Plane incompressible fluid flow past an array of arbitrary profiles vibrating with arbitrary phase shift

We consider the problem of the vibration of an array of arbitrary profiles with arbitrary phase shift. Account is taken of the influence of the vortex wakes. The vibration amplitude is assumed to be small. The problem reduces to a system of two integral Fredholm equations of the second kind, which are solved on a digital computer. An example calculation is made for an array of arbitrary form.A large number of studies have considered unsteady flow past an array of profiles. Most authors either solve the problem for thin and slightly curved profiles or they consider the flow past arrays of thin curvilinear profiles [1].In [2] a study is made of the flow past an array of profiles of arbitrary form oscillating with arbitrary phase shift in the quasi-stationary formulation. The results are reduced to numerical values. Other approaches to the solution of the problem of unsteady flow past an array of profiles of finite thickness are presented in [3–5] (the absence of numerical calculations in [3, 4] makes it impossible to evaluate the effectiveness of these methods, while in [5] the calculation is made for a symmetric profile in the quasi-stationary formulation).     

Plate cascade in subsonic unsteady gas flow

Accounting for fluid compressibility creates serious difficulties in solving the problem of oscillations of a grid of thin, slightly curved profiles in a subsonic stream. The problem has been solved in [1–3] for a widely-spaced cascade without stagger whose profiles oscillate in phase opposition. The phenomenon of aerodynamic (acoustic) resonance, which may arise in a grid in the direction transverse to the stream for definite values of the stream velocity and profile oscillation frequency, was discovered in [2]. An approximate solution of the problem in which account is not taken of the effect of the vortex trails on the gas flow has been obtained in [4]. In [5, 6] Meister studied in the exact linear formulation the problem of unsteady gas motion through an unstaggered cascade of semi-infinite plates. In [7] Meister considered a grid of profiles with finite chords, but the problem solution was not completed. The problem of subsonic gas flow through a staggered lattice whose profiles oscillate following a single law with constant phase shift was solved most completely in the studies of Kurzin [8, 9] using the method of integral equations. A method of solving the problem for the case of arbitrary harmonic oscillation laws for the lattice profiles was indicated in [10]. The results of the calculation of the unsteady aerodynamic forces for the particular case of a plate cascade without stagger are presented in [9,11], and the possibility of the occurrence of aerodynamic resonance in the cascade in the directions transverse to and along the stream is indicated.Another method of solving the problem is given in [12], in which the more general problem of unsteady subsonic gas flow through a three-dimensional cascade of plates is solved. In the present study this method is applied to the solution of the problem of oscillations of staggered plate cascades in a two-dimensional subsonic gas flow. The results are presented of an electronic computer calculation of the unsteady aerodynamic characteristics of the cascade profiles, which show the essential influence of fluid compressibility on these characteristics. In particular, a sharp decrease of the aerodynamic damping in the acoustic resonance regimes is obtained.     

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.     

Vortex flows of a viscous heat-conducting gas in a slightly divergent tube with volume energy supply

The results of a numerical investigation of viscous vortex flow in a slightly divergent tube with thermal energy supplied to the flow are presented. The initial stage of vortex flow development is considered for two different longitudinal velocity distributions simulating the velocity profiles in jet-like and wake-like vortex flows in the vicinity of the vortex axis. The first type of flow can be considered as a model for the near-axis region of the vortex formed in the flow around a delta wing at incidence. The second type can serve as a model for the near-axis region of the trailing vortex downstream of a high-aspect-ratio wing. The development of the two flows is studied for a constant area tube, a slightly divergent tube, and in the case of thermal energy supply from a volume energy source at a constant wall temperature.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 90–97, September–October, 1996.     

Aerodynamics of end-wall boundary layer in a vortex chamber

The need for the inclusion of end-wall boundary layers in the study of the aerodynamics of vortex chambers has been frequently mentioned in the literature. However, owing to limited experimental data [1–3] with reliable information on the wall layers, the existing computational methods for end-wall boundary layers are not well-founded. The question of which parameters determine the formation of end-wall flow remains debatable. In some studies [4, 5], the vortex chambers are conditionally divided into short and long chambers. However, there is no unique opinion on the role of end-wall flows in vortex chambers of different lengths. It has also not been established for what geometric and flow parameters the chamber could be considered long or short. In the present study, as in [1, 5–8], solution is obtained for the end-wall boundary-layer equations using integral methods, considering the boundary layer in the radial direction in the form of a submerged wall jet. Such an approach made it possible to use the laws for the development of wall jets [9], and obtain fairly simple relations for integral parameters, skin friction, mass flow in the boundary layer, and other characteristics. Results are compared with available experimental data and computations of others authors; turbulent flow is considered; results for laminar boundary layer are given in [10].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 117–126, September–October, 1986.     

Dynamics of thin vortex rings in a low-viscosity fluid

In order to describe the unsteady flow of a viscous fluid induced by a toroidal vorticity distribution we use the two-scale expansion method [6], By this means we obtain a vorticity distribution in the core of the thin vortex ring that is consistent with the external potential flow. The time dependence of the flow characteristics obtained confirms the experimental results for the inertial regime. The interaction of coaxial vortex rings is investigated as an example.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 52–59, September–October, 1992.     

Mathematical simulation of unsteady flow past a wing with jets

The various approximate approaches to the investigation of the unsteady aerodynamic characteristics of an airfoil with jet flap [1–3] are applicable only for an airfoil, low jet intensity, and low oscillation frequencies. In the present paper, the method of discrete vortices [4] is generalized to the case of unsteady flow past a wing with jets and arbitrary shape in plan. The problem is solved in the linear formulation; the conditions used are standard: no flow through the wing and jet, finite velocities at the trailing edges where there is no jet, and also a dynamical condition on the jet. The wing and jet are assumed to be thin and the medium inviscid and incompressible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 139–144, May–June, 1982.     

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.     

Aerodynamic interaction of two plane cascades of thin lightly loaded blades in relative motion in a subsonic gas flow

The problem of subsonic ideal-gas flow over two plane cascades of thin lightly loaded blades in relative motion is solved within the framework of the linear theory of small perturbations. By means of the method of integral equations [1] the problem is reduced to an infinite system of singular integral equations for the harmonic components of the oscillations in the distribution of the unknown aerodynamic load on the blades. The regularized system of integral equations for a finite number of harmonics is solved numerically by a collocation method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 168–175, May–June, 1987.     

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.     

The damped natural oscillations of a gas flowing past a cascade of flat plates

We solve the problem of the natural oscillations of a gas flowing past a cascade of flat plates under the Joukowsky-Chaplygin condition that the velocity at the trailing edge of the profiles is finite. In this case part of the energy of the oscillating gas is consumed in the formation of a trailing vortex. The corresponding eigenvalues of the problem are complex and so the natural oscillations of the gas are damped. The computational results are compared with the results of experimental investigation of acoustic resonance in flow past a cascade of flat plates obtained by Parker [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 84–88, September–October, 1970.     

Experimental investigation of flow in a trench

An experimental investigation was made of the flow of a viscous incompressible liquid in a trench of square transverse cross section, using a laser Doppler velocimeter. The investigation was made with two values of the Reynolds number Re, corresponding to laminar and turbulent flow conditions in the channel. The experimental data show that a core with a constant vorticity is formed in the trench, that a jet propagates near the walls of the trench, and that there are secondary eddies in the corners of the trench. The motion of a viscous liquid in a trench of rectangular cross section is part of a broad class of breakaway flows. Experimental data on the investigation of flow in trenches are extremely few. A majority of the existing information is limited to visual observations [1–4]. In [2, 5, 6] the question of the unstable character of flow in trenches was discussed. Quantitative measurements of stable eddy flows in trenches were made in [7–9] using a thermoanemometer, and in [7] measurements were made of the pressure at the bottom and walls of trenches; there are data on the distribution of the velocity in the middle sections of trenches. In [8] the mean velocity, the intensity of the turbulence, and the stress of the turbulent flow were obtained in several sections parallel to the side walls of the trench, In [9] a measurement was made of the velocities also in two cross sections of a trench in which one component of the velocity prevails. A brief analysis of the existing experimental results shows that these data are insufficient to form a detailed representation of the character of flow in a trench.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 76–86, March–April, 1976.     

Fluid flow in an arbitrary cascade on an axisymmetric stream surface in a variable thickness layer

A solution is given for the problem of flow past a cascade on an axisymmetric stream surface in a layer of variable thickness, which is a component part of the approximate solution of the three-dimensional problem for a three-dimensional cascade. Generalized analytic functions are used to obtain the integral equation for the potential function, which is solved via iteration method by reduction to a system of linear algebraic equations. An algorithm and a program for the Minsk-2 computer are formulated. The precision of the algorithm is evaluated and results are presented of the calculation of an example cascade.In the formulation of [1, 3] the problem of flow past a three-dimensional turbomachine cascade is reduced approximately to the joint solution of two-dimensional problems of the averaged axisymmetric flow and the flow on an axisymmetric stream surface in an elementary layer of variable thickness.In the following we solve the second problem for an arbitrary cascade with finite thickness rotating with constant angular velocity in ideal fluid flow: the solution is carried out on a Minsk-2 computer.Many studies have been devoted to this problem. A method for solving the direct problem for a cascade of flat plates in a hyperbolic layer was presented in [2]. Methods were developed in [1, 3] for constructing the flow for the case of a channel with variable thickness; these methods are approximately applicable for dense cascades but yield considerable error for small-load turbomachine cascades. The solution developed in [4], somewhat reminiscent of that of [2], is applicable for thin, slightly curved profiles in a layer with monotonically varying thickness. A solution has been given for a circular cascade for layers varying logarithmically [5] and linearly [6]. Approximate methods for slightly curved profiles in a monotonically varying layer with account for layer variability only in the discharge component were examined in [7–9]. A solution is given in [10] for an arbitrary layer by means of the relaxation method, which yields a roughly approximate flow pattern. The general solution of the problem by means of potential theory and the method of singularities presented in [11] is in error because of neglect of the crossflow through the skeletal line. The computer solution of [12] contains an unassessed error for the calculations in an arbitrary layer. The finite difference method is used in [13] to solve the differential equation of flow, which is illustrated by numerical examples for monotonie layers of axial turbomachines. The numerical solution of [13] is very complex.The solution presented below is found in the general formulation with respect to the geometric parameters of the cascade and the axisymmetric surface and also in terms of the layer thickness variation law.The numerical solution requires about 15 minutes of machine time on the Minsk-2 computer.     

Thin axisymmetric cavities in a subsonic compressible flow

The investigation of subsonic compressible flow past thin axisymmetric cavities carried out in [1–3] is continued by the method of asymptotic expansions. The dependence of the elongation of the cavity on the cavitation number and the Mach number is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 174–177, September–October, 1987.The author is grateful to Yu. L. Yakimov for discussing his results.     

Axially symmetric vortical flow of a nonviscous liquid in the semiinfinite gap between two coaxial circular cylinders

In the framework of the Hromek-Lamb equations we investigate the axially symmetric vortical flow of a nonviscous incompressible liquid in both semiinfinite and infinite gaps between two coaxial circular cylinders. The investigation is carried out for two circulation and flow functions and two different Bernoulli constants which are chosen in the form of a third-order polynomial in the flow function. This makes it possible to determine the effect of the azimuthal velocity component on the flow in an axial plane with radial and axial components of the velocity. It is shown that under certain circumstances wave oscillations in the flow are possible, in agreement with the results of [1–3] which investigated the flow in an infinite tube [1], in a semiinfinite tube with simpler circulation functions and Bernoulli constants [2], and in the two-dimensional case [3]. We determine the dependence of the formation of wave perturbations on the third term of the Bernoulli constant and on the azimuthal velocity component. The results of this work agree with investigations by other authors [1–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 38–45, September–October, 1977.The author thanks Yu. P. Gupalo and Yu. S. Ryazantsev for suggesting this problem and for their interest in the work. Thanks are also due to G. Yu. Stepanov for discussions and valuable comments.     

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.     

Effect of nonisothermality of the surface of a thin profile on the stability of a laminar boundary layer

Investigations of the stability of a subsonic laminar boundary layer have shown that, other things being equal, the stability of the laminar flow is considerably improved by cooling the entire surface of the body to a constant temperature T_w=const lower than the temperature of the free stream [1–3]. This is attributable to an increase in the critical Reynolds number of loss of stability and a decrease in the range of unstable perturbations of the Tollmien-Schlichting wave type when the surface is cooled. Recently, in the course of investigating the stability of laminar flow over a flat plate it was found [4, 5] that a similar improvement in flow stability can be achieved by raising the temperature of a small part of the surface near the leading edge of the plate. In this study we examine the possibility of delaying the transition to turbulent flow by creating a nonuniform temperature distribution along the surface of thin profiles, where the development of an adverse pressure gradient in the outer flow has a destabilizing effect on the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 36–42, September–October, 1986.In conclusion, the authors wish to thank M. N. Kogan for useful discussions of their results.     

Nonisothermal Couette flow of a non-Newtonian fluid under the action of a pressure gradient

Nonisothermal Couette flow has been studied in a number of papers [1–11] for various laws of the temperature dependence of viscosity. In [1] the viscosity of the medium was assumed constant; in [2–5] a hyperbolic law of variation of viscosity with temperature was used; in [6–8] the Reynolds relation was assumed; in [9] the investigation was performed for an arbitrary temperature dependence of viscosity. Flows of media with an exponential temperature dependence of viscosity are characterized by large temperature gradients in the flow. This permits the treatment of the temperature variation in the flow of the fluid as a hydrodynamic thermal explosion [8, 10, 11]. The conditions of the formulation of the problem of the articles mentioned were limited by the possibility of obtaining an analytic solution. In the present article we consider nonisothermal Couette flows of a non-Newtonian fluid under the action of a pressure gradient along the plates. The equations for this case do not have an analytic solution. Methods developed in [12–14] for the qualitative study of differential equations in three-dimensional phase spaces were used in the analysis. The calculations were performed by computer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 26–30, May–June, 1981.     

Experimental investigation of supersonic flow past blunt bodies with strong distributed injection

The entry of bodies into planetary atmospheres at high supersonic velocities is accompanied by intense evaporation of the surface due to radiative heat fluxes. A series of problems involving the conduction of investigations of such kind has been proposed by Petrov. In [1], in particular, the entry of a meteorite into an atmosphere was examined. The gasdynamic aspects of this problem have been approximately simulated by many authors by intense injection of gas in theoretical, e.g., [2–5], and experimental [6, 7] studies. The theoretical studies were based on two-layer [3, 4] or three-layer [5] schemes of gas flow between the shock wave and the surface of the body. The aim of the present work was an experimental investigation of the interaction of injection with a counter supersonic flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 84–95, May–June, 1978.