Nonlinear waves in a liquid film entrained by a turbulent gas stream

In the long-wavelength approximation and on the basis of a simplified system of equations analogous to the one considered by Shkadov and Nabil' [1, 2], an investigation is made into waves of finite amplitude in thin films of a viscous liquid on the walls of a channel in the presence of a turbulent gas stream. A bibliography on the linear stability of such plane-parallel flows can be found in [3–5]. The nonlinear stability is considered in [6]. A stationary periodic solution is sought in the form of a Fourier expansion whose coefficients are found near the upper curve of neutral stability by Newton's method and near the lower branch of the stability curve by the method of Petviashvili and Tsvelodub [7, 8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 2, pp. 37–42, March–April, 1981.I thank V. Ya. Shkadov for supervising the work and all the participants of G. I. Petrov's seminar for a helpful discussion.     

The stability of steady regimes in an isothermal chemical flow reactor

One of the fundamental problems in the theory of chemical reactors is the determination of the number of steady regimes and their stability. The problem of the number of steady regimes has been considered in many studies, for example, in [1–4]. The stability of a steady regime is usually established from an analysis of the behavior of small perturbations. The corresponding linear boundary-value problem for perturbations has been studied mainly in the limiting cases of ideal mixing and ideal displacement. When account was taken of longitudinal mixing, the only criteria obtained were ones which imposed fairly severe restrictions on the parameters [5]. In the present study numerical analysis is used in order to investigate the stability of steady concentration distributions in an isothermal chemical flow reactor with longitudinal mixing in the case of a single chemical reaction. The eigenvalues were obtained for the Sturm-Liouville problem, which fully characterize the stability for several laws of variation of the chemical reaction rate as a function of the concentration. A knowledge of the eigenvalues is essential, for example, in order to construct the stabilization system proposed in [6] for the unsteady regime.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 179–182, March–April, 1985.     

Axisymmetric motion of a viscous fluid in the finite annular cavity of a rotating cylindrical vessel

The method of nets is used to investigate unsteady axisymmetric viscous flow in a cylindrical gap of finite height. This situation is characterized by vortex motion in a plane passing through the axis of the coaxial cylinders. These flows have previously been studied in relation to the case of stepwise variation of the angular velocity of the cylinders [1]. In the present case the angular velocity is varied linearly in the acceleration stage and the acceleration interval is a parameter of the problem. After acceleration the rotation rate is determined from the ordinary differential equation describing the process of deceleration of the system as a whole.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 37–42, March–April, 1988.The authors are grateful to E. M. Zhukhovitskii for his interest and valuable comments.     

Development of finite-amplitude two-dimensional and three-dimensional disturbances in jet flows

The laminar-turbulent transition zone is investigated for a broad class of jet flows. The problem is considered in terms of the inviscid model. The solution of the initial-boundary value problem for three-dimensional unsteady Euler equations is found by the Bubnov-Galerkin method using the generalized Rayleigh approach [1–4]. The occurrence, subsequent nonlinear evolution and interaction of two-dimensional wave disturbances are studied, together with their secondary instability with respect to three-dimensional disturbances.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 8–19, September–October, 1985.     

Aspects of self-similar diffraction gas flows and their numerical simulation

A large number of papers has been devoted to the investigation of the interaction of a plane shock wave with bodies of various geometric shapes, and they have been generalized and classified for a stationary body in [1, 2]. Separate results of experimental and theoretical investigations of the interaction of a shock wave with a wedge, cone, sphere, and cylinder moving with supersonic velocities are contained in [3–9]. Analysis of the available results shows that the features of the unsteady gas flows formed in this case largely depend on the nature of the boundary-value problem that arises for the system of differential gas dynamic equations. The question of the wave structure of the unsteady gas flow and the accuracy of the obtained solution is central to the numerical investigation of the present class of problems. The most characteristic types of unsteady self-similar gas flows that arise on the interaction of a plane shock wave with bodies of a wedge or convex corner type are calculated on the basis of an explicit numerical continuous calculation method of the second order of accuracy. The accuracy of the numerical solutions is discussed on the basis of a comparison with the experimental data. The case of the interaction of a shock wave with the rarefaction wave that arises in a supersonic flow past a convex corner is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 146–152, July–August, 1986.     

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.     

Numerical simulation of shock wave intersections

A large number of papers, generalized and classified in [1, 2], have been devoted to unsteady gas flows arising in shock wave interaction. Experimental results [3–5] and theoretical analysis [6–9] indicate that the most interesting and least studied types of interaction arise in cases when there are several shock waves. At the same time, nonlinear effects, which depend largely on the nature of the shock wave intersections, become appreciable. Regions of existence of different types, of plane shock wave intersections have been analyzed in [10–13]. It has been shown that in a number of cases the simultaneous existence of different types of intersections is possible. The aim of the present paper is to study unsteady shock wave intersections in the framework of a numerical solution of the axisymmetric boundary-value problem that arises in the diffraction of a plane shock wave on a cone in a supersonic gas flow. Flow regimes that augment the experimental data of [3–5] and the theoretical analysis of [9] are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 134–140, September–October, 1986.     

Stability of quasistatic equilibrium state of an inhomogeneous viscous layer on an inclined plane

A study is made of the stability against small perturbations [1] of a slow flow of an incompressible inhomogeneous linearly viscous liquid under the influence of a force of gravity on an unbounded inclined plane. Problems of such kind arise in glaciology when one estimates the stability of snow on mountain slopes or determines the catastrophic movement of a glacier; the results can also be applied to solifluction phenomena [2, 3]. Equations for perturbations of parallel flows of linearly viscous fluids in the case of a continuous variation of the viscosity and density across the flow were derived in [4]. An attempt to solve the hydrodynamic problem with allowance for a perturbation of the viscosity was made in [5]; however, in the equations for the perturbations, simplifications resulted in the neglect of terms that take into account perturbations of the viscosity. In the quasistatic formulation considered here in the case when allowance is made for perturbation of the density and viscosity, the equation for the perturbation amplitudes is an ordinary differential equation with variable coefficients; analytic solution of the equation is very difficult, even for long-wave perturbations. In this connection a study is made of the stability of a laminar model; the viscosity and density are constant within each layer. A similar hydrodynamic problem in the long-wave approximation was considered in [6]. In the present paper an exact solution is constructed in the quasistatic formulation for a two-layer model; the solution shows that the viscosity of the lower layer has an important influence on the stability. Essentially, instability is observed when the lower layer acts as a lubricant.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 20–24, November–December, 1973.     

Quasi-one-dimensional approximation in two-dimensional problems of gas dynamics

A useful means of constructing approximate flow models is the hydraulic (for two-dimensional problems quasi-one-dimensional) approach, based on averaging the initial nonuniform flows over some direction or cross section [1]. In this case, at the expense of a rougher model it is possible to reduce the dimensionality of the problem. Here, this approach is extended to unsteady two-dimensional gas-dynamic processes; certain problems (flow around a cone or a blunt body, jet flows) are considered in the framework of the quasi-one-dimensional model obtained, and results are compared with the solutions of the corresponding two-dimensional problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 136–143, March–April, 1989.     

Far nonlinear field of a nonlifting airfoil for a generalised equation of transonic flow

The transonic flow equation [1] for plane unsteady irrotational idealgas flows is extended to the case of subsonic, transonic or supersonic flows in a region with an almost constant value of the velocity using orthogonal flow coordinates (family of equipotential lines and streamlines). A solution for the nonlinear far field of steady transonic flow past an airfoil has been obtained for the transonic equation [2]. In this paper it is obtained for a generalized transonic equation and its asymptotic expansion is given. In using difference methods of calculating the flow past an airfoil in the transonic regime a knowledge of the nonlinear field makes it possible to reduce the dimensions of the calculation region (near field) as compared with the region determined by the far field of the linear theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 87–91, January–February, 1986.     

Stability of two-dimensional curvilinear flows of an ideal barotropic fluid

It was established by Arnol'd [1] that the conservation laws for the energy and vorticity can be used to establish sufficient conditions of stability of two-dimensional curvilinear flows of an ideal incompressible fluid in the exact nonlinear formulation. It is shown below that one can obtain similarly conditions of stability of two-dimensional curvilinear steady flows of an ideal barotropic fluid in the linear approximation. One of the conditions has a significance similar to Rayleigh's criterion and its generalization by Arnol'd [1]; the other is the condition of subsonic flow. In addition, a variational principle is established and an expression found for the second variation of the corresponding functional; these can be used to prove the stability of these flows in the exact nonlinear formulation.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–25, September–October, 1981.I am sincerely grateful to V. L. Berdichevskii and A. G. Kulikovskii for constructive advice.     

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.     

Simple waves in a plasma with anisotropic pressure

Fast and slow simple waves are studied in the framework of the anisotropic magnetohydrodynamics of Chew, Goldberger, and Low [1]. Baranov [2] has constructed fields of integral curves for fast and slow waves and in two special cases has shown that such waves break in the compression section. The possibility of breaking of a slow wave in a rarefaction section was noted by Akhiezer et al. [3]. However, their general relations in simple waves [3] have been shown to be incorrect [2, 4]. In the present paper the nature of the variation of the longitudinal and transverse plasma pressures is determined, and the problem of the breaking of fast and slow waves is completely solved. Conditions under which a slow wave breaks in a rarefaction section are found. A fast wave always breaks in a rarefaction section.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 181–183, July–August, 1988.     

Nonlinear stability of two-dimensional steady flows of an ideal fluid with constant vorticity and their axisymmetric analogs

Flows with constant vorticity are widely used as local models of more complicated flows [1]. In many cases, such flows are stable against finite two-dimensional perturbations. In particular, the inviscid plane-parallel Couette flow has the property of nonlinear stability. Similar treatment of a class of axisymmetric flows yields nonlinear stability of a spherical Hill vortex and inviscid Poiseuille flow in a circular tube with respect to axisymmetric perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 16–21, October–December, 1981.     

Interaction between the boundary layer and a supersonic flow when part of the wall is rapidly heated

There have been many theoretical studies of aspects of the unsteady interaction of an exterior inviscid flow with a boundary layer [1–9]. The mathematical flow models obtained in these studies by the method of matched asymptotic expansions describe a wide range of phenomena observed experimentally. These include boundary layer separation near the hinge of a flap, the flow in the neighborhood of the trailing edge of an oscillating airfoil [1–2], and the development and propagation of perturbations in a boundary layer excited by an oscillating wall or some other way [3–5]. The present paper studies the interaction of an unsteady boundary layer with a supersonic flow when a small part of the surface of a body in the flow is rapidly heated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 66–70, January–February, 1984.     

Three-dimensional problem for entry of a disk into a compressible liquid

The unsteady problem for the oblique entry of a disk into water is solved. The water is assumed a perfect compressible liquid and the flow is assumed adiabatic. The flow and state parameters are determined during the numerical integration of the system of nonlinear equations which describe the given flow by means of a three-dimensional finite-difference scheme [1]. The variation in time of the drag coefficient as a function of the Mach number and the angles of entry and attack, the pressure distribution and the shape of the free surface formed behind the disk are investigated. The oblique entry of a disk into water and its subsequent motion have mainly been studied for velocities at which the compressibility of the water is negligible [2–4]. The influence of compressibility on the duration of the rise time and the impact load was investigated experimentally in the range of Mach numbers 0 < M0 <–0.3 [5]. Semiempirical dependences are obtained for the maximum of the drag coefficient and its rise time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 17–20, January–February, 1988.     

Region of applicability and the main features of the generalized Chapman-Enskog method

The present paper is made necessary by the publication of the foregoing paper in this issue by Kolesnichenko [1]. It considers the basic propositions of the generalized Chapman-Enskog method and analyzes the arguments put forward by Kolesnichenko [1] and the validity of the method. The position of the results obtained by Kolesnichenko [14–17] is indicated. Nonequilibrium flows of multiatomic gases in which there occur processes of exchange of internal energy of the molecules in collisions between them and chemical reactions (such processes are called inelastic) are encountered frequently in nature and technology. It is therefore naturally of interest to derive gas-dynamic equations for such flows. The methods of the kinetic theory of gases were first used to obtain equations describing the limiting cases of very fast inelastic processes that take place in times of the order of the molecule-molecule collision times (equilibrium case) and very slow inelastic processes that take place over times of the order of the characteristic flow time (relaxation case). In [2–5], an algorithm was proposed for deriving gas-dynamic equations valid for arbitrary ratios of the rates of the elastic and inelastic processes and reducing to the well-known equations for the limiting cases already mentioned. The algorithm is called the generalized Chapman-Enskog method (abbreviated to the generalized method). The development, modification, and analysis of its properties can be found in [4, 6–13]. In [1], Kolesnichenko has questioned the validity of this algorithm.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 126–136, May–June, 1984.We thank V. A. Rykov for helpful and constructive discussions of the work.     

Laminar boundary layer with uniform suction on flat plate in oscillating flow

We examine unsteady incompressible fluid flow in a laminar boundary layer with uniform suction for longitudinal flow over a flat plate when the external stream is a flow with constant velocity, on which there is superposed a sinusoidal disturbance convected by the stream, analogous to [1]. We study the stability of such flow in the boundary layer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 3, pp. 66–70, May–June, 1970.     

Limiting regimes of self-similar gas flows with account for finite chemical-reaction rate

One-dimensional unsteady flows of a combustible gas mixture with account for the finite chemical-reaction rate were studied in [1]. The conditions for self-similarity of such flows were indicated, mathematical formulation of the problem was given, and several numerical calculations were carried out.The authors pointed out the necessity for conducting additional studies, since they were not able to obtain numerically, by means of passages to the limit, self-sustaining detonation waves propagating with the Chapman-Jouguet (CJ) velocity.In this article we point out the reason why it was not possible to reach the CJ regime in [1], and a qualitative analysis is made, by means of the results of [2], of the system of equations describing the self-similar flows of a gas with finite chemical-reaction rate, and the passage to the limit is made to the self-sustaining CJ detonation waves in the presence of chemical reactions. It is also shown that the problem of unsteady flows of a combustible mixture of gases with finite chemical-reaction rate is analogous to the problem of the flow of a gas heated by radiation, examined in [3].In conclusion the authors wish to thank I. V. Nemchinov and A. G. Kulikovskii for discussions of this study.     

Quasi-three-dimensional calculation of flow in blade passages of turbines

The flow in turbomachines is currently calculated either on the basis of a single successive solution of an axisymmetric problem (see, for example, [1-A]) and the problem of flow past cascades of blades in a layer of variable thickness [1, 5], or by solution of a quasi-three-dimensional problem [6–8], or on the basis of three-dimensional models of the motion [9–11]. In this paper, we derive equations of a three-dimensional model of the flow of an ideal incompressible fluid for an arbitrary curvilinear system of coordinates based on averaging the equations of motion in the Gromek–Lamb form in the azimuthal direction; the pulsation terms are taken into account in the equations of the quasi-three-dimensional motion. An algorithm for numerical solution of the problem is described. The results of calculations are given and compared with experimental data for flows in the blade passages of an axial pump and a rotating-blade turbine. The obtained results are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 69–76, March–April, 1991.I thank A. I. Kuzin and A. V. Gol'din for supplying the results of the experimental investigations.