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.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 69–79, September–October, 1996.     

Aerodynamic characteristics of star-shaped bodies at Mach numbers M = 3–5

The results are given of an experimental investigation into the aerodynamic characteristics of star-shaped bodies with flat faces at Mach numbers M = 3–5 and angles of attack = 0–12 ° for different numbers of points of the star and different inner radii at the midsection. It is established that the star-shaped bodies have a much lower total drag than bodies of revolution of equivalent length and midsection area.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–93, July–August, 1981.     

Conservation of the resultant force vector in the plane of the angle of attack for slender star-shaped bodies in supersonic flows

At high supersonic flight speeds bodies with a star-shaped transverse and power-law longitudinal contour are optimal from the standpoint of wave drag [1–3]. In most of the subsequent experimental [4–6] and theoretical [6–9] studies only conical star-shaped bodies have been considered. For these bodies in certain flow regimes ascent of the Ferri point has been noted [10]. In [11] the boundary-value problem for elongated star-shaped bodies with a power-law longitudinal contour was solved for the case of supersonic flow. The present paper deals with the flow past these bodies at an angle of attack. It is found that for arbitrary star-shaped bodies with any longitudinal (in particular, conical) profile the aerodynamic forces can be reduced to a wave drag and a lift force, the lateral force on these bodies being equal to zero for any position of the transverse contour.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 135–141, November–December, 1989.     

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.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 34–40, May–June, 1982.     

On the center of pressure of conical bodies

The profiles of conical bodies for which the position of the center of pressure in a supersonic flow with symmetry plane does not depend on the flow parameters are considered. The theoretical investigation of the aerodynamic characteristics of circular cones [1] has shown that their center of pressure does not depend on the angle of attack when the shock wave is attached to the apex of the cone. It was established experimentally in [2, 3] for star-shaped bodies that the position of the center of pressure for such bodies hardly changes in a wide range of Mach numbers and angles of attack.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 99–104, January–February, 1980.I thank G. G. Chernyi for discussing the results.     

Dfag measurements on star-shaped body at m ≈ 6 and 8

Results have been obtained in recent years which make it possible to get an idea of the optimal shape of a three-dimensional body at high supersonic speeds. It has been shown [1–6] that bodies with a cross section in the form of a star with certain limitations have the least wave drag and remain optimal with respect to total drag with approximate account for the friction forces. The transition from the optimal body of revolution to the star-shaped body of equivalent volume and length makes a several-fold drag reduction. These theoretical results, initially obtained on the basis of the Newton drag law, were then confirmed by the exact solution [7] for bodies which were close in form to the optimal. Subsequent experimental studies investigated the flow pattern between two lobes representing an element of the star over a wide range of included angles. The experiments showed that there actually exists a flow between the rays corresponding to the solution [7], that this flow is stable, and that the wave drag calculated from the pressure distribution over the body surface is several fold less than for the equivalent cone. Although these results are encouraging, they do not prove the advantages of the star-shaped form for practical use. The point is that the star has considerably more wetted area; therefore the effect of the marked reduction of the wave drag may be compensated by an increase of the friction drag. The references above to the theory which considers friction are not convincing, since the friction estimates are approximate, while real friction is complicated by the presence of shock waves within the flow, the possibility of a turbulent boundary layer, separation, etc. Not all these factors are amenable to calculation, and it is clear that conclusions can be drawn on star drag only after making direct measurements of the total force acting on a model in a flow.In the following we describe the results of force tests conducted with a star model at M6 and 8. During the tests the flow pattern in the wake behind the body was photographed in addition to the force measurements.The authors wish to thank G. I. Petrov, G. G. Chernyi, M. Ya. Yudelovich, and A. A. Churilin for assistance in carrying out the experimentation.     

Aerodynamic coefficients of non-conical bodies with a star-shaped transverse cross section

It is well known that, in a supersonic flow, the wave resistance of a body of non-round transverse cross section can be less than the resistance of an equivalent body of revolution with the same length and volume. Starting from 1959, when an exact solution was obtained to the problem of supersonic flow around conical bodies with a pyramidal system of flat discontinuities [1], a number of publications have appeared [2–5] developing this direction. Article [3] pointed out the possibility of achieving a flow with reflected shock waves, normal to the faces of a pyramidal body, by selection of the form of the leading edge. In [6, 7], using the Newton resistance law, bodies were constructed with a transverse cross section of a star-shaped form, having a wave resistance several times less than for an equivalent body of revolution. Just such forms, with certain limitations, have the least wave resistance and retain optimality with respect to the total resistance, taking approximate account of friction forces. Still two more exact solutions were then found, corresponding to flow around star-shaped bodies with regular and Mach interaction between shock waves [8, 9]. At a seminar of the Institute of Mechanics of Moscow State University, G. G. Chernyi advanced the postulation of the existence of certain classes of three-dimensional bodies not having the property of similitude and retaining optimality with respect to determined characteristics, for example, the resistance, the aerodynamic quality, or the torque, and stated partial problems of finding various forms of optimal bodies. Classes of bodies, optimal with respect to the resistance, were obtained within the framework of the Newton theory; the bodies consisted of helical surfaces, as well as of sections of planes and conical surfaces, formed by straight lines connecting the leading edges with a round contour. As a result of calculations using the Newton theory and experimental investigations it was established that bodies with a wedge-shaped nose part, with determined geometric parameters, have greater values of the lifting and of the aerodynamic quality than round cones [10]. The possibility of lowering the resistance and increasing the aerodynamic quality of aircraft by giving them shapes of the transverse cross section in the form of a star [11–14] leads to new investigations of three-dimensional bodies which retain optimality with respect to their aerodynamic characteristics, and are used in conjunction with bodies of revolution. This latter factor is of decisive importance with the use of such configurations as the nose part of the aircraft, or of a multi-step diffusor. The present article gives the results of an experimental investigation of flow around two classes of such bodies: multi-wedge and helical.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 127–132, November–December, 1974.     

Multicomponent chemically-reacting turbulent viscous shock layer on a catalytic surface

Turbulent flows past blunt bodies at high supersonic speeds are mainly investigated within the framework of the boundary layer model. However, even at large Reynolds numbers owing to the strong entropy gradient on the lateral surface it becomes necessary to take boundary layer corrections into account in the higher approximations [1]. The use of viscous shock layer theory makes it possible to obtain fairly accurate results over a broad interval of variation of the Reynolds numbers without organizing iterations with respect to vorticity and displacement thickness. The nonequilibrium nature of both homogeneous and heterogeneous catalytic reactions is taken into account. The results obtained are compared with the experimental data [2, 3]. Previously, in [4, 5] turbulent flow was investigated within the framework of viscous shock layer theory in the case of equilibrium homogeneous reactions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 144–149, March–April, 1989.     

Aerodynamic drag of a pair of bodies in transonic and supersonic flow

The aerodynamic characteristics of models of pairs of bodies on the flow acceleration and deceleration intervals are investigated experimentally at transonic and supersonic flow velocities. The dependence of the drag coefficient of the pair model on the relative drag of the leading body is determined for supersonic velocities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 152–156, May–June, 1990.     

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.     

Investigation of supersonic three-dimensional flow past pointed axisymmetric bodies

The flow pattern near bodies of revolution with very long cylindrical and pointed nose sections is studied in the framework of an ideal gas model by means of a numerical method based on MacCormack's difference scheme. The existence of internal shock waves, oriented in both the longitudinal and the transverse directions, in the shock layer is established. The variation of the aerodynamic coefficients of the configuration with its length, angle of attack, and free stream Mach number is investigated. The calculated and experimental data are compared, and the connection between the flow parameters on the body surface and the position of the separation line of the boundary layer on its lateral face is established. A method of calculating the influence of the boundary layer on the values of the aerodynamic coefficients of bodies of revolution of large aspect ratio at small angles of attack is proposed. Axisymmetric flow near blunt bodies has been studied in detail in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–133, September–October, 1986.The author expresses his gratitude to A. N. Pokrovskii for his help in calculating the boundary layer parameters on the surfaces of the considered configurations.     

Optimum supersonic profile of given thickness

The problem of determining the profile possessing minimum wave drag in a uniform supersonic free gas stream is considered. The length, thickness, and angle of attack of the body are considered given. The analysis is limited to the class of bodies for which attached shocks originate during the flow. It is assumed that the flow is supersonic in the domains of influence of the components of the desired contour and there are no internal shocks. The problem is studied within the framework, of model and exact formulations. Results of numerical computations are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 89–96, January–February, 1976.The author is grateful to A. V. Shipilin for constant attention to the research.     

Three-dimensional thin viscous shock layer in the absence of planes of symmetry in the flow

The basic laws of viscous homogeneous gas flow at high supersonic speeds past smooth blunt bodies with a permeable surface are investigated within the framework of the thin viscous shock layer model. An efficient numerical method of solving these equations, which makes it possible to consider cases of flow past bodies at angles of attack and slip, when there are no planes of symmetry in the flow, is proposed. Some results of calculating the flow past a triaxial ellipsoid with an axial ratio of 103n73 at angles of attack =0–45° and slip angles =0–45° over a broad interval of Reynolds numbers are presented as an example. The effect of the principal determining parameters of the problem on the flow structure in the shock layer and the surface friction and heat transfer coefficients is analyzed. An expression for calculating the heat fluxes to the impermeable surface of smooth blunt bodies in a supersonic homogeneous viscous gas flow over a broad interval of Reynolds numbers is proposed on the basis of the solutions obtained and the results of other authors.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 150–158, March–April, 1989.     

Wave drag of slender star-shaped bodies at moderate supersonic flight velocities

At large supersonic flight velocities, star-shaped bodies have a significant advantage over axisymmetric bodies as regards the wave drag [1–3]. The majority of the more recent investigations on star-shaped bodies has been experimental (Conor, Zubin, and Ostapenko have given a review [4]; among the theoretical studies, those of Lapygin and Ostapenko [5–7] should be mentioned), but no new information about the wave drag of star-shaped bodies was obtained in these studies. In the present paper, the wave drag of star-shaped bodies at moderate supersonic flight velocities is considered.     

Flow of a supersonic ideal gas over a planar cascade in the case of a subsonic normal component in regimes with attached shocks

The supersonic flow of an inviscid gas that does not conduct heat over a cascade of planar pointed profiles is considered in the case when the component of the velocity vector of the undisturbed flow normal to the cascade front is subsonic. The investigation is restricted to regimes without separation and shock waves attached to the leading edges of the profiles and fairly dense cascades, for which the characteristics or shock waves leaving the trailing edges do not enter the region in front of the cascade. In such cases, the conditions behind the cascade do not influence the flow in front of it. In this sense, the flow in the cascade, as in a Laval nozzle in the case of supercritical gradients is trapped, In the hodograph plane, trapped regimes of flow over the cascade correspond to velocity vectors of the undisturbed flow that lie on a certain line (see, for example, [1–3]), which is constructed in the process of solution of the problem. This property has been called the directing influence of the cascade on the oncoming flow. Regimes with detached shocks can also be trapped if the separation of the shocks is due to the profiles being blunt. A method is proposed that for regimes with attached shocks makes it possible to calculate the entire flow field, including the wave structure at large distances from the cascade front; some results obtained by the method are also given. The study of regimes with attached shocks, for which the analysis is simplest, is, first, of interest in its own right and, second, is a stage in the creation of methods of calculation and subsequent investigation of cascades with arbitrary regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 108–113, July–August, 1979.We are grateful to M. Ya. Ivanov for assistance in updating the supersonic flow calculation program of [7], to G. Yu. Stepanov for helpful comments, and to E. V. Buganov and V. A. Vostretsov for assistance in preparing the paper.     

INTERFERENCE OF SHOCK AND BOUNDARY LAYER IN SUPERSONIC FLOW

激波与物面边界层的干扰涉及可压缩流动的稳定性、转捩、分离等问题,直接影响到飞行器的阻力、表面热防护和飞行性能等工程技术问题。首先总结了前人对于激波与边界层的干扰所做的工作,之后重点研究和对比分析了超声速与跨声速流动中,正激波、斜激波以及头部激波对于飞行器层流和湍流边界层的干扰影响。激波强度的不同对边界层干扰作用不同,在强干扰情况下将会引起边界层分离和翼型失速。     

Numerical modeling of the effect of geometric asymmetry on the damping aerodynamic characteristics of supersonic sectional flight vehicles

A method for calculating the steady inviscid supersonic flow past equivalent bodies is applied to the analysis of the time-dependent aerodynamic characteristics of sectional flight vehicles with an asymmetric rear stabilizer. The stabilizer asymmetry can be caused by either its deflection or a change in its shape due to heat-shield coating removal, boundary layer displacement thickness, developed separation flow zones, local deformations, or other distortions in the baseline form. Amathematical apparatus for modeling asymmetric sectional configurations by means of ruled surfaces with arbitrary contours of the reference cross-sections is developed. The uniform perfect-gas M_∞ = 6 flow past sectional vehicles performing plane oscillations about the zero angle of attack is calculated (the adiabatic exponent γ = 1.4). The calculated results demonstrate the effect of various asymmetries in the body shape on the aerodynamic coefficients.     

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.     

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.     

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.