Boundary layer interaction in three-dimensional flows

An approach known from the theory of matched asymptotic expansions involving the isolation of subregions with different scales is used to study flows which are assumed to be described by the boundary layer equations almost everywhere near the surface except for a fairly narrow zone in which the inflowing boundary layers interact. Two characteristic types of interaction are identified. An approximate theory describing the flow in the interaction zone, which makes it possible to locate the position of the interaction zone on the surface, is proposed. The interaction flow on the end wall of a vane channel is calculated subject to certain simplifications. The results of an experimental investigation of this flow are presented and it is shown that the theoretical model proposed describes the three-dimensional corner separation which occurs in the neighborhood of the line of intersection of the end wall and the convex edge of the vane.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 116–123, May–June, 1988.     

Three-dimensional separation in the neighborhood of roughness on the surface of an axisymmetric body

Steady-state viscous incompressible fluid flow past an axisymmetric slender body is considered at high Reynolds numbers in the regime with vanishing surface friction in a certain cross-section. In a small neighborhood of this cross-section interaction between the boundary layer flow and the external irrotational stream develops. In order to study the structure of the three-dimensional flow with local separation zones it is assumed that there is three-dimensional roughness on the surface of the body with the scale of the interaction zone. For this zone a numerical solution of the problem is obtained and its nonuniqueness is established. The surface friction line (limiting streamline) patterns with their inherent features are constructed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–79, May–June, 1995.Thus, on the basis of the asymptotic marginal separation theory it is possible to obtain fairly simple solutions describing flows with a complex surface friction line structure.     

Theory of self-induced separation of a turbulent boundary layer

Plane steady flow is considered in the case of an incompressible fluid in the neighborhood of the separation point of a turbulent boundary layer. It is shown that separation takes place in a region which lies at some small distance downstream from the region of interaction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 51–60, May–June, 1987.     

Asymptotic theory of turbulent separation

The flow of an incompressible fluid in the viscous wall sublayer of a turbulent boundary layer in the neighborhood of a point of separation is considered. On the basis of an asymptotic analysis of the Reynolds equations and without the use of any hypotheses about their closure it is shown that at large Reynolds numbers the velocity profile satisfies the well-known “half-power law,” and as a result of a large self-induced pressure gradient separation is preceded by the appearance of reverse flows in a thin viscous sublayer.     

Three-dimensional turbulent boundary layer on a body of complex shape

Flow and heat transfer problems associated with three-dimensional compressible gas flow past a body of complex shape at a small angle of attack are investigated on the basis of a finite-difference calculation. The results of a numerical solution of the equations of the three-dimensional turbulent boundary layer are presented. The effect of the leading parameters on three-dimensional flow development and heat transfer is analyzed. The characteristic flow regions in the boundary layer are found: lines of divergence and convergence on the surface, separation zones and flow interfaces. The location of the maximum values of the heat flux and friction on the surface is determined, the behavior of the limiting streamlines on the body is described, and the intensity of the secondary flows in the boundary layer is estimated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 25–35, September–October, 1986.     

Turbulent Channel Flow with an Artificial Two-Dimensional Wall Layer

Turbulent flow of an incompressible fluid in a plane channel with parallel walls is considered. The three-dimensional time-dependent Navier-Stokes equations are solved numerically using the spectral finite-difference method. An artificial force which completely suppresses lateral oscillations of the velocity is introduced in the near-wall zone (10 % of the channel half-width in the neighborhood of each wall). Thus, the three-dimensional flow zone, in which turbulent oscillations can develop, is separated from the wall by a fluid layer. It is found that the elimination of three-dimensionality in the neighborhood of the walls leads to a significant reduction in the drag. However, complete laminarization does not occur. The flow in the stream core remains turbulent and can be interpreted as a turbulent flow in a channel with walls located on the boundary of the two-dimensional layer and traveling at the local mean-flow velocity. The oscillations developing inside the two-dimensional layer, which have significant amplitude, distort the flow only in the adjacent zone. Beyond this zone the distributions of the mean characteristics and the structure of instantaneous fields completely correspond to ordinary turbulent flow in a channel with rigid walls. The results obtained confirm the hypothesis of the unimportance of the no-slip boundary conditions for the fluctuating velocity component in the mechanism of onset and self-maintenance of turbulence in wall flows.     

Flow of a viscous fluid in an annular gap with intensive blowing from the walls

Self-similar solutions are obtained in [1, 2] to the Navier-Stokes equations in gaps with completely porous boundaries and with Reynolds number tending to infinity. Approximate asymptotic solutions are also known for the Navier-Stokes equations for plane and annular gaps in the neighborhood of the line of spreading of the flow [3, 4]. A number of authors [5–8] have discovered and studied the effect of increase in the stability of a laminar flow regime in channels of the type considered and a significant increase in the Reynolds number of the transition from the laminar regime to the turbulent in comparison with the flow in a pipe with impermeable walls. In the present study a numerical solution is given to the system of Navier-Stokes equations for plane and annular gaps with a single porous boundary in the neighborhood of the line of spreading of the flow on a section in which the values of the local Reynolds number definitely do not exceed the critical values [5–8]. Generalized dependences are obtained for the coefficients of friction and heat transfer on the impermeable boundary. A comparison is made between the solutions so obtained and the exact solutions to the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–24, January–February, 1987.     

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.     

Heat transfer at the surface of a triangular body in a hypersonic viscous gas flow

The boundary layer on a semi-infinite triangular body of power-law shape is calculated for viscous interaction with an external hypersonic flow. The results of calculating the characteristics of the three-dimensional boundary layer are presented. The formation of secondary return flows and zones of intensified heat transfer on the surface of the body in the neighborhood of lines of flow divergence is noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 77–82, January–February, 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.     

Numerical investigation of fully developed turbulent fluid flow and heat transfer in a square duct

Three-dimensional fully developed turbulent fluid flow and heat transfer in a square duct are numerically investigated with the author's anisotropic low-Reynolds-number k-ε turbulence model. Special attenton has been given to the regions close to the wall and the corner, which are known to influence the characteristics of secondary flow a great deal. Hence, instead of the common wall function approach, the no-slip boundary condition at the wall is directly used. Velocity and temperature profiles are predicted for fully developed turbulent flows with constant wall temperature. The predicted variations of both local wall shear stress and local wall heat flux are shown to be in close agreement with available experimental data. The present paper also presents the budget of turbulent kinetic energy equation and the systematic evaluation for existing wall function forms. The commonly adopted wall function forms that are valid for two-dimensional flows are found to be inadequate for three-dimensional turbulent flows in a square duct.     

Properties of two- and three-dimensional separation flows at supersonic velocities

The results are given of experimental investigations into three-dimensional separation of a turbulent boundary layer in the neighborhood of oblique shock waves, wedge-shaped obstacles, and sweptback steps at Mach numbers M = 2, 2.25, 2.5, 3, 4 and Reynolds numbers Re = u/v = (30–36)· 10⁶ m^–1. The characteristic regimes of the separated flows are considered. There is a discussion of the results of comparison and generalization of the pressure distribution in the two- and three-dimensional separation regions, and empirical dependences are also given for determining some geometrical parameters of these regions. An analogy is found in the characteristic pressures, and pressure distribution for a number of two- and three-dimensional separation flows, which suggests that one could use some of the known methods of analysis of two-dimensional separation of a turbulent boundary layer to calculate estimates for the three-dimensional case. This is confirmed by a comparison of calculated and experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–50, May–June, 1979.I am grateful to A. M. Kharitonov and V. S. Dem'yanenko for helpful comments made during the work and for discussion of this paper, and also to V. M. Filatov for assistance in some of the experiments.     

Direction of streamlines near the wall in a three-dimensional turbulent boundary layer

Expressions are derived that, for a large class of flows, enable one to determine the direction of the streamlines near the wall in a three-dimensional turbulent boundary layer without integrating the system of boundary-layer equations. Comparison with available experimental data reveals a completely satisfactory agreement between the calculations and the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 19–25, March–April, 1980.     

Numerical investigation of flow past a system of two sweptback compression wedges

Supersonic flow past a three-dimensional configuration consisting of two neighboring wedges with sweptback leading edges mounted on a preliminary compression surface is numerically investigated. The case of sweptback wedges is considered, when their beveled surfaces deflect the compressed flows to opposite directions. The calculations are carried out on the basis of the averaged Navier-Stokes equations, together with the SST k-? turbulence model, at the freestream Mach number M = 6. The results obtained are compared with the data for inviscid flow calculated using the Euler equations. The flow pattern features, due to the interaction in the plane of symmetry between the shocks formed by the wedges and the shock-induced three-dimensional quasiconical separations of the turbulent boundary layer on the preliminary compression surface along the swept leading edges, are established. Within these separation zones the flow is directed away from the plane of symmetry of the configuration and is characterized by considerably greater values of the transverse velocity component, as compared with the flow outside of the separation zone.     

Quasihomogeneous model of cavitation flows in diffuser channels

Starting with the experiments carried out by Reynolds in 1894, the flow in Venturi tubes has traditionally been used to study and demonstrate various forms of cavitation. Numerous authors have carried out experimental research on the various flow regimes in diffuser channels [1–7] or have investigated theoretical models of such flows [6, 8]. The occurrence and development of cavitation is closely associated with the phenomenon of turbulent separation complicated by the presence of two-phase flow in the dissipation zone. For a long time these effects were considered separately, until Gogish and Stepanov [9] proposed a single model of cavitation and separation based on the theory of intense interaction of an incompressible potential flow and a turbulent cavitation layer of variable density and embracing the various stages of cavitation. The object of this study is to demonstrate the possibilities of this model with reference to the simple example of flows accompanied by cavitation and separation in plane and axisymmetric diffuser channels of the Venturi tube type with straight and curved walls. The dissipative flow near the walls is described by a quasihomogeneous model of turbulent two-phase flow, in which the presence of two phases is taken into account only by varying the mean density. The potential core of the flow is considered in the one-dimensional formulation. The displacement thickness serves as the flow interaction parameter. The conditions of ocurrence and development of circulatory flows are determined. Examples of symmetrical and nonsymmetrical flows are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 47–54, September–October, 1986.     

Calculation of the momentum and heat transfer in turbulent gas-particle jet flows

The equations for the second moments of the dispersed-phase velocity and temperature fluctuations are used for calculating gas-suspension jet flows within the framework of the Euler approach. The advantages of introducing the equations for the second moments of the particle velocity fluctuations has previously been quite convincingly demonstrated with reference to the calculation of two-phase channel boundary flows [9–11]. The flows considered below have a low solid particle volume concentration, so that interparticle collisions can be neglected and, consequently, the stochastic motion of the particles is determined exclusively by their involvement in the fluctuating motion of the carrier flow. In addition to the equations for the turbulent energy of the gas and its dissipation, the calculation scheme includes the equations for the turbulent energy and turbulent heat transfer of the solid phase; however, the model constructed does not contain additional empirical constants associated with the presence of the particles in the flow.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 69–80, May–June, 1992.     

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.     

“Non-free” interaction of plane shock waves and the boundary layer in the neighborhood of the leading edge of a plate with slip

The interaction between a normally impinging shock wave and the boundary layer on a plate with slip is studied in the neighborhood of the leading edge using various experimental methods, including special laser technology, to visualize the supersonic conical gas flows. It is found that in the “non-free” interaction, when the leading edge impedes the propagation of the boundary layer separation line upstream, the structure of the disturbed flow is largely identical to that in the developed “free” interaction, but with higher parameter values and gradients in the leading part of the separation zone. The fundamental property of developed separation flows, namely, coincidence of the values of the pressure “plateau” in the separation zone and the pressure behind the oblique shock above the separation zone of the turbulent boundary layer, is conserved. Moscow. e-mail: ostap@inmech.msu.su. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 57–69, May–June, 2000. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 97-01-00099).     

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.     

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.