to be reconsidered. The present note describes the classification and the general characteristics of the types of motion and gives a statistical example of the state of a nonuniformly heated gas.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 95–98, July–August, 1972.In conclusion the author wishes to thank V. V. Struminskii for a useful discussion of the results.     

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.     

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.     

Stability of one-dimensional gas flows in expanding regions

The stability of gas flows produced by the motion of a flat piston or the decay of an arbitrary discontinuity is considered. The boundaries of the region (or regions) in which the development of perturbations is considered are planes (shock wave, contact discontinuity, piston, etc.) which move away from each other.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 112–119, March–April, 1981.     

Calculation of unsteady natural concentration convection in binary mixtures of gases with arbitrary ratio of their densities

The approximate system of equations formulated by Nikulin, Potekhin, and Strelets [1] can, in contrast to the system in the Boussinesq approximation [2], be used to describe natural concentration convection in the presence of significant changes in the composition in gas mixtures with arbitrary ratio of the molecular weights of the components. In the present paper the possibilities opened up by the use of this system of equations are illustrated by the example of a numerical investigation of unsteady free convection of an isothermal binary gas mixture in a closed rectangular region in a wide range of variation of the Archimedes number and the ratio of the molecular weights of the components of the mixture.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 27–31, July–August, 1982.     

Numerical modeling of thermocapillary convection in a liquid layer with a nonlinear dependence of the surface tension on temperature

The principal characteristics of thermocapillary convection in a rectangular channel with one of the boundaries heated to a temperature higher and the other to a temperature lower than T₀ are investigated numerically on the basis of the Navier-Stokes equations. Certain convection characteristics corresponding to normal and anomalous thermocapillary effects are qualitatively compared. The conditions under which self-similar solutions of the type obtained in [10] can be used to describe the flow in a bounded region are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 138–143, January–February, 1991.     

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.     

The convective instability of Maxwell fluid-saturated porous layer using a thermal non-equilibrium model

A linear stability analysis is carried out to study viscoelastic fluid convection in a horizontal porous layer heated from below and cooled from above when the solid and fluid phases are not in a local thermal equilibrium. The modified Darcy–Brinkman–Maxwell model is used for the momentum equation and two-field model is used for the energy equation each representing the solid and fluid phases separately. The conditions for the onset of stationary and oscillatory convection are obtained analytically. Linear stability analysis suggests that, there is a competition between the processes of viscoelasticity and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary. Elasticity is found to destabilize the system. Besides, the effects of Darcy number, thermal non-equilibrium and the Darcy–Prandtl number on the stability of the system are analyzed in detail.     

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.     

Flow regimes with temperature extrema is the motion of a gas through a porous medium

Self-similar solutions are obtained for the problem of the motion of a heated gas through a porous medium with allowance for heat transfer between the gas and the solid phase in accordance with Newton's law. It is shown that there exist flow regimes in which the gas temperature increases with distance in the direction of motion.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 71–77, July–August, 1987.     

Self-similar solutions for energy liberation in a gas stream

By using dimensional analysis some possible kinds of nonstationary and stationary gas flows with energy liberation which result in self-similar problems are investigated. The cases of energy liberation in a gas at rest and in uniform supersonic and hypersonic streams are examined. The gas is assumed inviscid and perfect. Results of a computation of some hypersonic self-similar gas motions are presented. Three classes of self-similar gas motions have been well studied at this time: the strong explosion, the power-law flow caused by the expansion of a plane, cylindrical, or spherical piston [1], and conical flow (including combustion and detonation waves [2–4]). Some new self-similar motions caused by energy liberation on certain lines, surfaces, or in volumes will be examined below.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 106–113, November–December, 1974.