Intense evaporation of a gas from a two-dimensional periodic surface

Evaporation (or condensation) of a gas is said to be intense when the normal component of the velocity of the gas in the Knudsen layer has a value of the order of the thermal velocity of a molecule, c_T=(2kT/m)^1/2. In this case the distribution function of the molecules with respect to their velocities in the Knudsen layer differs from the equilibrium (Maxwellian) value by its own magnitude. As a result of this, over the thickness of the Knudsen layer the macroparameters also vary by their own magnitudes. So in order to obtain the correct boundary conditions for the Euler gas dynamic equations, it is necessary to solve the nonlinear Boltzmann equation in the Knudsen layer. The problem of obtaining such boundary conditions for the case of a plane surface was considered in [1–11]. In the present study this problem is solved for a two-dimensional periodic surface in the case when the dimensions of the inhomogeneities are of the order of the mean free path of the molecules and the inhomogeneities have a rectangular shape. The flow in the Knudsen layer becomes two-dimensional, and this leads to a considerable complication of the solution of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 132–139, March–April, 1985.In conclusion the author would like to express his gratitude to V. A. Zharov for his valuable advice, and also V. S. Galkin, M. N. Kogan, and N. K. Makashev for discussion of the results obtained.     

Influence of excitation of internal degrees of freedom of molecules on the process of weak evaporation or condensation

A study is made of the process of weak evaporation (or condensation) with allowance for excitation of vibrational and rotational degrees of freedom of diatomic molecules. The solution to the corresponding Knudsen layer problem is obtained on the basis of a model kinetic equation of the type of the Morse equation [1]. A relation is obtained that establishes the connection between the rate of evaporation (or condensation) and the parameters of the surface and the gas above it. The boundary conditions of slip for the equations of gas dynamics are analyzed. The results are compared with the evaporation or condensation in the case of a monatomic gas. The introduction of accommodation coefficients for an evaporating surface is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 98–110, November–December, 1979.     

Recovery factors for a permeable surface and a gas surface film in a supersonic turbulent boundary layer

The recovery factor on a permeable surface has been experimentally determined at various rates of injection of air into a supersonic turbulent boundary layer. On the basis of an analysis of the solutions of the integral momentum and energy equations for a turbulent boundary layer an expression is obtained for the recovery factor. The recovery factor in the region of a protective gas surface film in a supersonic external flow has been experimentally determined.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 131–136, March–April, 1972.     

Supersonic transverse rarefied gas flow past a strip

The supersonic flow of a monatomic gas consisting of hard spherical particles past a flat strip normal to the flow is investigated using the direct simulation Monte-Carlo (DSMC) method. The calculations are performed over the Knudsen and Mach number ranges 0.015–5 and 1.8–15, respectively. The structure of the compressed layer and the aerodynamic characteristics are systematically studied for the Mach number 5 and various Knudsen numbers. The dependences of the compressed-layer thickness in molecular free paths are found. The nonequilibrium processes in the neighborhood of the strip are described on the basis of the data on the temperature anisotropy with respect to three coordinates.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 159–167. Original Russian Text Copyright © 2005 by Maltsev and Rebrov.     

Strong blowing on the surfaces of a body immersed in a supersonic gas stream

This article considers the problem of strong blowing on the surface of a body immersed in a supersonic gas flow. It is not difficult to show that for intense blowing the motion of the blown gas can be described by the Euler equations, and viscosity and transport effects appear only in the neighborhood of the contact surface separating the oncoming flow and the blown gas. It is shown that to a first approximation the pressure is constant across the layer and equal to the pressure at the contact surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, pp. 97–104, No. 5, September–October, 1973.     

Supersonic gas flow over a semi-infinite plate with small harmonic spanwise disturbances

The method of matched asymptotic expansions is used to investigate the problem of supersonic perfect-gas flow over a semi-infinite surface with longitudinal ribbing formed by imposing small transverse harmonic perturbations on a flat plate. The ratio of the maximum amplitude of the surface perturbations to the thickness of the boundary layer is of the order of Re^–1/4. The problem is solved with allowance for four terms of the expansion.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 146–156, January–February, 1993.     

Three-dimensional laminar boundary layer on a permeable surface in the neighborhood of a symmetry plane

Analytical and numerical methods are used to investigate a three-dimensional laminar boundary layer near symmetry planes of blunt bodies in supersonic gas flows. In the first approximation of an integral method of successive approximation an analytic solution to the problem is obtained that is valid for an impermeable surface, for small values of the blowing parameter, and arbitrary values of the suction parameter. An asymptotic solution is obtained for large values of the blowing or suction parameters in the case when the velocity vector of the blown gas makes an acute angle with the velocity vector of the external flow on the surface of the body. Some results are given of the numerical solution of the problem for bodies of different shapes and a wide range of angles of attack and blowing and suction parameters. The analytic and numerical solutions are compared and the region of applicability of the analytic expressions is estimated. On the basis of the solutions obtained in the present work and that of other authors, a formula is proposed for calculating the heat fluxes to a perfectly catalytic surface at a symmetry plane of blunt bodies in a supersonic flow of dissociated and ionized air at different angles of attack. Flow near symmetry planes on an impermeable surface or for weak blowing was considered earlier in the framework of the theory of a laminar boundary layer in [1–4]. An asymptotic solution to the equations of a three-dimensional boundary layer in the case of strong normal blowing or suction is given in [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–48, September–October, 1980.     

Strong evaporation of a filler from a porous body with a plane surface

The problem of strong evaporation of matter filling periodic rectangular semi-infinite channels in a porous two-dimensional body is solved by a method of direct statistical modeling. The depths of the channels, the outer surface elements of the body, and the distance from the outer to the evaporation surface are assumed equal in order of magnitude to the mean free path of the molecules. Boundary conditions are obtained for the gas dynamics equations in Euler form, making it possible to describe adequately the flow outside the Knudsen layer. The flow structure in this last is investigated as a function of the determining parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 130–134, March–April, 1986.In conclusion I express my gratitude to V. S. Galkin and N. K. Makashev for their discussion of the results obtained.     

Some effects of strong blowing of gas into a supersonic flow

The intense evaporation of bodies moving in the atmospheres of planets at high supersonic velocities has been partly simulated both theoretically [1–5] (numerical calculations of strong blowing in the framework of the Navier-Stokes equations were also made at the Scientific-Research Institute of Mechanics at the Moscow State University by É. A. Gershbein and A. F. Kolesnikov [6]) as well as experimentally [7–9]. Below, the results are given of investigations of strong blowing of gas from the flat end of a cylinder into a supersonic flow at Reynolds numbers such that the mixing layer separating the blown and the oncoming gas is fairly thin. In this case, the mixing layer can be regarded as a contact surface, so that the problem of blowing can be solved in the framework of Euler's equations. The results of a numerical solution are compared with the results of experiments on the separation and profile of the shock wave, the thickness of the blowing layer on the axis, and also on the pressure distribution on the end of the cylinder. It was established experimentally, and then confirmed numerically that there is a downwash of the blown gas on the periphery of a porous end. It is shown that for the same blowing parameter K, which is equal to the ratio of the dynamic head of the blown gas to the dynamic head of the oncoming gas, and for a given distribution of K over the surface of the body the contact surface tends to a certain limiting position with increasing Mach number of the oncoming flow, i.e., the profile of the contact surface is stabilized. The influence of the adiabatic exponent on the thickness of the blowing layer is estimated. The present investigations continue earlier experimental studies, the main results of which have been presented in [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 91–98, January–February, 1980.     

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.     

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.     

Laminar boundary layer on swept-back wings of infinite span at an angle of attack

A study is made of the flow of a compressible gas in a laminar boundary layer on swept-back wings of infinite span in a supersonic gas flow at different angles of attack. The surface is assumed to be either impermeable or that gas is blown or sucked through it. For this flow and an axisymmetric flow an analytic solution to the problem is obtained in the first approximation of an integral method of successive approximation. For large values of the blowing or suction parameters, asymptotic solutions are found for the boundary layer equations. Some results of numerical solution of the problem obtained by the finite-difference method are given for wings of various shapes in a wide range of angles characterizing the amount by which the wings are swept back and also the blowing or suction parameters. A numerical solution is obtained for the equations of the three-dimensional mixing layer formed in the case of strong blowing of gas from the surface of the body. The analytic and numerical solutions are compared and the regions of applicability of the analytic expressions are estimated. On the basis of the solutions obtained in the present paper and studies of other authors a formula is proposed for the calculation of the heat fluxes to a perfectly catalytic surface of swept-back wings in a supersonic flow of dissociated and ionized air at different angles of attack. Flow over swept-back wings at zero angle of attack has been considered earlier (see, for example, [1–4]) in the theory of a laminar boundary layer. In [5], a study was made of flow over swept-back wings at nonzero angle of attack at small and moderate Reynolds numbers in the framework of the theory of a hypersonic viscous shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 27–39, May–June, 1980.We thank G. A. Tirskii for a helpful discussion of the results.     

Role of the Knudsen layer in the problem of droplet evaporation

A study is made of the part played by the Knudsen layer in the problem of weak unsteady evaporation of a spherical droplet in its own vapor. It is shown that use of the classical Hertz—Knudsen formula may lead to appreciable errors, in particular, in the determination of the time required by the droplet temperature to relax to the state corresponding to steady evaporation.Translated from Izvestiya Akademii Nauk SSSR; Mekhanika Zhidkosti i Gaza, No. 1, pp. 127–131, January–February, 1984.     

The evaporation and growth of spherical droplets at intermediate Knudsen numbers

Study of the condensation growth (and evaporation) of spherical particles in the intermediate region of Knudsen numbers leads to the theoretical prediction of the possibility of a maximum in the reduced number flux of vapor molecules at the surface of the droplet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 185–187, March–April, 1984.     

Intense Condensation of a Molecular Gas

The kinetic problem of intense subsonic condensation of a polyatomic gas on a plane surface is solved by the method of semispatial moments. The contribution of the internal degrees of freedom to the total heat capacity is taken into account. The domains of existence of the Boltzmann equation solution are determined for the given method. Analytic expressions for calculating the gas concentration behind the Knudsen layer depending on the Mach number and temperature are derived for the cases of mon-, di-, and triatomic gases.     

Thin weakly curved wings with maximum aerodynamic quality

Lifting wings that only slightly disturb the supersonic gas flow are considered. The plan shape and thickness distribution of the wing and the free-stream parameters are given. The flow problem is solved within the framework of the Prandtl model. The outer potential flow is determined in accordance with the linear theory. The turbulent boundary layer is found by the method of plane sections with allowance for the three-dimensional inviscid flow pattern. A numerical model of the flow is constructed in the class of piecewise-constant functions on characteristic calculation grids [1]. The variational problem of finding the weakly curved middle surface of the wing giving maximum aerodynamic quality is reduced, by analogy with [2], to a problem of nonlinear programming and is solved by the gradient projection method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 165–168, July–August, 1991.     

Diffusiophoresis of an aerosol particle in a binary gas mixture

The diffusion force and rate are calculated for the diffusiophoresis of a spherical particle in a binary gas mixture by solving the gas–kinetic equations. Two schemes of diffusiophoresis are considered: constant–pressure diffusion and diffusion of one mixture component through the other fixed component. The problem is solved by the integral–momentum method at arbitrary Knudsen numbers. Diffuse scattering of the gas molecules on the particle surface is assumed. The Lorentzian and Rayleigh models of a binary gas mixture are considered. The dependences of the force and rate of diffusiophoresis on the Knudsen number and the other determining parameters are analyzed. The results obtained are compared with well–known experimental data.     

Unsteady three-dimensional viscous shock layer in nonuniform gas flow in the neighborhood of a stagnation point

The combined influence of unsteady effects and free-stream nonuniformity on the variation of the flow structure near the stagnation line and the mechanical and thermal surface loads is investigated within the framework of the thin viscous shock layer model with reference to the example of the motion of blunt bodies at constant velocity through a plane temperature inhomogeneity. The dependence of the friction and heat transfer coefficients on the Reynolds number, the shape of the body and the parameters of the temperature inhomogeneity is analyzed. A number of properties of the flow are established on the basis of numerical solutions obtained over a broad range of variation of the governing parameters. By comparing the solutions obtained in the exact formulation with the calculations made in the quasisteady approximation the region of applicability of the latter is determined. In a number of cases of the motion of a body at supersonic speed in nonuniform media it is necessary to take into account the effect of the nonstationarity of the problem on the flow parameters. In particular, as the results of experiments [1] show, at Strouhal numbers of the order of unity the unsteady effects are important in the problem of the motion of a body through a temperature inhomogeneity. In a number of studies the nonstationary effect associated with supersonic motion in nonuniform media has already been investigated theoretically. In [2] the Euler equations were used, while in [3–5] the equations of a viscous shock layer were used; moreover, whereas in [3–4] the solution was limited to the neighborhood of the stagnation line, in [5] it was obtained for the entire forward surface of a sphere. The effect of free-stream nonuniformity on the structure of the viscous shock layer in steady flow past axisymmetric bodies was studied in [6, 7] and for certain particular cases of three-dimensional flow in [8–11].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 175–180, May–June, 1990.     

Effect of the coefficient of condensation on the flow of vapor and the condensation at sub- and supersonic speeds

A method is presented for calculating the process of one-dimensional steady condensation at an arbitrary value of the coefficient of condensation. The method is based on the use of the solutions of [1], which were obtained for the case of total absorption of incident molecules on an interphase surface, and on application of the rule of scaling proposed in [4]. An analysis is made of an experiment [3] on supersonic flow of a stream of air onto a surface cooled by liquid helium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 189–192, March–April, 1988.     

Condensation of an argon—monosilane mixture in a free jet

A mass–spectrometric study of the condensation of pure Ar and a 5% SiH₄ + 95% Ar mixture in a supersonic pulse free jet in a broad interval of stagnation pressures is performed. It is shown that a small content of monosilane in argon leads to the fact that condensation in the mixture begins at lower stagnation pressures than in pure argon; at high stagnation pressures, mixed argon—silane complexes are formed in the flow. The sequence of the stages of cluster formation in the mixture is determined.