Investigation of pre-separation boundary layer flows on the basis of various algebraic models of turbulence

The problem of the three-dimensional incompressible turbulent boundary layer developing ahead of a circular cylinder mounted at right angles on a flat plate is considered. The direction of the uniform approach stream is normal to the leading edge of the plate. The turbulence is simulated by means of five different isotropic algebraic models of eddy viscosity. The boundary layer equations are solved numerically by means of a second-order-accurate implicit finite-difference method. The principal characteristics of the flow obtained on the basis of the turbulence models selected are compared for a free-stream Reynolds number Re = 10⁷.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 36–43, November–December, 1991.     

Laminar boundary layer on blunt bodies with account for vorticity of the external flow

This paper presents the technique for and results from numerical calculations of the hypersonic laminar boundary layer on blunted cones with account for the vorticity of the external flow caused by the curved bow shock wave. It is assumed that the air in the boundary layer is in the equilibrium dissociated state, but the Prandtl number is assumed constant, =0.72. The calculations were made in the range of velocities 3–8 km/sec, cone half-angles _k=0°–20°. With account for the vortical interaction of the boundary layer with the external flow, the distribution of the thermal flux and friction will depend on the freestream Reynolds number (other conditions being the same). In the calculations the Reynolds number R, calculated from the freestream parameters and the radius of the spherical blunting, varies from 2.5·10³ to 5.10⁴. For the smaller Reynolds numbers the boundary layer thickness on the blunting becomes comparable with the shock standoff, and for R<2.5·10³ it is apparent that we must reconsider the calculation scheme. With R>5·10⁴ for cones which are not very long the vortical interaction becomes relatively unimportant. The results of the calculations are processed in accordance with the similarity criteria for hypersonic viscous gas flow past slender blunted cones [1, 2].     

Occurrence of Tolman-Schlichting waves in the boundary layer under the effect of external perturbations

Under small external perturbations, the initial stage of the laminar into turbulent flow transition process in boundary layers is the development of natural oscillations, Tolman-Schlichting waves, which are described by the linear theory of hydrodynamic stability. Subsequent nonlinear processes start to appear in a sufficiently narrow band of relative values of the perturbation amplitudes (1–2% of the external flow velocity) and progress quite stormily. Hence, the initial linear stage of relatively slow development of perturbations is governing, in a known sense, in the complete transition process. In particular, the location of the transition point depends, to a large extent, on the spectrum composition and intensity of the perturbations in the boundary layer, which start to develop according to linear theory laws, resulting in the long run in destruction of the laminar flow mode. In its turn, the initial intensity and spectrum composition of the Tolman-Schlichting waves evidently depend on the corresponding characteristics of the different external perturbations generating these waves. The significant discrepancy in the data of different authors on the transition Reynolds number in the boundary layer on a flat plate [1–4] is probably explained by the difference in the composition of the small perturbing factors (which have not, unfortunately, been fully checked out by far). Moreover, it is impossible to expect that all kinds of external perturbations will be transformed identically into the natural boundary-layer oscillations. The relative role of external perturbations of different nature is apparently not identical in the Tolman-Schlichting wave generation process. However, how the boundary layer reacts to small external perturbations, under what conditions and in what way do external perturbations excite Tolman-Schlichting waves in the boundary layer have practically not been investigated. The importance of these questions in the solution of the problem of the passage to turbulence and in practical applications has been emphasized repeatedly recently [5, 6], Only the first steps towards their solution have been taken at this time [4, 7–10], Out of all the small perturbing factors under the real conditions of the majority of experiments to investigate the flow stability and transition in the case of smooth polished walls, three are apparently most essential, viz.: the turbulence of the external flow, acoustic perturbations, and model vibrations. In principle, all possible mechanisms for converting the energy of these perturbations into Tolman-Schlichting waves can be subdivided into two classes (excluding the nonlinear interactions which are not examined here): 1) distributed wave generation in the boundary layer; and 2) localized wave generation at the leading edge of the streamlined model. Among the first class is both the possibility of the direct transformation of the external flow perturbations into Tolman-Schlichting waves through the boundary-layer boundary, and wave excitation because of the active vibrations of the model wall. Among the second class are all possible mechanisms for the conversion of acoustic or vortical perturbations, as well as the vibrations of the streamlined surface, into Tolman-Schlichting waves, which occurs in the area of the model leading edge.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 5, pp. 85–94, September–October, 1978.     

Investigation of the influence of the unit Reynolds number on the transition of a boundary layer on a sharp cone

To establish the influence of the unit Reynolds number on the transition of a boundary layer on the side surface of a cone, the transition was investigated on a model of a sharp cone with half-angle = 7.5 ° and lengths from 150 to 400 mm. The experiments were made in a shock tube at Mach number M = 6.1 in the wide range of Reynolds numbers Re_eL = 1.3·10⁶-5.5·10⁷. The position of the transition region was determined from the results of measurement of the local heat flux by calorimetric thermocouple converters. Data were obtained on the influence on the transition of the unit Reynolds number at large values. It was also shown that under the investigated conditions the base region does not influence the transition of the boundary layer on the surface of the cone.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 32–38, July–August, 1982.     

Structure of the flow of a viscous gas near the trailing edge of a flat plate

Solution of the complete system of Navier-Stokes equations forms the basis for a study of the nature of flow of a viscous heat-conducting gas in the neighborhood of a trailing edge of a flat plate. The problem was solved in accordance with a difference scheme of the third order of accuracy [1]. The calculation was carried out under the same conditions as the experiment of [2], in which a plate of finite dimensions (L = 12 cm) had supersonic M = 2, Re, = 1000 gas flow round it. In order to obtain a thickness of the boundary layer which was acceptable for the purpose of making the measurements (of the order of 2 cm), the unperturbed gas was slightly rarefied. In the study of such problems [3–7] it is necessary to use the complete system of Navier-Stokes equations, since in the immediate neighborhood of the trailing edge one of the important assumptions in the theory of the boundary layer, ²u/y^{2 2}u/x², does not hold. As a result the flow upstream near the trailing edge of the plate will depend on the flow immediately behind the edge, since the perturbations propagate both upstream and downstream in this case. The rarefaction of the gas creates additional difficulties in the formulation of the boundary conditions on the plate with flow round it when this problem is studied numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 27–30, March–April, 1987.     

Effect of nonisothermality of the surface of a thin profile on the stability of a laminar boundary layer

Investigations of the stability of a subsonic laminar boundary layer have shown that, other things being equal, the stability of the laminar flow is considerably improved by cooling the entire surface of the body to a constant temperature T_w=const lower than the temperature of the free stream [1–3]. This is attributable to an increase in the critical Reynolds number of loss of stability and a decrease in the range of unstable perturbations of the Tollmien-Schlichting wave type when the surface is cooled. Recently, in the course of investigating the stability of laminar flow over a flat plate it was found [4, 5] that a similar improvement in flow stability can be achieved by raising the temperature of a small part of the surface near the leading edge of the plate. In this study we examine the possibility of delaying the transition to turbulent flow by creating a nonuniform temperature distribution along the surface of thin profiles, where the development of an adverse pressure gradient in the outer flow has a destabilizing effect on the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 36–42, September–October, 1986.In conclusion, the authors wish to thank M. N. Kogan for useful discussions of their results.     

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.     

Cone in a supersonic flow near a surface with a turbulent boundary layer

The published information about the interaction of incident shocks and a turbulent boundary layer relate to cases of a thin boundary layer ( 1–3 mm) on a flat surface. The present study relates to supersonic flow with Mach number M = 3 and stagnation pressure p₀=1.2 MPa past cones near a surface with a thick boundary layer formed on a plate abutting the lower edge of a plane nozzle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 177–180, July–August, 1991.     

Interaction between an Inclined Shock and Boundary and High-Entropy Layers on a Flat Plate

The flow structure and heat exchange in the zone of interference between an inclined shock and the surface of a flat plate are investigated experimentally and theoretically as functions of many parameters, the interference being studied in both the presence and the absence of bluntness of the leading edge. The experiments were carried out at Mach numbers M = 6, 8, and 10 and the Reynolds numbers Re _L , calculated using the plate length L = 120 mm and the free-stream parameters, varied over the range from 0.24 ? 10⁶ to 1.31 ? 10⁶. The bluntness radius of the leading edge of the plate, the intensity of the impinging shock, and its location with respect to the leading edge were varied. The numerical simulation was carried out by solving the complete two-dimensional Navier-Stokes equations and averaged Reynolds equations using the q-ω turbulence model. The laminar boundary layer became turbulent inside the separation zone induced by the shock. It is shown that the plate bluntness significantly reduces the heat exchange intensity in the interference zone, this effect intensifying with increase in the Mach number.     

Some supersonic flow regimes on the windward side of V-shaped wings

The conditions of realization of regimes, detected in ideal gas theory [1, 2], with a floating Ferri point on the windward side of a wing with supersonic leading edges and breakdown of the conical flow in the presence of turbulent boundary layer separation are studied using experimental data on the flow over conical V-shaped wings. The experiments were carried out on three models of V-shaped wings with sharp leading edges having a convergence angle=40°, apex angles=30, 45, and 90° and lengths along the central chordL=100, 100, and 70 mm, respectively. The free-stream Mach numberM =3, and the unit Reynolds number Re=1.6 ·10⁸ m^–1. Boundary layer transition took place 10 mm from the leading edges of the models at a local Reynolds number Re=(1.5–2)·10⁶. Thus, on most of the wing surface the inner shock waves interacted with a turbulent boundary layer. In the experiments we employed; optical methods, which made it possible to observe shadow flow patterns in a plane normal to the rib of the V-shaped wing [3], as well as in the wake behind the wing and its leading edges (Töpler schlieren method); the oil-film visualization method for obtaining data on the position and dimensions of the separation zones and limiting streamline patterns on the surface of the model. The pressure distribution over the wing span was recorded by means of an automated data collection and processing system based on IKD6TD transducers. The errors of the pressure measurements did not exceed 1 %.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 137–150, March–April, 1992.     

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.     

Method of singular perturbations in the problem of free convection with constant heat flux on a vertical surface

The process of laminar free convection for medium values of the Grashof number is examined. An asymptotic solution which accounts for the effect of the leading edge and interaction of the boundary layer with the external flow is constructed, A comparison with experimental data shows that the solution obtained is applicable for Ra 10².Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 129–136, March–April, 1977.     

Two-dimensional interaction between an incident shock and a turbulent boundary layer in the presence of an entropy layer

The results of an experimental and numerical investigation of flow and heat transfer in the region of the interaction between an incident oblique shock and turbulent boundary layers on sharp and blunt plates are presented for the Mach numbers M_∞ = 5 and 6 and the Reynolds numbers Re_∞L = 27×10⁶ and 14×10⁶. The plate bluntness and the incident shock position were varied. It is shown that the maximum Stanton number St _m in the shock incidence zone decreases with increase in the plate bluntness radius r to a certain value and then varies only slightly with further increase in r. In the case of a turbulent undisturbed boundary layer heat transfer is diminished with increase in r more slowly than in the case of a laminar undisturbed flow. In the presence of an incident shock the bluntness of the leading edge of the flat plate results in a greater decrease in the Stanton number than in the absence of the shock. With increase in the bluntness of the leading edge of the plate the separation zone first sharply lengthens and then decreases in size or remains constant.     

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.     

Influence of the initial conditions on axisymmetric jets in a parallel flow

An experimental investigation was made of the initial-section flow of axisymmetric helium, air, and freon-12 jets in a parallel air flow for two different velocity profiles at the nozzle exit near the boundary of the jet. In one case, the velocity profile was determined by boundary layers on the nozzle walls; in the other case, it was produced artificially by means of a honeycomb of tubes of variable length. Measurements were made of the profiles of the mean and the pulsation velocity and the temperature. The flow was also photographed. The investigations showed that, depending on the initial conditions, the intensity of mixing of the jets in the initial section at Reynolds numbers Re 10⁴ (calculated using the jet diameter) can change from the level determined by molecular diffusion to the level characteristic of developed turbulent flow. The flow structure in the annular mixing layer also depends strongly on the initial conditions. The observed ordered structures in the mixing layer are related to a section of development of perturbations near the nozzle. The ordered structures are strongly influenced by the effect on the jet of acoustic vibrations from an external source. When the initial velocity profile is produced by the honeycomb, the transition to developed turbulence may be due to the development of long-wavelength perturbations or to the development of small-scale turbulence generated by the flow over the end of the honeycomb.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 18–24, July–August, 1980.We thank V. M. levlev and K. I. Artamonov for assistance and for discussing the work.     

Nonmonotonic relationship between the Reynolds number and the lift coefficient of a plate in a hypersonic rarefied gas flow

The aerodynamic coefficients of a plate in a hypersonic stream at low Reynolds numbers are studied over a wide range of similarity parameters. The dependence of the lift coefficientC _Y on the tangential force coefficient, the finite Mach number at the outer edge of the boundary layer and the velocity-slip and temperature-jump boundary conditions is taken into consideration. The nonmonotonic character of the relationship betweenC _Y and the Reynolds number, revealed previously in experiments, is explained within the framework of the viscous hypersonic interaction model.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 186–189, January–February, 1996.     

Stability of the boundary layer of a non-Newtonian liquid obeying a power-type rheological law

The stability of the stationary (steady-state) laminar boundary layer of a non-Newtonian liquid obeying a power-type rheological law at a semiinfinite plate situated in a longitudinal flow is analyzed. An approximate formula is derived for estimating the minimum Reynolds number at which the flow loses stability with respect to slight two-dimensional perturbations. Calculations of the point of stability loss for aqueous solutions of carboxyl methyl cellulose are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 121–124, March–April, 1971.     

Flow Structure in the Separation Zone in the Interaction between Shock Waves and the Boundary Layer on a Plate with Slip

A model which makes it possible to calculate the reverse-flow parameters in the separation zone is constructed on the basis of the results of an integrated experimental study of the characteristics of the separated flow developed in the transition from free to non-free interaction between plane shock waves and the boundary layer on a plate with slip. The effect of the Mach number of the reverse flow in the separation zone on the properties of inner boundary layer separation is analyzed. Features of the interference flow due to boundary layer transition are described. The present study is a continuation of investigations [1–3] devoted to the study of a new steady-state type of interaction between shock waves and the boundary layer on a plate with slip in which the separation line formed would propagate upstream beyond the sharp leading edge if no leading edge was present, i.e., so-called non-free interaction.     

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.     

Supersonic boundary layer transition induced by roughness on the attachment line of a yawed cylinder

The effect of a single two-dimensional irregularity and sandy roughness on boundary layer transition in supersonic flow over a yawed cylinder (M = 6)-has been experimentally investigated. The characteristic flow regimes beyond the roughness are identified, and their limits are determined as a function of the Reynolds number and the ratio of the height of the roughness to the characteristic thickness of the boundary layer. A qualitative comparison is made with the flow regimes induced by roughness on the attachment line in incompressible flow over a cylinder [1–3]. The thermal indicator coating method is used to measure the heat fluxes along the attachment line and a comparison is made with calculations carried out in accordance with the methods of other authors.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 28–35, November–December, 1991.The authors are grateful to A. F. Kiselev for helping to calculate the heat flux in the turbulent boundary layer.