Singularity in the solution of the equations of a three-dimensional boundary layer due to the collision of wall jets

The problem of the propagation of a three-dimensional jet of viscid incompressible fluid flowing from a narrow curved slot into a fluid-filled space along a rigid plane is considered within the framework of the equations of a steady laminar boundary layer. A class of initial conditions at the slot outlet which generates in the jet a velocity field without secondary flows is identified. Within this class the boundaryvalue problem for the three-dimensional boundary layer can be divided into two problems of lower dimensionality: a dynamic and a kinematic problem. As a result of the analysis of the kinematic problem the general structure of the regions of existence and uniqueness of the solution is determined. An investigation of the dynamic problem shows that as the boundaries of the region of existence are approached a singularity characterized by an infinite increase in the thickness of the jet is formed in the solution of the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 75–81, July–August, 1991.     

Flow of viscous gas out of a cylindrical channel into vacuum

A study is made of the exhausting of a jet of viscous gas from a cylindrical channel into vacuum in the presence of a flat bounding surface outside the channel in the plane of its exit section. The problem is solved numerically using the complete system of Navier—Stokes equations. The developed flow model makes it possible to take into account the influence of an external medium into which the jet exhausts on the structure of the flow in the exit section of the channel, and also the influence of the subsonic part of the boundary layer in the channel on the flow field of the jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–128, January–February, 1981.     

Singularities in the boundary layer on a cone at incidence

The singularities in the three-dimensional laminar boundary layer on a cone at incidence are studied. It is shown that these singularities are formed in the outer part of the boundary layer and described by linear equations whose solutions are obtained in analytic form. The known results for the plane of symmetry are classified on this basis. Two solutions of the non-self-similar problem are found, one of which has a singularity at zero incidence and in the sink plane. The second branch goes over continuously into the solution for axisymmetric flow. However, as the angle of attack increases, in the sink plane a singularity is formed and all the self-similar solutions existing here lose their meaning. Starting from the critical angle of attack, the flow in the vicinity of the sink plane is no longer described by the boundary layer equations, so that the results can be used to construct an adequate physical model.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 25–33, November–December, 1993.     

Thermal convection in a nonequilibrium turbulent medium with rotation

The equations of thermal convection in a rotating plane horizontal layer of nonequilibrium turbulent fluid are obtained, the system of equations is linearized and the boundary value problem is formulated. Some general properties of the perturbation spectrum are found and a solution, which includes the classical solution in the absence of turbulence as a limiting case, is obtained.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 62–70, November–December, 1994.     

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.     

Method of meridional sections in problems of a three-dimensional boundary layer

The difficulties and clumsiness of problems of calculating the heat transfer distribution over the surface of a body in a three-dimensional flow are well known. It is shown that this problem can be considerably simplified where the influence of the three-dimensionality of the flow, which in certain applications it is important to take into account, is only weak. In this case the three-dimensional problem can be reduced to a set of two-dimensional problems along the lines of meridional sections of the body. This has been demonstrated in detail with reference to the method of effective length or local similarity, which is widely used in engineering practice and is particularly justified in the the case of turbulent heat transfer law. However, in the three-dimensional case it is complicated by the need to calculate the distribution of the streamlines over the surface of the body [1–4]. In the presence of slight asymmetry of the flow the problem can be substantially simplified, mainly as a result of the demonstrated possibility of replacing (with quadratic accuracy) the streamlines by the lines of meridional sections. The possibility of an independent solution of the exact boundary layer equations along each meridional plane is demonstrated for the above-mentioned approximation (rule of meridional sections).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–73, May–June, 1986.     

Computation of the pressure on the surface of a fuselage with a wing in an ideal fluid

The flows around complex three-dimensional bodies by an ideal fluid are computed by methods [1–3] using approximation of the surface by a set of plane elements. A layer of surface singularities, whose intensity is found by solving a system of linear algebraic equations of very high order, is distributed continuously over each element. Evaluation of the system coefficients and its solution require significant machine time expenditures on powerful electronic computers. If in the method of [2] the total system of equations is separated successfully into several subsystems by simplifications and an approximate solution of the problem is obtained more rapidly than by the method in [1], then the same author practically used the method in [1] to design specific fuselages in [3]. A method [4] developed for a fuselage is expanded in this paper to design a wing-fuselage combination. This method turns out to be less tedious, without being inferior in accuracy, by being different from the method in [1] in the means of solving the fundamental integral equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 110–115, May–June, 1977.     

Magnetohydrodynamic lubrication theory for a cylindrical bearing

Liquid metal, which is a conductor of electric current, may be used as a lubricant at high temperatures. In recent years considerable attention has been devoted to various problems on the motion of an electrically conducting liquid lubricant in magnetic and electric fields (magnetohydrodynamic theory of lubrication), Thus, for example, references [1–3] study the flow of a conducting lubricating fluid between two plane walls located in a magnetic field. An electrically conducting lubricating layer in a magnetohydrodynamic bearing with cylindrical surfaces is considered in [4–8] and elsewhere.The present work is concerned with the solution of the plane magnetohydrodynamic problem on the pressure distribution of a viscous eletrically conducting liquid in the lubricating layer of a cylindrical bearing along whose axis there is directed a constant magnetic field, while a potential difference from an external source is applied between the journal and the bearing. The radial gap in the bearing is not assumed small, and the problem reduces to two-dimensional system of magnetohydrodynamic equations.An expression is obtained for the additional pressure in the lubricating layer resulting from the electromagnetic forces. In the particular case of a very thin layer the result reported in [4–8] is obtained. SI units are used.     

Development of three-dimensional disturbances in a boundary layer with pressure gradient

The results of an experimental investigation of the three-dimensional stability of a boundary layer with a pressure gradient are presented. A low-turbulence subsonic wind tunnel was employed. The development of a three-dimensional wave packet of oscillations harmonic in time in the boundary layer on a model wing is studied. The amplitudephase distributions of the pulsations in the wave packet are subjected to a Fourier analysis. Spectral (with respect to the wave numbers) decomposition of the oscillations enables the flow stability with respect to plane waves with different directions of propagation to be examined. The results are compared with the corresponding data obtained in flat plate experiments. The effect of the pressure gradient on the development of the three-dimensional spectral components of the disturbances and the dispersion properties of the flow is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 85–91, May–June, 1988.     

One-dimensional and two-dimensional analogs for three-dimensional viscous flows in the neighborhood of the plane of symmetry of a blunt body

An approximate method of determining the heat transfer and friction stress in three-dimensional flow problems using the two-dimensional and one-dimensional solutions is proposed. This method is applicable over a wide range of Reynolds numbers — from low to high. On the basis of a theoretical analysis of the approximate analytic solution of the equations of a three-dimensional viscous shock layer it is shown that the problem of determining the heat flux in the neighborhood of the plane of symmetry of bodies inclined to the flow at an angle of attack can be reduced, firstly, to the problem of determining that quantity for an axisymmetric body and, secondly, to the problem of determining the heat transfer to an axisymmetric stagnation point. On the basis of an analysis of the results of a numerical solution of the problem it is shown that corresponding analogs can also be used for the friction stress. The accuracy of the similarity relations established is estimated by solving the problem by a finite-difference method. A similarity relation of the same kind was previously obtained in [1] for a double-curvature stagnation point.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 117–122, January–February, 1990.     

Three-dimensional local separation

A solution to the problem of local separation of a three-dimensional boundary layer from an arbitrary smooth surface is constructed. Separation takes place along the limiting streamline at the points of which the component of the surface friction (calculated from the boundary-layer equations) that is orthogonal to this streamline has a break. An asymptotic expansion of the solution of the Navier-Stokes equations that describes the flow field in the separation region is found. The conclusions for the two-dimensional and self-similar theory of local separation are generalized to the three-dimensional case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 39–47, May–June, 1991.     

Plane filtration in a laminar medium in the presence of imperfect reservoirs

Exact and approximate solutions were obtained describing filtration flow in a plane region in sectors corresponding to imperfect reservoirs and suction gaps. These equations make it possible to obtain an approximate solution to the three-dimensional problem of the filtration of ground waters by solution of the two-dimensional problem; imperfect reservoirs are permitted in the filtration region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 76–83, May–June, 1973.     

Heat transfer in the neighborhood of a point of inflection in the leading edge of a plate in hypersonic flight

A study is made of the three-dimensional flow of a viscous gas around a flat plate with an inflection in the generator of the leading edge in the case of strong interaction between the exterior hypersonic flow and the boundary layer. Numerical solutions to the problem are obtained. It is shown that near points of inflection of the profile of the leading edge of a flat wing strong self-induced secondary flows can be formed together with associated local peaks of the heat fluxes and the friction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–45, May–June, 1980.     

Contact Interaction of the Faces of a Rectangular Crack under Normally Incident Tension–Compression Waves

A three-dimensional problem on the contact interaction between the faces of a rectangular crack under a normally incident harmonic tension–compression wave is considered. The problem is solved by using the method of boundary integral equations and an iterative algorithm. The contact forces and the discontinuity in the displacement of the crack faces are studied. The results obtained are compared with those for a finite plane crack.     

Influence of thermal inhomogeneity of the boundary on the forced motion of a viscous fluid in a plane layer

Numerical solution of the Navier-Stokes equations is used to investigate the laminar motion of an incompressible fluid in a plane layer when part of the lower boundary has a temperature different from the rest of the lower boundary. The problem is solved in the Boussinesq approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 171–173, January–February, 1982.     

Axisymmetric analogy for three-dimensional viscous flow problems

A method of solving three-dimensional flow problems with the aid of two-dimensional solutions, which can be used for any Reynolds numbers, is proposed. The method is based on the use of similarity relations obtained in the theoretical analysis of the approximate analytic solution of the equations of a three-dimensional viscous shock layer. These relations express the heat flux and the friction stress on the lateral surface of a three-dimensional body in terms of the values on the surface of an axisymmetric body. The accuracy is estimated by comparing the results with those of a numerical finite-difference calculation of the flow past bodies of various shapes. Similar similarity relations were previously obtained in [1] for the plane of symmetry of a blunt body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–118, November–December, 1991.The authors are grateful to G. A. Tirskii for his interest in their work.     

Three-dimensional laminar boundary layer on a permeable surface in the neighborhood of a symmetry plane

Analytical and numerical methods are used to investigate a three-dimensional laminar boundary layer near symmetry planes of blunt bodies in supersonic gas flows. In the first approximation of an integral method of successive approximation an analytic solution to the problem is obtained that is valid for an impermeable surface, for small values of the blowing parameter, and arbitrary values of the suction parameter. An asymptotic solution is obtained for large values of the blowing or suction parameters in the case when the velocity vector of the blown gas makes an acute angle with the velocity vector of the external flow on the surface of the body. Some results are given of the numerical solution of the problem for bodies of different shapes and a wide range of angles of attack and blowing and suction parameters. The analytic and numerical solutions are compared and the region of applicability of the analytic expressions is estimated. On the basis of the solutions obtained in the present work and that of other authors, a formula is proposed for calculating the heat fluxes to a perfectly catalytic surface at a symmetry plane of blunt bodies in a supersonic flow of dissociated and ionized air at different angles of attack. Flow near symmetry planes on an impermeable surface or for weak blowing was considered earlier in the framework of the theory of a laminar boundary layer in [1–4]. An asymptotic solution to the equations of a three-dimensional boundary layer in the case of strong normal blowing or suction is given in [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–48, September–October, 1980.     

Experimental investigation of the development of harmonic disturbances in the boundary layer on a flat plate at mach number M=4

The development of three-dimensional wave packets artificially introduced into a boundary layer has been experimentally investigated. The measurements were made by the hot-wire anemometer method in the boundary layer on a flat plate at a Mach number M = 4. The artificial disturbances were introduced into the boundary layer by means of an electric discharge. A Fourier analysis of the data made it possible to obtain the wave characteristics of the plane waves. The composition of the disturbances was analyzed and those most dangerous from the instability standpoint were identified. The data obtained are compared with the results of experiments carried out at M = 2. The differences in the data are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 54–58, November–December, 1990.     

Investigation of pre-separation boundary layer flows on the basis of various algebraic models of turbulence

The problem of the three-dimensional incompressible turbulent boundary layer developing ahead of a circular cylinder mounted at right angles on a flat plate is considered. The direction of the uniform approach stream is normal to the leading edge of the plate. The turbulence is simulated by means of five different isotropic algebraic models of eddy viscosity. The boundary layer equations are solved numerically by means of a second-order-accurate implicit finite-difference method. The principal characteristics of the flow obtained on the basis of the turbulence models selected are compared for a free-stream Reynolds number Re = 10⁷.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 36–43, November–December, 1991.     

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.