Flow investigation in separation zones ahead of supersonic jets in a subsonic stream

The flow in the three-dimensional separation zone of a turbulent boundary layer on a plate in front of a supersonic jet injected perpendicularly to the subsonic drifing flow is considered. The purpose of the investigation is to establish the physical singularities of subsonic flow around a supersonic jet obstacle and to obtain dependences of the geometric flow characteristics on the free-stream and injected-jet parameters. Results of an experimental investigation permitted proposing approximate dependences of the geometric three-dimensional separation-zone characteristics which appear in the subsonic stream ahead of a jet obstacle.     

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.     

Unsteady boundary layers with small cross flows

Despite the intensive development of computer technology and methods of solving the Navier-Stokes and Reynolds equations, the unsteady problems of the three-dimensional boundary layer are of significant interest in aerodynamics. So far these problems have been little studied as a result of objective difficulties related with the large dimensionality of the system of equations and the complexity of its investigation [{xc1}]. Therefore, analytic results in this field are important. In the present study the unsteady three-dimensional boundary layer equations are investigated in the case of small cross flows using the perturbation method. An intermediate system of equations, which includes the basic three-dimensional effects but is significantly simpler than the initial system is derived. The features of the formulation considered are studied in relation to the important practical problems of boundary layer flow past slender wings and weakly asymmetric bodies at small angles of attack.     

Approximate solution of the problem of the three-dimensional boundary layer in an incompressible fluid

A method of successive approximations is proposed for the solution of the equations of the three-dimensional incompressible boundary layer on bodies of arbitrary shape. A coordinate system connected with the streamlines of the external nonviscous flow is used. It is assumed that the velocity across the external streamlines is small. When the intensity of secondary flow is low the equations describing the boundary layer in an incompressible fluid are reduced to a form analogous to the equations for the boundary layer on axially symmetrical bodies. An approximate analytical solution is obtained for the velocity and for the friction in the form of equations which can be used for any problems of a three-dimensional incompressible boundary layer. The method developed was applied to the problem of the three-dimensional boundary layer at a plate with a cylindrical obstacle in the presence of a slip angle.     

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.     

Development of the boundary layer on a plate in a rotating system

A numerical investigation in the approximation of boundary layer theory has been made of the development of the flow near the surface of a rotating plate in a two-dimensional flow with rectilinear streamlines perpendicular to the leading edge in a rotating coordinate system attached rigidly to the plate. In an earlier investigation [1] using the approximate method of integral relations, Kurosaka obtained and described quantitatively a transition from a Blasius boundary layer to an Eckmann boundary layer in the form of three-dimensional oscillations. The solution described in the present paper confirms the oscillatory nature of the development of the boundary layer, but the quantitative results differ strongly from Kurosaka's.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 154–157, May–June, 1982.     

Law of transverse sections for the three-dimensional boundary layer on a thin wing in a hypersonic stream

The investigation of three-dimensional flows in boundary layers is important to determine the aerodynamic characteristics of wings such as the heat fluxes and friction drag. However, the circumstance that interaction of the boundary layer and the wake with an inviscid stream can play a governing role for the formation of the flow diagram as a whole is more important. The three-dimensional flow on a thin delta wing in a hypersonic stream is investigated in this paper. An important singularity of hypersonic flow is the low value of the gas density in the boundary layer as compared with the density on its outer boundary. It is shown that in the general case when the pressure in the wing span direction varies mainly by an order, high transverse velocities originate because of the smallness of the density within the boundary layer. This circumstance permits expansion of the solution for smallspan wings in a series in an appropriate small parameter. The equations in each approximation depend on two variables, while the third—longitudinal—variable enters as a parameter. The zero approximation can be considered as the formulation of the law of transverse plane sections for a three-dimensional boundary layer. As a comparison with the exact solutions calculated for delta wings with power-law distributions of the wing thickness has shown, the first approximation yields a very good approximation. Furthermore, flow modes with a different direction of parabolicity on the whole wing, as well as zones in which interaction with the external stream should absolutely be taken into account, are found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 75–84, March–April, 1976.     

The effect of anisotropic permeability on free convective boundary layer flow in porous media

The effect of an anisotropic permeability on thermal boundary layer flow in porous media is studied. The convective flow is induced by a vertical, uniformly heated surface embedded in a fluid-saturated medium. A leading-order boundary layer theory is presented. It is shown that the thickness of the resulting boundary layer flow is different from that obtained in an isotropic porous medium. In general, an anisotropic permeability induces a fluid drift in the spanwise direction, the strength of which depends on the precise nature of the anisotropy. Conditions are found which determine whether or not the boundary layer flow is three-dimensional.     

Experimental investigation of three-dimensional supersonic flow of a gas in the interference region of intersecting surfaces

A contemporary high-speed aircraft represents a complex three-dimensional configuration, where supersonic gas flow is accompanied by numerous local flow interaction zones, in particular, near the intersection of different surfaces. Such a flow is characterized by three-dimensional systems of shock and expansion waves, and close to the surfaces one finds interaction of boundary layers and, above all, interaction of shock waves with the boundary layer. In general, the angular configurations are formed by intersection or contact of nonplanar surfaces with swept-back or blunted leading edges. This makes it practically impossible to obtain a rigorous theoretiical solution to the problem of gas flow over these surfaces, and presents considerable difficulty in an experimental investigation. It is therefore of interest to study the physical features of gas flow in corner configurations of very simple form [1–3]. The present paper examines the results of an experimental investigation of typical features of symmetric and asymmetric interaction of compressive, expansive, and mixed flows in the interference region of planar surfaces intersecting at an angle of less than 180?.     

Linear evolution of controlled disturbances in the supersonic boundary layer on a swept wing

The linear development of controlled disturbances in the three-dimensional supersonic boundary layer on a swept model wing with a sharp leading edge is experimentally investigated at the Mach number 2. The spatial-temporal and spectral-wave characteristics of the wave train of unstable disturbances are obtained. The asymmetry of these characteristics, due to the secondary flow in the three-dimensional boundary layer, is confirmed.     

Generation of Three-Dimensional Disturbances in a Boundary Layer on Strong Interaction with a Hypersonic Flow

Laminar boundary layer flow over an infinite-span, finite-length flat plate is investigated in the regime of strong interaction with a hypersonic gas flow. Under the assumption that an additional condition dependent on the transverse coordinate can be imposed on the trailing edge of the plate the flow functions are expanded in power series in the vicinity of the leading edge. It is shown that these expansions include an indefinite function dependent on the transverse coordinate. The corresponding boundary value problems are formulated and solved and the eigenvalues are determined. It is established that in this case the two-dimensional boundary layer can rearrange itself into a three-dimensional boundary layer.     

Singularities of the boundary layer equations and the structure of the flow in the vicinity of the convergence plane on conical bodies

The singularities of the boundary layer equations and the laminar viscous gas flow structure in the vicinity of the convergence plane on sharp conical bodies at incidence are analyzed. In the outer part of the boundary layer the singularities are obtained in explicit form. It is shown that in the vicinity of a singularity a boundary domain, in which the flow is governed by the shortened Navier-Stokes equations, is formed; their regular solutions are obtained. The viscous-inviscid interaction effect predominates in a region whose extent is of the order of the square root of the boundary layer thickness, in which the flow is described by a two-layer model, namely, the Euler equations in the slender-body approximation for the outer region and the three-dimensional boundary layer equations; the pressure is determined from the interaction conditions. On the basis of an analysis of the solutions for the outer part of the boundary layer it is shown that interaction leads to attenuation of the singularities and the dependence of the nature of the flow on the longitudinal coordinate, but does not make it possible to eliminate the singularities completely.     

Criteria for the detachment of a three-dimensional boundary layer

On the assumption that the character of the flow in a specified section of the boundary layer depends solely on the behavior of the external flow in the immediate vicinity of the section under consideration, criteria are derived for the detachment of laminar and turbulent boundary layers in the case of three-dimensional flow. The detachment criteria so found are a consequence of the necessary condition for the detachment of a three-dimensional boundary layer deduced earlier [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 50–54, July–August, 1970.     

On the possibility of blowing a gas jet into a supersonic flow without formation of a three-dimensional boundary-layer separation zone

We consider the flow formed by the interaction of a supersonic flow and a transverse sonic or supersonic jet blown at right angles to the direction of the main flow through a nozzle whose exit section is in a flat wall. When a gas jet is blown through a circular opening [1] the pressure rises in front of the jet because of the stagnation of the oncoming flow. This leads to separation of the boundary layer formed on the wall in front of the blowing nozzle. The resulting three-dimensional separation zone leads to a sharp increase in the pressure and the heat fluxes to the wall in front of the blowing nozzle, which is undesirable in many modern applications. The aim of the present investigation was to find a shape of the exit section of the blowing nozzle for which there is no three-dimensional separation zone of the boundary layer in front of the blowing nozzle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 162–165, May–June, 1979.     

Acoustic Field Effect on Laminar Turbulent Transition on a Swept Wing in the Favorable Pressure Gradient Region

The development of disturbances in a three-dimensional boundary layer on a swept wing model is studied both under natural conditions and for artificial excitation of traveling waves by an acoustic field. It is found that steady-state streamwise structures are formed in the three-dimensional boundary layer; under natural conditions a wave packet leading to turbulence is detected. When the flow is exposed to the action of an acoustic field at a frequency from the wave packet, disturbances whose velocity along the streamwise structures is equal to 0.55 of the oncoming flow velocity are formed, while the laminar-turbulent transition is displaced upstream.     

Mechanism of three-dimensional boundary-layer receptivity

Boundary-layer receptivity is always a hot issue in laminar-turbulent transition. Most actual laminar-turbulent transitions belong to three-dimensional flows. An infinite back-swept flat-plate boundary layer is a typical three-dimensional flow. Study of its receptivity is important both in theory and applications. In this paper, a freestream turbulence model is established. A modified fourth-order Runge-Kutta scheme is used for time marching, and compact finite difference schemes are used for space discretization. On these bases, whether unsteady cross-flow vortices can be excited in the three-dimensional boundary layer(the infinite back-swept flat-plate boundary layer) by free-stream turbulence is studied numerically. If so, effects of the level and the direction of free-stream turbulence on the three-dimensional boundary-layer receptivity are further studied. Differences of the three-dimensional boundary-layer receptivity are then discussed by considering the non-parallel effect, influence of the leading-edge stagnation point of the flat plate, and variation of the back-swept angle separately. Intensive studies on the three-dimensional boundary-layer receptivity will benefit the development of the hydrodynamic stability theory, and provide a theoretical basis for prediction and control of laminar-turbulent transition.     

Uniqueness of the self-similar solutions of three-dimensional laminar boundary-layer equations

The equations of the three-dimensional laminar boundary layer on lines of flow outflow and inflow are studied for conical outer flow under the assumption that the Prandtl number and the productρμ are constant. It is shown that in the case of a positive velocity gradient of the secondary flow (α₁>0) the additional conditions which result from the physical flow pattern determine a unique solution of the system of boundary-layer equations. For a negative velocity gradient of the secondary flow (α₁≤0) these conditions are satisfied by two solutions. An approximate solution is obtained for the boundary layer equations which is in rather good agreement with the numerical integration results. Compressible gas flow in a three-dimensional laminar boundary layer is described by a system of nonlinear differential equations whose solution is not unique for given boundary conditions. Therefore additional conditions resulting from the physical pattern of the gas flow are imposed on the resulting solution. In the solution of problems with a negative pressure gradient these additional conditions are sufficient for a unique selection of the solution of the boundary-layer equations. However, in the case of a positive pressure gradient the solution of the boundary-layer equations satisfying the boundary and additional conditions may not be unique. In particular, in [1] in a study of a three-dimensional laminar boundary layer in the vicinity of the stagnation point it was shown that for $$c = {{\frac{{\partial v_e }}{{\partial y}}} \mathord{\left/ {\vphantom {{\frac{{\partial v_e }}{{\partial y}}} {\frac{{\partial u_e }}{{\partial x}}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial u_e }}{{\partial x}}}} > 0$$ the solution is unique, while for c<0 there are two solutions. In the present paper we study the question of the uniqueness of the self-similar solution of the three-dimensional laminar boundary-layer equations on lines of flow outflow and inflow for a conical outer flow.     

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.     

Three dimensional shock wave/boundary layer interactions

An investigation into a three-dimensional, curved shock wave interacting with a three-dimensional, curved boundary layer on a slender body is presented. Three different nose profiles mounted on a cylindrical body were tested in a supersonic wind tunnel and numerically simulated by solving the Navier–Stokes equations. The conical and hemispherical nose profiles tested were found to generate shock waves of sufficient strength to separate the boundary layer on the cylinder, while the shock wave generated by the ogival profile did not separate the boundary layer. For the separated flow, separation was found to occur predominantly on the windward side of the cylinder with the lee-side remaining shielded from the direct impact of the incident shock wave. A thickening of the boundary layer on the lee-side of all the profiles was observed, and in the conical and hemispherical cases this leads to the re-formation of the incident shock wave some distance away from the surface of the cylinder. A complex reflection pattern off the shock wave/boundary layer interaction (SWBLI) was also identified for the separated flow cases. For comparative purposes, an inviscid simulation was performed using the hemispherical profile. Significant differences between the viscous and inviscid results were noted including the absence of a boundary layer leading to a simplified shock wave reflection pattern forming. The behaviour of the incident shock wave on the lee-side of the cylinder was also affected with the shock wave amalgamating on the surface of the cylinder instead of away from the surface as per the viscous case. Test data from the wind tunnel identified two separation lines present on the cylindrical surface of the hemispherical SWBLI generator. The pair of lines were not explicitly evident in the original CFD simulations run, but were later identified in a high-resolution simulation.     

Three-dimensional coupled numerical model of creeping flow of viscous fluid

A three-dimensional coupled numerical model is developed to describe creeping flow in a computational domain that consists of a thick viscous layer overlaid with a thin multilayered viscous sheet. The density of the sheet is assumed to be lower than that of the layer. The model couples the Stokes equations describing the flow in the layer and the Reynolds equations describing the flow in the sheet. We investigate the long-time behavior of the flow in the sheet by using an asymptotic method and derive an ordinary differential equation for the sheet boundary displacements and the velocities at the interface between the sheet and the layer. The Stokes and Reynolds equations are coupled by applying the resulting equation as an internal boundary condition. Numerical implementation is based on a modified finite element method combined with the projection gradient method. The computational domain is discretized into rectangular hexahedra. Piecewise square basis functions are used. The model proposed enables different-type hydrodynamic equations to be coupled without any iterative improvements. As a result, the computational costs are reduced significantly in comparison with available coupled models. Numerical experiments confirm that the three-dimensional coupled model developed is of good accuracy.