Interaction between the boundary layer and a supersonic flow when part of the wall is rapidly heated

There have been many theoretical studies of aspects of the unsteady interaction of an exterior inviscid flow with a boundary layer [1–9]. The mathematical flow models obtained in these studies by the method of matched asymptotic expansions describe a wide range of phenomena observed experimentally. These include boundary layer separation near the hinge of a flap, the flow in the neighborhood of the trailing edge of an oscillating airfoil [1–2], and the development and propagation of perturbations in a boundary layer excited by an oscillating wall or some other way [3–5]. The present paper studies the interaction of an unsteady boundary layer with a supersonic flow when a small part of the surface of a body in the flow is rapidly heated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 66–70, January–February, 1984.     

Localized Coherent Structures in the Boundary Layer

A Blasius laminar boundary layer and a steady turbulent boundary layer on a flat plate in an incompressible fluid are considered. The spectral characteristics of the Tollmien—Schlichting (TS) and Squire waves are numerically determined in a wide range of Reynolds numbers. Based on the spectral characteristics, relations determining the three–wave resonance of TS waves are studied. It is shown that the three–wave resonance is responsible for the appearance of a continuous low–frequency spectrum in the laminar region of the boundary layer. The spectral characteristics allow one to obtain quantities that enter the equations of dynamics of localized perturbations. By analogy with the laminar boundary layer, the three–wave resonance of TS waves in a turbulent boundary layer is considered.     

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.     

Asymptotic analysis of the equations of a three-dimensional boundary layer

The asymptotic solutions of the self-similar equations of two- and three-dimensional boundary layers have been investigated by many authors (see, for example, [1–3]). In [4, 5], asymptotic solutions were found for non-self-similar equations for two-dimensional flow, and the propagation of perturbations near the external edge of the boundary layer was analyzed. In the present paper, asymptotic solutions are obtained for the non-self-similar equations of a three-dimensional laminar boundary layer of an incompressible fluid. It is shown that the conclusion drawn in [5] — that the boundary conditions can be transferred from infinity to a finite distance from the wall — is also true for three-dimensional flow. The obtained solutions explain the experimentally well-known phenomenon of the conservativeness of the secondary currents. The essence of this phenomenon is that a change in the sign of the transverse (along the normal to a streamline of the external flow) pressure gradient is accompanied by a very rapid change in the direction of the secondary flow near the wall, whereas in the upper layers of the boundary layer the direction remains unchanged for a substantial time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 155–157, September–October, 1979.     

Nonlinear waves in a viscous compressible fluid with relaxation

The propagation and stability of nonlinear waves in a viscous compressible fluid with relaxation that satisfies a Theological equation of state of Oldroyd type are investigated. An equation that describes the structure of the wave perturbations and its evolution is derived subject to the condition of balance of the nonlinear dissipative and relaxation effects, and its solutions of the solitary wave type are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 31–35, May–June, 1993.     

Transmission of waves in a liquid through absorbing layers and subsequent interaction of the waves with an obstacle

The propagation of waves in porous media is investigated both experimentally [1, 2] and by numerical simulation [3–5]. The influence of the relaxation properties of porous media on the propagation of waves has been investigated theoretically and compared with experiments [3, 4]. The interaction of a wave in air that passes through a layer of porous medium before interacting with an obstacle has been investigated with allowance for the relaxation properties [5]. In the present paper, in which the relaxation properties are also taken into account, a similar investigation is made into the interaction with an obstacle of a wave in a liquid that passes through a layer of a porous medium before encountering the obstacle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–53, March–April, 1983.     

Propagation of perturbations in the three-dimensional boundary layer on a delta wing in the viscid-inviscid interaction regime

The propagation of perturbations in the three-dimensional boundary layer on a planar delta wing in a hypersonic gas stream is investigated in the strong viscid-inviscid interaction regime. A characteristic associated with the induced pressure is found and an integral relation determining the velocity of its propagation is obtained. The directional diagrams of the propagation velocity of the characteristic surface in the boundary layer are determined for a series of constitutive parameters. This makes it possible to calculate the perturbation propagation velocities in the boundary layer when the velocity profiles in the longitudinal and transverse directions are known.     

Laminar boundary layer on an oscillating wedge

A study is made of the nonstationary laminar boundary layer on a sharp wedge over which a compressible perfect gas flows; the wedge executes slow harmonic oscillations about its front point. It is assumed that the perturbations due to the oscillations are small, and the problem is solved in the linear approximation. It is also assumed that the thickness of the boundary layer is small compared with the thickness of the complete perturbed region. Then in a first approximation the influence of the boundary layer on the exterior inviscid flow can be ignored, and the parameters on the outer boundary of the boundary layer can be taken equal to their values on the body for the case of inviscid flow over the wedge. They are determined from the solution to the inviscid problem that is exact in the framework of the linear formulation. The wall is assumed to be isothermal. The dependence of the viscosity on the temperature is linear. Under these assumptions, the problem of calculating the nonstationary perturbations in the boundary layer on the wedge is a self-similar problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 146–151, July–August, 1980.     

Structure of the disturbed boundary layer near a cold surface

A study is made of a laminar boundary layer near a cold plate in the regime of weak viscous -inviscid interaction in the limit in which the free-stream Mach number tends to infinity and the temperature factor to zero. Local disturbed flow regions formed due to the presence of small elements of roughness on the surface of the plate are investigated. It is found that thick regions of roughness elements induce disturbances of the frictional stresses and the heat flux of the same order as these quantities in the undisturbed boundary layer, while all thin regions of roughness elements induce only small, linear perturbations and, therefore, cannot cause separation of the boundary layer; the different regimes of flow past the roughness elements are described.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 19–27, January–February, 1991.     

Non-Darcian Effects on the Onset of Convection in a Porous Layer with Throughflow

The Brinkman extended Darcy model including Lapwood and Forchheimer inertia terms with fluid viscosity being different from effective viscosity is employed to investigate the effect of vertical throughflow on thermal convective instabilities in a porous layer. Three different types of boundary conditions (free–free, rigid–rigid and rigid–free) are considered which are either conducting or insulating to temperature perturbations. The Galerkin method is used to calculate the critical Rayleigh numbers for conducting boundaries, while closed form solutions are achieved for insulating boundaries. The relative importance of inertial resistance on convective instabilities is investigated in detail. In the case of rigid–free boundaries, it is found that throughflow is destabilizing depending on the choice of physical parameters and the model used. Further, it is noted that an increase in viscosity ratio delays the onset of convection. Standard results are also obtained as particular cases from the general model presented here.     

Stability of stationary convective flow of a mixture in a vertical porous layer

In contrast to the corresponding viscous flow, the convective flow of a homogeneous liquid in a planar vertical layer whose boundaries are maintained at different temperatures is stable [1]. When a porous layer is saturated with a binary mixture, in the presence of potentially stable stratification one must expect an instability of thermal-concentration nature to be manifested. This instability mechanism is associated with the difference between the temperature and concentration relaxation times, which leads to a buoyancy force when an element of the fluid is displaced horizontally. In viscous binary mixtures, the thermal-concentration instability is the origin of the formation of layered flows, which have been studied in detail in recent years [2–4]. The convective instability of the equilibrium of a binary mixture in a porous medium was considered earlier by the present authors in [5]. In the present paper, the stability of stationary convective flow of a binary mixture in a planar vertical porous layer is studied. It is shown that in the presence of sufficient longitudinal stratification the flow becomes unstable against thermal-concentration perturbations; the stability boundary is determined as a function of the parameters of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 150–157, January–February, 1980.     

Packets of wave perturbations in a Blasius flow

Asymptotic methods are used in order to study the development of quasiharmonic perturbations in the linear and slightly nonlinear problems of the stability of locally parallel Blasius flow. The connection between complex frequencies and wave numbers is analyzed in dependence on the initial spectrum and the local Reynolds numbers R, the law of propagation of the packets is determined, the equation of the amplitudes of their envelopes is constructed and solved, and an explanation is given of the conditions for the activation of regimes of the regular and irregular types. An interpretation is given of the observed properties of the initial stage in the evolution of the pulse in the boundary layer. It is concluded that the model has a limited ability to explain the mechanisms of stochastization.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 32–38, January–February, 1988.     

Influence of blowing on the characteristics of the nonstationary boundary layer on an oscillating wedge in a supersonic stream

A study is made of the influence of blowing on the characteristics of a nonstationary boundary layer. For this, we solve the equations of the laminar boundary layer on an infinite wedge moving with supersonic velocity and executing small oscillations about the tip. It is shown that increased blowing improves the damping of the oscillations of the wedge.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 171–175, January–February, 1981.     

Small perturbations spectrum in boundary-layer flow on a flat plate

The small perturbation spectrum of a number of flows has recently been analyzed carefully [1–3]. At the same time, investigations for the boundary layer have been limited within the framework of linear perturbation theory to the neighborhood of the neutral curve although a spectrum analysis is of indubitable interest not only to find the stability criterion of a laminar stream, but also to solve a problem with initial data about the time development of an arbitrary small perturbation. In particular, the possibility of representing an arbitrary perturbation in terms of a system of basis functions is related to the question of the completeness of the system. The finiteness was proved [4] and an estimate was obtained of the domain of eigenvalue existence in an investigation of the boundary-layer stability and a deduction has been made about the finiteness of the small perturbations spectrum for boundary-layer flow on this basis. A sufficiently complete survey of the investigation of the neutral stability of a laminar boundary layer can be found in the monograph [5]. The small perturbations spectrum in a boundary layer flow is obtained in this paper by methods of the linear theory of hydrodynamic stability by using the complete boundary conditions on the outer boundary. It is shown that the small perturbations spectrum is finite for each fixed value of the wave number . Singularities in the spectrum behavior are investigated for sufficiently small .Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 112–115, July–August, 1975.The author is grateful to M. A. Gol'dshtik and V. N. Shtern for useful discussions of the results of the research.     

Nonlinear internal waves in a fluid of infinite depth in the presence of a horizontal magnetic field

An investigation is made into the propagation of long nonlinear weakly nonone-dimensional internal waves in an incompressible stratified fluid of infinite depth in the presence of a horizontal magnetic field. It is shown that such waves are described by an equation representing the extension of the Benjamin-Ono equation to the weakly nonone-dimensional case. The equation obtained differs from that obtained in [4], which is attributable to the anisotropy of the medium resulting from the presence of a magnetic field. The stability of a soliton with respect to flexural perturbations is investigated. A particular case of the variation of the density with height at constant Alfvén velocity is examined in detail.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 65–72, November–December, 1987.     

Experimental investigation of the stability of a supersonic boundary layer on a plate with blunting of the leading edge

In an aerodynamic tube, an experimental investigation of the development of small natural perturbations in a laminar boundary layer on a plate was made. The measurements were made using a thermoanemometric method with a Mach number M = 2 and a unit Reynolds number Re₁ = 3.1·10⁶ m^–1. An investigation of the effect of blunting of the leading edge of the plate on the development of the perturbations was made. It is shown that blunting decreases the range of unstable frequencies and the region of instability and increases the critical Reynolds number of the loss of stability. In the initial section of the plate there is a rise in the perturbations of all frequencies up to a maximal value, whose coordinate is inversely proportional to the frequency. The development of perturbations in this region is insensitive to the state of the boundary layer. The position of the maximum does not change with blunting of the leading edge.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 65–70, July–August, 1977.     

Stability of a nonisothermal boundary layer in the presence of periodic perturbations of the exterior flow velocity

The stability of a laminar boundary layer in the presence of high-frequency time-periodic perturbations of the exterior flow velocity, in particular, acoustic vibrations, is investigated in a series of papers which are reviewed in detail in monograph [1]. The mechanisms by which such perturbations influence the stability and transition to the turbulent flow regime can vary. For example, they can lead to the deformation of the averaged field of the basic flow. However, there was good reason not to discuss the effect of this type of perturbation earlier, as it was considered that the change in the basic flow was very small even for perturbations of great amplitude. The aim of the present paper is to demonstrate how perturbations or pulsations in the exterior flow velocity can, by changing the basic flow, have a strong influence on the stability of the laminar boundary layer of a gas under appreciably nonisothermal conditions. Examples of calculations that support this assertion are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 183–186, September–October, 1984.The authors wish to thank A. A. Maslov for his help with the calculations.     

Receptivity of the inclined-cylinder attachment-line boundary layer to vortex perturbations

The receptivity of the boundary layer in the neighborhood of the attachment line of a cylinder inclined to the flow with respect to periodic vortex perturbations frozen into the stream is investigated. The problem considered simulates the interaction between external turbulence and the leading-edge swept wing boundary layer. It is shown that if the direction of the external perturbation vector is almost parallel to the leading edge, then the external perturbations are considerably strengthened at the outer boundary layer edge. This effect can cause laminar-turbulent transition on the attachment line at subcritical Reynolds numbers.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 72–85. Original Russian Text Copyright © 2004 by Ustinov.     

Calculation of the longitudinal velocity in the linear problem of a vibrator in a subsonic boundary layer

The linear problem is considered of a localized vibrator mounted on a flat plate in a subsonic boundary layer. The plate and the vibrator are assumed to be heat-insulated, and the dimensions of the vibrator and the frequency of the oscillations are such that the flow may be described by means of the equations of a boundary layer with self-induced pressure. The amplitude of the oscillations of the vibrator and the perturbations of the flow parameters corresponding to it are assumed to be small, and this makes it possible to linearize these equations. Integral transformations are used to construct a solution for values of the time greatly exceeding the period of the oscillations of the vibrator. The profiles of the perturbations of the longitudinal velocity are calculated in dependence on the transverse coordinate for various values of the longitudinal coordinate. A comparison is made with the profiles of the perturbations of the longitudinal velocity which have been obtained experimentally.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 61–67, May–June, 1987.     

Spin-down of a fluid in a low cylinder at large Reynolds numbers

The problem of the axisymmetric motion of a fluid between infinite disks is solved by the method of matched asymptotic expansions without introducing model assumptions. For the strongly nonlinear stage of spin-down solutions are found that correspond to initial states different from rigid-body rotation, when the boundary layer is not a Kármán layer. The experimental results obtained are in qualitative and quantitative agreement with the theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 39–46, May–June, 1986.The authors wish to thank A. M. Obukhov and F. V. Dolzhanskii for formulating the problem and for constructive discussion.