Parameters of flow near the axis in the shock layer in the interaction of a supersonic jet with a barrier

The axisymmetric interaction between a supersonic jet with a finite expansion ratio and a barrier is accompanied by the formation of complex sub- and supersonic flow in a shock layer whose thickness depends on the parameters of the jet and the position of the barrier. The main relationships of the interaction process have been established experimentally ([1–3] and others) and individual results of numerical calculations of such flows are known [4]. An analytical investigation of the parameters in the shock layer formed ahead of a plane barrier when an underexpanded jet impinges on it is presented below. The results of [5], where the region near the axis of a shock layer of arbitrary thickness is analyzed within the framework of a model of flow with a constant density, is placed at the basis of the analysis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 63–70, September–October, 1978.The author thanks Yu. M. Tsirkunov for useful discussions.     

Similarity solutions of the equations of a boundary layer in a nonuniform external flow

Similarity solutions of the equations of a laminar incompressible boundary layer, formed in a rotational external flow, are investigated. Such problems arise in the analysis of the flow in a boundary layer when there is an abrupt change in the boundary conditions (for example, in the case of a discrete inflation of the boundary layer, in hypersonic flow about blunt bodies, etc.). Various approaches to their solution have been proposed earlier in [1–4]. Solved below is the so-called inverse problem of boundary layer theory (see [3], p. 200), where the contour of the body that causes a given flow outside the boundary layer is unknown beforehand and is found during the course of solution of the problem in connection with the coupling of the longitudinal and transverse velocity components. The cases of a parabolic (u_e ~ y²) and a linear (u_e=a(x)+b(x)y) variation in the velocity of the external flow with distance along the transverse direction are considered in detail. The latter includes an investigation of the flow in the neighborhood of the critical point of a blunt body, taking account of the vorticity of the flow in the shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 78–83, March–April, 1971.     

On the formulation of the problem of the motion of a body in water

The selection of solutions describing steady irrotational flow of an ideal incompressible fluid over bodies is considered. The selection is based on restrictions that follow from the physical properties of a real fluid and from the presence of a boundary layer on the body. In particular, for any body one can specify a minimal Euler number below which flow without cavitation becomes physically impossible. In the limiting case of an Euler number equal to zero, only the Kirchhoff scheme is physical admissible, and the cavity section tends to a circle. An equation is derived for the limiting shapes of cavities at small cavitation numbers, and a comparison is made with known results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–26, September–October, 1980.     

Design of a wing airfoil in a flow with a separated turbulent boundary layer

The problem of designing the contour of an airfoil in a viscous (incompressible and compressible) flow with a separated turbulent boundary layer from a pressure distribution given on the separationless part of the contour is solved using the boundary layer theory together with the separated flow model proposed in [1]. Numerical calculations are carried out to demonstrate the possibilities of the method.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 83–91, May–June, 1994.     

Method of mean-mass parameters for a three-dimensional boundary layer in a flow with vorticity

When a gas flows with hypersonic velocity over a slender blunt body, the bow shock induces large entropy gradients and vorticity near the wall in the disturbed flow region (in the high-entropy layer) [1]. The boundary layer on the body develops in an essentially inhomogeneous inviscid flow, so that it is necessary to take into account the difference between the values of the gas parameters on the outer edge of the boundary layer and their values on the wall in the inviscid flow. This vortex interaction is usually accompanied by a growth in the frictional stress and heat flux at the wall [2, 3]. In three-dimensional flows in which the spreading of the gas on the windward sections of the body causes the high-entropy layer to become narrower, the vortex interaction can be expected to be particularly important. The first investigations in this direction [4–6] studied the attachment lines of a three-dimensional boundary layer. The method proposed in the present paper for calculating the heat transfer generalizes the approach realized in [5] for the attachment lines and makes it possible to take into account this effect on the complete surface of a blunt body for three-dimensional laminar, transition, or turbulent flow regime in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 80–87, January–February, 1981.     

Formation of a region of interaction and of a wake with the instantaneous start of a plate in an incompressible liquid

The flow arising in an incompressible liquid if, at the initial moment of time, a plate of finite length starts to move with a constant velocity in its plane, is discussed. For the case of an infinite plate, there is a simple exact solution of the Navier—Stokes equations, obtained by Rayleigh. The case of the motion of a semiinfinite plate has also been discussed by a number of authors. Approximate solutions have been obtained in a number of statements; for the complete unsteadystate equations of the boundary layer the statement was investigated by Stewartson (for example, [1–3]); a numerical solution of the problem by an unsteady-state method is given in [4]. The main stress in the present work is laid on investigation of the region of the interaction between a nonviscous flow and the boundary layer near the end of a plate. In passing, a solution of the problem is obtained for a wake, and a new numerical solution is also given for the boundary layer at the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–8, March–April, 1977.     

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.     

Wave flow of a liquid film together with a flow of gas

The wave flow of a thin layer of viscous liquid in conjunction with a flow of gas was considered in a linear formulation earlier [1, 2]. In this paper the problem of the wave flow of a liquid film together with a gas flow is solved in a nonlinear setting. On this basis relationships are derived for calculating the parameters of the film and the hydrodynamic quantities.Ivanovo. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 12–18, January–February, 1972.     

Excitation of Tollmien-Schlichting waves py a vibrating section of a surface with flow round it

In the context of the problem of describing the transition of a laminar boundary layer to a turbulent, great interest attaches to the study of susceptibility, i.e., of the reaction of the flow to various external influences, such as acoustic perturbations, surface roughness, vibration of the wall, turbulence of the unperturbed flow, etc. A general property of the effect of the factors mentioned above on the flow in a laminar boundary layer was discovered in experimental and numerical studies and is noted in [1]: in all cases an external forcing perturbation leads to the excitation of normal modes of oscillation in the boundary layer which propagate downstream, namely, Tollmien-Schlichting waves. There is an analytical calculation in [2, 3] of the amplitude of a wave excited by harmonic oscillations of a narrow band on the surface of a plane plate, the Reynolds number having been assumed to be infinitely large, and the frequency of the vibrator corresponding to the neighborhood of the lower branch of the neutral cuirve [4], In [5] the amplitude of the wave of instability generated is calculated by the method of expansion of the solution in a biorthogonal system of eigenfunctions. The amplitudes of the Tollmien-Schlichting waves are calculated below by means of a generalization of the method of [2] for the whole range of Reynolds numbers and frequencies of the vibrator corresponding to the region of instability: for moderate Reynolds numbers the problem is solved numerically, while for large Reynolds numbers an asymptotic solution is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 46–51, July–August, 1987.The author is grateful to M. N. Kogan and V. V. Mikhailov for useful discussions of the results of the study.     

Interaction of a three-dimensional boundary layer with an extensive obstacle

A special variant is considered of the theory of longitudinal—transverse interaction in which the pressure field in the perturbed region of flow forms under the influence of centrifugal forces which lead to a change in the pressure across the boundary layer. This regime of flow is realized in flow of an incompressible fluid, when the two-dimensional boundary layer developing along the smooth section of the contour of a solid body enters into interaction with a three-dimensional irregularity on the surface around which flow is taking place, a projection or a depression. On the assumption that the height of the irregularity is not great, a solution is constructed for the linearized problem of interaction. It is shown that the properties of the flow of fluid in the region of interaction, in particular the possibility of penetration of perturbations into the boundary layer in front of an irregularity, depend on the sign of the curvature exhibited by the contour of the body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 39–48, January–February, 1988.     

Motion of a gas shock layer with a free boundary under nonadiabatic conditions

This paper is a study of the effect of heat input (removal) on the characteristics of a shock layer produced by a gas at high supersonic velocity encountering a mobile boundary, which for generality is assumed to be free. We will use the Chernyi method, which was employed previously to solve the problem of a shock layer in an adiabatic flow [1, 2]. The results obtained can be useful for analysis of the effect of radiation (absorption) and processes involving the relaxation of internal degrees of freedom of molecules, condensation, chemical reactions, etc., whose effect on the gasdynamics of the flow in a shock layer may be similar to heat input or removal [3–5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 152–154, May–June, 1976.The author thanks A. K. Rebrov for discussion of the results.     

Stability of convective flow in a vibration field with respect to three-dimensional perturbations

A plane-parallel convective flow in a vertical layer between boundaries maintained at different temperatures becomes unstable when the Grashof number reaches a critical value (see [1]). In [2, 3] the effect of high-frequency harmonic vibration in the vertical direction on the stability of this flow was investigated. The presence of vibration in a nonisothermal fluid leads to the appearance of a new instability mechanism which operates even under conditions of total weightlessness [4]. As shown in [2, 3], the interaction of the usual instability mechanisms in a static gravity field and the vibration mechanism has an important influence on the stability of convective flow. In this paper the flow stability is considered for an arbitrary direction of the vibration axis in the plane of the layer and the stability characteristics with respect to three-dimensional normal perturbations are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 116–122, March–April, 1988.     

Development of the boundary layer on a plate in a rotating system

A numerical investigation in the approximation of boundary layer theory has been made of the development of the flow near the surface of a rotating plate in a two-dimensional flow with rectilinear streamlines perpendicular to the leading edge in a rotating coordinate system attached rigidly to the plate. In an earlier investigation [1] using the approximate method of integral relations, Kurosaka obtained and described quantitatively a transition from a Blasius boundary layer to an Eckmann boundary layer in the form of three-dimensional oscillations. The solution described in the present paper confirms the oscillatory nature of the development of the boundary layer, but the quantitative results differ strongly from Kurosaka's.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 154–157, May–June, 1982.     

Stability of stationary convective flow of a mixture in a vertical porous layer

In contrast to the corresponding viscous flow, the convective flow of a homogeneous liquid in a planar vertical layer whose boundaries are maintained at different temperatures is stable [1]. When a porous layer is saturated with a binary mixture, in the presence of potentially stable stratification one must expect an instability of thermal-concentration nature to be manifested. This instability mechanism is associated with the difference between the temperature and concentration relaxation times, which leads to a buoyancy force when an element of the fluid is displaced horizontally. In viscous binary mixtures, the thermal-concentration instability is the origin of the formation of layered flows, which have been studied in detail in recent years [2–4]. The convective instability of the equilibrium of a binary mixture in a porous medium was considered earlier by the present authors in [5]. In the present paper, the stability of stationary convective flow of a binary mixture in a planar vertical porous layer is studied. It is shown that in the presence of sufficient longitudinal stratification the flow becomes unstable against thermal-concentration perturbations; the stability boundary is determined as a function of the parameters of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 150–157, January–February, 1980.     

Vibrational convection in a plane layer with a longitudinal temperature gradient

An exact solution is obtained for the equations of vibrogravitational convection in an arbitrarily oriented plane fluid layer with a longitudinal component of the temperature gradient. It is shown that in the absence of a static field inclined vibration leads to the development of plane-parallel convective flow. On the other hand, gravitational plane-parallel convective flow can always be suppressed by an appropriate choice of the direction and amplitude of vibration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 12–15, July–August, 1990.     

Connective instability of a two-layer system with thermally insulated boundaries

The investigation of the convective stability of mechanical equilibrium of two horizontal layers of immiscible fluids has revealed the characteristics of such systems [1–3]. In particular, it has been found that, as distinct from a homogeneous horizontal layer, under certain conditions two-layer systems experience convective instability when uniformly heated from above and, moreover, oscillatory instability when heated from below. In [1–3] the problem was solved for a system with isothermal outer boundaries. In this paper the stability of equilibrium of two-layer systems is investigated for thermally insulated outer boundaries. Special attention is given to the study of the long wave instability mode.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 22–28, March–April, 1986.The authors wish to thank O. V. Kustova for assisting with the computations.     

Convective stability of a medium containing a heavy solid additive

Two problems of convective stability in a medium containing settling heavy solid particles are discussed. A study is made of the stability of steady convective flow of a medium containing an additive between vertical plates heated to different temperatures and also of the stability of a flat layer of a medium containing an additive which is heated from below. It is shown that the presence of settling solid particles has a significant stabilizing effect on convective stability.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 105–115, May–June, 1976.The author thanks E. M. Zhukhovitskii for directing the work, V. E. Nakoryakov and participants in the seminars directed by him, and also A. G. Kirdyashkin for providing valuable discussions of the results.     

Investigation of boundary layer stability in a gradient flow with a high degree of free-stream turbulence

The results of an experimental investigation of boundary layer stability in a gradient flow with a high degree of free-stream turbulence are presented. The question of the possible artificial generation, the further development and the effect on laminar-turbulent transition of instability waves (Tollmien-Schlichting waves) in the boundary layer on a wing profile is considered for a level of free-stream turbulence =1.75% of the free-stream velocity; the sensitivity of the flow to the disturbances and their control by means of boundary layer suction are investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 52–58, March–April, 1990.     

Theory of detachment of a laminar boundary layer

In this paper, we investigate the structure of the flow that arises in the case of uniformly distributed injection from a plane permeable surface, under conditions that ensure a transition from a flow in a boundary layer to a detached flow including a mixing layer and a region of inviscid wall flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 16–20, November–December, 1987.The author thanks V. Ya. Neiland for his discussion of the results and valuable advice.     

Model of a viscous nonequilibrium shock layer with a thin shock wave

It is shown that the concept of a viscous shock layer with boundary conditions specified in a thin shock wave is unsuitable for analyzing the flow of a chemically reacting gas, even in the case of high Reynolds numbers; it may produce a finite error when determining the parameters of the shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–178, September–October, 1973.