Theory of vortical helical ideal gas flows in laval nozzles

An asymptotic solution is found for the direct problem of the motion of an arbitrarily vortical helical ideal gas flow in a nozzle. The solution is constructed in the form of double series in powers of parameters characterizing the curvature of the nozzle wall at the critical section and the intensity of stream vorticity. The solution obtained is compared with available theoretical results of other authors. In particular, it is shown that it permits extension of the known Hall result for the untwisted flow in the transonic domain [1]. The behavior of the sonic line as a function of the vorticity distribution and the radius of curvature of the nozzle wall is analyzed. Spiral flows in nozzles have been investigated by analytic methods in [2–5] in a one-dimensional formulation and under the assumption of weak vorticity. Such flows have been studied by numerical methods in a quasi-one-dimensional approximation in [6, 7]. An analogous problem has recently been solved in an exact formulation by the relaxation method [8, 9]. A number of important nonuniform effects for practice have consequently been clarified and the boundedness of the analytical approach used in [2–7] is shown.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 126–137, March–April, 1978.The authors are grateful to A. N. Kraiko for discussing the research and for valuable remarks.     

Transonic flow of a gas jet from a vessel with plane walls

The problem of transonic flow of a gas jet with equilibrium excitation of the vibrational degrees of freedom of the molecules from an infinite symmetrical vessel with plane walls is reduced to a generalized Tricomi boundary-value problem for an equation of Chaplygin type. It is solved using a difference scheme [13] based on the decomposition of the difference operator in accordance with the type of differential operator. Calculation results are presented for a mixture of oxygen and nitrogen simulating air. The effect of the angle between the walls, the stagnation temperature and the back-pressure coefficient on the flow coefficient is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 128–135, November–December, 1987.The authors are grateful to F. Yu. Stepanov for his useful comments.     

On the optimal distribution of the flow rate of gas blown through the side walls of the channel of a de plasmotron (plasma generator)

It is well known that the blowing of cold gas through the side walls of the channel of a dc plasmotron (plasma generator) with longitudinal blowing over the arc leads to an increase in the useful power of the plasmotron [1]. The increase is due to the increase in the combustion voltage of the arc and also the decrease in the heat fluxes to the wall of the channel. The present paper solves the problem of the optimal distribution of the flow rate of gas blown through the side walls into the channel of a dc plasmotron of arbitrary shape F(x). The flow in the main channel and in the ducts in the side walls is described by the quasi-one-dimensional gas-dynamic equations investigated qualitatively in [2] and verified experimentally in [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 120–124, May–June, 1981.     

Planar gas flows with weak energy supply

Perfect gas flows in an unlimited space, which occur during rectilinear motion of a system of distributed heat sources, are investigated. The next modes in order of growth of the number M are examined: the heat conductive, convective, subsonic, transonic, supersonic, hypersonic. Examples of computations are presented. Flows with distributed heat sources attract ever-increasing attention. Such flows are important, e.g., in the problem of radiation propagation [1–5], in the analysis of a gasdynamic laser resonator and the optical characteristics of a ray [6]. Changes in the density because of absorption of the ray energy, which can result in an essential redistribution of the radiation intensity, are of great interest in these problems. Theoretical investigations of a general nature with distributed heat supply [7–10] are also important for the development of further applications. Gas flows for a given distribution of relatively weak heat sources switched on at a certain time are examined in this paper.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 95–102, September–October, 1978.     

Calculation of hydrogen-air combustion behind detached shock wave for supersonic flow past a sphere

Many of the published theoretical studies of quasi-one-dimensional flows with combustion have been devoted to combustion in a nozzle, wake, or streamtube behind a normal shock wave [1–6].Recently, considerable interest has developed in the study of two-dimensional problems, specifically, the effective combustion of fuel in a supersonic air stream.In connection with experimental studies of the motion of bodies in combustible gas mixtures using ballistic facilities [7–9], the requirement has arisen for computer calculations of two-dimensional supersonic gas flow past bodies in the presence of combustion.In preceding studies [10–12] the present author has solved the steady-state problem under very simple assumptions concerning the structure of the combustion zone in a detonation wave.In the present paper we obtain a numerical solution of the problem of supersonic hydrogen-air flow past a sphere with account for the nonequilibrium nature of eight chemical reactions. The computations encompass only the subsonic and transonic flow regions.The author thanks G. G. Chernyi for valuable comments during discussion of the article.     

Laminar flow between porous disks for intense homogeneous asymmetric inflow and outflow

In [1–3], a class of self-similar solutions was considered for the flow of incompressible fluid in a plane channel with porous walls, through which there is homogeneous symmetric inflow or outflow. An analogous class of self-similar solutions for flow between porous disks with natural homogeneous conditions at the periphery was considered in [4], where the asymptotic behavior of these solutions at a small Reynolds number of the outflow R was investigated, and the limiting form of the solution for symmetric outflow with R= was noted. In the present paper, the boundary-function method is applied to the singular problem corresponding to the flow between porous disks for asymmetric inflow and outflow characterized by large R.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 13–19, November–December, 1976.     

Convective stability of a weakly conducting liquid in an electric field

In an inhomogeneously heated weakly conductive liquid (electrical conductivity 10^–12–1 cm^–1) located in a constant electric field a volume charge is induced because of thermal inhomogeneity of electrical conductivity and dielectric permittivity. The ponderomotive forces which develop set the liquid into intense motion [1–6]. However, under certain conditions equilibrium proves possible, and in that case the question of its stability may be considered. A theoretical analysis of liquid equilibrium stability in a planar horizontal condenser was performed in [2, 4]. Critical problem parameters were found for the case where Archimedean forces are absent [2]. Charge perturbation relaxation was considered instantaneous. It was shown that instability is of an oscillatory character. In [4] only heating from above was considered. Basic results were obtained in the limiting case of disappearingly small thermal diffusivity in the liquid (infinitely high Prandtl numbers). In the present study a more general formulation will be used to examine convective stability of equilibrium of a vertical liquid layer heated from above or below and located in an electric field. For the case of a layer with free thermally insulated boundaries, an exact solution is obtained. Values of critical Rayleigh number and neutral oscillation frequency for heating from above and below are found Neutral curves are constructed. It is demonstrated that with heating from below instability of both the oscillatory and monotonic types is possible, while with heating from above the instability has an oscillatory character. Values are found for the dimensionless field parameter at which the form of instability changes for heating from below and at which instability becomes possible for heating from above.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 16–23, September–October, 1976.In conclusion, the author thanks E. M. Zhukhovitskii for this interest in the study and valuable advice.     

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.     

Transonic flows around an airfoil in wind tunnels with porous walls

A study is made of two-dimensional transonic flows of gas around an airfoil in the working part of a wind tunnel with porous walls. The values of the flow parameters are determined by the numerical solution of a boundary-value problem for the equation of the velocity potential; this problem simulates the gas flow around the profile in the tunnel with porous walls. The obtained results are then used to construct an asymptotic theory of the influence of the wind-tunnel height and the Mach number M of the flow in it on the characteristics of the flow around the airfoil.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–107, September–October, 1980.     

Nonisothermal instability of high-velocity elastoplastic flows

An important feature of the high-velocity deformation of solids is the localization of deformation, one of the causes of which may be the nonisothermal instability of plastic flow [1–6]. In connection with the intensive development of high-velocity technology in the treatment of materials, the investigation of the criteria for nonisothermal stability of processes of plastic deformation is of fundamental interest, since in certain cases they determine the optimum technological regimes [5]. The critical values of deformation velocities, above which the effects of thermal instability becomes decisive in the process of deformation of solids, are estimated by semiempirical methods in [1]. The non-boundary-value problem of the criteria for nonisothermal instability is analyzed in [2] for the point of view of flow stability in the so-called coupled formulation. The latter means that the heat-conduction equation is added to the basic equations determining the dynamics of an elastoplastic medium. The problem is solved in [6] in an analogous formulation, but for flow averaged over the spatial coordinate. The solution of the boundary-value problem for one-dimensional flow in this formulation is given in the present paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 133–138, May–June, 1986.     

Flow processes with change of regime in fractured reservoirs

Experimental and industrial observations indicate a strong nonlinear dependence of the parameters of the flow processes in a fractured reservoir on its state of stress. Two problems with change of boundary condition at the well — pressure recovery and transition from constant flow to fixed bottom pressure — are analyzed for such a reservoir. The latter problem may be formulated, for example, so as not to permit closure of the fractures in the bottom zone. For comparison, the cases of linear [1] and nonlinear [2] fractured porous media and a fractured medium [3] are considered, and solutions are obtained in a unified manner using the integral method described in [1]. Nonlinear elastic flow regimes were previously considered in [3–6], where the pressure recovery process was investigated in the linearized formulation. Problems involving a change of well operating regime were examined for a porous reservoir in [7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–73, May–June, 1991.     

Rarefied gas flow past a sphere with injection

Hypersonic rarefied gas flow over the windward face of a sphere is considered in the presence of distributed injection from the surface of the body. A similar problem was previously solved in [1–3] within the framework of continuum mechanics and in [4] on the basis of model kinetic equations. In the present study the calculations were carried out using the Monte Carlo method of direct statistical modeling [5, 6]. The injected gas was the same as the free-stream gas. A simple monatomic gas model with a rigid sphere interaction potential was employed. The reflection of the molecules from the surface of the body was assumed to be diffuse with total energy accommodation. The calculation procedure using weighting factors is described in [7]. The influence of injection on the mechanical and thermal effect of the gas flow on the body is investigated for various degrees of rarefaction of the medium and injection rates.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 175–179, July–August, 1990.     

Flow of a suspension in a flat channel with porous walls

In the flow of a suspension in a channel with porous walls, when the size of particles of a suspended phase is much less than the width of the channel but greatly exceeds the size of the pores (in particular, in the flow of blood in the plasma separator used in an artificial kidney system [1, 2]), phenomena are observed which apparently cannot be satisfactorily explained by means of the well-known solutions of problems on the motion of a Newtonian fluid [3]. For example, the flow rate of the liquid phase of the suspension through the walls depends on the velocity of the main flow and does not depend on the pressure drop on the wall at fairly high values of it [1, 2]. The present study gives below the formulation and an approximate solution, which explains this effect, of the problem of an incompressible two-phase suspension in a long slit with porous walls which are impermeable in relation to the suspended phase and through which the liquid phase is pumped. Certain effects are taken into account which are caused by the high volume concentration of the suspended phase.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 37–43, November–December, 1987.     

Unsteady flow of a viscous gas between two solid walls,one of which is free and vibrating at high frequency

The pressure field is investigated in a thin layer of a viscous compressible gas between two walls, one of which is free and executing high-frequency harmonic vibrations. Asymptotic methods are applicable to the case in which one wall vibrates at high frequencies [1]. The motion of the gas, as shown in [2], can be assumed to be nearly isothermal, and the influence of the inertial terms in the equation of motion for the gas can be neglected.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 111–116, March–April, 1971.     

Supersonic viscous perfect gas flow past a circular cylinder

The complete Navier-Stokes equations are used to calculate supersonic perfect gas flow past a circular isothermal cylinder by the method described in [1]. The effects of the Mach number M=2.5–10 and the Reynolds number Re=30-10⁵ on the flowfield structure and heat transfer to the cylinder wall are investigated. Special attention is paid to the study of the near wake and the local characteristics on the leeward side of the cylinder.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 107–115, November–December, 1993.     

Wave flow of a liquid film together with a flow of gas

The wave flow of a thin layer of viscous liquid in conjunction with a flow of gas was considered in a linear formulation earlier [1, 2]. In this paper the problem of the wave flow of a liquid film together with a gas flow is solved in a nonlinear setting. On this basis relationships are derived for calculating the parameters of the film and the hydrodynamic quantities.Ivanovo. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 12–18, January–February, 1972.     

Numerical investigation and self-similar solutions of the glacier-ocean interaction problem

The mathematical modeling of the heat and mass transfer processes in glaciers is an effective means of investigating and predicting their development. A full explanation of the problem of constructing appropriate mathematical models is given in [1–5]. By analyzing the equations involved [3, 6] it is possible to establish the principal factors and dimensionless numbers determining glacier dynamics and provide justification for neglecting the secondary terms. In particular, a simplified closed system of differential equations for the detailed calculation of all the hydrodynamic characteristics of the glacier can be obtained for K_hj « 1 up to O(K _h ² ), where K_h is the ratio of the vertical and horizontal scales of the ice mass investigated (K_h 10^–4–10^–6). In this case many of the qualitative characteristics of glacier dynamics are preserved even in one-dimensional models within the subisothermal approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–7, September–October, 1986.     

A case of jet flow of a gravity fluid

The problem of plane steady ideal heavy fluid flow bounded by an impermeable polygonal section, a curvilinear arc section, and a finite section of free surface is investigated in an exact nonlinear formulation. Hydrodynamic singularities may exist in the stream. A large class of captation problems of jet theory reduces to studying this kind of flow. The unique solvability of the problem under investigation is proved for sufficiently large Froude numbers and small arc curvature. A method of solution is given and an example is computed. Such problems have been solved earlier by numerical methods [1–3]. Some problems about jet flows of a gravity fluid with polygonal solid boundaries have been investigated by an analogous method in [4, 5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 140–143, May–June, 1975.     

Spherical expansion of vapor during evaporation of a droplet

The problem of the evaporation of a spherical particle is solved by a numerical finnite-difference method for the stationary and nonstationary cases on the basis of the generalized Krook kinetic equation [1]. Evaporation into a vacuum and into a flooded space are considered taking into account the reduction in size and cooling of the droplet. The minimum mass outflow is determined for stationary evaporation into a vacuum at small Knudsen numbers. The results are compared with those of other authors for both the spherical and plane problems. Most previous studies have used different approximations which reduce either to linearizing the problem [2, 3] or to use of the Hertz-Knudsen equation [4]. The inaccurate procedure of matching free molecular and diffusive flows at some distance from the surface of the droplet [5] is completely unsuitable in the absence of a neutral gas. Equations for the rate of growth of a droplet in a slightly supercooled vapor were obtained in [6] from a solution of the ellipsoidal kinetic model by the method of (expansion of) moments.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 97–102, March–April, 1976.     

The study of the induction of subsonic wind tunnels with an axisymmetric working part

For the case of small perturbations of the velocity on the boundary of the flow and arbitrary degree of perforation of the wall of the wind tunnel which is constant along its length, a solution has been found for the axisymmetric boundary-value problem of subsonic flow around a thin body of revolution in perforated boundaries. The boundary condition connects the tangential component of the perturbed velocity and the component normal to the wall, and has a general form for the whole boundary. From the solution obtained, the optimum degree of perforation of the wall is found for which distortion of the pressure coefficient on the surface of the model is a minimum in comparison with the unbounded flow round the body. The questions of the induction of the walls of the working part of the wind tunnel, the formulation of the boundary condition, and the determination of the corrections to the aerodynamic characteristics of the model are considered in a number of studies [1–5]. Lately a promising method was been worked out for reducing the influence of the walls directly in the process of experiment by regulating the parameters of the gas near the boundary in such a way that they correspond to the parameters in flow round the same body by an unbounded flow [3, 5]. However, the design properties of many working tunnels do not permit this method to be applied. The induction of such tunnels as these can be reduced only by the choice of an optimum degree of perforation of the walls. In the articles [6–8] there is a study of the effect of perforated boundaries on the flow round a profile, the degree of perforation in [7] being assumed to be small.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 150–154, January–February, 1985.The author is grateful to V. M. Neiland for his constant attention to the study and for discussion of the results.