A case of jet flow of a gravity fluid

The problem of plane steady ideal heavy fluid flow bounded by an impermeable polygonal section, a curvilinear arc section, and a finite section of free surface is investigated in an exact nonlinear formulation. Hydrodynamic singularities may exist in the stream. A large class of captation problems of jet theory reduces to studying this kind of flow. The unique solvability of the problem under investigation is proved for sufficiently large Froude numbers and small arc curvature. A method of solution is given and an example is computed. Such problems have been solved earlier by numerical methods [1–3]. Some problems about jet flows of a gravity fluid with polygonal solid boundaries have been investigated by an analogous method in [4, 5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 140–143, May–June, 1975.     

Method of narrow strips for nonlinear problems of a stratified fluid

Plane steady flow is considered for an ideal incompressible stratified fluid in a gravitational field of force. It is a characteristic feature of these flows that the density is constant and Bernoulli's constant remains the same along a streamline. Internal waves arise because of ponderability in the stratified fluid; they are not due to the presence of a free surface. These wave motions are studied in detail in the linear formulation, but flows of the solitary wave type can be described only by nonlinear equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–178, March–April, 1986.     

Stability of unsteady viscous flows

A study is made of the linear stability of plane-parallel unsteady flows of a viscous incompressible fluid: in the mixing layer of two flows, in a jet with constant flow rate, and near a wall suddenly set in motion [1]. The slow variation of these flows in time compared with the rate of change of the perturbations makes it possible to use the method of two-scale expansions [2]. The stability of nonparallel flows with allowance for their slow variation with respect to the longitudinal coordinate was investigated, for example, in [3–6]. The unsteady flows considered in the present paper have a number of characteristic properties of non-parallel flows [1], but in contrast to them are described by exact solutions of the Navier-Stokes equations. In addition, for unsteady planeparallel flows a criterion of neutral stability can be uniquely established by means of the energy balance equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, 138–142, July–August, 1981.I thank G. I. Petrov for suggesting the problem, and also S. Ya. Gertsenshtein and A. V. Latyshev for assisting in the work.     

Similarity of flows in strongly underexpanded jets of viscous gas

The fulfillment of the conditions formulated in [1] for the similarity of flows in strongly under-expanded jets of a viscous, thermodynamically ideal gas is studied. The limits of applicability of these conditions are established on the basis of exact solutions of the one-dimensional Navier —Stokes equations and experimental investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika zhidkosti i Gaza, No. 6, pp. 117–125, November–December, 1978.     

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

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

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.     

Asymptotic analysis of a pulsating flow of viscous fluid in a symmetric deformed channel

The time-periodic flow of a viscous incompressible fluid in a two-dimensional symmetric channel with slightly deformed walls is considered. The solution of the Navier-Stokes equations is constructed by means of the method of matched asymptotic expansions [1] at large characteristic Reynolds numbers. It is shown that in an unsteady flow a region of nonlinear perturbations surrounds the line of zero velocity inside the fluid. The formation and development of such nonlinear zones with respect to time is considered. An alternation of the topological features of the streamline pattern in the nonlinear perturbation zone is discovered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 17–23, July–August, 1987.The author is deeply grateful to V. V. Sychev for his formulation of the problem and his attentive attitude to my work.     

Electrosolenoidal flows in a cylindrical cavity

The main difficulties in investigating three-dimensional magnetohydrodynamic (MHD) flows with vorticity arise, first, because it is necessary to solve an independent boundary-value problem in order to find the field of the electromagnetic forces and, second, because the regimes of these flows are strongly nonlinear for the majority of high-power technological MHD processes and a number of natural phenomena. Particular importance attaches to MHD flows generated by the interaction of an electric current applied to the fluid with the magnetic self-field. This class of MHD flows has become known as electrosolenoidal flows [1]. The presence of a definite symmetry in the distribution of the electromagnetic forces and the geometry of the region of the liquid conductor makes it possible to find a solution in self-similar form. The present paper is devoted to exact solutions of the nonlinear equations for axisymmetric electrosolenoidal flows of a conducting incompressible fluid in infinite cylindrical cavities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 48–53, May–June, 1991.     

Hydrodynamics in weak gravitational fields two-dimensional oscillations of an ideal fluid in a rectangular channel

The two-dimensional problem of determining the frequencies and modes of small natural osciliations of an ideal fluid in a rectangular channel under near-weightless conditions is considered. It is assumed that a weak gravitational field acts parallel to the vertical walls of the channel. The Ritz method is employed for the variational problem, which is equivalent to the problem of oscillations of a fluid under weightless conditions [1, 2].Khar'kov. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–13, September–October, 1972.     

Impact of a strip of compressible liquid on a barrier

The exact solution of the plane problem of the impact of a finite liquid strip on a rigid barrier is obtained in the linearized formulation. The velocity components, the pressure and other elements of the flow are determined by means of a velocity potential that satisfies a two-dimensional wave equation. The final expressions for them are given in terms of elementary functions that clearly reflect the wave nature of the motion. The exact solution has been thoroughly analyzed in numerous particular cases. It is shown directly that in the limit the solution of the wave problem tends to the solution of the analogous problem of the impact of an incompressible strip obtained in [1]. A logarithmic singularity of the velocity parallel to the barrier in the corner of the strip is identified. A one-dimensional model of the motion, which describes the behavior of the compressible liquid in a thin layer on impact and makes it possible to obtain a simple solution averaging the exact wave solution, is proposed. Inefficient series solutions are refined and certain numerical data on the impact characteristics for a semi-infinite compressible liquid strip, previously considered in [2–4] in connection with the study of the earthquake resistance of a dam retaining water in a semi-infinite basin, are improved. The solution obtained can be used to estimate the forces involved in the collision of solids and liquids. It would appear to be useful for developing correct and reliable numerical methods of solving the nonlinear problems of fluid impact on solids often examined in the literature [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 138–145, November–December, 1990.The results were obtained by the author under the scientific supervision of B. M. Malyshev (deceased).     

Instability due to three-dimensional disturbances

In the linear theory of the stability of parallel flows of a viscous fluid, most attention is usually given to plane-wave disturbances. The reason is the validity in many cases of the Squire theorem, which states that the critical Reynolds number R is determined by two-dimensional disturbances [1]. It is shown in the present paper that for large R the region generating the turbulence in the initial stage of its development is formed by three-dimensional disturbances. This feature applies both to the generating range of wave numbers and the dimension of the wall layer, where the fluctuating energy is produced. The consequences of the Squire transformations for parallel flows are analyzed. The contribution of resonant nonlinear triad coupling to the rapid growth of fluctuating energy is studied for the case of an explosive instability in an extended laminar mode. It is shown that the rate of turbulent energy production is not governed by the small derivatives of linear theory, but by nonlinear triad coupling of neutral and growing disturbances, with their three-dimensional nature playing an important role.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 29–34, September–October, 1976.The author thanks M. A. Gol'dshtik for his interest in the work and for discussion of the results.     

Theory of vortical helical ideal gas flows in laval nozzles

An asymptotic solution is found for the direct problem of the motion of an arbitrarily vortical helical ideal gas flow in a nozzle. The solution is constructed in the form of double series in powers of parameters characterizing the curvature of the nozzle wall at the critical section and the intensity of stream vorticity. The solution obtained is compared with available theoretical results of other authors. In particular, it is shown that it permits extension of the known Hall result for the untwisted flow in the transonic domain [1]. The behavior of the sonic line as a function of the vorticity distribution and the radius of curvature of the nozzle wall is analyzed. Spiral flows in nozzles have been investigated by analytic methods in [2–5] in a one-dimensional formulation and under the assumption of weak vorticity. Such flows have been studied by numerical methods in a quasi-one-dimensional approximation in [6, 7]. An analogous problem has recently been solved in an exact formulation by the relaxation method [8, 9]. A number of important nonuniform effects for practice have consequently been clarified and the boundedness of the analytical approach used in [2–7] is shown.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 126–137, March–April, 1978.The authors are grateful to A. N. Kraiko for discussing the research and for valuable remarks.     

Impact of a figure of revolution in a stream of incompressible fluid with separation of a jet

The impact interaction of bodies with a fluid in a flow with jet separation has been considered in [1–3], for example. This investigation was in the two-dimensional formulation. The present paper considers the three-dimensional problem of impact of a figure of revolution in a stream of an ideal incompressible fluid with separation of a jet in accordance with Kirchhoff's scheme. A boundary-value problem is formulated for the impact flow potential and solved by the Green's function method. A method for constructing the Green's function is described. Expressions are given for the coefficients of the apparent masses. The results are given of computer calculations of these coefficients in the case of a cone using the flow geometry of the corresponding two-dimensional problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 176–180, November–December, 1980.     

Form of a free surface during steady flow of a capillary fluid in a rectangular channel

The two-dimensional problem of the form of a free surface of an ideal incompressible fluid during steady flow from a rectangular channel through a thin slot with simultaneous uniform delivery of fluid through the side walls is examined. Forces of gravity and surface tension are taken into account. The nonlinear problem of the simultaneous determination of the free surface and velocity field of the fluid is solved by the iteration method. Convergence of the iterations to the solution of the problem for small values of the parameters is investigated. The solution of the linearized problem is obtained in a closed form for a small depth of the discharge and small width of the channel, which is compared with the solution of the problem in a complete formulation. Graphs of the free surface of the fluid for different values of the parameters, obtained as a result of numerical solution of the nonlinear problem, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 67–75, January–February, 1977.     

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

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

Effect of high-frequency vibration on the development of secondary convective conditions

An investigation is made of the development of convective flows of a viscous incompressible liquid, subjected to high-frequency vibration. The nonlinear equations of convection are used in the Boussinesq approximation, averaged in time. The amplitude of the perturbations is assumed to be small, but finite. For a horizontal layer with solid walls the existence of both subcritical and supercritical stable secondary conditions is established. In a linear statement, the problem of stability in the presence of a modulation has been discussed in [1–3]. Articles [4–6] were devoted to investigation of the nonlinear problem. In [4], the method of grids was used to study secondary conditions in a cavity of square cross section. In the case of a horizontal layer with free boundaries [5, 6], the character of the branching is established by the method of a small parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 90–96, March–April, 1976.The authors thank I. B. Simonenko for his useful evaluation of the work.     

Stability of flows of the Hamel type in channels with permeable walls

The stability of nonparallel flows of a viscous incompressible fluid in an expanding channel with permeable walls is studied. The fluid is supplied to the channel through the walls with a constant velocity v₀ and through the entrance cross section, where a Hamel velocity profile is assigned. The resulting flow in the channel depends on the ratio of flow rates of the mixing streams. This flow was studied through the solution of the Navier—Stokes equations by the finite-difference method. It is shown that for strong enough injection of fluid through the permeable walls and at a distance from the initial cross section of the channel the flow approaches the vortical flow of an ideal fluid studied in [1]. The steady-state solutions obtained were studied for stability in a linear approximation using a modified Orr—Sommerfeld equation in which the nonparallel nature of the flow and of the channel walls were taken into account. Such an approach to the study of the stability of nonparallel flows was used in [2] for self-similar Berman flow in a channel and in [3] for non-self-similar flows obtained through a numerical solution of the Navier—Stokes equations. The critical parameters *, R*, and Cr* at the point of loss of stability are presented as functions of the Reynolds number R₀, characterizing the injection of fluid through the walls, and the parameter , characterizing the type of Hamel flow. A comparison is made with the results of [4] on the stability of Hamel flows with R₀ = 0.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 125–129, November–December, 1977.The author thanks G.I. Petrov for a discussion of the results of the work at a seminar at the Institute of Mechanics of Moscow State University.     

Steady plane-parallel flow of a magnetic fluid

In the hydrodynamics of a Newtonian fluid, nonlinear effects are connected only with the presence of convective derivatives in the equations and therefore disappear when plane-parallel flows are considered. Non-Newtonian effects are usually taken into account either phenomenologically in the expression for the stress tensor or by explicitly considering additional degrees of freedom. A theory of the effective viscosity of a magnetic fluid is constructed in [1] by regarding a magnetic fluid as a medium with internal rotation. It was shown that the flow of fluid in a magnetic field is non-Newtonian. Later, many authors (see, for example, [2, 3]) studied one- and two-dimensional flows under the influence of a pressure difference. However, the study was usually limited to continuous and smooth solutions. In the present work, we study the plane-parallel flow of a magnetic fluid in a homogeneous magnetic field under the influence of a longitudinal pressure gradient. We also consider discontinuous solutions. It is shown that for large longitudinal pressure gradients and sufficiently great intensities of the magnetic field, the problem has an infinite number of steady solutions which differ in the number and position of discontinuities of the magnetization and the associated abrupt changes in the velocity profile. Steady regimes and their stability are studied numerically with allowance for weak diffusion of magnetization and internal angular momentum. It is shown that the degeneracy is then lifted; however, in a certain region of parameters several stable steady regimes still exist.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 57–64, November–December, 1984.The authors would like to thank M. I. Shliomis for his constant interest in this work.     

Mathematical construction of a Reissner–Mindlin plate theory for composite laminates

A Reissner–Mindlin theory for composite laminates without invoking ad hoc kinematic assumptions is constructed using the variational-asymptotic method. Instead of assuming a priori the distribution of three-dimensional displacements in terms of two-dimensional plate displacements as what is usually done in typical plate theories, an exact intrinsic formulation has been achieved by introducing unknown three-dimensional warping functions. Then the variational-asymptotic method is applied to systematically decouple the original three-dimensional problem into a one-dimensional through-the-thickness analysis and a two-dimensional plate analysis. The resulting theory is an equivalent single-layer Reissner–Mindlin theory with an excellent accuracy comparable to that of higher-order, layer-wise theories. The present work is extended from the previous theory developed by the writer and his co-workers with several sizable contributions: (a) six more constants (33 in total) are introduced to allow maximum freedom to transform the asymptotically correct energy into a Reissner–Mindlin model; (b) the semi-definite programming technique is used to seek the optimum Reissner–Mindlin model. Furthermore, it is proved the first time that the recovered three-dimensional quantities exactly satisfy the continuity conditions on the interface between different layers and traction boundary conditions on the bottom and top surfaces. It is also shown that two of the equilibrium equations of three-dimensional elasticity can be satisfied asymptotically, and the third one can be satisfied approximately so that the difference between the Reissner–Mindlin model and the second-order asymptotical model can be minimized. Numerical examples are presented to compare with the exact solution as well as the classical lamination theory and the first-order shear-deformation theory, demonstrating that the present theory has an excellent agreement with the exact solution.     

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.