Laminar boundary layer with uniform suction on flat plate in oscillating flow

We examine unsteady incompressible fluid flow in a laminar boundary layer with uniform suction for longitudinal flow over a flat plate when the external stream is a flow with constant velocity, on which there is superposed a sinusoidal disturbance convected by the stream, analogous to [1]. We study the stability of such flow in the boundary layer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 3, pp. 66–70, May–June, 1970.     

Influence of blowing into the boundary layer on supersonic flow past an oscillating cone

The influence of blowing on the unsteady characteristics of a boundary layer is studied for the example of supersonic flow past a sharp cone oscillating about zero angle of attack. The problem of the interaction of the inviscid exterior flow with the laminar boundary layer is solved. It is shown that blowing proportional to the heat flux improves the damping of the oscillations of the cone. If the blowing anticipates the heat flux in phase this effect is strengthened.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 43–46, July–August, 1983.     

Method of mean-mass parameters for a three-dimensional boundary layer in a flow with vorticity

When a gas flows with hypersonic velocity over a slender blunt body, the bow shock induces large entropy gradients and vorticity near the wall in the disturbed flow region (in the high-entropy layer) [1]. The boundary layer on the body develops in an essentially inhomogeneous inviscid flow, so that it is necessary to take into account the difference between the values of the gas parameters on the outer edge of the boundary layer and their values on the wall in the inviscid flow. This vortex interaction is usually accompanied by a growth in the frictional stress and heat flux at the wall [2, 3]. In three-dimensional flows in which the spreading of the gas on the windward sections of the body causes the high-entropy layer to become narrower, the vortex interaction can be expected to be particularly important. The first investigations in this direction [4–6] studied the attachment lines of a three-dimensional boundary layer. The method proposed in the present paper for calculating the heat transfer generalizes the approach realized in [5] for the attachment lines and makes it possible to take into account this effect on the complete surface of a blunt body for three-dimensional laminar, transition, or turbulent flow regime in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 80–87, January–February, 1981.     

Cascaded thin lightly loaded airfoils vibrating in a subsonic separated flow

The problem of calculating the nonstationary aerodynamic characteristics of a cascade of thin lightly loaded airfoils in a subsonic flow with the formation of thin separation zones of finite extent is solved approximately. As in [1–5], an approach based on a linear small-perturbation analysis, within which the flow is assumed to be inviscid, is employed and the boundaries of the unsteady separation zones are simulated by oscillating lines of contact discontinuity. However, instead of the requirement of a given fixed pressure at the boundary of the separation zone, used in [1–5], this study proposes a more general condition according to which on each element of length of the thin separation layer the pressure oscillates with an amplitude proportional to the local value of the amplitude of its thickness oscillations. The problem is reduced to a system of two singular integral equations which can be solved numerically.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 181–191, January–February, 1995.     

Stability of laminar boundary layer on an elastic surface in an incompressible fluid

Interest in the present problem arose after the publication of the results of the experiments of Kramer [1–3]. In addition to the studies indicated in [4], the articles [5–8] are devoted to the question of the interaction of a flexible elastic surface with the boundary layer. In the present paper the problem of the interaction of an elastic surface with disturbances arising in the boundary layer is posed as in [4]. The approximate nature of the methods of solving the problem of the hydrodynamic stability of the laminar boundary layer leads to a difference in the final computational formulas even in the case when authors use the same Heisenberg-Tollmien-Schlichting-Lin scheme. Therefore, in what follows we present a comparison of the data on the stability of the boundary layer on a solid wall obtained by several authors with the calculations using the formulas, which are then generalized to the case of the elastic surface.The author wishes to thank G. I. Petrov and V. A. Medvedev for discussions of the present study.     

Boundary layer in the vicinity of the stagnation point of a body of axial symmetry in a turbulent stream

The flow in the boundary layer in the vicinity of the stagnation point of a flat plate is examined. The outer stream consists of turbulent flow of the jet type, directed normally to the plate. Assumptions concerning the connection between the pulsations in velocity and temperature in the boundary layer and the average parameters chosen on the basis of experimental data made it possible to obtain an isomorphic solution of the boundary layer equations. Equations are obtained for the friction and heat transfer at the wall in the region of gradient flow taking into account the effect of the turbulence of the impinging stream. It is shown that the friction at the wall is insensitive to the turbulence of the impinging stream, while the heat transfer is significantly increased with an increase in the pulsations of the outer flow. These properties are confirmed by the results of experimental studies [1–4].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 83–87, September–October, 1973.     

Effect of viscosity on the lift-drag ratio of a thin blunt wing at hypersonic rates of flow round it

The effect of viscosity on the carrying properties of hypersonic aircraft appears at great flight altitudes, where an important factor is the interaction of the laminar boundary layer with inviscid flow. In the present study the method of bands is used to make an approximate calculation of this effect for a regime of weak viscous interaction [1]. The results of [2] are used for conditions of inviscid flow round a body. The local coefficient of friction and coefficients of the additional pressure induced by the boundary layer are determined from the data for a plate of infinite width [3]. Simple relationships are obtained which make it possible to estimate the effect of viscosity on the magnitude of the maximum lift-drag ratio and the value of the angle of attack corresponding to it. The results are given of an experimental study of hypersonic flow round a plane triangular wing in a broad range of Reynolds numbers, and these confirm the relationships obtained and indicate the region in which they are applicable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 149–152, November–December, 1988.     

The structure of the supersonic turbulent boundary layer in its interaction with a shock wave

An experimental study is made of the turbulent boundary layer in its interaction with a shock wave, the purpose being to clarify questions connected with the increase in the fullness of the velocity profiles. New systematic data are obtained on the development of the boundary layer, and its structure and asymptotic behavior beyond the interaction region. These results are for axisymmetric flow in the range of Mach numbers M=2–4 and angles of rotation of the flow 10–25°. Conditions of developed separation are included. Extensive information about the general properties of flows with separation has been obtained in a number of studies. A survey of these may be found, for example, in [1, 2]. Certain questions about the separation and reattachment of the boundary layer are clarified. The dimensions of the separation region are determined and its structure studied in detail for various shapes of the surface around which the flow takes place. Nevertheless it has not yet proved possible to reach a complete understanding of this complex phenomenon. Usually plane models have been used for the investigations, but in this case it is evidently impossible to exclude completely the influence of end effects on the flow in the interaction zone. Therefore it is preferable to study such flows in axisymmetric models; this considerably eases the task of analyzing and interpreting the results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 75–82, September–October, 1985.     

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.     

Calculation of the viscosity effect on the aerodynamic characteristics of slender bodies at hypersonic flow velocities

Approximating dependences of the local coefficients of friction, heat transfer, and pressure induced by a boundary layer on the generalized similarity parameters, including the inviscid flow characteristics, are obtained on the basis of the results of a numerical calculation of hypersonic flow past a number of plane and axisymmetric bodies. If the inviscid flow characteristics are known, these relations can be used to take the viscosity approximately into account under conditions of interaction between the laminar boundary layer and the hypersonic inviscid stream [1].Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 142–150, July–August, 1995.     

Determination of hydrodynamic reactions acting on an oscillating profile

Sedov's equations [1], which make it possible to calculate the total hydrodynamic reactions exerted by an ideal fluid on an oscillating profile, are well known. These equations are expressed by contour integrals containing the complex potential of the unsteady flow past the profile. Certain modifications of these equations are proposed in paper [2]. In this paper, other equations are proposed for the calculation of the same quantities, based on the specification of the tangential velocity component of the fluid along the contour of the oscillating profile. In a number of cases, the application of these equations can be more useful than that of Sedov's equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 189–192, July–August, 1985.     

Transition flow of a fibrous suspension in a pipe as the flow of an anisotropic fluid

The transition flow is considered of a fibrous suspension in a pipe. The flow region consists of two subregions: at the center of the flow a plug formed by interwoven fibers and fluid moves as a rigid body; between the solid wall and the plug is a boundary layer in which the suspension is a mixture of the liquid phase and fibers separated from the plug [1–3]. In the boundary region the suspension is simulated as an anisotropic Ericksen—Leslie fluid [4, 5] which satisfies certain additional conditions. Equations are obtained for the velocity profile and drag coefficient of the pipe, which are both qualitatively and quantitatively in good agreement with the experimental results [6–8]. Within the framework of the model, a mechanism is found for reducing the drag in the flow of a fibrous suspension as compared to the drag of its liquid phase.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 91–98, September–October, 1985.     

Boundary layer on a blunt body in a flow of dusty gas

The problem of interaction of gas-dust flows with solid surfaces arose in connection with the study of the motion of aircraft in a dusty atmosphere [1–2], the motion of a gas suspension in power generators, and in a number of other applications [3]. The presence of a disperse admixture may lead to a significant increase in the heat fluxes [4] and to erosion of the surface [5]. These phenomena are due to the joint influence of several factors — the change in the structure of the carrier-phase boundary layer due to the presence of the particles, collisions of the particles with the surface, roughness of the ablating surface, and so forth. This paper continues an investigation begun earlier [6–7] into the influence of particles on the structure of the dynamical and thermal two-phase boundary layer formed around a blunt body in a flow. The model of the dusty gas [8] has an incompressible carrier phase. The method of matched asymptotic expansions [9] is used to obtain the equations of the two-phase boundary layer. In the frame-work of the refined classification made by Stulov [6], it is shown that the form of the boundary layer equations is different in the presence and absence of inertial precipitation of the particles. The equations are solved numerically in the neighborhood of the stagnation point of the blunt body. The temperature and phase velocity distributions in the boundary layer, and also the friction coefficients and the heat transfer of the carrier phase are found for a wide range of the determining parameters. In the case of an admixture of low-inertia particles that are not precipitated on the body, it is shown that even when the mass concentration of the particles in the undisturbed flow is small their accumulation in the boundary layer can lead to a sharp increase in the thermal fluxes at the stagnation point.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–107, September–October, 1985.I thank V. P. Strulov for a discussion.     

REDUCED-ORDER MODELS OF UNSTEADY TRANSONIC VISCOUS FLOWS IN TURBOMACHINERY

The proper orthogonal decomposition (POD) technique is applied in the frequency domain to obtain a reduced-order model of the unsteady flow in a transonic turbomachinery cascade of oscillating blades. The flow is described by a inviscid—viscous model, i.e. a full potential equation outer flow model and an integral equation boundary layer model. The nonlinear transonic steady flow is computed first and then the unsteady flow is determined by a small perturbation linearization about the nonlinear steady solution. Solutions are determined for a full range of frequencies and validated. The full model results and the POD method are used to construct a reduced-order model in the frequency domain. A cascade of airfoils forming the Tenth Standard Configuration is investigated to show that the reduced-order model with only 15–75 degrees of freedom accurately predicts the unsteady response of the full system with approximately 15 000 degrees of freedom.     

Asymptotic behavior of solutions of the Orr-Sommerfeld equations in regions adjacent to two branches of the neutral curve

The asymptotic behavior of the upper and lower branches of the neutral stability curve of a boundary layer found by Lin [1] was determined more accurately by various authors [2–4], who, on the basis of the linearized Navien-Stokes equations, analyzed the higher approximations in the Reynolds number R. In the limit R , neutral perturbations have wavelengths that exceed in order of magnitude the boundary layer thickness. The long-wavelength asymptotic behavior of the Orr-Sommerfeld equation is, in particular, of interest because the characteristic solutions of the linearized equations of free interaction (triple-deck theory) [5–7] are a limiting form of Tollmierr-Schlichting waves in an incompressible fluid with critical layers next to the wall [8–9]. At the same time, the dispersion relation, which is identical to the secular equation of the Orr-Sommerfeld problem, contains an entire spectrum of solutions not considered in the earlier studies [2–4]. The first oscillation mode in the spectrum may be either stable or unstable. In the present paper, solutions are constructed for each of the subregions (including the critical layer) into which the perturbed velocity field in the linear stability problem is divided at large Reynolds numbers. Dispersion relations describing the neighborhood of the upper and lower branches of the neutral curve for the boundary layer are derived. These relations, which contain neutral solutions as a special case, go over asymptotically into each other in the unstable region between the two branches.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–11, July–August, 1984.     

Numerical prediction of locally forced turbulent boundary layer

An unsteady numerical simulation was performed to analyze flow structure behind a local suction/blowing in a flat-plate turbulent boundary layer. The local forcing was given to the boundary layer flow by means of a sinusoidally oscillating jet. A version of the unsteady k––f_μ model [Fluid Dyn. Res. 26 (6) (2000) 421] was employed. The Reynolds number based on the momentum thickness was about Re_θ=1700. The forcing frequency was varied in the range 0.011f⁺0.044 with a fixed forcing amplitude A_o=0.4. The predicted results were compared and validated with the experimental data. It was shown that the unsteady locally forced boundary layer flow is predicted well by the k––f_μ model. The time-dependent numerical flow visualizations were demonstrated during one period of the local forcing. The effect of the pitch angle of local forcing on the reduction of skin friction was examined.     

Aerodynamics of end-wall boundary layer in a vortex chamber

The need for the inclusion of end-wall boundary layers in the study of the aerodynamics of vortex chambers has been frequently mentioned in the literature. However, owing to limited experimental data [1–3] with reliable information on the wall layers, the existing computational methods for end-wall boundary layers are not well-founded. The question of which parameters determine the formation of end-wall flow remains debatable. In some studies [4, 5], the vortex chambers are conditionally divided into short and long chambers. However, there is no unique opinion on the role of end-wall flows in vortex chambers of different lengths. It has also not been established for what geometric and flow parameters the chamber could be considered long or short. In the present study, as in [1, 5–8], solution is obtained for the end-wall boundary-layer equations using integral methods, considering the boundary layer in the radial direction in the form of a submerged wall jet. Such an approach made it possible to use the laws for the development of wall jets [9], and obtain fairly simple relations for integral parameters, skin friction, mass flow in the boundary layer, and other characteristics. Results are compared with available experimental data and computations of others authors; turbulent flow is considered; results for laminar boundary layer are given in [10].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 117–126, September–October, 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.     

Supersonic flow around the trailing edge of a thin profile

The method of mergeable asymptotic expansions has recently been used effectively in investigations devoted to the study of boundary layer interaction with an external inviscid flow at high subcritical Reynolds numbers Re. The asymptotic analysis permits obtaining a limit pattern of the flow around a solid as Re þ, and determining the similarity and quantitative regularity laws which are in good agreement with experimental results. Thus by using the method of mergeable asymptotic expansions it is shown in [1–4] that near sites with high local curvature of the body contour and flow separation and attachment points, an interaction domain appears that has a small length on the order of Re^-3/8. In this flow domain, which has a three-layer structure, the pressure distribution in a first approximation already depends on the change in boundary-layer displacement thickness, while the induced pressure gradient, in turn, influences the flow in the boundary layer. An analogous situation occurs in the neighborhood of the trailing edge of a flat plate where an interaction domain also appears [5, 6]. The flow in the neighborhood of the trailing edge of a flat plate around which a supersonic viscous gas flows was examined in [7]. Numerical results in this paper show that the friction stress on the plate surface remains positive everywhere in the interaction domain, and grows on approaching the trailing edge. The supersonic flow around the trailing edge of a flat plate at a small angle of attack was investigated in [8, 9], Supersonic flow of a viscous gas in the neighborhood of the trailing edge of a flat plate at zero angle of attack is examined in [10], but with different velocity values in the inviscid part of the flow on the upper and lower sides of the plate. The more general problem of the flow around the trailing edge of a profile with small relative thickness is investigated in this paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 36–42, May–June, 1981.     

Characteristics of an unstably localized disturbance in a compressed boundary layer

Experiments have demonstrated [1] that the transition of streamline-type flow into turbulent flow in a boundary layer occurs as a result of the formation and development of turbulent spots apparently arising from small natural disturbances. A study of the nonlinear evolution and interaction of localized disturbances requires knowledge of their characteristics to a linear approximation [2]. In the current work, results are presented of calculations of such characteristics for the first two unstable modes in a supersonic boundary layer on a two-dimensional plate (M = 4.5, T_w = 4.44).Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 50–53, January–February, 1976.