Conical gas flows with shock waves and turbulent boundary layer separation

The results of a theoretical and experimental investigation of nonsymmetric flow around a V-wing with supersonic leading edges are presented. The range of the angles of attack and yaw, on which additional singular lines are formed on the windward cantilever of the wing, are experimentally determined using different techniques of flow diagnostics. These are one convergence line and two divergence lines in the transverse flow which were not previously observable in the calculations of ideal-gas flow around wings. It is established that the appearance of the three new alternating singular lines located between the central chord of the wing and a convergence line, exterior to them and occurring within the framework of the ideal gas model, is associated with the relation between the intensities of two contact discontinuities. One of these proceeds from the branching point of the bow shock above the leeward cantilever, while the second issues out of the triple point of a λ-shaped shock configuration accompanying developed turbulent-boundary-layer separation generated by an internal shock incident on the leeward cantilever surface. If the intensity of the contact discontinuity proceeding from the branching point of the bow shock is large as compared with that of the contact discontinuity of the λ-configuration, then the flow pattern realized on the windward cantilever is analogous to that obtained within the framework of the ideal gas model, that is, it includes one convergence line on the wing surface. Under these conditions, the results of the calculations within the framework of the ideal gas model are applicable for understanding the phenomena occurring in the wing shock layer in a considerable part of the control parameter range, including the regimes with intense internal shocks generating turbulent boundary layer separation from the leeward cantilever. Corrections should be made only for a carachteristic pressure distribution in the separation zone and, as a consequence of separation, for an elevated pressure level in the vicinity of the central chord which is the stagnation line of the transverse flow that has passed across the oblique and terminating shocks of the λ-configuration and possesses a higher stagnation pressure than the flow that has passed in an ideal gas across the internal shock incident normally on the leeward cantilever. This is possible only when the divergence line, at which the stream surface enclosing the turbulent boundary layer separation zone enters, does not go over from the leeward onto the windward cantilever.     

Submerged shocks in conical gas flows in the presence of a mach shock-wave configuration

The results of a numerical study of a new type of singularities in the Mach shock-wave structure realized in supersonic nonsymmetric conical flows over V-wings with a bow shock attached to the leading edges are presented. Within the framework of the ideal gas model we study the changes in the shock system on transition, with increase in the sweep angle, from the region of nonsymmetric Mach interaction of the shocks attached to the leading edges of the wing to the region of special flow patterns, where on the windward cantilever surface a rarefaction flow is realized rather than a flow with an internal shock. It is shown, in particular, that in the region with special wing flow patterns a Mach system of shocks with a submerged shock proceeding from the branch point above the windward cantilever may exist.     

Supersonic Flow Around a V-Shaped Wing at Incidence and Yaw

A solution of the problem of the flow around a V-wing with supersonic leading edges at low angles of attack and yaw is obtained within the framework of the linear theory. Possible patterns of nonsymmetric flow around the wing are analyzed as functions of the wing geometry and the freestream velocity direction, and the ranges of angles of attack and yaw on which these patterns are realized are established. Some previously undescribed shock wave configurations are found to exist in the wing-induced conical flows.     

Inviscid vortex structures in the shock layers of conical flows around V wings

The applicability of the criteria of existence of inviscid vortex structures (vortex Ferri singularities) is studied in the case in which a contact discontinuity of the corresponding intensity proceeds from the branching point of the λ shock wave configuration accompanying turbulent boundary layer separation under the action of an inner shock incident on the leeward wing panel. The calculated and experimental data are analyzed, in particular, those obtained using the special shadow technique developed for visualizing supersonic conical streams in nonsymmetric, Mach number 3 flow around a wing with zero sweep of the leading edges and the vee angle of 2π /3. The applicability of the criteria of existence of inviscid vortex structures is established for contact discontinuities generated by the λ shock wave configuration accompanying turbulent boundary layer separation realized under the action of a shock wave incident on the leeward wing panel. Thus, it is established that the formation of the vortex Ferri singularities in a shock layer is independent of the reason for the existence of the contact discontinuity and depends only on its intensity.     

The problem of supersonic flow over the lower surface of a triangular wing

In formulating the problem we make no assumption of smallness of the angle of attack; the attached three-dimensional compression shock which arises under the lower surface of the wing may be of arbitrary intensity, and in form is assumed to differ little from a plane shock; a finite yaw angle is allowed. We consider linear supersonic conical flow which is realized, with the exception of a characteristic linear dimension, in the portion of space bounded by the shock, the plane of the wing, and the surface of a disturbance cone with vertex at the discontinuity of the supersonic leading edge and which is a disturbance of the uniform flow behind the plane shock wave.The problem studied reduces to the homogeneous Hilbert boundary-value problem for an analytic function of a complex variable, whose real and imaginary parts are the partial derivatives of the unknown pressure disturbance with respect to the similarity coordinates.In the solution of the boundary-value problem, the effective method of Lighthill, developed with application to diffraction problems [1, 2], is generalized to the problem of an asymmetric region.The particular case of hypersonic flow about an unyawed triangular wing has been studied by Malmuth [3]; the author obtains the problem considered by Lighthill in [2] and writes out the solution contained in that work.     

The flow structure near the windward side of V-shaped wings with an attached shock wave on the leading edges

The results of a numerical calculation of a symmetric flow of supersonic gas with the Mach number M=3 past the windward side of V-shaped wings with an opening angle =40° and apex angles =30, 45, and 90° are given. The possibility of the ascent of one or two Ferri points from the break point of the transverse contour of the wing is discovered and explained. It is shown that conical flow near wings of finite length need not exist in flow regimes corresponding to angles of attack at which a Ferri point ascends, while at angles of attack smaller and larger than a certain interval, conical flow will exist. The investigation is conducted by means of a numerical method of stabilization with an artificial viscosity. The longitudinal coordinate, relative to which the steady system of equations is hyperbolic, played the part of the time variable, usual for methods of stabilization. The numerical method constructed using the scheme of [1] is described in [2] and was successfully applied to the calculation of different regimes of supersonic flow past conical wings with supersonic leading edges [2–6]. In. the present investigation the calculation algorithm of [2] is modified and makes it possible to realize motion with respect to the parameter a, this being particularly important for the stabilization of the solution in the calculation of flow regimes for which regions with a total velocity Mach number close to unity arise in the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–131, January–February, 1986.     

Aerodynamic Characteristics of Slender Star-Shaped Bodies with Supersonic Leading Edges

A theory of the flow around slender conical bodies with a star-shaped cross-section in regimes with shock attachment along the length of the leading edges is developed. The aerodynamic characteristics of the slender star body, a single cycle of which represents a V-shaped wing, are obtained for different numbers of reflections of the disturbances proceeding from the leading edges of the cycles from the wing cantilevers, using superposition of the solutions of linear problems of flow around the V-wing.     

Hypersonic flow past a wing at large angles of attack with detached shock wave

The thin shock layer method [1–3] has been used to solve the problem of hypersonic flow past the windward surface of a delta wing at large angles of attack, when the shock wave is detached from the leading edge (but attached to the apex of the wing) and the velocity of the gas in the shock layer is of the same order as the speed of sound. A classification of the regimes of flow past a delta wing at large angles of attack has been made. A general solution has been obtained for the problem of three-dimensional hypersonic flow past the wing allowing for nonequilibrium physicochemical processes of thermal radiation of the gas at high temperatures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 149–157, May–June, 1985.     

Vortex structure of separated flows on model wings at low freestream velocities

Flow past model wings is experimentally investigated in a subsonic wind tunnel at large angles of attack at which the laminar boundary layer separates near the leading edge of the wing (flow stall). The object of the study was the flow structure within the separation zone. The carbon-oil visualization of surface streamlines used in the experiments showed that in the separation zone there exist one or more pairs of large-scale vortices rotating in the wing plane. Certain general properties of the vortex structures in the separation zone are found to exist, whereas the flow patterns may differ depending on the model aspect ratio, the yaw angle, and other factors.     

Separated flow over a strongly cambered conical wing at small angles of attack

The interference of supersonic flows on the concave surface of conical wings was experimentally investigated in [1] for various values of the camber and angles of attack. In order to establish the detailed structure of the interference flow the laminar flow past a wing model in the form of half the surface of a circular cone with vertex angle 2_k = 34° was numerically modeled within the framework of the quasiconical approximation for the three-dimensional Navier-Stokes equations [2]. Under this assumption, confirmed by analysis of the experimental data [1], it was found that the displacement of the external inviscid flow as a result of intense flow separation beyond the leading edges leads to flow patterns similar to those realized on V wing's with a break in the transverse contour [3]. At nonzero angles of attack weak secondary separation was detected beneath the flattened regions of primary separation located in the shaded parts of the concave surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 130–136, July–August, 1989.     

Supersonic flow past circular cones at large angles of attack

Flow past sharp-nosed circular cones is investigated for a broad range of freestream Mach numbers M>1 and cone half-angles _c at angles of attack from zero to the value at which conical flow breaks down. Several new results are obtained with regard to the position of the Ferri point, the shape of the local supersonic zones and internal shock wave, and the nonmonotonicity of the windward shock slope as a function of the angle of attack. The existence of flow regimes in which the radial velocity on the windward side is directed toward the apex of the cone is demonstrated. The investigation is carried out numerically with relaxation of the solution in a fictitious time coordinate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 6, pp. 79–84, November–December, 1973.     

Numerical analysis of heat transfer on the surface of a delta wing in a hypersonic air flow at large angles of attack

The heat transfer on a delta wing with blunt edges and various catalytic surface properties in a hypersonic air flow at 40 ° and 60 ° angles of attack has been investigated by a numerical flow model for a viscous reacting gas in the shock layer near the windward side of blunt elongated bodies.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 196–199, September–October, 1984.     

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.     

Hypersonic flow past conical V-shaped wings

Hypersonic flow past delta wings with a V-shaped cross-section has been investigated both theoretically and experimentally. Much attention is given to the examination and classification of possible conical flow patterns in the vicinity of the windward surface using the thin shock layer method. Solutions for shocks both attached to and detached from the leading edges them are obtained. It is shown that qualitatively new flow patterns can appear in the flow past V-shaped wings as compared with the case of a planar wing.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 170–178, September–October, 1996.     

Supersonic Flow Past a Bent Delta Wing and Its Aerodynamic Characteristics

The supersonic flow past a bent delta wing and its aerodynamic characteristics, both local and integral, are studied using approximate analytic estimates and numerical calculations. The flow regimes with the shock attached to the leading edges are considered. It is established that at M = 4–6 and angles of attack as high as 6° the lift-drag ratio of the wing can be increased by 10% by deflecting the wing nose downwards. This is confirmed by the results obtained earlier in the hypersonic thin-shock-layer approximation. The effect is also obtained in calculations made within the framework of the Navier-Stokes equations.     

Steady Streamwise Structures in the Boundary Layer on a Swept Wing with Elevated Turbulence of the Incoming Flow

Results for a turbulized flow past the windward side of a swept wing model are presented. Origination of steady disturbances in the form of streamwise structures is found. The greatest effect on the formation of these disturbances is exerted by the curvature of the external flow streamlines. The secondary flow in the boundary layer leads to an increase in the characteristic scale of disturbances in the transverse direction, as compared to the flow around the model at a zero yaw angle.     

Entropy effects in hypersonic flow over blunt delta wings

The problem of hypersonic flow over blunt delta wings is considered. It is shown that in the case of large wing lengths x -100, where x is the longitudinal coordinate measured in blunt nose radii, extremal flow regimes characterized by an essentially nonuniform distribution of the gas dynamic parameters (density, entropy, Mach number) may be realized in the shock layer near the windward surface of the wing. The location of the zones of flow convergence or divergence on the surface of a delta wing with sweep angle x=75° is established. For the same wing the ranges of Mach numbers M and angles of attack leading to extremal flow regimes are indicated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, 178–181, March–April, 1991.     

Prediction of asymmetric vortical flows around slender bodies using Navier-Stokes equations

Steady and unsteady asymmetric vortical flows around slender bodies at high angles of attack are solved using the unsteady, compressible, this-layer Navier-Stokes equations. An implicit, upwind-biased, flux-difference splitting, finite-volume scheme is used for the numerical computations. For supersonic flows past point cones, the locally conical flow assumption has been used for efficient computational studies of this phenomenon. Asymmetric flows past a 5° semiapex-angle circular cone at different angles of attack, free-stream Mach numbers, and Reynolds numbers has been studied in responses to different sources of disturbances. The effects of grid fineness and computational domain size have also been investigated. Next, the responses of three-dimensional supersonic asymmetric flow around a 5° circular cone at different angles of attack and Reynolds numbers to short-duration sideslip disturbances are presented. The results show that flow asymmetry becomes stronger as the Reynolds number and angles of attack are increased. The asymmetric solutions show spatial vortex shedding which is qualitatively similar to the temporal vortex shedding of the unsteady locally conical flow. A cylindrical afterbody is also added to the same cone to study the effect of a cylindrical part on the flow asymmetry. One of the cases of flow over a cone-cylinder configuration is validated fairly well by experimental data.     

The nature of the triple point singularity in the case of stationary reflection of weak shock waves

The structure of flow in the vicinity of a triple point in the problem of stationary irregular reflection of weak shock waves is numerically investigated within the framework of the Euler model, including the von Neumann paradox range. To improve the accuracy of the solution near singular points a new technology including a grid contracted toward the triple point and the discontinuity fitting is applied. It is shown that in the four-wave flow pattern the curvatures of the tangential discontinuity and the Mach front at the triple point are finite. The singularity is concentrated only in a sector between the reflected wave front and the expansion fan. When the three-wave flow pattern is realized, the curvatures of the tangential discontinuity and both wave fronts at the triple point are infinite. On the range of weak and moderate shock waves the logarithmic singularity in subsonic sectors near the triple point conserves up to transition to the regular reflection.     

Experimental study of flow about flat delta wings at m=5 and angles of attack from 0 to 70°

Results are presented of an experimental study of the flow about flat delta wings with sharp and blunt leading edges and sweep angles =60, 70, 80° for a freestream Mach number of 5 in the range of angles of attack from 0 to 70°. Comparison of the experimental data with calculation results indicated good agreement. The patterns of the limiting streamlines on the body surface obtained with the aid of special paints indicate that the calculated design flow scheme is realized in these regimes.