Choice of a self-similar solution in boundary-layer theory

If the speed of the outer flow at the edge of the boundary layer does not depend on the time and is specified in the form of a power-law function of the longitudinal coordinate, then a self-similar solution of the boundary-layer equations can be found by integrating a third-order ordinary differential equation (see [1–3]). When the exponent of the power in the outerflow velocity distribution is negative, a self-similar solution satisfying the equations and the usually posed boundary conditions is not uniquely determinable [4], A similar result was obtained in [5] for flows of a conducting fluid in a magnetic field. In the present paper we study the behavior of non-self-similar perturbations of a self-similar solution, enabling us to provide a basis for the choice of a self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–46, July–August, 1974.     

Numerical solution of a boundary layer problem on a nonconducting surface of an MHD channel

The problem of boundary layer flow on a nonconducting wall has been considered in [1–3]. Therein, it was assumed that either the problem is self-similar [1], or the solution was found in the form of a power series in a small parameter [2,3]. The objective of these assumptions is to reduce the boundary layer equations to ordinary differential equations. In the present work the problem is solved without making these assumptions. The distribution along the channel length of the frictional resistance and heat transfer coefficients on the wall are obtained, and the variation of these coefficients with the load parameter is studied.     

The coefficient of regeneration of the enthalpy in a three-dimensional laminar boundary layer

Self-similar solutions of the equations of a three-dimensional laminar boundary layer are of interest from two points of view. In the first place, they can be used to construct approximate calculating methods, making it possible to analyze several variants and to consider complex flows, in which it is impossible to neglect the interaction between the boundary layer and the external flow (for example, in the region of hypersonic interaction [1–3]). In the second place, the analysis of self-similar solutions permits clarifying the effect of individual parameters on one or another characteristic of the boundary layer and representing this effect in predictable form. One of the principal characteristics of a three-dimensional boundary layer, as also of a two-dimensional, is the coefficient of regeneration of the enthalpy. The value of this coefficient is needed for determining the temperature of a thermally insulated surface, as well as for finiing the real temperature (or enthalpy) head, which determines the value of the heat flux from a heated gas to the surface of the body around which the flow takes place. The article presents the results of calculations of the coefficient of regeneration of the enthalpy for locally self-similar solutions of the equations of a three-dimensional boundary layer, forming with flow around a cylindrical thermally insulated surface at an angle. It is clarified that the dependence of the coefficient of regeneration of the enthalpy on the determining parameters is not always continuous.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 60–63, January–February, 1973.     

Self-similar solutions of nonstationary equations of a plane laminar magnetohydrodynamic boundary layer

Self-similar solutions of nonstationary equations of the boundary layer in ordinary hydrodynamics are discussed in [1, 2]. In this paper self-similar solutions of nonstationary equations of a plane magnetohydrodynarnic boundary layer are sought. In this case, a transformation to curvilinear coordinates of a certain special form is employed. Its choice is determined by the requirements essential to reducing the equations of the boundary layer to a system of ordinary equations. H. Weyl's iterative method is used to solve the equations describing the flow over a plate suddenly set in motion.     

Calculation of supersonic viscid flow near stagnation line of a blunt body

A numerical study is made of supersonic flow of a viscous gas in the vicinity of the stagnation line of plane and axisymmetric blunt bodies (cylinder, sphere). As in [1–5], which consider the compressed layer of a viscous gas in the vicinity of the stagnation point, use is made of the locally self-similar approximation, which is used to transform the Navier-Stokes equations into a system of ordinary differential equations. In the present paper the solution is sought with the simplifications of [5] and with more general conditions, which makes it possible to study a broad class of flows. The proposed numerical algorithm permits obtaining the structure of the compressed layer near the stagnation line, including the shock wave and the boundary layer. The calculations made on a computer for different flow conditions are illustrated by graphs.The author wishes to thank G. I. Petrov, G. F. Telenin, and L. A., Chudov for their interest in the study and for their helpful discussions. discussions.     

An invariant solution of the turbulent boundary-layer equations

The problem of the group stratification of the system of equations describing motion in the laminar sublayer and the turbulent core is considered. The fundamental group admissible by the initial system is constructed; invariant solutions constructed on one of the subgroups lead to a system of ordinary differential equations. Joining of the solutions and interchange of the equations occur at the boundary of the laminar sublayer. A class of power-law flows of a turbulent boundary layer is investigated. In the region of decelerated motion a double-valued solution is found corresponding to attached or separated flow. The commonly used integral characteristics are calculated and presented in the form of an interpolation polynomial.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 4, pp. 126–132, July–August, 1975.     

Similarity method for imploding strong shocks in a non-ideal relaxing gas

Self-similar solutions arise naturally as special solutions of system of partial differential equations (PDEs) from dimensional analysis and, more generally, from the invariance of system of PDEs under scaling of variables. Usually, such solutions do not globally satisfy imposed boundary conditions. However, through delicate analysis, one can often show that a self-similar solution holds asymptotically in certain identified domains. In the present paper, it is shown that self-similar phenomena can be studied through use of many ideas arising in the study of dynamical systems. In particular, there is a discussion of the role of symmetries in the context of self-similar dynamics. We use the method of Lie group invariance to determine the class of self-similar solutions to a problem involving plane and radially symmetric flows of a relaxing non-ideal gas involving strong shocks. The ambient gas ahead of the shock is considered to be homogeneous. The method yields a general form of the relaxation rate for which the self-similar solutions are admitted. The arbitrary constants, occurring in the expressions for the generators of the local Lie group of transformations, give rise to different cases of possible solutions with a power law, exponential or logarithmic shock paths. In contrast to situations without relaxation, the inclusion of relaxation effects imply constraint conditions. A particular case of the collapse of an imploding shock is worked out in detail for radially symmetric flows. Numerical calculations have been performed to determine the values of the self-similarity exponent and the profile of the flow variables behind the shock. All computations are performed using the computation package Mathematica.     

A spatial laminar boundary layer on a streamline in conical external flow of a uniform gas in the absence of sources and sinks

Theoretical study of a three-dimensional laminar boundary layer is a complex problem, but it can be substantially simplified in certain particular cases and even reduced to the solution of ordinary differential equations.One such particular case is the flow of a compressible gas on a streamline in conical external flow. The case is of considerable practical importance because the local heat fluxes may take extremal values on such lines.Such flow, except for the conical case, has been examined [1–4], and an approximate method has been given [1] on the basis of integral relationships and a special form for the approximating functions. A numerical solution has been given [2, 3] for such flow around an infinite cylinder. It was assumed in [1–3] that the Prandtl number and the specific heats were constant, and that the dynamic viscosity was proportional to temperature. Heat transfer has been examined [4] near a cylinder exposed to a flow of dissociated air.Here we give results from numerical solution of a system of ordinary differential equations for the flow of a compressible gas in a laminar boundary layer on streamlines in conical external flow, with or without influx or withdrawal of a homogeneous gas. It is assumed that the gas is perfect and that the dynamic viscosity has a power-law temperature dependence.     

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.     

Use of the generalized similarity method for calculating the inner zone of a turbulent boundary layer

The use of the generalized similarity method for calculating laminar boundary layers has been fully justified (see [1, §113, 114, 148]). The replacement of the partial differential equations by ordinary differential equations, their universality and the possibility of physically interpreting the solutions in the first, parametric stage of the calculations, which distinguish the generalized similarity method from direct numerical integration methods, are preserved in the case of a turbulent boundary layer also. A comparison of the calculated and experimental velocity profiles in the inner zone of the turbulent boundary layer suggests that the generalized similarity method could be used for calculating the turbulent layer as a whole.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 25–34, September–October, 1990.     

Self-similar problem of the motion of a plane layer of heated material with an arbitrary equation of state

The self-similar problem of the nonstationary motion of a plane layer of material in which energy from an external source is released for values of the flux density q₀ on the boundary which are constant in time is considered. The self-similar variable is = m/t, where m is the Lagrangian mass coordinate and t is the time. The characteristic values of the velocity, density, and pressure do not vary with time. For a self-similar problem the energy flux density q must also depend only on the self-similar variable. In this case q() can be an arbitrary function of its argument and can be given by a table. Examples are presented of actual physical processes in which the mass of the energy-release zone increases linearly with time. The equation of state can have an arbitrary form, including specification by a table. The gaseous state of matter for an arbitrary variable adiabatic exponent, the condensed state, and a two-phase state can be described. A solution of the self-similar problem is presented for the heating of a half-space bounded by a vacuum for a certain specific equation of state and various flux densities q₀ and velocities M of the advance of the energy-release zone.Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskoi Fiziki, No. 5, pp. 136–145, September–October, 1975.     

Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials

In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary constitutive function that are likely to yield such solutions are also indicated.     

Self-similar solutions in a plasma with axial magnetic field (<Emphasis Type="Italic">θ</Emphasis>-pinch)

In this paper, Lie group of transformation method is used to investigate the self-similar solutions for the system of partial differential equations describing a plasma with axial magnetic field (θ–pinch). The arbitrary constants occurring in the expressions for the infinitesimals of the local Lie group of transformations give rise to two different cases of possible solutions i.e. with a power law and exponential shock paths. A particular solution to the problem in one case has been found out.     

Spin-down of a fluid between infinite cones

This study is concerned with the spin-down of a fluid between stationary cones. It follows on from [7], where solutions were obtained for a fluid spinning down between two infinite disks and where it was shown that under various initial conditions the dependence of the velocity on radius and time tends to a universal Kármán stage. In the case of cones the analogous universal stage is not of the Kármán type, which makes possible an experimental check of the applicability of the self-similar boundary layer equations generalizing the Karman equations previously considered in [11–13]. The experiments confirm the conclusions of the theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 37–44, July–August, 1986.In conclusion, the authors wish to thank A. M. Obukhov and F. V. Dolzhanskii for formulating the problem and constructive discussions.     

Calculation of the boundary layer on the electrically conducting wall of a plane channel

There are presently available quite a large number of works devoted to the study of the motion of an electrically conducting fluid in boundary layers formed on electrodes or on the nonconducting walls of various MHD devices. However, the methods of solving the boundary layer equations in these studies are based on various simplifying assumptions which allow the problem to be reduced to the solution of a system of ordinary differential equations. Thus, in [1] there is imposed on the flow the special magnetic fieldH1/x, which enables the problem to be reduced to the self-similar form, while in the studies of other authors [2, 3] either the solution is sought in the form of expansions in x, or it is assumed that the problem is locally self-similar [4]. In the present paper we construct the solution of the MHD boundary layer equations which is obtained by one of the numerical methods which has long been used for solving the boundary layer equations for a nonconducting fluid.     

On some new solutions obtainable by means of invariant transformations

With reference to the example of the equations of monoenergetic nonrelativistic beam of particles of like charge, it is shown how new noninvariant solutions can be obtained by means of invariant transformations (§ 1). The conditions under which Lorentz forces can be ignored and the electric field considered a potential field are obtained for nonstationary flows. Solutions that describe the passage through a plane diode of high-frequency current from the emitter in a high-frequency electric field for an arbitrary relationship between the constant component of the collector potential and the amplitude of the ac voltage across it are derived (§2). Multivelocity (the velocity vector is a multivalued function) beams, and also electrostatic beams that can be described by Vlasov's equations are examined (§3).Given a system of differential equations (S) for m 1 unknown functions u^k (k=1,.,m) of n – m 1 independent variables xⁱ (i=1, ., n – m). The set of values (x, u) is considered as the set of coordinates of a point in n-dimensional space E_n. Any solution of this system u=u(x) defines some manifold in E_n. All possible solutions of (SI specify in E_n some set M. Any invariant transformation of system (S) has the property that it does not lead out of M. In a number of cases, this makes it possible to obtain new solutions by means of invariant transformations, no limitations being imposed on the solutions transformed. For a given system (S), all transformations that preserve (Si and form a continuous group, can be obtained by the method developed by L. V. Ovsyannikov [1–3]. Note that new solutions arise only when the principal group G of system (S) allows other than merely elementary transformations: magnifications, rotations, and translations are, as a rule, useless.Below, solutions of the equations of a monoenergetic nonrelativistic beam of particles of like charge are examined as an example [6–8].     

Nonuniqueness of the solution of the problem of viscous interaction on an axisymmetric body

The article discusses solutions of the equations of the hypersonic boundary layer on an axisymmetric offset slender body (with a power exponent equal to 3/4), taking account of interactions with a nonviscous flow. It is shown that, in this case, the equations of the boundary layer have solutions differing from the self-similar solution corresponding to flow around a semi-infinite body. The solutions obtained are analogous to solutions for a strong interaction on a plate with slipping and triangular vanes [1–4], but are obtained over a wide range of values of the parameter of viscous interaction. An asymptotic solution is given to the problem with the approach to zero of the interaction parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 41–47, September–October, 1973.The authors thank V. V. Mikhailova for discussion of the work and useful advice.     

SIMILARITY SOLUTIONS OF BOUNDARY LAYER EQUATIONS FOR A SPECIAL NON-NEWTONIAN FLUID IN A SPECIAL COORDINATE SYSTME

Two dimensional equations of steady motion for third order fluids are expressed in a special coordinate system generated by the potential flow corresponding to an inviscid fluid. For the inviscid flow around an arbitrary object, the streamlines are the phicoordinates and velocity potential lines are psi-coordinates which form an orthogonal curvilinear set of coordinates. The outcome, boundary layer equations, is then shown to be independent of the body shape immersed into the flow. As a first approximation, assumption that second grade terms are negligible compared to viscous and third grade terms. Second grade terms spoil scaling transformation which is only transformation leading to similarity solutions for third grade fluid. By ~sing Lie group methods, infinitesimal generators of boundary layer equations are calculated. The equations are transformed into an ordinary differential system. Numerical solutions of outcoming nonlinear differential equations are found by using combination of a Runge-Kutta algorithm and shooting technique.     

Asymptotic analysis of turbulent boundary layer flow on a flat plate at high Reynolds numbers

The equations of the turbulent boundary layer contain a small parameter — the reciprocal of the Reynolds number, which makes it possible to carry out an asymptotic analysis of the solutions with respect to that small parameter. Such analyses have been the subject of a number of studies [1–5]. In [2, 5] for closing the momentum equation algebraic Prandtl and turbulent viscosity models were used. In [1, 3, 4] the structure of the boundary layer was analyzed in general form without formulating specific closing hypothesis but under additional assumptions concerning the nature of the asymptotic behavior of the limiting solutions in the various regions.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 106–117, May-June, 1993.     

Self-similar solutions of three-dimensional laminar magnetohydrodynamic boundary-layer equations

Self-similar solutions of three-dimensional boundary-layer equations of an incompressible fluid in ordinary hydrodynamics were considered in [1–3] et al. The present work looks for self-similar solutions of three-dimensional magnetohydrodynamic boundary-layer equations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 10–17, July–August, 1968.