Occurrence of periodic regimes in steady supersonic mid flows due to loss of electrical conductivity of medium

A study is made of the features of supersonic magnetohydrodynamic (MHD) flows due to the vanishing of the electrical conductivity of the gas as a result of its cooling. The study is based on the example of the exhausting from an expanding nozzle of gas into which a magnetic field (Re_m 1) perpendicular to the plane of the flow is initially frozen. It is demonstrated analytically on the basis of a qualitative model [1] and by numerical experiment that besides the steady flow there is also a periodic regime in which a layer of heated gas of electric arc type periodically separates from the conducting region in the upper part of the nozzle. A gas-dynamic flow zone with homogeneous magnetic field different from that at the exit from the nozzle forms between this layer and the conducting gas in the initial section. After the layer has left the nozzle, the process is repeated. It is established that the occurrence of such layers is due to the development of overheating instability in the regions with low electrical conductivity, in which the temperature is approximately constant due to the competition of the processes of Joule heating and cooling as a result of expansion. The periodic regimes occur for magnetic fields at the exit from the nozzle both greater and smaller than the initial field when the above-mentioned Isothermal zones exist in the steady flow. The formation of periodic regimes in steady MHD flows in a Laval nozzle when the conductivity of the gas grows from a small quantity at the entrance due to Joule heating has been observed in numerical experiments [2, 3]. It appears that the oscillations which occur here are due to the boundary condition. The occurrence of narrow highly-conductive layers of plasma due to an initial perturbation of the temperature in the nonconducting gas has previously been observed in numerical studies of one-dimensional flows in a pulsed accelerator [4–6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 138–149, July–August, 1985.     

Some characteristics of supersonic flow of an electrically conductive gas in an MHD-channel

We study supersonic flows of an electrically conductive gas in crossed electric and magnetic fields [1] in the presence of shock waves. It is shown that three steady flow regimes can exist, and that these are defined by the electrical conductivity of the gas as a function of temperature and density.

1. The normal regime is characterized by a tendency for the shock to move toward the channel entrance on increase of the static pressure at the channel exit. The steady regime of this type exists and is stable.
2. The anomalous regime (formally constructed) is characterized by a tendency for the shock to move toward the exit on increase of the static pressure at the channel exit. This regime is unstable and the flow in the MHD-channel may be either entirely supersonic or entirely subsonic.
3. The limiting (boundary) regime is intermediate between the normal and anomalous regimes and is characterized by the fact that the stationary position of the shock wave and its amplitude are not uniquely defined. Steady flow in this case is not unique.

This study involves formal construction both of the solution to the steady-state problem and the corresponding nonsteady-state problem [4]. The establishment of a steady regime in the solution of the unsteady problem, is at the same time, a verification of its stability.     

Solution of the direct problem of the mixed flow of a conducting gas in a laval nozzle with a meridional magnetic field

We consider the direct problem in the theory of the axisymmetric Laval nozzle (including sonic transition) for the steady flow of an inviscid and nonheat-conducting gas of finite electrical conductivity. The problem is solved by numerical integration of the equations of unsteady gas flow using an explicit difference scheme that was proposed by Godunov [1,2], and was used to calculate steady and unsteady flows of a nonconducting gas in nozzles by Ivanov and Kraiko [3]. The subsonic and the supersonic flows of a conducting gas in an axisymmetric channel when there is no external electric field, the magnetic field is meridional, and the magnetic Reynolds numbers are small have previously been completely investigated. Thus, Kheins, Ioller and Élers [4] investigated experimentally and theoretically the flow of a conducting gas in a cylindrical pipe when there is interaction between the flow and the magnetic field of a loop current that is coaxial with the pipe. Two different approaches were used in the theoretical analysis in [4]: linearization with respect to the parameter S of the magnetogasdynamic interaction and numerical calculation by the method of characteristics. The first approach was used for weakly perturbed subsonic and supersonic flows and the solutions obtained in analytic form hold only for small S. This is the approach used by Bam-Zelikovich [5] to investigate subsonic and supersonic jet flows through a current loop. The numerical calculations of supersonic flows in a cylindrical pipe in [4] were restricted to comparatively small values of S since, as S increases, shock waves and subsonic waves appear in the flow. Katskova and Chushkin [6] used the method of characteristics to calculate the flow of the type in the supersonic part of an axisymmetric nozzle with a point of inflection. The flow at the entrance to the section of the nozzle under consideration was supersonic and uniform, while the magnetic field was assumed to be constant and parallel to the axis of symmetry. The plane case was also studied in [6]. The solution of the direct problem is the subject of a paper by Brushlinskii, Gerlakh, and Morozov [7], who considered the flow of an electrically conducting gas between two coaxial electrodes of given shape. There was no applied magnetic field, and the induced magnetic field was in the direction perpendicular to the meridional plane. The problem was solved numerically in [7] using a standard process. However, the boundary conditions adopted, which were chosen largely to simplify the calculations, and the accuracy achieved only allowed the authors [7] to make reliable judgments about the qualitative features of the flow. Recently, in addition to [7], several papers have been published [8–10] in which the authors used a similar approach to solve the direct problem in the theory of the Laval nozzle (in the case of a nonconducting gas).Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 5, pp. 14–20, September–October, 1971.In conclusion the author wishes to thank M. Ya. Ivanov, who kindly made available his program for calculating the flow of a conducting gas, and also A. B. Vatazhin and A. N. Kraiko for useful advice.     

Acoustic,entropy, and vorticity perturbations in a channel of variable cross section

A study is made of the problem of the propagation of infinitesimally small perturbations in a gas stream moving in a channel of variable cross section when the flow cannot be regarded as isentropic and irrotational. The solution is found in the framework of the linear theory of the flow of an ideal gas and the quasi-one-dimensional hydraulic approximation for the steady regime. For irrotational and isentropic perturbations in a nozzle, this problem was considered in [1–4]. In [1], the problem is generalized to take into account entropy perturbations in the nozzle for the case of longitudinal oscillations. The present paper treats arbitrary modes in a nozzle and takes into account not only entropy but also vorticity perturbations in the moving stream. For each of the three perturbation types — acoustic, entropy, and vorticity — the solutions are expanded in series in cylindrical functions. It is shown that in the considered approximation each oscillation mode can be analyzed independently of the others. In the special case of flow in a Laval nozzle, the concept of impedance (admittance), which is widely used in acoustics, is generalized to take into account entropy and vorticity perturbations. The contribution to the flow dynamics of the acoustic, entropy, and vorticity perturbations is estimated numerically for longitudinal and transverse modes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 91–98, January–February, 1982.     

Boundary layer on a rotating disk with account for the Hall effect

The steady rotation of a disk of infinite radius in a conducting incompressible fluid in the presence of an axial magnetic field leads to the formation on the disk of a three-dimensional axisymmetric boundary layer in which all quantities, in view of the symmetry, depend only on two coordinates. Since the characteristic dimension is missing in this problem, the problem is self-similar and, consequently, reduces to the solution of ordinary differential equations.Several studies have been made of the steady rotation of a disk in an isotropically conductive fluid. In [1] a study was made of the asymptotic behavior of the solution at a large distance from the disk. In [2] the problem is linearized under the assumption of small Alfven numbers, and the solution is constructed with the aid of the method of integral relations. In the case of small magnetic Reynolds numbers the problem has been solved by numerical methods [3,4]. In [5] the method of integral relations was used to study translational flow past a disk. The rotation of a weakly conductive fluid above a fixed base was studied in [6,7], The effect of conductivity anisotropy on a flow of a similar sort was studied approximately in [8], In the following we present a numerical solution of the boundary-layer problem on a disk with account for the Hall effect.     

Calculation of the boundary layer on the electrically conducting wall of a plane channel

There are presently available quite a large number of works devoted to the study of the motion of an electrically conducting fluid in boundary layers formed on electrodes or on the nonconducting walls of various MHD devices. However, the methods of solving the boundary layer equations in these studies are based on various simplifying assumptions which allow the problem to be reduced to the solution of a system of ordinary differential equations. Thus, in [1] there is imposed on the flow the special magnetic fieldH1/x, which enables the problem to be reduced to the self-similar form, while in the studies of other authors [2, 3] either the solution is sought in the form of expansions in x, or it is assumed that the problem is locally self-similar [4]. In the present paper we construct the solution of the MHD boundary layer equations which is obtained by one of the numerical methods which has long been used for solving the boundary layer equations for a nonconducting fluid.     

Effect of hall currents on the flow of a conducting gas at high flow velocities

In [1] the flow of a compressible fluid was examined for the case when the conductivity = with account for the Hall effect. Oates [2] solves the problem of the influence of Hall currents on the flow in an accelerator for channels having a very small ratio of height to length when the velocity component in the direction of the channel height may be assumed to be zero. The problem of the influence of Hall currents on the flow of a conducting gas of finite conductivity is solved below for the case when the gas is accelerated to high velocities ( 50–100 km/sec) with account for the presence of two velocity components.     

Rising vortex ring in a viscous fluid

Most theoretical results for thermals, whose motion is determined by the complex interaction between dynamics and buoyancy, have been obtained numerically [1–4]. The analytic solutions for a convection element have been limited to consideration of the self-similar regime [5]. At the same time, the preself-similar stage of development of a vortex ring of dynamic origin has been described analytically [6]. This approach is now extended to a rising vortex ring. In this case a modification of the traditional formulation of the problem makes it possible to obtain an analytic solution of the problem of a weak thermal in the form of unsteady temperature, vorticity and stream function fields that tend in the limit to the self-similar regime. The rate of ascent of the convective vortex ring is found. A solution is obtained for the two-dimensional analog of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–48, May–June, 1989.     

Extremal nozzle contours for gas flows with particle lag

The determination of the extremal nozzle contour for gas flow without foreign particles has been carried out in several studies [1–6], based on the calculation of the flow field using the method of characteristics.In [7, 8] the equations are derived for the characteristics and the relations along the streamlines which are required for calculating two-dimensional gas flow with foreign particles. The variational problem for two-phase flow in the two-dimensional formulation may be solved by the method of Guderley and Armitage [9] with the use of equations given in [7] or [8]; however this method is very tedious, even with the use of high-speed computers.In [10, 11] studies are made of two-phase one-dimensional flows by expanding the unknown functions in series in a small parameter, defined by the particle dimensions. In [12] a solution is given for the variational problem (in the one-dimensional formulation) of designing the contour of a nozzle with maximal impulse. However that study does not take account of the static term appearing in the impulse and the solution is obtained in relative cumbersome form. Moreover, the question of account for the losses due to nonparallelism and nonuniformity of the discharge was not considered.The present paper considers in the one-dimensional formulation the flow of a two-phase medium in a Laval nozzle with small particle lags (in velocity and temperature). The variational problem of determining the maximal nozzle impulse is formulated along the nozzle contour for fixed geometric expansion ratio. The impulse losses due to nonparallelism of the discharge are simulated by a function which depends on the ordinates which are variable along the contour and on the slope of the tangent to the contour.The author wishes to thank Yu. D. Shmyglevskii and A. N. Kraiko for helpful discussions and V. K. Starkov for carrying out the calculations on the computer.     

Propagation of planar shock waves in an exponential atmosphere

We consider planar explosions in a medium with an exponential density distribution. In contrast to the so-called sectorial approximation [1] we take into account the overflow of energy from a lower region to an upper region, so that our solution of the problem considered here gives a truer picture of the flow of the gas at a later stage of a point explosion in a nonhomogeneous atmosphere. The numerical solution in both upper and lower regions of the flow merges into the corresponding limiting self-similar regime [2, 3]. The calculations are carried out up to a gap in the atmosphere [4]. The computational method is based on implicit difference approximations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 5, pp. 31–35, September–October, 1971.The authors are deeply grateful to L. A. Chudov for his constant interest in our work and for useful discussions, and they also wish to thank É. I. Andriankin for meaningful discussions of the paper.     

Numerical solution of the direct-problem of mixed nozzle flow

A large part of the known results of Laval nozzle theory relates to the inverse problem, in which the velocity distribution on some line (usually the axis of symmetry) is given rather than the nozzle contour. Many important properties of transonic flows have been disclosed as a result of numerous studies, whose basic results were presented together with an extensive bibliography in Ryzhov's monograph [1]. The solution of the inverse problem has recently been used not only to analyze the qualitative characteristics but also to construct nozzles with rather marked variation of the slope of the generator, which are of practical interest. In this connection we note the work of Pirumov [2] and also the studies of Hopkins and Hill [3, 4]. The latter authors, in addition to the classical Laval nozzle, studied several nozzle schemes with a centerbody. Pirumov used a specially developed numerical method for the solution of the inverse problem (we note that in the subsonic part of the nozzle the corresponding Cauchy problem is incorrect), while Hopkins and Hill used a series expansion which was preceded by a change of variables.There are considerably fewer studies devoted to the solution of the direct problem of mixed nozzle flow. Numerical methods have been used by Alikhashkin, Favorskii, and Chushkin [5], Favorskii [6], and Danilov [7], with the method of integral relations being used in the first two studies. Finally, there has recently been extensive development of the method of expansion in powers of ^1/2, where is the ratio of the radius (or half-width of the nozzle to the radius of curvature of the wall, calculated at the throat section. Such expansions have been used by Hall [8] and Kliegel and Quan [9] to study flow in classical Laval nozzles, and by Moore [10] and Moore and Hall [11] to study flow in nozzles with a centerbody. We note that the ^1/2-expansion method is suitable only in those cases in which the wall radii of curvature are large.In the following the asymptotic method is used to solve the direct problem of mixed flow in nozzles. This reduces the very complex boundary value problem for an elliptic-hyperbolic system of equations with two unknown variables to the Cauchy problem (more precisely, to a mixed problem with initial conditions in a bounded two-dimensional region and boundary conditions which are independent of the third variable) for a hyperbolic system with three unknown variables. The integration of the equations describing the two-dimensional (plane of axisymmetric) nonsteady flow was accomplished with the aid of the Godunov-Zabrodin-Prokopov difference scheme [12]. Several types of nozzles with centerbody are calculated as well as the classical Laval nozzle. The contours of the subsonic parts of the nozzles were either closed (finite combustion chamber) or open (nozzle joins an infinite cylindrical tube). In the first case the flow is provided by three-dimensional mass and energy sources which are introduced at some fixed part of the combustion chamber. In the second case there are no mass and energy sources, but a boundary condition is established at a plane perpendicular to the nozzle axis and located at a finite distance from the throat section, and this condition becomes the flow uniformity condition as this plane moves away to infinity.The authors wish to thank I. Yu. Brailovskii for valuable advice in the selection of the difference scheme, U. G. Pirumov for the kind offer of the results of his calculations, and A. M. Konkina and L. P. Frolova for assistance in the calculations.     

Asymptotic theory of the flow around an obstacle by a sonic flow

The first investigation of the problem of the flow around an obstacle by a gas flow whose velocity is equal to the speed of sound at infinity was carried out in [1, 2], where it is shown in particular that the principal term of the appropriate asymptotic expansion is a self-similar solution of Tricomi's equation, to which the problem reduces in the first approximation upon a hodographic investigation. The requirement that the stream function be analytic as a function of the hodographic variables on the limiting characteristic was an important condition determining the selection of the self-similarity exponent n (xy^–n is an invariant of the self-similar solution). The analytic nature of the velocity field everywhere in the flow above the shock waves, which arise from necessity upon flow around an obstacle, follows from this condition. The latter was found in [3], where one of the branches of the solution obtained in [1] was used in the region behind the shock waves. The principal and subsequent terms of the asymptotic expansion describing a sonic flow far from an obstacle were discussed in [4], where the author restricted himself to Tricomi's equation. Each term of the series constructed in [4] contains an arbitrary coefficient (we will call it a shape parameter) which is not determined within the framework of a local investigation, and consideration of the problem of flow around a given obstacle as a whole is necessary in order to determine these shape parameters. It follows from the results of [4] that the problem of higher approximations to the solution of [1] coincides with the problem, of constructing a flow in the neighborhood of the center of a Laval nozzle with an analytic velocity distribution along the longitudinal axis (a Meyer-type flow). Along with the Meyer-type flow in the vicinity of the nozzle center, which corresponds to a self-similarity exponent n=2, two other types of flow are asymptotically possible with n=3 and 11, given in [5]. The appropriate solutions are written out in algebraic functions in [6]. The results of [5] show that the condition that the velocity vector be analytic on the limiting characteristic in the flow plane is broader than the condition that the stream function be analytic as a function of the hodographic variables, which is employed in [1, 2, 4]. Therefore, the necessity has arisen of reconsidering the problem of higher approximations for the obstacle solution of F. I. Frankl'. It has proved possible for the region in front of the shock waves to use a series which is more general than in [4], which implies the inclusion of an additional set of shape parameters. The solution is given in the hodograph plane in the form of the sum of two terms; the series discussed in [4] corresponds to the first one, and the series generated by the self-similar solution with n=3 or with n=11 corresponds to the second one.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 99–107, May–June, 1979.The authors thank S. V. Fal'kovich for a useful discussion.     

The problem of a gas bubble in plane ideal fluid flow

We study the problem of two-dimensional fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall.Two-dimensional ideal fluid flow past a gas bubble on whose boundary surface-tension forces act (or a gas bubble bounded by an elastic film) has been studied by several authors. Zhukovskii, who first studied jet flows with consideration of the capillary forces, constructed an exact solution of the problem of symmetric flow past a gas bubble in a rectilinear channel [1]. However, Zhukovskii's solution is not the general solution of the problem; in particular, we cannot obtain the flow past an isolated bubble from his solution. Slezkin [2] reduced the problem of symmetric flow of an infinite fluid stream past a bubble to the study of a nonlinear integral equation. The numerical solution of this problem has recently been found by Petrova [3]. McLeod [4] obtained an exact solution under the assumption that the gas pressure p₁ in the bubble equals the flow stagnation pressure p₀. Beyer [5] proved the existence of a solution to the problem of flow of a stream having a given velocity circulation provided p₁p₀.We examine the problem of two-dimensional ideal fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall. The solution depends on the value of the contact angle . The existence of a solution is proved in some range of variation of the parameters, and a technique for finding this solution is given. The situation in which =1/2 is studied in detail.     

Two-dimensional flow of a perfect conducting gas in crossed electric and magnetic fields

Reference [1, 2] give a solution of the problem of the two-dimen-, sional flow of an inviscid thermally-nonconducting gas with constant conductivity in a channel of constant cross section for particular forms of the given applied magnetic field. The present paper obtains a solution of the problem of the two-dimensional flow of a gas with variable conductivity in crossed electric and arbitrary magnetic fields by means of the small parameter method. The magnetic Reynolds number R_m and the magnetohydrodynamic interaction parameter S are chosen as parameters. The international system of units is employed.Notation V flow velocity - j electric current density - p pressure in the flow - E electric field strength - gas density - electrical conductivity of the gas - T gas temperature - ratio of specific heats at constant pressure and volume - L channel half-height - ] permeability (magnetic) - B magnetic induction vector - B₀ applied magnetic field     

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.     

Cylinder in viscous incompressible fluid flow

We present a technique for calculating the temperature field in the vicinity of a cylinder in a viscous incompressible fluid flow under given conditions for the heat flux or the cylinder surface temperature. The Navier-Stokes equations and the energy equation for the steady heat transfer regime form the basis of the calculations. The numerical calculations are made for three flow regimes about the cylinder, corresponding to Reynolds numbers of 20, 40, and 80. The pressure distribution, voracity, and temperature distributions along the cylinder surface are found.It is known that for a Reynolds number R>1 the calculation of cylinder drag within the framework of the solution of the Oseen and Stokes equations yields a significant deviation from the experimental data. In 1933 Thom first solved this problem [1] on the basis of the Navier-Stokes equations. Subsequently several investigators [2, 3] studied the problem of viscous incompressible fluid flow past a cylinder.It has been established that a stable solution of the Navier-Stokes equations exists for R40 and that in this case the calculation results are in good agreement with the experimental data. According to [2], a stable solution also exists for R=44. The possibility of obtaining a steady solution for R>44 is suggested.Analysis of the results of [2] permits suggesting that the questions of constructing a difference scheme with a given order of approximation of the basic differential relations which will permit obtaining the sought solution over the entire range of variation of the problem parameters of interest are still worthy of attention.Calculation of the velocity field in the vicinity of a cylinder also makes possible the calculation of the cylinder temperature regime for given conditions for the heat flux or the temperature on its surface. However, we are familiar only with experience in the analytic solution of several questions of cylinder heat transfer with the surrounding fluid for large R within the framework of boundary layer theory [4].     

Interaction of a jet exhausting from a body with an opposing supersonic stream of rarefied gas

The experimental investigation of supersonic flow past a sphere with a jet exhausting from the front point of the sphere into the flow at large [1] and moderate [2] Reynolds numbers Re has revealed an effect of shielding from the oncoming stream, this leading to a decrease in the drag coefficient of the sphere and of the energy flux to it. A numerical simulation of the flow has been made in the case of supersonic flow past a sphere with a sonic jet from a nozzle situated on the symmetry axis in the continuum regime [3]. In the present paper, this problem is investigated for flow of a rarefied gas on the basis of numerical solution of a model kinetic equation for a monatomic gas.     

On the Quasi‐Steadiness Hypothesis as Applied to Gas Exhaustion from a Receiver

A semiempirical method of determining the stabilization time for a quasisteady mode of gas exhaustion from a receiver after sudden opening of the nozzle and the time evolution of the real flow rate at the stage of the transitional process are considered. The numerical solution of the equations of exhaustion gas dynamics in a twodimensional formulation and the results of model experiments demonstrated that the method can be used to estimate the conditions of applicability of the quasisteadiness hypothesis and to determine the discharge coefficient of the nozzle with controlled accuracy.     

Application of unsteady mixing equations to some aerodynamic problems

Tangential discontinuities [1] are introduced in solving several transient and steady-state problems of gas dynamics. These discontinuities are unstable [2] as a result of the effects of viscosity and thermal conductivity. Therefore it is advisable to replace the tangential discontinuity by a mixing region and account for its interaction with the inviscid flows, establishing on the boundaries of this region the conditions of vanishing friction stress and equality of the velocity and temperature components to the corresponding velocity and temperature components of the inviscid flows. This formulation improves the accuracy of the solution of such problems by posing them as problems with irregular reflection and intersection of shock waves [1].The consideration of the interaction of unsteady turbulent mixing regions with the inviscid flow also permits the formulation of several problems in which the effects of viscosity lead to complete rearrangement of the flow pattern (the lambda-configuration) with the interaction of the reflected shock wave with the boundary layer in the shock tube [3,4], the formation of zones of developed separation ahead of obstacles, etc.).In this connection, §1 presents an analysis of the self-similar solutions of the unsteady turbulent mixing equations (a corresponding analysis of the laminar mixing equations which coincide with the boundary layer equations is presented in [1]). It is shown that these self-similar solutions describe, along with the several problems noted above, the problems of the formation of steady jets and mixing zones in the base wake.As an example, §2 presents, within the framework of the proposed schematization, an approximate solution of the problem of the interaction of a shock wave reflected from a semi-infinite wall with the boundary layer on a horizontal plate behind the incident shock wave. The results obtained are applied to the analysis of reflection in a shock tube. Computational results are presented which are in qualitative agreement with experiment [3, 4].     

Dimensionless numbers in the problem of the interaction of a freely expanding jet with a plate

An approximate solution is obtained to the problem of the force interaction of a jet of ideal gas exhausting from an axisymmetric nozzle onto a plate. Dimensionless numbers are found together with variables in which the solution to this problem for the general case of lateral interaction of the jet with the plate has a self-similar form.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 169–173, March–April, 1981.