Influence of magnetic reynolds number on the current distribution in a magnetohydrodynamic channel with allowance for the Hall effect

Stationary plane flow of a conducting gas across a magnetic field in a magnetohydrodynamic channel of constant cross section made up of electrodes of finite length and insulators is considered in the linear approximation. It is assumed that the electromagnetic forces are small. It is shown that the current density increases near the exit from the interelectrode gap with increasing magnetic Reynolds number. The mutual influence of the Hall parameter and of the magnetic Reynolds number on the distribution of the currents in the channel is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 148–152, May–June, 1971.     

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.     

Stationary flow of a conducting liquid in a rectangular channel with two nonconducting walls and two walls having an arbitrary conductivity parallel to an external magnetic field

The flow of a conducting liquid in a channel of rectangular cross section with two walls (parallel to the external magnetic field) having an arbitrary conductivity, the other two being insulators, is considered. The solution of the problem is presented in the form of infinite series. The relationships obtained are used for numerical calculations of the velocity distribution and the distribution of the induced magnetic field over the cross section for several modes of flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkostt i Gaza, No. 5, pp. 46–52, September–October, 1970.     

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.     

Stationary supersonic conducting gas flow in a channel with nonconducting walls under weak magnetohydrodynamic interaction

A solution of the problem of flow in a channel with nonconducting walls for a small magnetohydrodynamic interaction parameter N is obtained by numerical methods. In the 0–10 range of variation of the Hall and magnetic Reynolds number parameters the distributions of the electrical parameters and the average (over the cross section) and local gasdynamic flow parameters are computed for two different geometries of the applied magnetic field. It is shown that an increase in the Hall and magnetic Reynolds number parameters is accompanied by a diminution in the Joule dissipation and the perturbation of the average (over the cross section) gasdynamic flow characteristics. It is disclosed that the distribution of the gasdynamic parameters over the channel cross section is extremely nonmonotonic in the end current zones.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 20–29, July–August, 1970.In conclusion, the author is grateful to A. B. Vatazhin for useful comments and constant attention to the research and to I. U. Tolmach for valuable comments.     

Joule loss due to compressibility of gas in a channel of variable section

It is known that closed electric currents arise in a conducting medium moving in a non-uniform magnetic field. These currents lead to additional energy loss and adversely affect the characteristics of magnetohydrodynamic channels. (The numerous investigations of these effects are dealt with in the review [2, 3].) Eddy electric currents are also formed, however, when a medium flows in a uniform magnetic field perpendicular to the to the plane of motion if the channel has a variable cross section and the medium is compressible [1], This paper is devoted to an investigation of some features of these flows. It is assumed in the analysis that the gas flows in channels whose geometry varies slightly.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 3–9, July–August, 1968.     

Calculation of the flow of an electrically conducting liquid in a duct of rectangular cross section with electrodes

A numerical solution is given for the problem of the flow of an electrically conducting liquid in a duct of rectangular cross section whose walls in the direction at right angles to the applied magnetic field are nonconducting, whereas those parallel to the field are perfect conductors. It is assumed that all the quantities except the pressure are independent of the coordinate along the axis of the duct, that the applied magnetic field is homogeneous, and that the induced current is diverted into an external circuit. The total current in the external circuit and the difference of the potentials of the conducting walls are found as functions of the external load, the Hartmann number, and the ratio of the lengths of the sides of the duct. It should be noted that problems of this kind have already been considered on many occasions and by many different approximate methods. The most complete bibliography on this question can be found in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 41–45, September–October, 1970.     

Electric field in the transverse cross section of the channel of an MHD generator

During the motion of a partially ionized gas in magnetohydrodynamic channels the distribution of the electrical conductivity is usually inhomogeneous due to the cooling of the plasma near the electrode walls. In Hall-type MHD generators with electrodes short-circuited in the transverse cross section of the channel the development of inhomogeneities results in a decrease of the efficiency of the MHD converter [1]. A two-dimensional electric field develops in the transverse section. Numerical computations of this effect for channels of rectangular cross section have been done in [2, 3], At the same time it is advisable to construct analytic solutions of model problems on the potential distribution in Hall channels, which would permit a qualitative analysis of the effect of the inhomogeneous conductivity on local and integral characteristics of the generators. In the present work an exact solution of the transverse two-dimensional problem is given for the case of a channel with elliptical cross section stretched along the magnetic field. The parametric model of the distribution of the electrical conductivity of boundary layer type has been used for obtaining the solution. The dependences of the electric field and the current and also of the integral electrical characteristics of the generator on the inhomogeneity parameters are analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 3–10, January–February, 1973.     

Linear approximation for the flow of a conducting gas with a high magnetic reynolds number through a channel

In this paper, we discuss the flow of a nonviscous and non-heat-conducting gas through a channel of variable cross section under the influence of a transverse magnetic field. For high magnetic Reynolds numbers, the flow is shown to consist of a core and current layers at the electrodes and at the fixed channel walls. The distributions of currents and other parameters in the core and in the current layers are found analytically, in a linear approximation. The Joule dissipation in the current layers may be more intense than that in the core. The longitudinal currents and Joule dissipation increase with increasing Hall parameter in the electrode layers. Zhigulev [1] has shown that magnetic boundary layers may form in the flow of a conducting gas when there is a high magnetic Reynolds number (R_m«1). He illustrated this situation by the shielding of a plasma flow from the magnetic fields produced near a plate which is electrically isolated from the plasma and through which a current is flowing. In an incompressible fluid, the layer thickness is proportional to R_m ^?1/2. Morozov and Shubin [2] have offered a linear-approximation treatment of the structure of the electromagnetic near-electrode layers which arise during the flow of a nonviscous plasma with a high R_m and a small “exchange” parameter ξ≈H/R_m, for flow transverse to a magnetic field and near a corrugated wall. They pointed out the possible formation of “dissipationless” near-electrode layers with thicknesses on the order of the Debye or electron Larmor radii, and a “dissipative” layer whose thickness increases along the length of the electrodes and is proportional to (R_mc_B ²/c_T ²)^?1/2, where c_B and c_T are the magnetic and thermal sound velocities. Morozov and Shubin studied the properties of dissipationless and dissipative electromagnetic layers at segmented accelerator electrodes through which a current is passing, for an arbitrary “exchange” parameter, in [2] and [3], respectively. The exchange parameter ξ was found in [4]. Such layers should also exist at solid electrodes and at the nonconducting walls of an accelerator channel. Study of the two-dimensional flow in a channel is significantly simplified when such layers are present.     

Nonstationary flow of a conducting gas in crossed electric and magnetic fields

The plasma is inviscid, cool, and not thermally conducting; it flows in a channel of constant cross section. The solution is derived by the small parameter method, for which purpose the magnetic interaction N is used. There have been previous studies of the transient-state flow of an inviscid and thermally nonconducting plasma in crossed electric and magnetic fields [1–3]. A plasma of infinite conductivity has been considered [1], as well as flow involving entropy change in an MHD system with strong electromagnetic fields [2, 3].     

The stability of a free rotating layer in an electrically conducting fluid in an axial field

The subject considered is a homogeneous electrically conducting incompressible medium with a current in a homogeneous external magnetic field and bounded by parallel insulating planes normal to the induction vector. When the current is fed by means of a system of coaxial electrodes located on one or both of the insulating planes, regions arise in which the medium is in rotational motion. If the lateral wall is at a sufficient distance from the electrodes, the rotating layer which forms as a result of the interaction of the axial magnetic field and the radial component of the electric current has free lateral boundaries. A study is made of the way in which the Reynolds number for the loss of stability in such a layer depends on the Hartmann number and on the geometric parameter for high values of the Hartmann number and low values of the magnetic Reynolds number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 166–173, September–October, 1984.     

Slow steady flows of a conducting fluid at large Hartmann numbers

We consider slow steady flows of a conducting fluid at large values of the Hartmann number and small values of the magnetic Reynolds number in an inhomogeneous magnetic field. The general solution is obtained in explicit form for the basic portion (core) of the flow, where the inertia and viscous forces may be neglected. The boundary conditions which this solution must satisfy at the outer edges of the boundary layers which develop at the walls are considered. Possible types of discontinuity surfaces and other singularities in the flow core are examined. An exact solution is obtained for the problem of conducting fluid flow in a tube of arbitrary section in an inhomogeneous magnetic field.The content of this paper is a generalization of some results on flows in a homogeneous magnetic field, obtained in [1–8], to the case of arbitrary flows in an inhomogeneous magnetic field. The author's interest in the problems considered in this study was attracted by a report presented by Professor Shercliff at the Institute of Mechanics, Moscow State University, in May 1967, on the studies of English scientists on conducting fluid flows in a strong uniform magnetic field.     

Two-dimensional spreading of a cloud of conducting gas in a magnetic field self-similar solution

A self-similar solution of the problem on the spreading in a magnetic field of a cloud of conducting gas, having the shape of a cylinder of noncircular cross section, is constructed. The cylindrical surface of the gas is restrained by a nonconducting sheath that spreads according to a prescribed law. The shape of the transverse cross section of the cylindrical cloud is determined from the solution. Cross sections obtained for a concrete case are represented in graphic form.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 29–36, March–April, 1973.In conclusion the author thanks V. I. Khonichev for assistance with the numerical calculations.     

End effects in magnetohydrodynamic channels at finite magnetic reynolds numbers

The effects of the magnetic Reynolds number have been examined via the distribution of the magnetic fields induced by the motion of a medium in a rectangular channel with conducting walls in the presence of an inhomogeneous magnetic field; the effects of wall conductivity and geometry of the external field are also examined as regards the distribution of the induced currents, the Joule loss, and the electric and magnetic fields over the cross section. The problem has previously been considered for a channel with insulating walls [1].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 19–27, May–June, 1971.We are indebted to A. B. Vatazhin for his interest.     

The boundary layer of a compressible electrically conducting gas at the electrode of a magnetohydrodynamic channel in the absence of ionization equilibrium

We consider the laminar boundary layer of a compressible electrically conducting gas formed at the conducting wall of the channel. We assume that the charged particle concentration in the field of the flow is distinct from the equilibrium distribution. We take into account the destruction of the quasi-neutrality of the gas in a narrow layer at the wall. We assume that the Debye length is much greater than the mean free path length of the charged particles. We investigate the case when the emission currents are substantial.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 164–169, May–June, 1971.     

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.     

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.     

Approximate solution of the problem of the flow of a conducting liquid in a duct with rectangular cross section

An approximate analytic solution is constructed for the problem of the fully developed stationary flow of a viscous incompressible liquid with a finite isotropic conductivity in a duct of rectangular cross section in the presence of an external magnetic field at right angles to nonconducting walls. An investigation is made of the extent to which flows in ducts with two electrodes parallel to the field resemble flows with four nonconducting walls. Theoretical and experimental investigations devoted to this problem are reviewed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 33–40, September–October, 1970.     

A qualitative investigation of the equations of quasi-one-dimensional magnetohydrodynamic channel flow

A qualitative investigation of the system of differential equations describing the quasi-one-dimensional flow of an electrically conducting medium at small magnetic Reynolds numbers gives an idea of the different possible flow patterns occuring when the electromagnetic field and channel shape are given in different ways. Such a treatment is essential for the calculation of one-dimensional flows, and also for the solution of variational problems [1].In the literature devoted to this question studies have been made of flow in a one-dimensional electromagnetic field and a channel of constant cross section [2], as well as of the flow when the magnetic field is described by specially given functions of the flow velocity [3]. These cases reduce to the analysis of integral curves in a plane.In the present paper the investigation is carried out for an arbitrary distribution of the electric and magnetic fields and channel shape, which leads to a consideration of the behavior of integral curves in three-dimensional space. The qualitative results are illustrated by examples.     

Effect of an inhomogeneous magnetic field on the structure of an ionized gas flow

Results of an experimental study of the flow of an ionized gas produced by a shock wave through an inhomogeneous magnetic field are presented. Braking of the gas flow produced by the end currents is determined at two fixed sections of the magnetogasdynamic channel from the value of the isolated shock wave formed in the vicinity of the hemispherical model over which the flow passes. Maximum recorded reduction in Mach number was 30%, and with a magnetogasdynamic interaction parameter greater than 1.5, a transition of supersonic flow to infrasonic at the exit of the magnetic zone was observed. Experimental results were compared with a solution of a model problem which assumed one-dimensional flow in the flow core. The gas used was argon, with a maximum magnetic field induction of 1.5 T.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 174–178, September–October, 1976.