Turbulent flow of a fluid in the neighborhood of the trailing edge of a plate at zero angle of attack

The laminar flow regime of an incompressible fluid at the trailing edge of a plate was studied by Stewartson and Messiter [1, 2] by means of the method of matched asymptotic expansions. In. the present paper, this method is used to analyze the same problem, but in the case of turbulent flow in the boundary layer and the wake. A system of linear equations of elliptic type with variable coefficients is obtained for the averaged values of the flow parameters in the main part of the boundary layer and the wake that is responsible for the change in the displacement thickness. A solution of this system is constructed by the Fourier method in the case of a power law of the velocity in front of the interaction region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–23, November–December, 1983.     

The effect of vertical vibrations on the onset of convection in a planar rotating layer

The effect of vertical vibrations on the convection in a rotating planar fluid layer heated from below was studied. In this case a modulation parameter, the acceleration due to gravity, appears in the problem. The modulation of the parameter may have a significant effect on the onset of convective instability. Parameter modulation in nonrotating layers has been investigated in earlier work [1–3]. The presence of rotation significantly increases the complexity of the mathematical problem, introducing an additional dependence of the solution on the Taylor number Ta and the Prandtl number Pr. Furthermore, an oscillatory convection regime can occur at the stability limit in rotating fluids with Pr < 1. Parameter modulation in the rotating fluid may not only lead to a change in the stability limit and critical wavelength but also to a change in the eigenfrequency of the oscillatory convection. Rauscher and Kelly [4] examined the effect of parameter modulation on the convective stability of a rotating fluid only for the particular case of a sinusoidal variation in the temperature gradient with a small amplitude for Pr = 1, i.e., the effect of modulation was studied on only a steady convection regime.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 12–22, July–August, 1984.     

Flow between a porous rotating disk and a plane

It is shown that when a viscous incompressible fluid is sucked through a stationary porous disk spontaneous rotation of the fluid sets in at a certain Reynolds number. This is consistent with the results of a specially designed experiment. Another unusual result is the existence of multicell regimes, corresponding to suction, when the force acting on the porous, rapidly rotating disk is a lift force and, moreover, anomalously large. Charts of the possible steady-state flow regimes, stable and unstable, have been constructed. In the case of fairly intense suction and rotation a stable self-oscillating regime is observed. In the limit of vanishingly small viscosity unusual boundary layer properties associated with suction are noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 53–65, November–December, 1989.     

Unsteady processes in the neighborhood of a well in the case of a nonlinear flow law and determination of stratum parameters

A study is made of the propagation of disturbances in the neighborhood of a well in the case of a linear elastic regime for a flow law with limiting gradient, and also for a nonlinear elastic regime for different forms of the flow law. The obtained results are used to investigate the difference between two forms of flow anomaly — nonlinearity of the flow law associated with non-Newtonian behavior of the fluid and a pressure dependence of the parameters of the nonlinear flow law due to nonlinear elastic deformations of strata [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 66–70, July–August, 1983.I thank V. M. Entov and L. A. Chudov for helpful advice and discussion.     

Existence and nonuniqueness of local separation zones in viscous jets

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

Convective stability of a horizontal rotating fluid layer with helical turbulence

The problem of the corrective stability of a horizontal layer of turbulent fluid rotating about a vertical axis with a fixed heat flow at the boundaries is investigated in the case in which the intensity of the helical background does not depend on the rate of rotation and the degree of heating.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 33–39, January–February, 1992.The author is grateful to S. S. Moiseev for proposing the subject and to G. Z. Gershuni and D. V. Lyubimov for useful discussions.     

A class of exact solutions of a nonlinear boundary-value problem for fluid flow in a rotating cylinder

The flow of an ideal incompressible weightless fluid that fills a rotating cylinder is investigated. The rotation axis of the cylinder is outside it and parallel to the cylinder generator, and the form of the cylinder section is determined in the process of solution of the problem. In the paper, a class of exact solutions of the problem is obtained in terms of elementary functions for different angular velocities of the cylinder. In these solutions, the flow field is formed by two straight vortex filaments parallel to the cylinder generator. The intensities of the vortex filaments are determined by the angular velocity . Investigations of ideal fluid flow in rotating vessels were begun already in the last century by Stokes and Zhukovskii [1]. The subject has been reviewed in monographs [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 1, pp. 71–75, January–February, 1984.     

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.     

The boundary layers of a fully ionized two-temperature plasma with given component temperatures at an electrode

The flow of a plasma with different component temperatures in the boundary layers at the electrodes of an MHD channel is investigated without any assumptions as to self-similarity. For the calculation of the electron temperature, the full energy equation for an electron gas [1] is solved with allowance for the estimates given in [2]. In contrast to [3, 4], the calculation includes the change in temperature of electrons and ions along the channel caused by the collective transport of energy, the work done by the partial pressure forces, and the Joule heating and the energy exchange between the components. The problem of the boundary layers in the flow of a two-temperature, partially ionized plasma past an electrode is solved in simplified form by the local similarity method in [5–7]. In these papers, either the Kerrebrock equation is used [5, 6] or the collective terms are omitted from the electron energy equation [7].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–10, September–October, 1972.The author thanks V. V. Gogosov and A. E. Yakubenko for interest in this work.     

Viscous flow in a partially-filled horizontal cylinder rotating at constant speed

The problem under consideration is that of the stationary shape of the free surface of a viscous fluid in a steadily rotating horizontal cylinder. In the majority of investigations of this problem the thickness of the fluid layer coating the inner surface of the cylinder is assumed to be small [1–3]. The case of a near-horizontal free surface, with the bulk of the fluid at the cylinder bottom, was considered in [4], where, after considerable simplification, the governing equations were reduced to ordinary differential equations. In the present study the behavior of the free surface is investigated using a creeping flow approximation. The controlling parameters vary over a wide range. In the numerical computations a boundary element method was used. The numerical results have been confirmed experimentally.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–30, May–June, 1993.     

Character of the boundary layer near its origin on a body rotating in a fluid at rest

The evolution of the boundary layer on bodies of revolution rotating about the symmetry axis in a fluid at rest is largely determined by the position of its origin with respect to the axis of rotation. If the origin of the boundary layer coincides with a pole of the rotating body, then under fairly general assumptions as to the shape of the body the boundary layer has a nonzero thickness in the vicinity of the pole, and the flow in it is described by a particular self-similar solution of the boundary-layer equations [1, 2]. The applicability of existing approximate methods for calculation of the boundary layer [2, 3] is restricted to this case. The results of the present article refer to the case in which the boundary originates at the leading edge at a finite distance from the rotation axis. The behavior of the solution of the boundary-layer equations near the edge is determined. A transformation of variables that reduces the system of boundary-layer equations to a form suitable for analysis and solution is derived. This transformation is used to obtain universal equations determining the local behavior of the boundary layer in the vicinity of its origin in both of the cases indicated above.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–140, July–August, 1976.     

Calculation of auto-oscillations resulting from the loss of stability of a nonisothermal Couette flow

A study is made of the periodic (in time) auto-oscillatory regime (cycle) of azimuthal wave type which branches from a stationary nonisothermal Couette flow between rotating cylinders. Yudovich's method [1, 2] is used. It is shown that the cycle is unique (up to a shift along the axis z of the cylinders and rotation through any angle) and stable against spatially periodic three-dimensional perturbations. The results are given of a numerical calculation of the first two terms in the expansion of the cycle in Lyapunov-Schmidt series. The torque acting on the surface of the inner cylinder is calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–32, May–June, 1981.I thank V. I. Yudovich for constant interest in the work.     

Method of calculation of interaction of wall flows

A flow of viscous compressible fluid in the neighborhood of the line of interaction of wall flows is considered. A method of calculating the line of interaction and the direction of the self-induced secondary flow is developed. Papers [1–3] are devoted to the simulation of a separation flow with singularities in the neighborhood of singular lines and points, where boundary-layer equations are invalid. However, the theories of local separation used at present have mainly been developed only for two-dimensional problems, while the models of viscous-inviscid interaction have restrictions in application for turbulent flows with developed separation. The interaction of three-dimensional wall turbulent flows is considered below. It is assumed that the thickness of the boundary layers and the scales of the interaction zones are small in comparison with the characteristic dimension of the system, while the line of discontinuity of the solutions of the three-dimensional boundary layer equations is the same as the line of interaction of the wall flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–59, March–April, 1987.The author is grateful to G. Yu. Stepanov and V. N. Ershov for their interest in my work and their valuable remarks.     

Convection in rotating spherical layers with allowance for centrifugal forces

The investigation of convection in rotating spherical layers with a central gravitational field g(r) is very important for the study of the global motions in the atmospheres of large planets and the convective zones of stars. In recent years, many studies of these questions have been made (they have been reviewed, for example, by Yavorskaya and Belyaev [1]), but the centrifugal convective force has been ignored in all the numerical and analytic investigations. In some cases, for example, for large planets, the centrifugal force may reach an appreciable value, O.1g, and have a strong influence on the convective motion. The present paper studies the occurrence of convection in slowly rotating spherical layers with allowance for centrifugal forces. It is shown that the centrifugal force leads to the appearance in a layer of an axisymmetric flow, at the stability limit of which convective cells of banana or toroidal shape can develop. The latter are possible only in layers with undeformable boundaries at sufficiently large values of the Froude number. Irrespective of the form of the layer and the magnitude of the centrifugal force, the banana-shaped cells propagate in a wavelike manner in the direction opposite to the rotation. In the case of undeformable boundaries, the centrifugal force stabilizes the motion of the fluid as compared with the case of a layer at rest. Deformation of one or both of the boundaries under the influence of the centrifugal force leads to destabilization of the basic flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 14–21, March–April, 1984.     

Flow of a viscous fluid in an annular gap with intensive blowing from the walls

Self-similar solutions are obtained in [1, 2] to the Navier-Stokes equations in gaps with completely porous boundaries and with Reynolds number tending to infinity. Approximate asymptotic solutions are also known for the Navier-Stokes equations for plane and annular gaps in the neighborhood of the line of spreading of the flow [3, 4]. A number of authors [5–8] have discovered and studied the effect of increase in the stability of a laminar flow regime in channels of the type considered and a significant increase in the Reynolds number of the transition from the laminar regime to the turbulent in comparison with the flow in a pipe with impermeable walls. In the present study a numerical solution is given to the system of Navier-Stokes equations for plane and annular gaps with a single porous boundary in the neighborhood of the line of spreading of the flow on a section in which the values of the local Reynolds number definitely do not exceed the critical values [5–8]. Generalized dependences are obtained for the coefficients of friction and heat transfer on the impermeable boundary. A comparison is made between the solutions so obtained and the exact solutions to the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–24, January–February, 1987.     

Drag associated with transitional flow regime of a fiber suspension in a tube

A semlempirical model is constructed of the flow of a fiber suspension of low and medium concentration in regimes that are usually called mixed and undeveloped turbulent regimes [1–4]. It is shown that although the flow of fiber suspensions in these regimes has features similar to those of the turbulent flow of a Newtonian fluid, for example, a logarithmic velocity profile, the characteristic features of the flow in both regimes can be better explained, not by turbulence of the flow, but by orientation of the fibers in it and by plastic flow of the fiber continuum. For this reason, to distinguish the mixed and undeveloped turbulent regimes from a truly turbulent regime it is proposed here to describe them by a general name — transitional flow. The obtained expressions agree qualitatively and quantitatively with the experimental results of Lee and Duffy [2], Sanders and Meyer [3], and Mih and Parker [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–45, May–June, 1984.I thank V. N. Nikolaevskii and A. N. Golubyatnikov for interest in the work and helpful comments.     

Rotational inviscid flow with circulation zones in a channel of variable cross section

The development of the theory of rotational motion of inviscid fluids for the purposes of describing channel flow encounters certain difficulties in connection with the appearance of viscosity effects near the walls. In the potential-rotational model [1], in which the vorticity is nonzero only in a closed circulation zone surrounded by potential flow, it is assumed that the separation and attachment points are known in advance. For example, for flow around a cavity these points coincide with the extreme corner points of the contour. The problem of determining the vorticity in a closed zone for the potential-rotational model has been investigated in a number of studies [2, 3], etc. In the case of an incompressible fluid the vorticity in the circulation zone is constant for two-dimensional flow and proportional to the distance from the axis for axisymmetric flow. The value of the constant is found from the steady-state condition for the adjoining viscous layers. If the channel walls have a smooth profile without corner points, then for determining the boundaries of the circulation zones additional conditions must be used. This study employs another scheme, in which the vorticity is formed outside the region of flow and in a particular problem is specified in the form of a boundary condition. An analytic solution describing the rotational flow of an inviscid fluid in a channel with a slightly varying cross section is obtained. Three types of entrance flow nonuniformity are considered: 1) uniform shear flow, 2) wake-type flow, and 3) potential flow with a narrow wall boundary layer. Streamline patterns with circulation zones are constructed for flows in diffuser channels with the above-mentioned types of entrance nonuniformity. A model of flow separation in a channel with a turbulent boundary layer on the walls is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 31–37, March–April, 1985.In conclusion the author wishes to thank E. Yu. Shal'man, A. N. Kraiko, and A. B. Vatazhin for useful discussions and advice.     

Universal equations of the laminar boundary layer on a rotating blade

The article discusses the problem of the motion of an incompressible liquid in a boundary layer on a blade rotating uniformly around an axis perpendicular to the swing of the blade. A parametric method is used to solve the problem, and three series of parameters are introduced, on which depend the characteristics of the boundary layer. A corresponding system of universal equations is set up, which is integrated over a broad range of change in the parameters. The results obtained permit investigating the principal laws governing flow in a boundary layer on a rotating blade. The effect of rotation on breakaway and other characteristics of the boundary layer is clarified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 84–93, March–April, 1971.     

Two-phase couette flow

A study is made of plane laminar Couette flow, in which foreign particles are injected through the upper boundary. The effect of the particles on friction and heat transfer is analyzed on the basis of the equations of two-fluid theory. A two-phase boundary layer on a plate has been considered in [1, 2] with the effect of the particles on the gas flow field neglected. A solution has been obtained in [3] for a laminar boundary layer on a plate with allowance for the dynamic and thermal effects of the particles on the gas parameters. There are also solutions for the case of the impulsive motion of a plate in a two-phase medium [4–6], and local rotation of the particles is taken into account in [5, 6]. The simplest model accounting for the effect of the particles on friction and heat transfer for the general case, when the particles are not in equilibrium with the gas at the outer edge of the boundary layer, is Couette flow. This type of flow with particle injection and a fixed surface has been considered in [7] under the assumptions of constant gas viscosity and the simplest drag and heat-transfer law. A solution for an accelerated Couette flow without particle injection and with a wall has been obtained in [6]. In the present paper fairly general assumptions are used to obtain a numerical solution of the problem of two-phase Couette flow with particle injection, and simple formulas useful for estimating the effect of the particles on friction and heat transfer are also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–46, May–June, 1976.