Two-phase couette flow

A study is made of plane laminar Couette flow, in which foreign particles are injected through the upper boundary. The effect of the particles on friction and heat transfer is analyzed on the basis of the equations of two-fluid theory. A two-phase boundary layer on a plate has been considered in [1, 2] with the effect of the particles on the gas flow field neglected. A solution has been obtained in [3] for a laminar boundary layer on a plate with allowance for the dynamic and thermal effects of the particles on the gas parameters. There are also solutions for the case of the impulsive motion of a plate in a two-phase medium [4–6], and local rotation of the particles is taken into account in [5, 6]. The simplest model accounting for the effect of the particles on friction and heat transfer for the general case, when the particles are not in equilibrium with the gas at the outer edge of the boundary layer, is Couette flow. This type of flow with particle injection and a fixed surface has been considered in [7] under the assumptions of constant gas viscosity and the simplest drag and heat-transfer law. A solution for an accelerated Couette flow without particle injection and with a wall has been obtained in [6]. In the present paper fairly general assumptions are used to obtain a numerical solution of the problem of two-phase Couette flow with particle injection, and simple formulas useful for estimating the effect of the particles on friction and heat transfer are also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–46, May–June, 1976.     

A spatial laminar boundary layer on a streamline in conical external flow of a uniform gas in the absence of sources and sinks

Theoretical study of a three-dimensional laminar boundary layer is a complex problem, but it can be substantially simplified in certain particular cases and even reduced to the solution of ordinary differential equations.One such particular case is the flow of a compressible gas on a streamline in conical external flow. The case is of considerable practical importance because the local heat fluxes may take extremal values on such lines.Such flow, except for the conical case, has been examined [1–4], and an approximate method has been given [1] on the basis of integral relationships and a special form for the approximating functions. A numerical solution has been given [2, 3] for such flow around an infinite cylinder. It was assumed in [1–3] that the Prandtl number and the specific heats were constant, and that the dynamic viscosity was proportional to temperature. Heat transfer has been examined [4] near a cylinder exposed to a flow of dissociated air.Here we give results from numerical solution of a system of ordinary differential equations for the flow of a compressible gas in a laminar boundary layer on streamlines in conical external flow, with or without influx or withdrawal of a homogeneous gas. It is assumed that the gas is perfect and that the dynamic viscosity has a power-law temperature dependence.     

Ignition of hydrogen in a turbulent boundary layer on a plate

The ignition of hydrogen blown into a turbulent supersonic boundary layer on a flat plate is investigated numerically. It is assumed that the mixture consists of six chemically active components H, O, OH, H₂O, O₂, H₂ and inert nitrogen N₂. The boundary layer is divided into outer and inner regions, for which different expressions for the coefficients of turbulent transport are used. The influence of pulsations on the rates of the chemical reactions, and also the back reaction of the chemical processes on the mechanism of turbulent transfer are not taken into account. The surface of the plate is assumed to be absolutely catalytic with respect to the recombination reactions of the H and O atoms. The influence of the blowing intensity, the Mach number in the outer flow, and the pressure on the ignition delay is analyzed. The possibility of effective porous cooling of the surface when there is combustion in the boundary layer is demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 33–40, November–December, 1979.I thank V. G. Gromov and V. A. Levin for their interest in the work.     

Theory of a catalytic calorimeter

An analytic solution is found to the problem of heat transfer to a catalytic calorimeter on a plate, the catalytic properties of the plate and the calorimeter being different. The solution is obtained under the assumption of a chemically frozen flow of a binary mixture in the boundary layer. The method of mass-average variables is used [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 192–195, January–February, 1980.     

Theory of flame propagation in a gas and drop mixture

The equations of a reacting multiphase continuous medium [1] are used to investigate the problem of steady-state flame front propagation in a gas mixture with evaporating drops. A simple model for ignition of the liquid drops is proposed which is based on the application of the method of equally accessible surfaces [2] to the heat and mass exchange processes between the microflames surrounding the separate drops, the drops, and the carrying gas medium. The parameter distributions in the macroscopic flame front as well as the dependences of the flame propagation velocity in the gas suspension on a number of parameters governing the process under investigation are represented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 101–108, July–August, 1973.     

Conjugated heat and mass transfer between a reactive solid and a gas in the presence of nonequilibrium chemical reactions

An investigation of a multicomponent boundary layer taking account of nonequilibrium chemical reactions has been made in a number of publications [1–3]; here, the temperature of the solid was assumed to be known or was determined from the condition of the conservation of energy at the interface between the gas and the solid, taking account of the solution of the equation of thermal conductivity in the solid phase. At the same time, heating of the material of a coating is an unavoidable step in any mechanism of thermokinetic decomposition and, in view of this, it is necessary to take account of the lag of the heat-transfer process inside the solid. Therefore, it is necessary to solve the equation of the energy balance in the solid phase simultaneously with the system of the equations of the boundary layer, i.e., the conjugate problem. The present article discusses the problem of flow around a solid in the vicinity of a frontal critical point, taking account of the dependence of the processes taking place in the solid body on the time, in the presence of two heterogeneous and one homogeneous reactions. The distributions of the velocity, the temperature, and the concentrations in the boundary layer are obtained, as well as the mass rate of entrainment of the material at different moments of time. The time of the change between kinetic and diffusion conditions of the course of the heterogeneous chemical reactions (the ignition time) is determined. It is established that, in the presence of a homogeneous chemical reaction, the mass rate of entrainment is less than with a frozen flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 121–128, March–April, 1974.     

Multicomponent diffusion and heat exchange during the flow of an ionized gas in chemical equilibrium about a body

In the present paper, we investigate the flow of a multicomponent, partially ionized gas mixture in the boundary layer around an impermeable surface; the state in the boundary layer being one of chemical equilibrium. As an example, the flow of partially ionized air about a sphere is considered. The chemical composition at every point of the boundary layer with running values of the pressure, temperature, and concentrations of the chemical elements was found by Newton's method, parallel with the integration of the fundamental system of nonself-similar differential equations by the curve-fitting method. Also discussed is another method of finding the composition, which is more economical from the point of view of expenditure of machine time. It is found that the dimensionless heat flux to the wall at the front critical point of the sphere can be 26% larger for constant concentrations of the chemical elements than it is for variable concentrations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 53–59, May–June, 1972.The author thanks G. A. Tirskii for discussing the work.     

Blowing into a supersonic stream through a permeable surface

Distributed blowing of gas into a supersonic stream from flat surfaces using an inviscid flow model was studied in [1–9]. A characteristic feature of flows of this type is the influence of the conditions specified on the trailing edge of the body on the complete upstream flow field [3–5]. This occurs because the pressure gradient that arises on the flat surface is induced by a blowing layer whose thickness in turn depends on the pressure distribution on the surface. The assumption of a thin blowing layer makes it possible to ignore the transverse pressure gradient in the layer and describe the flow of the blown gas by the approximate thin-layer equations [1–5]. In addition, at moderate Mach numbers of the exterior stream the flow in the blowing layer can be assumed to be incompressible [3]. In [7, 8] a solution was found to the problem of strong blowing of gas into a supersonic stream from the surface of a flat plate when the blowing velocity is constant along the length of the plate. In the present paper, a different blowing law is considered, in accordance with which the flow rate of the blown gas depends on the difference between the pressures on the surface over which the flow occurs and in the reservoir from which the gas is supplied. As in [8, 9], the solution is obtained analytically in the form of universal formulas applicable for any pressure specified on the trailing edge of the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 108–114, September–October, 1980.I thank V. A. Levin for suggesting the problem and assistance in the work.     

Numerical study of flows of a viscous compressible gas and of an equilibrium gas mixture in a flat nozzle in the presence of strong blowing

The problem of the interaction of a viscous supersonic stream in a flat nozzle with a transverse gas jet of the same composition blown through a slot in one wall of the nozzle is examined. The complete Navier-Stokes equations are used as the initial equations. The statement of the problem in the case of the absence of blowing coincides with [1]. The conditions at the blowing cut are obtained on the assumption that the flow of the blown jet up to the blowing cut is described by one-dimensional equations of ideal gasdynamics. The proposed model of the interaction is generalized to the case of flow of a multicomponent gas mixture in chemical equilibrium. The exact solutions found in [2] are used as the boundary conditions at the entrance to the section of the nozzle under consideration. The results of numerical calculations of the flows of a homogeneous nonreacting gas and of an equilibrium mixture of gases consisting of four components (H₂, H₂O, CO, CO₂) are given for different values of the parameters of the main stream and of the blown jet. In the latter case it is assumed that the effect of thermo- and barodiffusion can be neglected.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 55–63, July–August, 1974.     

Formation of a region of interaction and of a wake with the instantaneous start of a plate in an incompressible liquid

The flow arising in an incompressible liquid if, at the initial moment of time, a plate of finite length starts to move with a constant velocity in its plane, is discussed. For the case of an infinite plate, there is a simple exact solution of the Navier—Stokes equations, obtained by Rayleigh. The case of the motion of a semiinfinite plate has also been discussed by a number of authors. Approximate solutions have been obtained in a number of statements; for the complete unsteadystate equations of the boundary layer the statement was investigated by Stewartson (for example, [1–3]); a numerical solution of the problem by an unsteady-state method is given in [4]. The main stress in the present work is laid on investigation of the region of the interaction between a nonviscous flow and the boundary layer near the end of a plate. In passing, a solution of the problem is obtained for a wake, and a new numerical solution is also given for the boundary layer at the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–8, March–April, 1977.     

Similar solution of laminar boundary layer equations with a flame front

This paper is devoted to the general question of the effect of chemical reactions taking place in the boundary layer with the release of heat on the heat transfer, mass transfer, and friction. In the most general formulation the problem is associated with extremely complex computations because of the presence of many components in the mixture of gases. Moreover, to date the chemical kinetics have received little study, and there are simply no reliable data that can be used in the calculations. Therefore we consider in the following a simplified case in which the chemical reaction proceeds in an infinitely narrow region-a flame front. Mathematically this front is a surface of discontinuity of the concentration and temperature derivatives. Physically this corresponds to the limiting case of chemical equilibrium, in which equilibrium occurs with complete combustion and is realized at moderate temperatures and quite high pressures.This formulation of the problem simplifies the calculation considerably, since the gas mixture may be considered everywhere binary. In this case it is not necessary to introduce into consideration either the chemical kinetics or the chemical equilibrium constants. Moreover, in the case of this formulation we may make use of a similar solution of the boundary layer equations. The extreme simplification of the computations does not alter the physical essence of the problem-the effect of the heat sources within the boundary layer on friction, heat transfer and mass transfer. A large number of computations was made to permit clarification of the specific nature of the dependences of the coefficients of friction, and heat and mass transfer on the rate of fuel injection.Several studies have been published in which a similar assumption was made. Avduevskii and Obroskova [1] consider the case when the flame front is located right at the wall. Harnett and Eckert [2] calculated the boundary layer for a flame front on the assumption that the physical parameters do not depend on temperature. This assumption gives a picture which is inaccurate both quantitatively and qualitatively. Finally, when the present study was nearly completed, the article of Libby and Pierucci [3] appeared. In that study the external flow (air) is a mixture of oxygen and an inert gas (argon). Therefore there is a three-component mixture in the boundary layer at every point. This significantly complicates the computation. At the same time, the dependence of the physical parameters on temperature and the concentrations of the mixture components is taken in simplified form (an unjustified simplification, in our view). The authors of [3] carried out only a limited number of computations, which did not make it possible to clarify the characteristic features of the variation of the friction, heat transfer, and mass transfer coefficients as a function of the rate of combustible gas injection.Notation x coordinate along plate in flow direction - y coordinate along normal to plate - u, v velocity components - T absolute temperature - c mass concentration - volume concentration - density - c_p specific heat at constant pressure - viscosity coefficient - D diffusion coefficient - h specific enthalpy - heat conductivity - M Mach number in external stream - k adiabatic exponent in external stream The author wishes to thank V. S. Avduevskii for guiding this investigation.     

Boundary layer in the vicinity of the stagnation point of a body of axial symmetry in a turbulent stream

The flow in the boundary layer in the vicinity of the stagnation point of a flat plate is examined. The outer stream consists of turbulent flow of the jet type, directed normally to the plate. Assumptions concerning the connection between the pulsations in velocity and temperature in the boundary layer and the average parameters chosen on the basis of experimental data made it possible to obtain an isomorphic solution of the boundary layer equations. Equations are obtained for the friction and heat transfer at the wall in the region of gradient flow taking into account the effect of the turbulence of the impinging stream. It is shown that the friction at the wall is insensitive to the turbulence of the impinging stream, while the heat transfer is significantly increased with an increase in the pulsations of the outer flow. These properties are confirmed by the results of experimental studies [1–4].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 83–87, September–October, 1973.     

Boundary layer and heat transfer in a two-phase gas-evaporating droplet medium

An asymptotic model of the flow in the laminar boundary layer of a gas-evaporating droplet mixture is constructed within the framework of the two-continuum approximation. The case of evaporation of the droplets into an atmosphere of their own vapor is examined in detail with reference to the example of longitudinal flow over a hot flat plate. Numerical and asymptotic solutions of the boundary layer equations constructed are found for a number of limiting situations (low droplet concentration, no droplet deposition, significant droplet deposition). The development of the flow with respect to the longitudinal coordinate is studied and it is shown that in the absence of droplet deposition a region of pure vapor may be formed near the surface. Similarity criteria are established and the mechanism of surface heat transfer enhancement is studied for a low evaporating droplet concentration in the boundary layer. In the inertial deposition regime the results of calculating the integral heat transfer coefficient are found to correspond with the experimental data [1].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 42–50, May–June, 1992.     

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.     

Supersonic flow of a combustible gas mixture past a sphere

In recent years considerable interest has developed in the problems of steady-state supersonic flow of a mixture of gases about bodies with the formation of detonation waves and slow combustion fronts. This is due in particular to the problem of fuel combustion in a supersonic air stream.In [1] the problem of supersonic flow past a wedge with a detonation wave attached to the wedge apex is solved. This solution is based on using the equation of the detonation polar obtained in [2]-the analog of the shock polar for the case of an exothermic discontinuity. In [3] a solution is given of the problem of cone flow with an attached detonation wave, and [4] presents solutions of the problems of supersonic flow past the wedge and cone with the formation of attached adiabatic shocks with subsequent combustion of the mixture in slow combustion fronts. In the two latter studies two different solutions were also found for the problem of flow past a point ignition source, one solution with gas combustion in the detonation wave, the other with gas combustion in the slow combustion front following the adiabatic shock. These solutions describe two different asymptotic pictures of flow of a combustible gas mixture past bodies.In an experimental study of the motion of a sphere in a combustible gas mixture [5] it was found that the detonation wave formed ahead of the sphere splits at some distance from the body into an ordinary (adiabatic) shock and a slow combustion front. Arguments are presented in [6] which make it possible to explain this phenomenon and in certain cases to predict its occurrence.The present paper presents examples of the calculation of flow of a combustible gas mixture past a sphere with a detonation wave in the case when the wave does not split. In addition, the flow near the point at which the detonation wave splits is analyzed for the case when splitting occurs where the gas velocity behind the wave is greater than the speed of sound. This analysis shows that in the given case the flow calculation may be carried out without any particular difficulties. On the other hand, the calculation of the flow for the case when the point of splitting is located in the subsonic portion of the flow behind the wave (or in the region of influence of the subsonic portion of the flow) presents difficulties. This flow case is similar to the problem of the supersonic jet of finite width impacting on an obstacle.     

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.     

Thermal effect of a freely expanding gas jet on a flat plate

An experimental investigation was made into the thermal effect of a single gas jet on a plate at Mach numbers of the nozzles in the range 2–6.1, specific heat ratio = 1.4, total pressure difference up to 6·10⁷, gas temperature 450–520 °K in the forechamber, and pressure in the forechamber (10–20)· 10⁵ Pa. The proposed dimensionless numbers made it possible to obtain generalized dependences of the distribution of the heat flux to the plate on the conditions of the problem. A method of approximate calculation of the heat fluxes is proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 119–126, July–August, 1981.     

Motion of a gas shock layer with a free boundary under nonadiabatic conditions

This paper is a study of the effect of heat input (removal) on the characteristics of a shock layer produced by a gas at high supersonic velocity encountering a mobile boundary, which for generality is assumed to be free. We will use the Chernyi method, which was employed previously to solve the problem of a shock layer in an adiabatic flow [1, 2]. The results obtained can be useful for analysis of the effect of radiation (absorption) and processes involving the relaxation of internal degrees of freedom of molecules, condensation, chemical reactions, etc., whose effect on the gasdynamics of the flow in a shock layer may be similar to heat input or removal [3–5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 152–154, May–June, 1976.The author thanks A. K. Rebrov for discussion of the results.     

Exact solution of the problem of convective heat exchange for an elliptic cylinder and a plate in a fluid with small Prandtl number

The heat exchange problem is solved for an elliptic cylinder and a plate in an incompressible fluid flow with small Prandtl number Pr1. For flow along a plate, the values of the Nusselt number Nu obtained by solving the complete energy equation and the heat boundary layer equation are compared.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 26–31, May–June, 1996.