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.     

Transient motion of groundwater (subsurface water) in the presence of evaporation

The spread of a mound of groundwater in the region between two parallel channels with different water levels (H₁ at x = 0 and H₂ at x = L) during irrigation is studied with due allowance for evaporation. Evaporation is taken into account in relation to the depth of the groundwater h(x, t); its intensity is regarded as zero when h < h₀ (where h₀ is the critical level of the ground-water), while varying linearly or remaining constant when h > h₀. The intensity or irrigation is regarded as constant. This problem is solved by using the thermal potentials of a double layer and reduces to the solution of a nonlinear integral equation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 176–182, March–April, 1975.     

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.     

Effect of flow acceleration and initial turbulence level on velocity fluctuations

The effect of various levels of initial turbulence on the decay of the velocity fluctuations in convergent channels with different contraction ratios has been experimentally investigated. The results of the experiments indicate that the level of turbulence has a strong influence on the longitudinal velocity fluctuations and a weaker effect on the transverse fluctuations. The experimental data demonstrate the inapplicability of the rapid distortion theory and the need for that theory to take into account the energy dissipation, the integral scale and the degree of anisotropy.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 42–49, September–October, 1993.     

Coaxial turbulent jets with strong central blowing

The results of solving the problem of the initial section of isothermal coaxial jets with strong central blowing, when the transverse pressure gradient has only a slight effect and there is no circulation zone in the central jet are given. The problem is solved by the integral relation method with allowance for jet interference and the presence of a cocurrent flow. The results of an experimental investigation of these jets over a wide range of the geometric and regime parameters are also given. The results of the calculations made using the formulas obtained are compared with the experimental data.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 52–59, May–June, 1996.     

Instability of combined convective flow in a vertical layer

In the present paper, a study is made of the stability of plane-parallel flow induced by a transverse temperature difference between the boundaries of a layer and a longitudinal pressure gradient. This problem was solved earlier by the author [3] in a purely hydrodynamic formulation without allowance for thermal factors; the results then obtained correspond to the limiting case of small Prandtl numbers. In the paper, a numerical solution to the problem with the complete formulation is given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–9, May–June, 1982.I thank G. Z. Gershuni for supervising the work, and also M. A. Gol'dshtik and V. N. Shtern for a helpful discussion.     

Boundary layer development in supersonic MHD generators

A numerical model of the turbulent boundary layers in the gas dynamic channel of a supersonic MHD generator is constructed. This model describes the development and structure of the layers in the nozzle, on the electrode and insulating walls of the duct, in the two-dimensional approximation. The characteristics of the boundary layers in various generator operating regimes are investigated numerically. The integral boundary layer thicknesses characterizing the nonuniformity of the gas dynamic and electrodynamic quantities are calculated. The limits of applicability of the integral calculation method are determined for typical MHD generator operating conditions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 35–41, May–June, 1990.The authors wish to thank A. B. Vatazhin, V. A. Bityurin, and V. A. Zhelnin for discussing the formulation of the problem, A. A. Yakushev for participating in the discussion of the results, and Yu. V. Rakseeva and L. V. Yashina for preparing the article.     

The boundary layers of a fully ionized two-temperature plasma with given component temperatures at an electrode

The flow of a plasma with different component temperatures in the boundary layers at the electrodes of an MHD channel is investigated without any assumptions as to self-similarity. For the calculation of the electron temperature, the full energy equation for an electron gas [1] is solved with allowance for the estimates given in [2]. In contrast to [3, 4], the calculation includes the change in temperature of electrons and ions along the channel caused by the collective transport of energy, the work done by the partial pressure forces, and the Joule heating and the energy exchange between the components. The problem of the boundary layers in the flow of a two-temperature, partially ionized plasma past an electrode is solved in simplified form by the local similarity method in [5–7]. In these papers, either the Kerrebrock equation is used [5, 6] or the collective terms are omitted from the electron energy equation [7].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–10, September–October, 1972.The author thanks V. V. Gogosov and A. E. Yakubenko for interest in this work.     

Calculation of base pressure and enthalpy behind a blunt trailing edge separating two supersonic streams with allowance for boundary layers and heat fluxes

A relatively simple method of calculating the parameters of the flow behind a blunt trailing edge separating two supersonic streams is developed. The method is based on the use of the boundary layer approximation and integral laws of mass and energy conservation (viscous-inviscid interaction model). It makes it possible to determine the base pressure and base enthalpy with allowance for the effect of Mach numbers, Reynolds numbers, initial boundary layer thicknesses, specific heat ratios and wall enthalpies for various ratios of the total pressures and enthalpies of the two streams.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 167–176, September–October, 1991.     

Discontinuous solutions of an interacting three-dimensional hypersonic boundary layer

The paper is a mathematical study of the three-dimensional flow of viscous gas in a hypersonic boundary layer that develops along a flat wing whose leading edge has a step shape. The flow interacts with a flap on the wing set at a small angle. A linear solution to the problem is constructed under the assumption that the deflection angle of the flap is small and the difference between the length of the plates is of order unity. It is shown that an important part in the formation of the flow near and behind the flap may be played by the change in the pressure along the span of the wing due to the step shape of the leading edge. It is significant that although the pressure and displacement thickness are continuous functions of the transverse coordinate, the longitudinal and transverse components of the friction force have discontinuities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 19–26, March–April, 1991.I thank V. V. Sychev and A. I. Ruban for suggesting the problem, for valuable advice, and assistance.     

Motion of a suspension in the laminar boundary layer on a flat plate

The boundary layer motion of a weak suspension is investigated with allowance for the effect on the particles not only of the Stokes force but also of the additional transverse force resulting from the transverse nonuniformity of the flow over the individual particle. As distinct from studies [1–3], in which the limiting values of the transverse force (Saffman force) were used [4], the velocity and density of the dispersed phase have been determined with allowance for the dependence of the Saffman force on the ratio of the Reynolds numbers calculated from the velocity of the flow over the individual particle and the transverse velocity gradient of the undisturbed flow, respectively [5, 6].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 66–73, January–February, 1992.In conclusion the authors wishes to thank M. N. Kogan, N. K. Makashev, and A. Yu. Boris for useful discussions of the results.     

The effect of transverse pressure on the stability of a plate

The stability problem of a centrally compressed infinite plate is solved with allowance for the transverse normal deformation caused by uniform load for various boundary conditions at the edges. The linearized nonlinear equations of elastic deformation of thin plates taking into account transverse shear and transverse normal deformation are used. The obtained critical loads are compared with existing solutions.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 170–178, March–April, 2005.     

Two-dimensional magnetic fields in magnetohydrodynamic channels with steel walls at finite magnetic reynolds numbers

Plane problems on the distribution of a two-dimensional magnetic field in magnetohydrodynamic channels with ferromagnetic walls at appreciable magnetic Reynolds numbers and prescribed flow hydrodynamics are studied. An integral representation for the total magnetic induction is constructed with the use of a complex influence function describing the field resulting from a unit current. This makes it possible to obtain arbitrarily close approximations to exact solutions of the problems on a digital computer. Influence functions for various channels can be determined by mirror reflections and conformai mappings. The method is illustrated by numerical calculations of the distribution of the magnetic field for the flow of a conducting fluid along a plane ferromagnetic wall and the flow of a fluid in the space between ferromagnetic walls. Calculations are carried out on the effect of an external circuit and an inhomogeneous transverse velocity profile on the distribution of the magnetic field.Translated from Zhumal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–11, September–October, 1971.In conclusion the authors thank G. A. Lyubimov, A. B. Vatazhin, V. V. Gogosov, and A. E. Yakuberiko for useful discussion of the formulation of the problem and of results of the study.     

Steady-State MHD Flows in Nozzles with an External Longitudinal Magnetic Field

Steady-state MHD flows in channels of the nozzle type in the presence of an external longitudinal magnetic field can be divided into two significantly different classes. Subcritical flows, in which the Alfvén velocity calculated from the longitudinal magnetic field is less than the plasma velocity, have mainly the same properties as flows in a transverse magnetic self-field and their quantitative characteristics depend only slightly on the longitudinal magnetic field strength. Supercritical flows with the opposite inequality for the velocities correspond to strong longitudinal magnetic field. The main difference is the transitions between different forms of energy (kinetic, thermal, and electromagnetic). The present study contains a classification of possible flows, namely, sub- and supercritical and sub-, super-, and transonic flows with respect to the fast and slow magnetosonic and Alfvén velocities. Examples of these flows are given. The effect of the problem parameters on the flow properties is investigated.     

Dispersed phase motion in laminar boundary layer flow over a wedge

The motion of a dispersed phase in the laminar boundary layer on a wedge is considered with allowance for the effect of not only the Stokes force, which coincides in direction with the flow velocity, but also the transverse force (Saffman force) resulting from the transverse nonuniforrnity of the flow over the individual particle [1–3].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 34–42, November–December, 1993.In conclusion, the authors wishes to thank S. V. Manuilovich for assisting with the numerical calculations.     

Effect of surface temperature on the stability of the swept attachment line boundary layer

At fairly high Reynolds numbers instability may develop on the line of attachment of the potential flow to the leading edge of a swept wing and lead to a transition to boundary layer turbulence directly at the leading edge [1, 2]. Although the first results relating to the stability and transition of laminar flow at the leading edge of swept wings were obtained almost 30 years ago, the problem remains topical. The stability of the swept attachment line boundary layer was recently investigated numerically with allowance for compressibility effects [3, 4]. Below we examine the effect of surface temperature on the stability characteristics of the laminar viscous heat-conducting gas flow at the leading edge of a side slipping wing.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 78–82, November–December, 1990.     

On the stability of certain nonparallel flows of a viscous incompressible liquid in a channel

The stability of very simple nonparallel flows of a viscous incompressible liquid in an infinite plane channel described by the exact solutions of the Navier-Stokes equations is studied. Such solutions are realized between two parallel porous plates when the liquid (or gas) is forced in at one wall and drawn out at the same velocity at the other, with a steady flow of liquid along the channel. In this case the transverse velocity component is constant, and the profile of the longitudinal velocity component is independent of the longitudinal con-ordinate x, being an asymmetric function of the transverse coordinate y. A study of the hydrodynamic stability then reduces to the solution of an equation differing from the Orr-Sommerfeld equation by virtue of the presence of additional terms containing the transverse velocity component of the main flow. By numerically solving both this equation and the ordinary Orr-Sommerfeld equation and comparing the corresponding results for various inflowing Reynolds numbers R₀=v₀h/ (v₀ is the inflow velocity, h is the width of the channel), the effect of the nonparallel and asymmetrical nature of such flows on their stability is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 125–129, July–August, 1970.     

Hall effect in the MHD model of plasma channel flow

The steady flow regime with quasi-radial electric current and nonequipotential boundaries is obtained in calculating axisymmetric plasma flow in a coaxial channel with an azimuthal self-magnetic field. The calculations were carried out using the MHD model with allowance for the Hall effect. The regime obtained describes the flow in the stream core of the channel of a modern quasi-steady heavy-current plasma accelerator outside thin near-electrode layers. Estimates of the Hall corrections relating to these layers are given.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 56–65, September–October, 1995.     

The influence of the hall effect on the current structure in a plasma flow with a spatially periodic magnetic field

The structure of the electric fields and current is studied for stationary plasma flow in an axially symmetric, spatially periodic magnetic field. The problem is solved in the magnetohydrodynamic approximation with allowance for the Hall term in the generalized Ohm's law equation. It is assumed that the magnetic Reynolds number and the interaction parameter are small.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 11–16, September–October, 1972.The author thanks N. A. Khizhnyak and A. A. Kalmykov for useful discussions.     

Effect of dissipative processes on compression flows in a plasma accelerator channel

Plasma flows in coaxial channels with a truncated central electrode are accompanied by compression and heating of the plasma on the channel axis [1–4]. Such flows were calculated in [1, 4] within the framework of a simple MHD model and by simple numerical methods and, accordingly, the results reflect only the basic qualitative characteristics of compression flows. Below, these flows are investigated in greater detail on the basis of a more accurate physical model with allowance for the finite conductivity, heat conduction and radiation of the plasma and impurities. The cases of anisotropic and classical isotropic heat conduction are considered. The numerical method employed is based on two finite-difference schemes: SHASTA-FCT [5–7] and TVD [8, 6]. The main advantage of these methods is the high resolution of the shock waves and contact discontinuities, which is highly desirable in describing compression flows. The calculations relate to the case of a fully ionized hydrogen plasma.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 102–109, May–June, 1991.In conclusion, the author wishes to express his gratitude to K. V. Brushlinskii and A. I. Morozov for frequent discussions and to K. P. Gorshenin for the use of his calculation results.