Model of Formation of the Electric Current in Aircraft Jet Engine Ducts

A possible mechanism of formation of electric currents (engine currents) in aircraft engine jets (with subsequent charging of the aircraft) is investigated. This mechanism is a result of the presence of electrons and ions at concentrations of 10⁷–10¹⁰cm^-3 in the gas flow in the engine duct. The electrons and ions are formed in the fuel combustion chamber as a result of chemo- ionization reactions. The wall flow zones in which the electrical quasi-neutrality of the medium is violated are considered. In these zones a nonzero normal component of the electric current is formed on the wall surface and a streamwise electric current develops in the duct. A general functional dependence of the engine current on the basic dimensionless parameters is obtained on the basis of similarity theory and dimensional analysis. Within the framework of electrical diffusion boundary layer theory a model problem of formation of the maximum possible engine current is formulated. A universal system of equations and boundary conditions, which contains no dimensionless parameters, is obtained and investigated. The engine current is qualitatively estimated for real engines and the calculation results are compared with the experimental data obtained under airfield conditions.     

The turbulent boundary layer of dissociated gas in the initial section of a tube

Many theoretical and experimental papers [1–4] have been devoted to investigating the turbulent boundary layer in the initial section of a channel. For the most part, however, the flow of an incompressible fluid with constant parameters is considered. There are many practical cases in which it is of interest to treat the development of the turbulent boundary layer of gas in the initial section of a pipe when conditions are strongly nonisothermal. A solution of a problem of this type, based on the theory of limit laws, is given in paper [1]. The present article extends this solution to the case of the flow of a high-enthalpy gas when the effect of gas dissociation on the turbulent boundary layer characteristics must be taken into account. We shall consider the flow of a mixture of i gases which is in a frozen state inside the boundary layer, and in an equilibrium state on its boundaries. Formulas are derived for the laws of friction and heat exchange, and a solution is given for the turbulent boundary layer equations in the initial section of the pipe when the wall temperature is constant and the gas flows at a subsonic velocity.Finally the authors are grateful to S. S. Kutateladze for discussing the paper.     

Breakaway self-similar flows in a laminar magnetohydrodynamic boundary layer with blowing and suction

A study was made of self-similar flows in a magnetohydrodynamic boundary layer in the presence of a pressure gradient and with the blowing and suction of a conducting liquid. The region of the existence of self-similar solutions for breakaway flow conditions, characterized by a friction at the wall equal to zero, was determined. The regions of a change in the determining parameters, with which nonbreakaway flows are established at impermeable and permeable surfaces, are indicated. It is noted that under diffusion flow conditions the self-similar equation of a magnetohydrodynamic boundary layer with fixed boundary conditions at the wall and at infinity permits an innumerable set of solutions. The article proposes a method for selecting a solution and indicates a calculating method for determining it. It demonstrates the possibility of a considerable broadening of the region of flows without breakaway with the imposition of electric and magnetic fields.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 38–46, May–June, 1972.     

Structure of a collisionless boundary layer and turbulent damping of ions

The flow of a low-pressure plasma in a MHD channel is unstable in a number of cases. The instability can be caused by a current flowing across the magnetic field. In this study we investigate an unstable, turbulent flow of a rarefied plasma near the magnetized electrodes, representing plane magnetic dipoles. Owing to the growth of microscopic turbulence near the electrodes, the maximum density of the current that is induced in the plasma is localized and turbulent damping of the incoming flow occurs. The energy of damping goes into the turbulent heating of the plasma. Under these conditions a structure of the boundary layer is found for a stationary flow. The characteristic transverse dimension of the boundary layer is considerably less than the particle mean free path; therefore, such a boundary layer can be called collisionless.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 17–28, September–October, 1972.     

Heat transfer enhancement with deformation of the velocity profile

The problem of enhancing the heat transfer in channels and boundary layers by the appropriate deformation of the fluid velocity profile is considered. The resulting additional hydraulic losses, the price of heat transfer enhancement, are determined. The possibilities of controlling heat transfer by redistributing the fluid velocity in channels are demonstrated with reference to flows at low Prandtl numbers. Laminar and turbulent liquid and gas flows with heat transfer in channels and boundary layers are numerically modeled on the basis of modern models of turbulence (flow development in channels with different initial velocity profiles, flows with wall roughness and boundary layer flows with forces acting on the flow to cause deformation of the velocity profile). In all cases it is found that the heat transfer can be enhanced only at the expense of a considerable increase in the hydaulic losses. A class of self-similar thermal problems for flows in plane diffusers is formulated. The eigenfunctions — temperature modes — for various velocity profiles are determined with allowance for the nonuniqueness of the solution of the classical dynamical problem for a plane diffuser and the corresponding heat transfer coefficients are found.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 94–105, May–June, 1993.The authors are grateful to A. Yu. Klimenko for useful discussions.     

Electrode layers in flows of an inviscid plasma with good conductivity

The structure of the electromagnetic electrode layers that are produced in flows across a magnetic field by a completely ionized and inviscid plasma with good conductivity and a high magnetic Reynolds number is examined in a linear approximation. Flow past a corrugated wall and flow in a plane channel of slowly varying cross section with segmented electrodes are taken as specific examples. The possibility is demonstrated of the formation of nondissipative electrode layers with thicknesses on the order of the Debye distance or electron Larmor radius and of dissipative layers with thicknesses on the order of the skin thickness, as calculated from the diffusion rate in a magnetic field [2].In plasma flow in a transverse magnetic field, near the walls, along with the gasdynamie boundary layers, which owe their formation to viscosity, thermal conductivity, etc. (because of the presence of electromagnetic fields, their structures may vary considerably from that of ordinary gasdynamic layers), proper electromagnetic boundary layers may also be produced. An example of such layers is the Debye layer in which the quasi-neutrality of the plasma is upset. No less important, in a number of cases, is the quasi-neutral electromagnetic boundary layer, in which there is an abrupt change in the frozen-in parameter k=B/p (B is the magnetic field and p is the density of the medium). This layer plays a special role when we must explicitly allow for the Hall effect and the related formation of a longitudinal electric field (in the direction of the veloeiryv of the medium). We will call this the magnetic layer. The magnetic boundary layer can be dissipative as well as noudissipative (see below). The dissipative magnetic layer has been examined in a number of papers: for an incompressible medium with a given motion law in [1], for a compressible medium with good conductivity in [2], and with poor conductivity in [3]. In the present paper, particular attention will be devoted to nondissipative magnetic boundary layers.     

Theory of Disappearance of the Electric Signal Induced by an Aircraft Jet Engine in the Afterburner Regime

Electrical aspects of aircraft jet engine operation are considered. A phenomenon previously observed experimentally, namely, the disappearance of the engine current and alternating electric signal induced by the engine jet in the afterburner regime, is explained. It is shown that this phenomenon results from the detachment of electrons from negative ions when the gas temperature in the afterburner increases. This leads to an increase in the effective conductivity of the gas. As a result, the engine current circuit is closed on the internal duct walls and engine charging becomes insignificant.A physico-mathematical model of the electrical processes in the afterburner is formulated and model problems are solved.     

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.     

Analysis of turbulent boundary layer interaction with an external supersonic flow behind a step

An integral method of analyzing turbulent flow behind plane and axisymmetric steps is proposed, which will permit calculation of the pressure distribution, the displacement thickness, the momentum-loss thickness, and the friction in the zone of boundary layer interaction with an external ideal flow. The characteristics of an incompressible turbulent equilibrium boundary layer are used to analyze the flow behind the step, and the parameters of the compressible boundary layer flow are connected with the parameters of the incompressible boundary layer flow by using the Cowles-Crocco transformation.A large number of theoretical and experimental papers devoted to this topic can be mentioned. Let us consider just two [1, 2], which are similar to the method proposed herein, wherein the parameter distribution of the flow of a plane nearby turbulent wake is analyzed. The flow behind the body in these papers is separated into a zone of isobaric flow and a zone of boundary layer interaction with an external ideal flow. The jet boundary layer in the interaction zone is analyzed by the method of integral relations.The flow behind plane and axisymmetric steps is analyzed on the basis of a scheme of boundary layer interaction with an external ideal supersonic stream. The results of the analysis by the method proposed are compared with known experimental data.Notation x, y longitudinal and transverse coordinates - X, Y transformed longitudinal and transverse coordinates - , ^*, ^** boundary layer thickness, displacement thickness, momentum-loss thickness of a boundary layer - , ^*, ^** layer thickness, displacement thickness, momentum-loss thickness of an incompressible boundary layer - u, velocity and density of a compressible boundary layer - U, velocity and density of the incompressible boundary layer - , stream function of the compressible and incompressible boundary layers - , dynamic coefficient of viscosity of the compressible and incompressible boundary layers - r₁ radius of the base part of an axisymmetric body - r radius - R transformed radius - M Mach number - friction stress - p pressure - a speed of sound - s enthalpy - v Prandtl-Mayer angle - P Prandtl number - P_t turbulent Prandtl number - r₂ radius of the base sting - b step depth - =0 for plane flow - =1 for axisymmetric flow Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–40, May–June, 1971.In conclusion, the authors are grateful to M. Ya. Yudelovich and E. N. Bondarev for useful comments and discussions.     

A particular solution for the chaplygin equation

In the theoretical studies of several gasdynamic problems a major role is played by the hodograph plane, where the equations in terms of velocity component variables are linear. In these studies a primary role is played by the Chaplygin equation for the stream function . Chaplygin [1] obtained a general solution for the equation of motion in the hodograph plane. Particular exact solutions of the hodograph are also known [2]: radial flow, spiral flow, etc. Below we consider a particular solution of the Chaplygin equation.     

Self-similar problem of the motion of a plane layer of heated material with an arbitrary equation of state

The self-similar problem of the nonstationary motion of a plane layer of material in which energy from an external source is released for values of the flux density q₀ on the boundary which are constant in time is considered. The self-similar variable is = m/t, where m is the Lagrangian mass coordinate and t is the time. The characteristic values of the velocity, density, and pressure do not vary with time. For a self-similar problem the energy flux density q must also depend only on the self-similar variable. In this case q() can be an arbitrary function of its argument and can be given by a table. Examples are presented of actual physical processes in which the mass of the energy-release zone increases linearly with time. The equation of state can have an arbitrary form, including specification by a table. The gaseous state of matter for an arbitrary variable adiabatic exponent, the condensed state, and a two-phase state can be described. A solution of the self-similar problem is presented for the heating of a half-space bounded by a vacuum for a certain specific equation of state and various flux densities q₀ and velocities M of the advance of the energy-release zone.Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskoi Fiziki, No. 5, pp. 136–145, September–October, 1975.     

Laminar boundary layer in axisymmetric swirling ducted gas flow

Several studies have been published [1–3] in which the authors solve the problem of the laminar boundary layer in an incompressible fluid on the walls of an axisymmetric duct in the presence of swirl in the outer flow. In [3], Loitsyanskii's parametric method of [4, 5], generalized to the case of three-dimensional flow, is used to solve this problem.In this article the parametric method for integrating the universal equations is extended to the solution of the problem of the laminar boundary layer on the wall of an axisymmetric channel in the case of swirling gas flow.     

Calculation of the rate of coalescence of an emulsion in a turbulent flow

The central moment of the theory describing the merging (coalescence) of the drops of an emulsion is determination of the time of the approach of a drop or a number of drops colliding with a given drop in unit time. In the stage immediately preceding the merging of the drops the forces of the hydrodynamic braking of the approaching drops are found to be considerable. The role of these forces has been analyzed earlier for the case of the capture of small drops by large drops in an oncoming flow in the presence of an external electrical field [1] and for the problem of the Brownian coalescence of drops, taking account of the effect of the electric double layer and of surface forces of interaction [2–4]. The present article considers the approach of drops with turbulent diffusion in an electrical field. Of the greatest interest is the sharp slowing of the approach due to the hydrodynamic interaction of the drops, considerably sharper than in the case of molecular diffusion [2]. As a result, the sharp acceleration of the approach and coalescence of drops with the action of an electrical field on an emulsion in a turbulent flow becomes understandable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 47–55, May–June, 1976.The authors are grateful to G. I. Barenblatt, A. I. Leonov, V. I. Loginov, and L. P. Smirnov for their evaluation and criticism of the work.     

Asymptotic solution of the equations of a laminar multicomponent boundary layer with intensive suction

An asymptotic solution is obtained for the equations of the laminar multicomponent boundary layer encountered in the plane-parallel and axially symmetrical flow of a gas with large values of the suction parameter. It is shown that the roots of the characteristic equation to which the solution of the diffusion equations reduce in the first approximation may be found in the form of radicals when the external gas flow contains chemical components capable of being combined into r5 groups as regards their diffusion properties. The number of components in the groups and the number of components in the boundary layer may be arbitrary. Asymptotic equations are obtained for the coefficient of friction, the temperature and concentration gradients, and the diffusion flows of the components on the surface of the body. By way of example, formulas are given for the thermal flux passing to a body during the flow of dissociated air or a dissociated mixture of N₂ and CO₂. A numerical solution is given for the equations of the boundary layer in the case of the flow of dissociated air. The asymptotic solution is compared with the numerical result, and the range of applicability of the asymptotic equations is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 66–74, November–December, 1970.The author wishes to thank G. A. Tirskii for discussion of this analysis.     

Flow of a sonic free gas stream around a profile

The stream function far from a profile in a two-dimensional sonic free gas jet is constructed. The stream function satisfies the Tricomi equation and is constructed on the , plane by the method of singular integral equations. With unlimited increase of the jet width and by satisfying a certain condition, the stream function transforms to the self-similar solution of Frank [1, 2] and Guderley, which describes an unbounded sonic stream far from the profile. In conclusion, flow of a sonic jet issuing from a duct about a profile is considered.The author wishes to thank S. V. Fal'kovich for valuable suggestions in discussing this article.     

Subsonic electrically quasi-neutralweakly-ionized gas flow past a spherical probe

The electric characteristics of a sphere located in a flow of viscous, electrically quasi-neutral weakly-ionized gas containing electrons and monovalent ions are investigated theoretically and numerically. As in the majority of applications, the electrogasdynamic (EGD) interaction parameter is assumed to be small. This makes it possible to solve the gasdynamic and electric equations successively. The spherical surface is assumed to be conducting and heat-insulated. At low free-stream Mach numbers the gas temperature is almost constant in the region of flow past the sphere. This makes it possible to use the model of a viscous incompressible medium. The flow past a sphere is analyzed for gasdynamic Reynolds numbers varying over the interval 0 ≤ Re ≤ 1000. The electrodynamic equations in which the convection and diffusion of the electrons and ions and their electrical drift are taken into account are reduced to three elliptic equations for the electron and ion concentrations and the electric potential. A constant potential is assigned on the boundary of the computation region simulating infinity. The entire problem is simulated numerically using specially constructed grids. The charged-component, potential, and electric current fields are determined and the volt-ampere characteristics of the sphere are constructed for various gas velocities. The results obtained generalize the available data on the voltampere characteristics of a sphere (probe) in a weakly-ionized medium at rest.     

Cone in a supersonic flow near a surface with a turbulent boundary layer

The published information about the interaction of incident shocks and a turbulent boundary layer relate to cases of a thin boundary layer ( 1–3 mm) on a flat surface. The present study relates to supersonic flow with Mach number M = 3 and stagnation pressure p₀=1.2 MPa past cones near a surface with a thick boundary layer formed on a plate abutting the lower edge of a plane nozzle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 177–180, July–August, 1991.     

Use of turbulent energy balance equation in the theory of turbulent flows along walls

The turbulent flow of an incompressible fluid is considered in a plane channel, a circular tube, and the boundary layer on a flat plate. The system of equations describing the motion of the fluid consists of the Reynolds equations and the mean kinetic energy balance equation for turbulent fluctuations. On the basis of an analysis of experimental data, hypotheses are formulated with respect to the eddy kinematic viscosity and lengthl entering into the expression for specific dissipation of turbulent energy into heat. It is assumed that in the central (outer) region of the flow in a channel, andl are constants, and expressions are taken for them which are used for a free boundary layer; near the walll varies linearly and almost linearly. Results of calculations of the turbulent energy distribution, the mean velocity, and the drag coefficient are in good agreement with the existing experimental data. The values of two empirical coefficients, which enter into the system of equations as the result of the hypotheses, are close to those obtained for a free boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–33, May–June, 1973.     

Bifurcation and transition to turbulence in the gap between rotating and stationary parallel disks

The flow in the gap between rotating and stationary parallel disks is an attractive object for studying the transition characteristics in three-dimensional internal flows. Firstly, in this case a large region of the basic motion is satisfactorily described by a self-similar solution to the Navier-Stokes equations [1]; secondly, as the parameter = h²/v ( is the. angular velocity of rotation of one of the disks and h is the gap width) varies, there is an evolution of the basic motion, so that it is easy to produce different types of initial and subsequent instabilities. The basic steady regime for axially symmetric flow has been studied by many authors (see [1, 2]). Questions of the transition in the gap between disks have been considered [3, 4]. This paper presents a methodology and the results of experimental investigations for different types of initial and subsequent instabilities in the gap between disks enclosed by a cylindrical cover. It was found that as a result of the loss of stability of the basic regime one of two steady vortex regimes is developed depending on the value of the relative gap width. The subsequent stages of soft excitation of the turbulent regime are described and the corresponding boundaries established. It is shown that in very narrow gaps the excitation of turbulence has a hard nature of the type realized in Couette flow. The stability limit for a laminarized boundary layer on a rotating disk and the boundary for complete turbulence of the layer were determined for relatively wide gaps. A comparison was made with known data for an unenclosed rotating disk.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 28–36, September–October, 1984.