Physical properties of the flow in the separation region for three-dimensional interaction of a boundary layer with a shock wave

In a supersonic stream we consider the three-dimensional flow in the plane of symmetry in the region of interaction of a boundary layer with a shock wave which arises ahead of an obstacle mounted on a plate. The principal characteristic of this flow is the penetration of a filament of the ideal fluid within the separation zone and the formation on the surface of the plate and obstacle of narrow segments with high pressures, high velocity gradients, and large heat transfer coefficients.Pressure distribution measurements were made, shadow and schlieren photos were taken, and photographs of the flow pattern on the surface were made using dye coatings and low-melting models. The basic physical characteristics of the separation flow are established. The independence of the separation zone length of the boundary layer thickness is shown. Local supersonic flows are detected in the separation region, flow regimes are identified as a function of the angle of encounter of the separating flow with the obstacles, characteristic flow zones in the interaction region are identified.Notation s coordinate of separation point on the plate - l length of separation zone - H obstacle height - d obstacle transverse dimension - u freestream velocity - velocity gradient on stagnation line of obstacle - b jet width - compression shock standoff from the body - p static pressure - p_* pressure at stagnation point on obstacle - density - viscosity coefficient - boundary-layer thickness - compression shock angle - effective angle of separation zone - setting angle of obstacle on plate - M Mach number - R Reynolds number - P Prandtl number     

Regime diagram of the development of long-wave near-wall disturbances in a hypersonic boundary layer

A regime diagram of the development of slow near-wall disturbances induced by an unsteady self-induced pressure perturbation in a hypersonic boundary layer is constructed for a disturbance wavelength greater than the boundary layer thickness. It is shown that the main factors shaping the perturbed flow are the gas enthalpy near the body surface, the intensity of the viscous-inviscid interaction, and the nature (sub- or supersonic) of the main part of the boundary layer. Nonlinear boundary-value problems are formulated for regimes in which the near-wall boundary layer region plays a decisive role. Numerical and analytical solutions are obtained in the linear approximation. It is shown that intensification of the viscous-inviscid interaction or an increase in the role of the supersonic main region of the boundary layer impart generally supersonic properties to the main part of the boundary layer, i.e. the upstream propagation of the disturbances is damped and the disturbance growth downstream becomes more intense. Damping of the viscous-inviscid interaction and an increase in the role of the subsonic main part of the boundary layer have the opposite effect. Surface cooling increases the effect of the main part of the boundary layer on the formation of pressure disturbances and surface heating leads to an increase in the effect of the near-wall boundary layer region. It is also shown that for the regimes considered disturbances propagate in a direction opposite to that of the free stream from the turbulent flow region located downstream of the local disturbance development region.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 59–71. Original Russian Text Copyright © 2004 by Bogolepov and Neiland.     

Analysis of turbulent boundary layer interaction with an external supersonic flow behind a step

An integral method of analyzing turbulent flow behind plane and axisymmetric steps is proposed, which will permit calculation of the pressure distribution, the displacement thickness, the momentum-loss thickness, and the friction in the zone of boundary layer interaction with an external ideal flow. The characteristics of an incompressible turbulent equilibrium boundary layer are used to analyze the flow behind the step, and the parameters of the compressible boundary layer flow are connected with the parameters of the incompressible boundary layer flow by using the Cowles-Crocco transformation.A large number of theoretical and experimental papers devoted to this topic can be mentioned. Let us consider just two [1, 2], which are similar to the method proposed herein, wherein the parameter distribution of the flow of a plane nearby turbulent wake is analyzed. The flow behind the body in these papers is separated into a zone of isobaric flow and a zone of boundary layer interaction with an external ideal flow. The jet boundary layer in the interaction zone is analyzed by the method of integral relations.The flow behind plane and axisymmetric steps is analyzed on the basis of a scheme of boundary layer interaction with an external ideal supersonic stream. The results of the analysis by the method proposed are compared with known experimental data.Notation x, y longitudinal and transverse coordinates - X, Y transformed longitudinal and transverse coordinates - , ^*, ^** boundary layer thickness, displacement thickness, momentum-loss thickness of a boundary layer - , ^*, ^** layer thickness, displacement thickness, momentum-loss thickness of an incompressible boundary layer - u, velocity and density of a compressible boundary layer - U, velocity and density of the incompressible boundary layer - , stream function of the compressible and incompressible boundary layers - , dynamic coefficient of viscosity of the compressible and incompressible boundary layers - r₁ radius of the base part of an axisymmetric body - r radius - R transformed radius - M Mach number - friction stress - p pressure - a speed of sound - s enthalpy - v Prandtl-Mayer angle - P Prandtl number - P_t turbulent Prandtl number - r₂ radius of the base sting - b step depth - =0 for plane flow - =1 for axisymmetric flow Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–40, May–June, 1971.In conclusion, the authors are grateful to M. Ya. Yudelovich and E. N. Bondarev for useful comments and discussions.     

Flow behind the boundary layer separation point in a supersonic stream

The flow structure behind the separation point of a laminar boundary layer in a supersonic stream has been investigated. Analytic and numerical solutions are obtained for simple semiinfinite separation zones starting from the leading edge or a point on the smooth surface. The question of the pressure plateau in a separation zone of finite length is discussed and its value is calculated on the basis of asymptotic theory. The asymptotic theory of flow [1, 2] in the neighborhood of the separation point of the laminar boundary layer in a supersonic gas stream (region of free interaction) is employed. The local solution obtained is subsequently used to construct the flow pattern in the separation zone [3]. An analysis is made of the behavior of the solution for the free-interaction region on transition to the region of reverse flows. The results make it possible actually to compute (in the first approximation) the pressure in the plateau region after establishing the mathematical significance of this concept, previously introduced on the basis of the experimental results. At the same time relatively simple solutions are obtained for semiinfinite separation zones.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 19–25, May–June, 1971.     

Boundary-layer separation on a cooled body and interaction with a hypersonic flow

In previous papers, e.g., [1, 2], boundary-layer separation was investigated by analyzing solutions of the boundary-layer equations with a given external pressure distribution. In general, this kind of solution cannot be continued after the separation point. Study of the asymptotic behavior of solutions of the Navier-Stokes equations [3–5] shows that, in boundarylayer separation in supersonic flow over a smooth surface, the main effect on the flow in the immediate vicinity of the separation point is a large local pressure gradient induced by interaction with the external flow. The solution can be continued beyond the separation point and linked to the solutions in the other regions, located downstream [5]. Similar results for incompressible flow were recently obtained in [6]. We can assume that in general there is always a small region near the separation point in which separation is self-induced, and where the limiting solution of the Navier-Stokes equations does not contain unattainable singular points. However, this limiting slope picture can be more complex and can contain more regions where the behavior of the functions differed from that found in [3–6]. The present paper investigates separation on a body moving at hypersonic speed, where the ratio of the stagnation temperature to the body temperature is large. It is shown that both. for hypersonic and supersonic speeds the flow near the separation point is appreciably affected by the distribution of parameters over the entire unperturbed boundary layer, and not only in a narrow layer near the body, as was true in the flows studied earlier [3–6]. Regions may appear with appreciable transverse pressure drops within the zone occupied by layers of the unperturbed boundary layer. Similarity parameters are given, the boundary problems are formulated, and the results of computer calculation are presented. The concept of subcritical and supercritical boundary layers is refined, and the dependence of pressure coefficients responsible for separation on the temperature factor is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 99–109, November–December 1973.     

Axisymmetrical bodies with minimal resistance and with minimal flow toward the surface of the body,with different characters of the flow in the boundary layer

Approximate formulas are obtained which permit calculating the distribution of the friction stress and of the local heat flux over the surface of a body of arbitrary form, with a given pressure distribution, both with laminar and with turbulent flow conditions in the boundary layer. These approximate equations were used to solve the variational problems of determination of the form of axisymmetrical bodies with a minimal resistance or with a minimal total flow of heat to the surface (in the class of bodies consisting of a flat leading edge and a lateral surface) in a hypersonic flow of viscous gas. In the solution of variational problems of determination of the optimal form of a body from the conditions of minimal resistance or of a minimal total heat flux toward the surface, we must be able to determine the distribution of the pressure, the friction stress, and the local heat flux along the surface of a body of arbitrary form. At large Reynolds numbers, the problem of determining the pressure distribution comes down to solving the Euler equation with corresponding boundary conditions. However, at the present time there are no effective methods for solving this problem (at least from the point of view of using the methods for solution of variational problems); in the solution of variational problems, to determine the pressure distribution this forces us to use various approximate methods (for example, the method of tangential wedges or cones, the Newton method, etc.). The use of such approximate formulas renders unfeasible on exact solution of the equations of the boundary layer, for which the distribution of the gas-dynamic parameters at the outer limit of the boundary layer (including also the pressure distribution) must be known previously. This makes it necessary to construct approximate formulas to determine the friction stress, _w, and the local heat flux, q_w, whose accuracy in an arbitrary body will be determined by the accuracy of the assignment of the gas-dynamic parameters at the outer limit of the boundary layer. We give below the derivation of such dependences for laminar and turbulent flow in a boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 94–102, March–April, 1971.     

The Effect of Time-Periodic Surface Temperature Oscillations on Free Convection from a Vertical Surface in a Porous Medium

In this paper, we consider the unsteady free convection boundary layer flow which is induced by time-periodic variations in the surface temperature of a vertical surface embedded in a porous medium. The basic steady flow is that of a power-law distribution where the surface temperature varies as the nth power of the distance from the leading edge. Small-amplitude time-periodic disturbances are added to this basic distribution. Both the low- and high-frequency limits are considered separately, and these are compared with a full numerical solution obtained by using the Keller-box method. Attention is restricted to the cases n1; when n=1, the flow is locally self-similar for any prescribed frequency of modulation.     

Experimental investigation of the influence of the flow structure on the aerodynamic coefficients of the IXV vehicle

In the framework of the ESA Future Launchers Preparatory Program (FLPP) an experimental study on the aerodynamic behavior during the re-entry phase of the Intermediate eXperimental Vehicle (IXV) configuration was conducted in the DLR hypersonic wind tunnel H2K in Cologne. Tests were carried out at Mach 6.0 and 8.7 with different flap deflection angles and the angle of attack varied continuously between 20° and 55° to investigate the flow topology as well as the aerodynamic forces and moments and the surface pressure distribution. The experimental data show that depending on the combination of the flap deflection angle (δ _L/R) and angle of attack (α) the complex flow structure in the vicinity of the flaps significantly influences the vehicle’s aerodynamic coefficients. An analysis of this shock/shock and shock/boundary layer interaction causing flow separation with reattachment is performed.     

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.     

Iterative method for supersonic flow laminar boundary layer interaction in the presence of separation

The classical two-dimensional compressible boundary-layer equations supplemented by a relation describing the interaction of boundary layer with external inviscid flow (see, e.g., [1]) are treated as the governing equations in one of the methods to study the viscous-inviscid interaction. It is then necessary in the case of supersonic flow to specify certain downstream boundary conditions for the closure of the governing system, i.e., it is a boundary-value problem (e.g., [2]). The shooting technique for parameters at the beginning of the computational region to obtain the solution satisfying such a condition usually requires large computer time since the integral curves are highly sensitive to small changes in upstream boundary conditions. A more effective method is the algorithm of global relaxations of pressure distribution along the entire computational region [1]. A numerical method to compute supersonic interacting boundary layer in the presence of separation is presented in this paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 89–93, January–February, 1984.     

On near-wall turbulence-generating events in a turbulent boundary layer on a riblet surface

The boundary layer over a drag reducing riblet surface is investigated using hot-wire anemometry and flow visualisation. The concept of a riblet sublayer is introduced, and a definition is proposed in terms of a region of reduced turbulence energy production formed near the wall by the addition of riblets. The hot wire records are examined using a modified form of quadrant analysis, and results obtained over plain and riblet surfaces are compared. Close to the wall, the addition of riblets produces a marked reduction in the occurrence of ejection (2nd quadrant) events. A corresponding increase in the incidence of sweep (4th quadrant) events is accompanied by the development of a strong tendency toward a preferred event duration, and a preferred interval between events. These changes diminish rapidly with distance from the surface, becoming almost undetectable beyondy ⁺=40. They are discussed in the light of flow visualisation results, and interpreted in terms of mechanisms associated with the interaction between the riblets and the inner boundary layer flow structures. A conceptual model of the flow mechanisms in the riblet sublayer is proposed.     

Asymptotic Solutions of Problems of One-Dimensional Time-Dependent Combustible Gas Flows in the Presence of Thermal Action

In a development of studies [1, 2], asymptotic solutions of the Navier-Stokes equations are found for one-dimensional combustible gas flows in the presence of various forms of thermal action on a moving surface (x=x _w(t)). In the problems considered, the temperature or the heat flux q _w(t) is specified on the surface or the surface is the interface between a combustible gas and a moving heated piston or another gas (for example, in a shock tube). Use is made of the fact that, as t , in many cases the values of v _w=(dx/dt)_w and q _w are bounded. This leads to a steady-state flow in the flame zone in the coordinate system moving with its front and homogeneous uniform flow ahead of and behind it. Solutions of all these problems are given for the burnt-gas boundary layer region adjacent to the surface. The numerical calculations performed confirm the results obtained. A velocity law leading to time invariability of the flow pattern obtained with allowance for the interaction between the boundary layer and the burnt-gas homogeneous flow is found, including in the problem of the breakdown of an arbitrary discontinuity. The results are generalized to include the case of motion at an angle of incidence with an additional velocity component aligned with the surface.     

Asymptotic theory for calculating heat fluxes near the corner of a body

A method is presented for calculating the distribution of the thermal fluxes, friction stresses, and pressure near the corner point of a body contour in whose vicinity the outer supersonic flow passes through an expansion wave. The method is based on a study of the asymptotic solutions of the Navier-Stokes equations as the Reynolds number R approaches infinity for the flow region in which the longitudinal gradients of the flow functions are large, invalidating conventional boundary layer theory. This problem was examined in part in [1], in which the distribution of the friction and pressure in a region with length on the order of a few thicknesses of the approaching boundary layer was obtained in the first approximation. The leading term of the expansion for the thermal flux to the surface of the body vanishes for a value of the Prandtl number equal to unity and for other values of the Prandtl number does not match directly with its value in the undisturbed boundary layer.The thermal-flux distribution is obtained for values of the Prandtl number approaching unity. For this purpose it was necessary to consider a more general double passage to the limit as 1 and 0 for a finite value of the parameter B=[(–1)/] [–ln ^1/4/]^1/4 characterizing the ratio of the effects of thermal conduction, viscous dissipation, and convection. The solution obtained previously [1] corresponds to the particular case B and therefore for actual values of R=10⁴–10⁶, ~ 0.7 overestimates considerably the effect of the dissipative term on heat transfer, although even in first approximation it describes the pressure distribution well and the friction distribution satisfactorily. For smooth matching of the solutions with the corresponding flow functions in the undisturbed boundary layer it was necessary to introduce a flow region with free interaction for the expansion flow. Equations and boundary conditions which describe the flow as a whole are presented. Examples are given of numerical calculations and comparison with experiment.     

Study of laminar separation in the vicinity of the point of piecewise-constant pressure distribution over the wall

A family of two-dimensional divergent channels with piecewise-constant velocity and pressure distributions over the wall is considered. The method of matched asymptotic expansions is applied to study the two-dimensional viscous incompressible flow at high but subcritical Reynolds numbers in the vicinity of a pressure jump point on the channel wall. It is shown that if the pressure difference is of the order O(Re^?1/4), then in the vicinity of this point a classical region of interaction between the viscous boundary layer on the wall and the outer inviscid flow occurs. The problem formulated for the interaction region is solved numerically. The asymptotic values of the pressure difference corresponding to separationless flow are determined and the separation flow patterns are constructed.     

Effect of Boundary-Layer Thickness on the Structure of a Near-Wall Flow with a Two-Dimensional Obstacle

Results of physical and numerical experiments on investigating the effect of the depth of immersion of a two-dimensional obstacle with a square cross section into a developed turbulent boundary layer on the length of the separated flow region are presented. The numerical simulation is based on solving averaged Navier–Stokes equations with the use of the k– model of turbulence. The near-wall flow is visualized in the experiments, and the fields of mean and fluctuating velocities are measured. Flow regions where the results of numerical simulation agree with experimental data are determined. It is shown that the length of the recirculation flow region in the near wake increases with decreasing depth of immersion of the two-dimensional obstacle into the turbulent boundary layer.     

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.     

The Laminar Boundary Layer over a Permeable Wall

An analysis is given of the laminar boundary layer over a permeable/porous wall. The porous wall is passive in the sense that no suction or blowing velocity is imposed. To describe the flow inside and above the porous wall a continuum approach is employed based on the Volume-Averaging Method (S. Whitaker The method of volume averaging). With help of an order-of-magnitude analysis the boundary-layer equations are derived. The analysis is constrained by: (a) a low wall permeability; (b) a low Reynolds number for the flow inside the porous wall; (c) a sufficiently high Reynolds number for the freestream flow above the porous wall. Two boundary layers lying on top of each other can be distinguished: the Prandtl boundary layer above the porous wall, and the Brinkman boundary layer inside the porous wall. Based on the analytical solution for the Brinkman boundary layer in combination with the momentum transfer model of Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38 (1995) 2635). for the interface region, a closed set of equations is derived for the Prandtl boundary layer. For the stream function a power series expansion in the perturbation parameter is adopted, where is proportional to ratio of the Brinkman to the Prandtl boundary-layer thickness. A generalization of the Falkner–Skan equation for boundary-layer flow past a wedge is derived, in which wall permeability is incorporated. Numerical solutions of the Falkner–Skan equation for various wedge angles are presented. Up to the first order in wall permeability causes a positive streamwise velocity at the interface and inside the porous wall, but a wall-normal interface velocity is a second-order effect. Furthermore, wall permeability causes a decrease in the wall shear stress when the freestream flow accelerates, but an increase in the wall shear stress when the freestream flow decelerates. From the latter it follows that separation, as indicated by zero wall shear stress, is delayed to a larger positive pressure gradient.     

Experimental and Numerical Study of a Hypersonic Separated Flow in the Vicinity of a Cone‐Flare Model

An axisymmetric laminar separated flow in the vicinity of a coneflare model is studied experimentally and numerically for a Mach number M = 6. The distributions of pressure and Stanton numbers along the model surface and velocity profiles in the region of shock wave–boundary layer interaction are measured and compared with the calculated data. The influence of the laminar–turbulent transition on flow parameters is studied numerically.     

Wall pressure fluctuations in the reattachment region of a supersonic free shear layer

A study was made of the wall pressure fluctuations in the reattachment region of a supersonic free shear layer. The free shear layer was formed by the separation of a Mach 2.9 turbulent boundary layer from a backward facing step. Reattachment occurred on a 20° ramp. By adjusting the position of the ramp, the base pressure at the step was set equal to the freestream pressure, and the free shear layer formed in the absence of any turning. An array of flush-mounted, miniature, high-frequency pressure transducers was used in the vicinity of the reattachment region to make multichannel measurements of the fluctuating wall pressure. Contrary to previous observations of this flow, the reattachment region was found to be highly unsteady, and the pressure fluctuations were found to be large. The overall behavior of the wall pressure loading is similar in scale and magnitude to the unsteadiness of the wall pressure field in compression ramp flows at the same Mach number. Rayleigh scattering was used to visualize the instantaneous shock structure in the streamwise and spanwise direction. Spanwise wrinkles on the order of half the boundary layer thickness were observed on the shock sheet.