Secondary convective motions in a plane inclined layer

A linear theory of stability of a plane-parallel convective flow between infinite isothermal planes heated to different temperature was developed in [1–6]. At moderate Pr values the instability is monotonic and leads to the development of steady secondary motions. These motions for the case of a vertical layer have been investigated by the net [7, 8] and small-parameter [9] methods. In this paper steady secondary motions in an inclined layer are investigated. The small-parameter and net methods are used. The hard nature of excitation of secondary motions in a defined range of tilt angles is established. There are two types of secondary motions, whose regions of existence overlap — vortices at the boundary of countercurrent streams and convection rolls; the hard instability is due to the development of convection rolls. The analog of the Squire transformation obtained in [4] for infinitely small disturbances of a plane-parallel convective flow is extended to secondary motions of finite amplitude.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–9, May–June, 1977.I thank G. Z. Gershumi, E. M. Zhukhovitskii, and E. L. Tarunin for interest in the work and valuable discussion.     

Model of two-dimensional vortex motion with an entrainment mechanism

A model for describing the vertically averaged vortex motions of an incompressible viscous fluid with an arbitrary vertical structure of the bottom Ekman boundary layer is proposed. An approach similar to that adopted in [1] is used: the second moments of the deviations from the average velocities required in order to close the vorticity equation are calculated by means of the Ekman solution for gradient flows, which makes it possible to take the integral bottom boundary layer effect into account. As a result, these terms lead to a specific form of nonlinear friction with a coefficient that depends on the vorticity of the average flow. In the particular case of a constant vertical turbulent transfer coefficient the inaccuracies of the model described in [1] can be eliminated. The generalized vorticity equation obtained has solutions of the vorticity spot type with a uniform internal vorticity distribution, which can be effectively investigated by means of appropriate algorithms [2]. The mechanism of entrainment at the vorticity front is illustrated with reference to the example of the evolution of vorticity spots. An exact solution of the problem of the evolution of an elliptic vortex (generalized Kirchhoff vortex), which in the case of fairly strong anticyclonic vorticity degenerates first into a line segment (vortex sheet) and then into a point vortex, is constructed. Equations describing the dynamics of an elliptic vorticity spot in an external field with a linear dependence of the velocity on the horizontal coordinates and generalizing the classical Chaplygin-Kida model [3, 4] are constructed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 49–56, November–December, 1992.     

Linear theory of unsteady gas flow under the influence of nonpotential external forces

We determine in the linear formulation the velocity and pressure fields excited in a compressible medium by a lifting filament that displaces and deforms arbitrarily. For general unsteady motion of such a filament we give explicit formulas that express the velocity at a given point in terms of the intensity of the free vortices entering the audio signal audibility zone constructed for this point. We examine gas flow caused by an arbitrary external body force field.Studies devoted to the determination of gas velocity fields for flow past slender bodies relate primarily to translational motion of a body with a dominant constant velocity [1–3]. Gas velocities for helical motion of a rectilinear lifting filament within the gas have been examined in [4].     

Conditions for nonlinear stability of flows of an ideal incompressible liquid

Stability of steady state flows of an ideal incompressible liquid with homogeneous density with some type of symmetry (translational, axial, rotational, or helical) is considered. Two types of sufficient conditions for nonlinear stability are obtained, which can be proven by constructing two types of functionals which have absolute minima at the given steady state solutions. Each of the functionals used is the sum of the kinetic energy and some other integral, specific to the given class of motion. The first type of stability conditions are a generalization to the case of finite perturbations and a new class of flows of the well known Rayleigh criterion [1] for centrifugal stability of rotating flows relative to perturbations with rotational symmetry. In the same sense the second type of stability conditions generalize another result, also originally proposed by Rayleigh, according to which plane-parallel flow of a liquid is stable in the absence of an inflection point in the velocity profile [1]. A nonlinear variant of the latter condition for the class of planar motions was first obtained in [2]. To systematize the results extensive use is made of the analogy between the effects of density stratification and rotation in the form of [3], The results to be presented relate to stability of a wide class of hydrodynamic flows having the required symmetry. For example, they relate to flows in tubes and channels which rotate or are at rest, and flows with concentrated annular or spiral vortices.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 70–78, May–June, 1986.     

Analysis of the hydrodynamic interaction between cascades of thin profiles taking account of vortex wake evolution

The papers [1–5] are devoted to an investigation of aspects of the hydrodynamic interaction of cascades of profiles in a nonlinear formulation: it is shown experimentally in [1] and theoretically in [2] that the free vortex sheet ruptures upon meeting a profile; taking account of the evolution of vortex wakes, the flows around two cascades of solid profiles of infinitesimal [3] and finite [4] density are computed; results of an experimental investigation of the dynamic reactions of the flow on two mutually moving cascades of thin profiles are presented in [5]. The interference between two cascades of thin profiles in an inviscid, incompressible fluid flow is examined in this paper, where a modified method from [6] is used.Translated from Zhurnal Prikladnoi MekhaniM i Tekhnicheskoi Fiziki, No. 4, pp. 61–65, July–August, 1976.The author is grateful to D. H. Gorelov for discussing the research.     

Impact on a plate on the surface of a fluid

Questions of the interaction between solid and elastic structures with an ideal fluid which are associated with the initial stage of the impact and penetration of bodies in the fluid were considered in [1–4]. Results are presented below of an analysis of a central impact on a solid weightless plate which is on the surface of a compressible fluid. The impact velocity is much less than the speed of sound in the medium. Computations are performed by a finite-difference Lagrange method according to a program for plane motions of a continuous medium [5] by using a volume artificial viscosity of Neumann-Richtmayer type [6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–145, May–June, 1978.     

Calculation of the flow around wings of arbitrary planform over a wide range of angles of attack

A numerical method is described for the calculation of the distributed and total aerodynamic characteristics of a thin wing of any planform. We use only the generally accepted hypotheses-smoothness of flow around the wing and the Chaplygin-Zhukovskii condition of finite velocity at the trailing edges. The medium is considered ideal and incompressible.The development of a nonlinear theory for the wing of small aspect ratio in a compressible medium is one of the most important and difficult problems of wing theory. It has long attracted the attention of the aerodynamiscists. Chaplygin touched on this question in his 1913 report On the vortex theory of the finite span wing, presented to the Moscow Mathematical Society. Several interesting ideas and schemes were proposed by Golubev (see, for example, [2]). The first adequately correct and effective attempt to determine theoretically the nonlinear variation of wing normal force with angle of attack was that of Bollay [3]. In this work he studied rectangular wings of very small aspect ratio. The circulation variation law along the span was taken to be constant, and along the chord it was taken the same as for a flat plate of infinite span. It was also assumed that the centerlines of the free vortices trailing from the wing tips are straight lines and form the same angle with the plane of the wing. The magnitude of this angle was calculated from the average value of the relative velocity. The boundary condition at the wing was satisfied at a single point.In several later studies [4–8] attempts have been made to extend this approach somewhat. In [7] the circulation variation law along the wing chord is calculated, and the boundary conditions are satisfied more exactly. However, attempts to convert to the study of wings of more complex planform, when the circulation can no longer be considered constant along the span, are hydrodynamically incorrect [5, 6, 8]. In these studies schemes are used in which with smooth flow around the wing the free vortices stand off from the wing surface. The angles which the vortex centerlines form with the wing surface are assumed or are calculated on the basis of very arbitrary hypotheses.In the present paper the vortex layer which simulates the wing surface, just as in the linear theory [9, 10], is replaced by a system of discrete vortices. The free vortices away from the wing then are discrete curvilinear vortex filaments. Each of them is replaced by a series of rectilinear vortex segments. The number of bound and free discrete vortices may be increased without limit. The position of the free vortex segments is determined in the computation process, which is carried out sequentially for a series of angles of attack , beginning with 0 when the linear theory scheme holds. We note that the question of accounting for the effect of the leading-edge free vortex sheet is not considered here, although the method described may also be used to obtain results for this problem.The proposed method turned out to be very general, flexible and convenient for the digital computer. It permits studying the practical convergence of the solution, and also permits obtaining not only the total and distributed characteristics of the wing of arbitrary planform, but also studying such delicate questions as the rollup of the vortex sheet behind the wing.The author wishes to thank O. N. Sokolov and T. M. Muzychenko for the example calculations.     

Motion of two vortex rings

It is well-known [1] that two coaxial rings which are moving in the same direction pass through each other alternately. In the case of thin vortex rings this phenomenon was first considered qualitatively in [2]. The assumption that the vortex rings are thin means that when their interaction is considered they can be assumed to be annular vortex filaments. In the present paper, on the basis of the approach suggested in [2], certain new properties are determined for a system of two coaxial vortex rings of the same intensity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 176–177, September–October, 1985.The authors express their sincere thanks to A. A. Aleksandrov for his interest in their work.     

Three-dimensional instability of an elliptic Kirchhoff vortex

The instability of a Kirchhoff vortex [1–3] with respect to three-dimensional perturbations is considered in the linear approximation. The method of successive approximations is applied in the form described in [4–6]. The eccentricity of the core is used as a small parameter. The analysis is restricted to the calculation of the first two approximations. It is shown that exponentially increasing perturbations of the same type as previously predicted and observed in rotating flows in vessels of elliptic cross section [4–9] appear even in the first approximation. As distinct from the case of plane perturbations [1-3], where there is a critical value of the core eccentricity separating the stable and unstable flow regimes, instability is predicted for arbitrarily small eccentricity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–45, May–June, 1988.     

Calculation of unsteady supersonic flow past a two-dimensional cascade of plates ween vorticity inhomogeneities of the flow act on it

In the framework of the linear theory of small perturbations the problem of unsteady subsonic flow past a two-dimensional cascade of plates has been considered in a number of papers. Thus, the unsteady aerodynamic characteristics of a cascade of vibrating plates were calculated in [1] by the method of integral equations, while the same method was used in [2, 3] to calculate the sound fields that are excited when sound waves Coming from outside or vorticity inhomogeneities of the oncoming flow act on the cascade. The problem of a two-dimensional cascade of vibrating plates in a supersonic flow was solved in [4, 5]. In [4] the solution was constructed on the basis of the well-known solution of the problem of vibrations of a single plate, while in [5] a variant of the method of integral equations was used which differed slightly from the usual formulation of this method [1–3]. The approach proposed in [5] is used below to calculate the unsteady flow past a two-dimensional cascade of plates in the case when vorticity inhomogeneities of a supersonic oncoming flow act on it. Equations are obtained for the strength of the unsteady pressure jumps arising in such a flow and the vortex wakes shed from the trailing edges of the plates. Examples of the calculations illustrating the accuracy of the method and its possibilities are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp, 152–160, May–June, 1986.     

Calculation of the flow of an ideal fluid past a wing of finite thickness

The problem of irrotational flow past a wing of finite thickness and finite span can be reduced by Green's formula to the solution of a system of Fredholm equations of the second kind on the surface of the wing [1]. The wake vortex sheet is represented by a free vortex surface. Besides panel methods (see, for example, [2]) there are also methods of approximate solution of this problem based on a preliminary discretization of the solution along the span of the wing in which the two-dimensional integral equations are reduced to a system of one-dimensional integral equations [1], for which numerical methods of solution have already been developed [3–6]. At the same time, a discretization is also realized for the wake vortex sheet along the span of the wing. In the present paper, this idea of numerical solution of the problem of irrotational flow past a wing of finite span is realized on the basis of an approximation of the unknown functions which is piecewise linear along the span. The wake vortex sheet is represented by vortex filaments [7] in the nonlinear problem. In the linear problem, the sheet is represented both by vortex filaments and by a vortex surface. Examples are given of an aerodynamic calculation for sweptback wings of finite thickness with a constriction, and the results of the calculation are also compared with experimental results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 124–131, October–December, 1981.     

Experimental study of vortex flow about low aspect ratio wings and circular cones at M = 2

Characteristic of the flow about wings of low aspect ratio with subsonic leading edges and bodies of revolution at angles of attack is the formation of spiral vortices as a result of rolling-up of the transverse flow, which separates near the wing leading edge and on the lateral generator of the body. The vortices, concentrated in a pair of free vortex cores, interact with the boundary layer, causing a complex flow pattern on the surface of the model in question.There are several methods which make it possible to study the flow about the model. Pickups may be used to measure the pressure field or the velocity field near the model. This technique has found wide application and was used for studying the flow pattern about wings and bodies of revolution at both subsonic and supersonic speeds (see, for example, [1–4]). However, this method is very tedious and, in addition, the probes always introduce disturbances into the flow, particularly for supersonic speeds.A visual picture of the vortex flow may be obtained in a towing basin by adding to the water metal powder in the suspended state, or by introducing filaments of colored liquid [1, 5].The vapor screen [6] and smoke [3] methods are also used for flow visualization.The boundary layer flow on the model may be studied with the aid of oil or evaporting coatings. These methods have been used in [1, 7] to study flow about wings and in [8] to study flow about circular cones.According to the studies presented in [9] of an electric discharge with the application of high voltage to electrodes located in an air stream, a stable glow occurs as a result of the prebreakdown discharge.The properties of the prebreakdown discharge have been used by the authors of the present paper to study visually the vortex flows (high voltage electric discharge method). This technique was used to obtain the trajectories of the vortex trails for low aspect ratio wings and circular cones mounted at various angles of attack in a stream with Mach number M=2 and Reynolds number R=0.9·10⁶.In conclusion the author wishes to thank B. V. Kalachev, R. V. Bertyn, and E. D. Korolev for assistance in carrying out the experiments.     

Instabilities in multi-hole converging flow of viscoelastic fluids

Studies of the onset of instabilities were conducted on single hole and multi-hole contractions using laser speckle visualization. A well characterized elastic fluid was used with constant viscosity of 13.1 Pa · s and elasticity characterized by a longest relaxation time constant of 2.233 s. The onset of instabilities was characterized in terms of the Deborah number and the contraction ratio. Three types of instabilities were observed: pulsing vortices, azimuthally rotating vortices, and swirling vortices. For the single hole contractions the critical Deborah number for instability increased from 4.4 to 5.07 to 5.25 as the contraction ratio increased from 4: 1 to 8: 1 to 12: 1. The magnitude of the instabilities was much greater for the 4: 1 contraction than for the other two contraction ratios. For the multi-hole contraction a square array of nine holes was used and the ratio of the hole diameter to hole spacing was varied. The height of the vortices is very similar for the single hole and multi-hole contractions at low Deborah numbers. At high Deborah numbers the effect of adjacent holes is to reduce the height of the vortices by a factor of three. For the 4: 1 spacing no secondary vortex was observed below a Deborah number of De = 3.7. Secondary vortices occurred for the 8:1 and 10:1 spacing at all Deborah numbers. Unstable pulsing vortices appeared for all spacings at a critical Deborah number around 5.5. Adjacent holes decreased the strength of the unsteady vortex motions. The centerline velocities were measured for the multi-hole contraction at shear rates of 5, 30, and 300 s^–1. The elongational strain rates are similar at a low shear rate of 5 s^–1. As shear rate is increased the onset of stretching occurs closer to the plane of the contraction for the smaller contraction ratios.     

Dynamics of vorticity at a sphere

The article discusses the two-dimensional flow of an incompressible liquid between two infinitely close concentric spheres, due to an initial distribution of the vorticity differing from zero. The concept of point singularities (vortices, sources, and sinks) at a sphere is introduced. Equations of motion are obtained for point vortices, as well as invariants of the motion, known for the plane case [1]. The simplest case of the mutual motion of a pair of vortices is considered. Equations are obtained for the motion of point vortices at a rotating sphere. Integral invariants for the continuous distribution of the vorticity are obtained, having the dynamic sense of the total kinetic energy and the momentum of the liquid at the sphere. The effect of the topology of the sphere on the dynamics of the vorticity is noted, and a comparison is made with the plane case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 57–65, November–December, 1977.     

Pressure pulsations on a rigid wall caused by a vortex street

A topic of obvious interest is the interaction between the vortex wake behind bluff bodies and nearby objects, in particular plane surfaces. At present the most detailed studies in flows such as these have been on the characteristics of the velocity field of the vortices [1, 2]. The main topic of attention in what follows is pressure pulsations on a wall caused by separation of vortices from an adjacent cylinder. This problem is especially important for the determination of vibrations and noises due to bodies located on the rigid boundary of the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 167–169, March–April, 1986.The author wishes to express his appreciation to M. I. Rabinovich for discussion of the results, and to P. R. Gromov and A. B. Zobnin for help in conducting the experiment.     

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.     

Numerical simulation of the dynamics of a liquid in a moving axisymmetric cavity

The problem of the motion of an ideal liquid with a free surface in a cavity within a rigid body has been most fully studied in the linear formulation [1, 2]. In the nonlinear formulation, the problem has been solved by the small-parameter method [3] and numerically [4–7]. However, the limitations inherent in these methods make it impossible to take into account simultaneously the large magnitude and the threedimensional nature of the displacements of the liquid in the moving cavity. In the present paper, a numerical method is proposed for calculating such liquid motions. The results of numerical calculations for spherical and cylindrical cavities are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–177, March–April, 1984.     

Conservation of the resultant force vector in the plane of the angle of attack for slender star-shaped bodies in supersonic flows

At high supersonic flight speeds bodies with a star-shaped transverse and power-law longitudinal contour are optimal from the standpoint of wave drag [1–3]. In most of the subsequent experimental [4–6] and theoretical [6–9] studies only conical star-shaped bodies have been considered. For these bodies in certain flow regimes ascent of the Ferri point has been noted [10]. In [11] the boundary-value problem for elongated star-shaped bodies with a power-law longitudinal contour was solved for the case of supersonic flow. The present paper deals with the flow past these bodies at an angle of attack. It is found that for arbitrary star-shaped bodies with any longitudinal (in particular, conical) profile the aerodynamic forces can be reduced to a wave drag and a lift force, the lateral force on these bodies being equal to zero for any position of the transverse contour.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 135–141, November–December, 1989.     

Effect of homogeneous stratification of the fluid on the dynamics of a kármán vortex street behind a circular cylinder

The experimental results of studying the effect of homogeneous stratification of the fluid on the conditions of generation of a Kárman vortex street [1] developing in the wake of a cylinder in steady horizontal motion are described. In a homogeneous medium at Reynolds numbers Re >5 two symmetrical regions of vorticity of opposite sign are formed behind the cylinder and move together with the latter. As the speed of the cylinder increases, the link between the vortices and the cylinder grows weaker, the vortices are stretched out along the flow and at Re > 40 begin to separate alternately, forming a vortex street in the wake. At first, the frequency of vortex separation increases sharply with increase in Re, but then levels off. It is found that in a uniformly stratified fluid the onset of vortex separation from the moving cylinder is delayed. The dependence of the critical Reynolds number (onset of vortex separation) on the internal Froude number is obtained. The effect of stratification of the fluid on the frequency of separation of the vortices in the Kármán street is investigated. The effect of the Froude number on the dependence of the Strouhal number on the Reynolds number is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 83–86, January–February, 1986.In conclusion the authors wish to thank A. T. Onufriev for his interest in their work and useful discussions of the results.