M. A. Zubin

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.

Theoretical and experimental investigation into the structure of supersonic flow past bodies of star-shaped form and their aerodynamic characteristics

The results are given of a theoretical and experimental investigation into supersonic flow over bodies with star-shaped transverse section and flat faces having an equivalent circular cone of elongation 1.3 as a function of the number of “petals” of the star-shaped body and the interior radius at its midsection. Data are given on the coefficient of wave drag of such bodies, and the total drag calculated using a semiempirical theory is compared with the results of weight measurements.

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 angleγ=40°, apex anglesβ=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 %.

Structure of supersonic inviscid nonsymmetric conical flow around a V-wing

The shock wave structure of flow around a V-wing and its properties determining the conical flow topology are numerically investigated within the framework of the inviscid gas model on a wide range of the angles of attack and yaw when in the disturbed supersonic flow either nonsymmetric Mach interaction between the shocks attached to the leading edges of the wing or a shockless flow in the compressed layer on the windward cantilever is realized. The subranges of the angles of attack and yaw with the disturbed flow properties characteristic of the wing of the given geometry are determined. It is found that at high angles of attack, when the branching point of the bow shock beneath the leeward cantilever generates an intense contact discontinuity, the structure of the conical flow in the shock layer on the windward cantilever involves a singularity of a new type which can be characterized as a “vortical” Ferri singularity. It is located above the point of convergence of the streamlines proceeding from the leading edges of the wing, at the vertex of the corresponding contact discontinuity. Flow patterns with the point of convergence of the streamlines proceeding from the leading edges located in the elliptical flow region, which is placed at a local maximum of the pressure distribution over the surface are also found. The range of the angles of attack and yaw on which this new property of supersonic conical flows is realized in the presence of a branched shock system is determined.

Geometrical characteristics of the separation of a turbulent boundary layer in the case of interaction with a normal shock wave in conical flows

Published experimental data [1–5] are used to analyze the influence of the determining parameters on the size of the separation region formed when a normal shock wave impinges on a turbulent boundary layer in conical flows. Empirical dependences are proposed that make it possible to calculate the size of the region and its position relative to the incident shock wave or the direction of the undisturbed flow.

Simulation of the aerodynamic drag of three-dimensional star-shaped bodies at hypersonic velocities

The domain of the parameters in which the aerodynamic drag of hypersonic pyramidal bodies, whose wave component is calculated within the framework of conical flows with the boundary layer displacement thickness taken into account, agrees satisfactorily with the experimental data is found. The calculation model is also applicable in the region of minimum aerodynamic drag of star-shaped bodies in the class of conical bodies equivalent in length and mid-sectional area.

Flow structure on the windward side of V-wings with separation of the turbulent boundary layer

The qualitative characteristics of shock-layer flow associated with separation of the turbulent boundary layer under the influence of one or several successive shock waves are explored with reference to a number of examples typical of 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.

Conditions of the existence of vortex Ferri singularities in supersonic conical flows

The parameters determining the properties of the branching points on shock waves in conical flows and responsible for the existence of inviscid vortex structures in shock layers are established. The quantitative data for these parameters corresponding to the generation of the vortex Ferri singularities are presented. They are based on the results of an analysis of the numerical calculations of inviscid flow around V-wings of different geometries at the Mach numbers from the 3 ≤ M ≤ 10 range with the bow shock corresponding to the Mach-type interaction between the shocks attached to the leading edges. It is shown that at about M ≤ 2.35 the vortex structures no longer exist in the shock layers of conical flows. It is established that in the particular flow regimes characterized by regular-to-Mach-type transition of the pattern of interaction between the shocks attached to the leading edges an increase of the “blow-up” type in the shock layer thickness can be observable. The experimental data confirm the existence of the vortex Ferri singularities at the corresponding values of the relevant parameters.

Distinctive features of the shock layer flow structure in conical gas flows

The results of a numerical and experimental investigation of the flow structure in symmetric and nonsymmetric flows around V-wings with attached shocks on the leading edges are presented. Emphasis is placed on the appearance of new critical points, including vortex Ferri singularities, in the shock layer and transformations in the flow structure with increase in the angles of attack and yaw. In particular, it is established that the flow structure in the plane of symmetry of the flow around V-wings without yaw, which involves the Mach-type shock configuration, undergoes a jumpwise variation with increase in the angle of attack. Additionally to one Ferri singularity of the node type located at the corner point of the transverse wing contour in the plane of symmetry of the flow, there arise two more critical points, those of flow divergence and convergence. The latter point is the second Ferri singularity; it is located nearer to the bridge-shaped shock of the Mach-type shock system and can be of both the node and the saddle type. In the latter case there appear two vortex Ferri singularities located at the vertices of the contact discontinuities proceeding from this critical point on both sides of the plane of symmetry. Certain data on the position of the critical points relative to the wing contour bend are presented as functions of the wing geometry, together with the transformation of the topological shock-layer flow pattern in the presence of yaw. The comparison of the results calculated within the framework of the Euler model with the experimental data on the shock-layer flow structure obtained using a special optical method for visualizing conical flows showed their good agreement.