Supersonic flow over a slender wing with subsonic edges

The method of integral equations is generalized to calculate steady flow past wings with an arbitrary shape in plan with subsonic leading and trailing edges. The determination of the velocity potential in the leading part of the wing, where there is no influence of the vortex sheet, is reduced to the solution of a two-dimensional integral equation of the second kind. The trailing part, which is subject to the influence of the vortex sheet, is divided into a number of subregions, in which the calculation of the acceleration potential reduces to the solution of one-dimensional equations of the type of Fredholm equations of the second kind and to quadrature. The unique solvability of the obtained integral equations is investigated; it is shown that they can be solved by successive approximation. As an example, the solution to the problem of flow past a flat delta-shaped wing is found and compared with the exact solution to the problem found by the method of conic flows [4, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 119–127, September–October, 1981.I thank G. Yu. Stepanov for discussing the paper.     

Calculation of aerodynamic characteristics of arrays of profiles of arbitrary shape

It is well known that the problem on nonseparating potential flow of an incompressible fluid about an array of profiles reduces to an integral equation for a certain real function, determined on the contours of the profiles of the array. As such a function one can take, as was done, for instance, in [1–5], the relative velocity of the fluid on the profiles of the array. For arrays of profiles of arbitrary shape it is necessary to solve the corresponding integral equation numerically. In the particular examples of the calculation of aerodynamic arrays that are available [1–3] the numerical methods used were based on the approximate evaluation of contour integrals by rectangle formulas. As investigations showed, sizeable errors arose thereby in the approximate solution obtained, these being especially significant in the case of curved profiles of relatively small bulk. In the present paper a method for the numerical solution of the integral equation obtained in [5] is proposed. The method is based on the replacement of a profile of the array with an inscribed N polygon, the length of whose sides is of the order N^–1 and whose internal angles are close to . Convergence with increasing N of the numerical solution to an exact solution of the integral equations at the reference points is demonstrated. Examples of the calculation are given.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 105–112, March–April, 1972.     

Weakly rarefied gas flows between nonuniformly heated parallel plates

The aim of this research is to establish the validity of the predictions of the theory of slow nonisothermal flows, to study the limits of applicability (with respect to the Knudsen number) of the conclusions reached and to determine the effect of the Knudsen layers on these flows on the basis of a numerical investigation of slow nonisothermal weakly rarefied gas flow in a plane infinite channel with weakly nonequilibrium heating of the walls and a finite wall temperature difference. The gas flow is described by a relaxation transport equation. The results obtained show how quickly, as the Knudsen number decreases, the solutions of the transport equation outside the Knudsen layers tend to the solution of the equations of gas dynamics of slow nonisothermal flows (and not to the solution of the Navier-Stokes equations).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 115–121, January–February, 1988.     

Electrically charged two-phase flow past a sphere with a high electrostatic surface potential

A two-phase medium with a carrier phase in the form of an incompressible electrically neutral fluid and a dispersed phase in the form of inertial charged particles flows past an electrically charged sphere. It is assumed that the electrohydrodynamic interaction parameter is insignificant and that the flow conditions correspond to potential unseparated flow of the carrier medium over the sphere. The motion of the dispersed phase is described by continuum dynamic equations incorporating the electric field, which is the sum of the external field created by the sphere and the field induced by the dispersed particles. The electric field is determined by means of the equations of electrodynamics, which must be considered together with the dynamic equations. In the case considered a large electrostatic potential is applied to the sphere. This prevents the particles striking the surface of the sphere and leads to the intersection of the particle trajectories. In order to solve this problem within the framework of the two-velocity continuum we introduce a surface of discontinuity of the parameters to replace the zone of multiphase flow. The location of the surface of discontinuity, the distribution of the velocity and density of the dispersed phase and the distribution of electrostatic potential are found as a result of solving a system of elliptic and hyperbolic equations in two regions separated by the surface of discontinuity. The results of numerically integrating the system of equations formulated are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 90–95, March–April, 1987.     

Existence and nonuniqueness of local separation zones in viscous jets

The plane steady problem of the flow of a viscous wall jet past a smoothed break in the contour of a body is considered. For convenience, the flow in the neighborhood of the junction between two flat plates inclined at an angle to each other is chosen for study. As a result of the small extent of the region investigated the flow field is divided into two layers: the main part of the jet, which undergoes inviscid rotation, and a thin sublayer at the wall, which ensures the satisfaction of the no-slip condition. Particular interest attaches to the flow regime in which the solution in the sublayer satisfies the Prandtl boundary layer equations with a given pressure gradient. A similar problem was studied in [1–4]. The present case is distinguished by the structure of the free interaction region in a small neighborhood of the point of zero surface friction stress. By means of the method of matched asymptotic expansions, applied to the analysis of the Navier-Stokes equations, it is established that the interaction mechanism is that described in [5–7]. As a result, an integrodifferential equation describing the behavior of the surface friction stress function is obtained. A numerical solution of this equation is presented. The range of plate angles on which solutions of the equation obtained exist and, therefore, flows of this general type are realized is determined. The essential nonuniqueness of the possible solutions is established, and in particular attention is drawn to the possible existence of six permissible friction distributions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 38–45, January–February, 1986.The author wishes to thank V. V. Sychev and A. I. Ruban for their useful advice and discussion of the results.     

The zonal linearization method for multidimensional mass transfer problems in porous media

A number of methods have been proposed in recent years for calculating the combined flows of immiscible and miscible liquids in strata to systems of boreholes. We propose a method which can naturally be called the zonal linearization method [1]. It is more compact than the usual finite-difference method and has high accuracy, in particular, in the neighborhood of a borehole, since it is closely similar to the method of characteristics. The method can be applied to both continuous and discontinuous flows and in principle makes it possible to investigate the formation and breakdown of discontinuities. As distinct from the method of characteristics, it is well suited to programming and implementation on a computer, and it also makes it possible to obtain an approximate analytic solution of the problem in many cases and to estimate the accuracy of the solution. The method is based on the zonal linearization of the equation for mass conservation in the total flow between chosen surfaces or contour lines (lines of equal saturation or concentration). Determination of the dynamics of the contour surfaces leads to a Cauchy problem for a system of integrodifferential equations involving partial derivatives. The zonal linearization method is a development of the scheme described in [2–4], and the method of solving the Cauchy problem is a generalization of the methods described in [4–13]. The essence of the method and its convergence are illustrated by means of two-dimensional problems in two-phase filtration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 66–80, July–August, 1973.     

Nonlinear Electrohydrodynamic Stability of a Poiseuille Two–Layer Flow

The long–wave stability of the Poiseuille two–layer flow of homogeneous viscous dielectrics between plate electrodes under a constant potential difference is studied in an electrohydrodynamic approximation. A linear asymptotic stability analysis shows that surface polarization forces are a destabilizing factor, in addition to viscous stratification. The method of many scales is used to obtain the Kuramoto—Sivashinsky equation governing the weakly nonlinear evolution of the interface between the dielectrics. Within the framework of the approaches used, it is shown that nonlinear interactions limit perturbation growth and the interface does not fail even for a rather large potential difference.     

The steady-state flow of a rarefied gas from a spherical source or sink

The method of characteristics is used to solve problems in the steady-state flows of a rarefied gas on the basis of approximating the kinetic equations. Numerical results are given for the solution of the problem of the flow from a spherical source or sink using the generalized Kruk equation [1] and approximating the Boltzmann equation by the method proposed by the author [2, 3], Various flow conditions are discussed: flow into a vacuum, flow into a flooded volume, flow from a sink.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 58–66, March–April, 1971.     

Choice of a self-similar solution in boundary-layer theory

If the speed of the outer flow at the edge of the boundary layer does not depend on the time and is specified in the form of a power-law function of the longitudinal coordinate, then a self-similar solution of the boundary-layer equations can be found by integrating a third-order ordinary differential equation (see [1–3]). When the exponent of the power in the outerflow velocity distribution is negative, a self-similar solution satisfying the equations and the usually posed boundary conditions is not uniquely determinable [4], A similar result was obtained in [5] for flows of a conducting fluid in a magnetic field. In the present paper we study the behavior of non-self-similar perturbations of a self-similar solution, enabling us to provide a basis for the choice of a self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–46, July–August, 1974.     

Integral equations for subsonic gas flow in turbines

Conclusions Integral equations provide an, exact formulation for the flow in a turbine lattice; solution of the integral equation by successive approximation gives the potential and flow speed in analytical form. If (3.16) and (4.7) are met, the process also converges to the exact value. This algorithm is comparatively readily, implemented with a medium-power computer.The result from the integral equations goes with standard results [3, 4, 7] to show that the technique is of some value in research on subsonic flows.Zaporozhe Engineering Institute. Translated from Prikladnaya Mekhanika, Vol. 14, No. 9, pp. 110–117, September, 1978.     

Some exact solutions of the ferrohydrodynamics equations

The theoretical study of nonisothermal flows of magnetizable liquids presents serious matheical difficulties, which are intensified as compared to to the study of normal liquids by the necessity of simultaneous solution of both the hydrodynamics and Maxwell's equations, with corresponding boundary conditions for the magnetic field. Thus, in most cases problems of this type are solved by neglecting the effect of the liquid's nonisothermal state on the field distribution within the liquid, and also, as a rule, with use of an approximate solution for Maxwell's equations and fulfillment of the boundary conditions for the field [1–3]. The present study will present easily realizable practical formulations of the problem which permit exact one-dimensional solutions of the complete system of the equations of thermomechanic s of electrically nonconductive incompressible Newtonian magnetizable liquids with constant transfer coefficients. A common feature of the formulations is the presence of a longitudinal temperature gradient along the boundaries along which liquid motion is accomplished. Plane-parallel convective flows of this type in a conventional liquid and their stability were considered in [4–6].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 126–133, May–June, 1979.     

Some problems of turbulent electrohydrodynamic jets

The problem of an axisymmetric turbulent electrohydrodynamic jet exhausting from a nozzle into an interelectrode gap is formulated. A numerical method of integrating the system of equations describing this flow is developed. This method is used to investigate three-dimensional effects in the jet (expansion of the jet, reverse flows). The influence on the jet characteristics (currents of the charge carried out of the nozzle, jet diameter, etc.) of the geometrical and electrical parameters and also of purely hydrodynamic factors (level of turbulence, relative velocity of parallel flow, etc.) is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 144–149, September–October, 1980.     

Air-steam mixture flow with condensation on ions and electrokinetic processes

A physical model of air-steam flow with homogeneous condensation, condensation on ions, mass exchange between droplets and surrounding medium, and charge exchange between droplets and ion component is presented. A kinetic equation for the droplet distribution over sizes and charges is used in the model. On the basis of this equation, the moment equations are obtained and various approximate ways of closing them are proposed. The electric self-fields produced by the ion component and the charged dispersed phase are taken into account. Modifications of the equations for the case of turbulent flow are given. A one-dimensional flow model taking into account certain special features of the condensation and electrophysical processes in real flows is realized numerically.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 67–77, January–February, 1996.     

The potential approximation in the theory of conical flows

The use of potential theory to describe external flows at intermediate supersonic velocities makes it possible to construct very fast algorithms for calculating the flow even in the presence of subsonic regions [1, 2]. However, this approach involves errors associated with the neglect of the increase of entropy in the bow shock. The magnitude of these errors and their effect on the values of the various flow parameters are most easily estimated with reference to examples of conical flows. The shock-capturing projection-grid method [3] is used for integrating the conical potential equation. The results of calculating the flow past circular and elliptical cones, a triangular plate and a V-wing are compared with the corresponding solutions of the system of Euler equations. The region of applicability of the potential model is determined and it is shown that the satisfaction of the Hugoniot shock polar equation at the bow shock increases the error of the pressure calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 112–118, May–June, 1990.In conclusion the authors wish to thank V. V. Kovalenko for calculating the Euler equations and A. N. Kraiko for discussing the results.     

Nonlinear problem of flow past a source with the formation of a stagnant zone given different Bernoulli constants

The problem of free flow past a point source is considered for two streams with different Bernoulli constants whose encounter creates a bounded region of constant pressure. The theory and method of solving problems of plane ideal jet flows with different Bernoulli constants in the jets were developed in [1]. Here, in conformity with [1], a nonlinear system of equations is derived, the question of the construction of a high-accuracy numerical solution is considered, and certain calculation results are presented for various values of the Bernoulli and cavitation numbers, which are dimensionless parameters of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 55–60, May–June, 1986.     

Formation of one-dimensional flows of viscous heat-conducting gas for small parameter perturbations

The formation of unsteady one-dimensional flows is studied, using the solution of the problem of reflection of a normally incident plane shock wave from a heat-conducting wall as an example. The process is considered for low intensities of the incident wave, behind which the gas temperature hardly differs from the initial wall temperature. The flows with complicated internal structures that arise are investigated on the basis of Navier-Stokes equations linearized near the initial state. An analytic solution of the problem describing the discontinuous structure of the reflected flow is constructed, which can serve as a test in the numerical solution of the original nonlinearized Navier-Stokes equations. The influence of the Prandtl numbers, the specific heat ratio, and dissipative and other factors is considered. The features of the effects of viscosity, thermal conductivity, and accommodation on the formation of flows and ideal (inviscid, nonheat-conducting) and dissipative zones are traced. It is shown that the solution of the linearized system agrees with the solution for asymptotic flow regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 105–111, September–October, 1986.     

Potential model of the coanda effect

Three schemes of potential flow around a convex contour are examined: free jet; in the presence of a jet-directing wall extending to the contour (without a gap); and in the presence of a jet-directing wall not extending to the contour (with a gap). It is assumed that the jet, not branching, flows around the contour and on one side. The existence and (with certain limitations) uniqueness of the solution of integral equations corresponding to the first two schemes are proved. Their analytic solutions are given for the case of a flow of a sufficiently thin jet around a circle. The first problem is calculated numerically in the entire region of the parameters. The scheme with a gap is examined for the case of flow around a half-plane. The problem of closure of the given potential models is dicsussed. In the case of the scheme without a gap the situation is typical for potential flow problems: The position of the separation point is a free parameter of the model. At the same time, in the presence of a gap (in particular, for the case of a free jet) the flow is determined completely by a system ofa priori assigned geometric parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 10–22, July–August, 1978.The author is grateful to M. A. Gol'dshtik for attention to the work and discussion of the results.     

Sonic gas flow around a nonsymmetric profile

A particular solution of the transonic equations describing two-dimensional flow of an ideal gas is obtained for nonsymmetric flow around a certain profile. The aerodynamic characteristics of the profile are determined.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 76–81, September–October, 1984.The authors are grateful to S. V. Fal'kovich for the useful discussion of the results.     

Two-dimensional free boundary layer in a stratified medium

Expressions for the fluctuation characteristics of shear flow in a stratified medium are obtained on the basis of the equations for the single-point second-order moments of the velocity and temperature fields and then closure of those equations by means of semiempirical hypotheses. The Prandtl equation, with the influence or Archimedean forces taken into account, is used to analyze plane jet flows and wake flows of a body, Numerical computations are carried out for a plane wake, and the results are compared with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 71–79, July–August, 1977.     

Numerical study of two-dimensional separation on the basis of the Navier-Stokes equations

The phenomenon of the separation of a flow from the surface of a body, and the transfer of fluid which is slowed down in the boundary layer to the exterior flow, is of primary importance both in practice and in theory. From the practical point of view, flows with separation are important primarily because the separation of the boundary layer usually sets the upper limit of the efficiency, and therefore of the application, of many aerodynamic devices. From the theoretical point of view, the greatest importance lies in the problem of selecting the unique solution and the problem of elaborating effective numerical methods of studying flows with separation. The complexity of experimental research and the variety of problems connected with flow past bodies where separation occurs make the theoretical study of their general laws important. The aim of this work is to study separation zones and certain processes of controlling them on the basis of the full Navier—Stokes equations in the case of two-dimensional steady flows of a viscous incompressible fluid for moderately low Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 26–32, January–February, 1985.