Shock stability in magnetogasdynamic channel flows

The question of shock stability in a perfect-gas channel flow was examined in [1] in the onedimensional approximation under various assumptions: the disturbances are not reflected from the channel exit section, weak shock, etc. The results were found to coincide for two specific forms of the boundary conditions at the channel exit, from which it was concluded that the shock was not sensitive to the exit boundary condition. In [2] the question of shock stability was studied numerically in relation to a conducting-gas flow in a flat channel of constant cross section in the presence of a magnetic field (zero electric field intensity). It was established that the shock stability is significantly affected by the form of the conductivity law. A condition for the limiting regime between the stable and unstable regions was also given for flow with a shock wave. It was assumed that the pressure in the channel exit section is given. In this paper the effect of the exit boundary condition on shock stability in gasdynamic and magnetogasdynamic flows is demonstrated for small magnetic Reynolds numbers. Stability criteria are obtained for shocks near the channel exit for a specific exit condition. The influence of electromagnetic effects (conductivity law, electric load factor) on shock stability is investigated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, pp. 16–23, January–February, 1970.The author is grateful to A. G. Kulikovskii for discussing his work.     

Effect of the form of the electrodes on the current distribution in a magnetogasdynamic channel

A solution is given to the plane problem of the flow of a conducting gas across a homogeneous magnetic field in a magnetogasdynamic channel taking account of the Hall effect at small magnetic Reynolds numbers. The channel is formed by two long electrodes, and the cross section of the channel varies slightly and periodically along the gas flow. It is assumed that the electromagnetic forces are small. It is shown that the current distribution in the channel is nonuniform to a consider able degree and that inverse currents can form at the electrodes, with both subsonic and supersonic flows of the conducting gas. Transverse motion of the gas, due to a change in the cross section of the channel, leads to an increase of Joule energy losses. In [1] the current distribution was obtained in a flat channel formed by infinite dielectric walls, with the flow of a steady-state stream of plasma through the channel across a homogeneous magnetic field. With interaction between the flow and the magnetic field, closed current loops develop in the channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 26–33, November–December, 1970.     

Stability of an ionizing shock wave in an electromagnetic field

The stability of a plane shock wave, which is ionizing a gas, against small two-dimensional perturbations in the presence of a uniform electromagnetic field is analyzed. The applied electric field is assumed to be normal to the wave front while the magnetic field is parallel to the front and perpendicular to the plane in which the perturbations are propagating. The medium is assumed to be a two-parameter gas. The case of an ideal gas is analyzed in detail. The presence of an electric field component normal to the front leads to the formation of a surface charge at the front whose interaction with the electromagnetic field considerably affects the stability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 134–142, November–December, 1978.In conclusion, the authors thank A. G. Kulikovskli for useful discussions of the present work.     

Simple stationary waves in an inclined magnetic field

One component of the solution to the problem of flow around a corner within the scope of magnetohydrodynamics, with the interception or stationary reflection of magnetohydrodynamic shock waves, and also steady-state problems comprising an ionizing shock wave, is the steady-state solution of the equations of magnetohydrodynamics, independent of length but depending on a combination of space variables, for example, on the angle. The flows described by these solutions are called stationary simple waves; they were considered for the first time in [1], where the behavior of the flow was investigated in stationary rotary simple waves, in which no change of density occurs. For a magnetic wave, of parallel velocity, the first integrals were found and the solution was reduced to a quadrature. The investigations and the applications of the solutions obtained for a qualitative construction of the problems of streamline flow were continued in [2–8]. In particular, problems were solved concerning flow around thin bodies of a conducting ideal gas. The general solution of the problem of streamline flow or the intersection of shock waves was not found because stationary simple waves with the magnetic field not parallel to the flow velocity were not investigated. The necessity for the calculation of such a flow may arise during the interpretation of the experimental results [9] in relation to the flow of an ionized gas. In the present paper, we consider stationary simple waves with the magnetic field not parallel to the flow velocity. A system of three nonlinear differential equations, describing fast and slow simple waves, is investigated qualitatively. On the basis of the pattern constructed of the behavior of the integral curves, the change of density, magnetic field, and velocity are found and a classification of the waves is undertaken, according to the nature of the change in their physical quantities. The relation between waves with outgoing and incoming characteristics is explained. A qualitative difference is discovered for the flow investigated from the flow in a magnetic field parallel to the flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1976.The author thanks A. A. Barmin and A. G. Kulikovskii for constant interest in the work and for valuable advice.     

Investigation of the properties of a shock wave arising in a supersonic flow of plasma,passing through a transverse magnetic field

In a flow of plasma, set up by an ionizing shock wave and moving through a transverse magnetic field, under definite conditions there arises a gasdynamic shock wave. The appearance of such shock waves has been observed in experimental [1–4] and theoretical [5–7] work, where an investigation was made of the interaction between a plasma and electrical and magnetic fields. The aim of the present work was a determination of the effect of the intensity of the interaction between the plasma and the magnetic field on the velocity of the motion of this shock wave. The investigation was carried out in a magnetohydrogasdynamic unit, described in [8]. The process was recorded by the Töpler method (IAB-451 instrument) through a slit along the axis of the channel, on a film moving in a direction perpendicular to the slit. The calculation of the flow is based on the one-dimensional unsteady-state equations of magnetic gasdynamics. Using a model of the process described in [9], calculations were made for conditions close to those realized experimentally. In addition, a simplified calculation is made of the velocity of the motion of the above shock wave, under the assumption that its front moves at a constant velocity ahead of the region of interaction, while in the region of interaction itself the flow is steady-state.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 86–91, January–February, 1975.     

Experimental investigation of the interaction of a shock wave with a magnetic field

We present certain results of an experimental investigation of the propagation of a shock wave S through a magnetohydrodynamic channel of the Faraday type. Under conditions of short circuiting of the induced currents in the supersonic stream that follows the shock wave, we registered the occurrence of a shock front T. The x-t diagrams of the motion of the S and T shock waves in the channel and behind it are determined. For a number of fixed sections in the channel, we have measured the density and degree of ionization of the gas and determined their time dependence. The investigations were performed in argon, and the ionizing shock wave propagated with Mach numbers 12–13. The magnetic field intensity was 1.5 T.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 153–183, May–June, 1971.     

Stability of a radiation-absorbing ionizing shock wave in an electromagnetic field

The linear stability of a radiation-absorbing ionizing shock wave (light detonation waves) in the presence of a uniform electromagnetic field is investigated. The applied electric field is considered to be normal to the wave front and the magnetic field to be parallel to the front and perpendicular to the plane in which perturbations propagate. The medium satisfies a two-parameter equation of state. Analytic stability criteria are obtained. For a perfect gas the effect of the electromagnetic field and radiation on shock wave stability is determined.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 23–30, January–February, 1996.     

Regular reflection of a magnetohydrodynamic shock wave from a conducting wall

In two-dimensional supersonic gasdynamics, one of the classical steady-state problems, which include shock waves and other discontinuities, is the problem concerning the oblique reflection of a shock wave from a plane wall. It is well known [1–3] that two types of reflection are possible: regular and Mach. The problem concerning the regular reflection of a magnetohydrodynamic shock wave from an infinitely conducting plane wall is considered here within the scope of ideal magnetohydrodynamics [4]. It is supposed that the magnetic field, normal to the wall, is not equal to zero. The solution of the problem is constructed for incident waves of different types (fast and slow). It is found that, depending on the initial data, the solution can have a qualitatively different nature. In contrast from gasdynamics, the incident wave is reflected in the form of two waves, which can be centered rarefaction waves. A similar problem for the special case of the magnetic field parallel to the flow was considered earlier in [5, 6]. The normal component of the magnetic field at the wall was equated to zero, the solution was constructed only for the case of incidence of a fast shock wave, and the flow pattern is similar in form to that of gasdynamics. The solution of the problem concerning the reflection of a shock wave constructed in this paper is necessary for the interpretation of experiments in shock tubes [7–10].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 102–109, May–June, 1977.The author thanks A. A. Barmin, A. G. Kulikovskii, and G. A. Lyubimov for useful discussion of the results obtained.     

Unsteady-state flow of a gas in a channel,resulting from the motion of a piston,with magnetohydrodynamic offtake of energy and luminescence of the gas

The present paper discusses the one-dimensional unsteady-state flow of a gas resulting from the motion of a piston in the presence of weak perturbing factors, with which the investigation of the perturbed (with respect to the usual self-similar conditions) motion reduces to the solution of ordinary differential equations, is indicated. The distributions of the parameters of the gas between the piston and the shock wave are found. The conditions under which there is acceleration or slowing down of the shock front are clarified. As an example, this paper considers the unsteady-state motion of a conducting gas in a channel with solid electrodes under conditions where electrical energy is generated, and the flow of a gas taking radiation into account, under the assumption of optical transparency of the medium. The theory developed is used to solve the problem of the motion of a thin wedge with a high supersonic velocity in an external axial magnetic field, taking account of the luminescence of the layer of heated gas between the wedge and the shock wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 17–25, September–October, 1970.     

Pressure of an arbitrary acoustical wave on a plane

A study is made of the perturbed flow of a gas, brought about by a weak shock wave, falling on a fixed surface at an arbitrary angle. A solution determining the field of the velocities behind the front of the wave in an initially boundary-value problem with movable boundaries for a three-dimensional wave equation is obtained in the form of a double integral, containing an arbitrarily given function determining the parameters of the gas in the incident wave. The region of integration is a region included within an ellipse, whose relative eccentricity is equal to the sine of the angle of inclination of the front of the incident wave. A formula is obtained for the distribution of the pressure at the plane.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 114–116, January–February, 1975.     

Structure of detonation waves,ionizing a gas,in the presence of an electromagnetic field. Cases of low magnetic viscosity

If behind a detonation wave, ionizing a gas, the magnetic Reynolds number is much greater than unity, then in order to describe such waves (just as for ionizing shock waves) complementary relations [1, 2] are necessary. These complementary relations are not the consequence of the basic integral laws, but can be found from a consideration of the wave structure. In [2], the structure of detonation waves, ionizing a gas, was investigated in an oblique magnetic field. It was supposed that the flow in a layer representing the structure is determined by the finite rate of the chemical reaction and the finite electrical conductivity. In the case when the characteristic length of the chemical reaction is much less than the characteristic dissipation length of the magnetic field, the complementary relations which ensure the existence of the structure are obtained in explicit form. The case is considered below when the characteristic length of the chemical reaction is much greater than the dissipation length of the magnetic field. In this case, the complementary relations are obtained in the explicit form.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 95–101, May–June, 1976.     

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.     

Distribution of radiant thermal flux over the surface of a sphere for hypersonic flow of a nonviscous radiating gas past the sphere

In the present study using the Newtonian approximation [1] we obtain an analytical solution to the problem of flow of a steady, uniform, hypersonic, nonviscous, radiating gas past a sphere. The three-dimensional radiative-loss approximation is used. A distribution is found for the gasdynamic parameters in the shock layer, the withdrawal of the shock wave and the radiant thermal flux to the surface of the sphere. The Newtonian approximation was used earlier in [2, 3] to analyze a gas flow with radiation near the critical line. In [2] the radiation field was considered in the differential approximation, with the optical absorption coefficient being assumed constant. In [3] the integrodifferential energy equation with account of radiation was solved numerically for a gray gas. In [4–7] the problem of the flow of a nonviscous, nonheat-conducting gas behind a shock wave with account of radiation was solved numerically. To calculate the radiation field in [4, 7] the three-dimensional radiative-loss approximation was used; in [5, 6] the self-absorption of the gas was taken into account. A comparison of the equations obtained in the present study for radiant flow from radiating air to a sphere with the numerical calculations [4–7] shows them to have satisfactory accuracy.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 44–49, November–December, 1972.In conclusion the author thanks G. A. Tirskii and É. A. Gershbein for discussion and valuable remarks.     

Nonregular interaction of shock waves in magnetohydrodynamics

The nonregular quasisteady interaction of plane-polarized shock waves in a magnetic field is considered. Within the framework of the magnetohydrodynamic approximation, a Mach interaction pattern is proposed and investigated. This pattern contains the Mach shock wave moving between the initial shocks and connecting their branch points. Unlike the gasdynamic case, in the vicinity of both branch points the flow contains, in addition to the Mach shock, up to five waves whose qualitative structure cannot be predicted in advance. Numerical methods of solving the problem are elaborated. The numerical simulations of the phenomenon under consideration were carried out by means of a specially designed software package with the entire range of all the upstream parameters being covered. It was found that the velocity and inclination of the Mach shock depend significantly on the vector of the magnetic induction in the initial state. The wave pattern of the developing flow is a function of all the key parameters, being particularly sensitive to the inclination of the magnetic field. The values of the parameters for which catastrophic restructurings of the wave pattern occur are determined. Sudden changes in the flow and the magnetic field occur in a zone behind the colliding shock waves. The interaction of the shock waves at different angles and for different magnetic fields are analyzed in detail.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 184–198, May–June, 1993.The authors wish to thank G. G. Chernyi, A. B. Vatazhin, A. G. Kulikovskii, and G. A. Lyubimov for helpful discussions of their results.     

Effect of electrical conductivity of flow on current distribution in a magnetogasdynamic channel with allowance for the Hall effect

The effect of a magnetic field on the current distribution on a plane continuous anode situated opposite the cathode in a rectangular magnetogasdynamic channel with an external magnetic field was experimentally investigated. The distributions of the charged-particle density and the electron temperature near the outlet end of the electrodes were measured. The distribution of electrical conductivity in the flow was calculated. The electron density distribution along the channel is attributed to ambipolar diffusion of plasma to the walls. For an interpretation of the current distribution results, the method of integral relations in a linear approximation was used to solve the problem of a constant-velocity flow of a gas with variable electrical conductivity across a magnetic field in a plane magnetogasdynamic channel of constant cross section formed by electrodes of finite length and insulators. The Hall effect was taken into account. Experiments in which the effect of an external magnetic field on the current distribution on plane sectioned short electrodes in a magnetogasdynamic accelerator was investigated were described in [1]. In the present investigation, continuous long electrodes were used. These electrodes prevented the side effects due to coupling of the current to the ends of the electrode sections and helped to reveal some features of the current density distribution on the anode.     

Investigation into the interaction of a shock wave with an acute cone

A numerical investigation has been made into the nonstationary axisymmetric flow which arises from the interaction of a shock wave with a fixed acute cone. A solution has been obtained in self-similar coordinates for regimes in which a shock wave attached to the point of the cone is formed in the gas flow. The region of existence of these regimes has been established and the method of stabilization used to calculate the gas-dynamic functions in the disturbed region. The main attention is devoted to studying the transition from regular reflection to Mach reflection, a study of the features of double Mach reflection, and comparison of the results for the axisynuaetric and two-dimensional cases. The pressure distributions along the surface of the cone are given in a form which makes it possible to obtain dependences valid in a wide range of variation of the intensities of the incident shock wave and for different values of the adiabatlc exponent of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 98–104, May–June, 1980.     

Interaction of a supersonic jet exhausting into vacuum with an obstacle

A study is made of the interaction between an axisymmetric supersonic jet exhausting into vacuum and an obstacle of a fairly complicated configuration and positioned relative to the nozzle in such a way that in the interaction region behind the detached shock wave there is a three-dimensional flow possessing a symmetry plane. The flow in the interaction region is described by the system of equations of motion of an inviscid perfect gas with boundary conditions on the shock wave (Rankine-Hugoniot relation) and on the surface of the obstacle (no-flow condition). The other boundaries of the region are the symmetry plane of the flow and an arbitrarily chosen surface in the supersonic part of the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti Gaza, No. 1, pp. 156–161, January–February, 1981.     

Flow of a conductive gas in a circular tube in a strong axiosymmetric magnetic field

The flow of a conductive gas along a channel in an external axiosymmetric magnetic field with a finite value of the magnetogasodynamic parameter N is examined. Numerical flow calculations are performed for a circular tube in such a field. Gas dynamic parameter fields, total pressure losses, and electric current intensities with the presence of transsonic zones and highly compressed regions are determined. Through comparison of the results obtained with linear theory data, the range of applicability of the latter is determined. Of the works dedicated to study of flow in external magnetic fields with N1, we should take note of [1], in which the process of entry of the gas into a transverse magnetic field was examined; [2], which studied one-dimensional transient motion with shock waves; and [3], where mixed flow in a Laval nozzle with an axiosymmetric homogeneous magnetic field was studied. Flow in a circular tube was examined in [4]; but the analysis performed by the characteristic method permitted calculation of only the initial supersonic flow zone. Motion in circular tubes in the presence of an axiosymmetric, magnetic field was studied in the linear formulation in [4, 5].Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–155, September–October, 1972.     

Effect of electron cyclotron waves on the structure of a weak collisionless shock wave

An analytic study is made of the structure of a weak collisionless shock wave propagating in a magnetized plasma at right angles to the magnetic field. Dissipation is produced by an instability associated with electron cyclotron oscillations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti 1 Gaza, No. 2, pp. 187–190, March–April, 1982.I thank V. B. Baranov for suggesting the problem and constant interest in the work, and also A. V. Ershov for discussing the results.     

Stabilization of plasma flow in a transverse magnetic field

Results are given of a theoretical and experimental investigation of the intensive interaction between a plasma flow and a transverse magnetic field. The calculation is made for problems formulated so as to approximate the conditions realized experimentally. The experiment is carried out in a magneto-hydrodynamic (MHD) channel with segmented electrodes (altogether, a total of 10 pairs of electrodes). The electrode length in the direction of the flow is 1 cm, and the interelectrode gap is 0.5 cm. The leading edge of the first electrode pair is at x = 0. The region of interaction (the region of flow) for 10 pairs of electrodes is of length 14.5 cm. An intense shock wave S propagates through argon with an initial temperature T_o = 293 °K and pressure p_o = 10 mm Hg. The front S moves with constant velocity in the region x < 0 and at time t = 0 is at x = 0. The flow parameters behind the incident shock wave are determined from conservation laws at its front in terms of the gas parameters preceding the wave and the wave velocity W_S. The parameters of the flow entering the interaction region are as follows: temperature T ₀ ¹ = 10,000 °K, pressure P ₀ ¹ = 1.5 atm, conduction ₀ ¹ = 3000 ^–1·m^–1, velocity of flow u ₀ ¹ = 3000 m·sec^–1, velocity of sounda ₀ ¹ = 1600 m·sec^–1, degree of ionization = 2%, 0.4. The induction of the transverse magnetic field B = [0, B_y(x), 0] is determined only by the external source. Induced magnetic fields are neglected, since the magnetic Reynolds number Re_m 0.1. It is assumed that the current j = (0, 0, jz) induced in the plasma is removed using the segmented-electrode system of resistance R_e. The internal plasma resistance is R_i = h(A)^–1 (h = 7.2 cm is the channel height; A = 7 cm² is the electrode surface area). From the investigation of the intensive interaction between the plasma flow and the transverse magnetic field in [1–6] it is possible to establish the place x* and time t* of formation of the shock discontinuity formed by the action of ponderomotive forces (the retardation wave R_T), its velocity W_T, and also the changes in its shape in the course of its formation. Two methods are used for the calculation. The characteristic method is used when there are no discontinuities in the flow. When a shock wave R_T is formed, a system of nonsteady one-dimensional equations of magnetohydrodynamics describing the interaction between the ionized gas and the magnetic field is solved numerically using an implicit homogeneous conservative difference scheme for the continuous calculation of shock waves with artificial viscosity [2].Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 112–118, September–October, 1977.