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.     

Aerodynamics and static stability of star-shaped conical bodies in supersonic flow

The aerodynamics of conical bodies with a star-shaped cross section have been experimentally investigated over a broad range of variation of the parameters determining their geometry at a free-stream Mach number M=6. The position of the center of pressure of star-shaped bodies with an optimum trailing edge shape is investigated in relation to similarity parameters previously obtained theoretically. A correspondence is established between the derivatives of the normal force with respect to the angle of attack for pyramidal star-shaped bodies and bodies with the optimum trailing edge.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 142–150, November–December, 1992.     

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.     

Comparative analysis of viscous flows in cavities and channels containing axisymmetric obstacles

The flow past axisymmetric bodies of various shapes in expanding cavities and cylindrical channels is studied on the basis of a numerical solution of the Navier-Stokes equations. For each body shape velocity, pressure and shear stress distributions are obtained. These data are then used for the purposes of a comparative flow analysis in terms of body shape and Reynolds number. The properties characteristic of flows around bodies in channels and cavities with moving boundaries are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 60–66, May–June, 1988.     

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.     

Lifting bodies using the flow behind axisymmetric conical shock waves

One of the methods of designing aircraft with supersonic flight speeds involves solving an inverse problem by means of the well-known flow schemes and the substitution of rigid surfaces for the flow surfaces. Lifting bodies using the flows behind axisymmetric shock waves belong to these configurations. All lifting bodies using the flow behind a conical shock wave can be divided into two types [1]. Bodies whose leading edge passes through the apex of the conical shock wave pertain to the first type and those whose leading edge lies below the apex of the conical shock wave, to the second. For small apex angles of the basic cone at hypersonic flow velocities an approximate solution of the variation problem was obtained, which showed that the lift-drag ratio of lifting bodies of the second type is higher than that of the first [2]. The present paper gives a numerical solution of the problem for flow past lifting bodies of the second type using the flow behind axisymmetric conical shock waves with half-angles of the basic cone _S=9.5 and 18° The upper surfaces of the bodies are formed by intersecting planes parallel to the velocity vector of the oncoming flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 135–138, March–April, 1986.     

Supersonic axisymmetric flow past a flat-ended cylinder in the wake of bodies of various shape

The pattern of the flow past a pair of bodies under conditions of direct and reverse flow restructuring is established and the aerodynamic and thermodynamic parameters at the surface of a flat-ended cylinder located in a supersonic wake are determined in relation to the leading body shape, drag, and permeability, and the distance between the bodies.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 125–132, November–December, 1996.     

Supersonic flow over elongated blunt bodies with a cross section of elliptical shape

Up to now computational algorithms have been developed for, and systematic studies have been made of, supersonic flow over axisymmetric bodies both by a stream of ideal gas and by an air stream with equilibrium and nonequilibrium physicochemical transformations [1–6]. Conical flows around bodies having cross sections of different shapes and in a wide range of angles of attack have been studied in detail [7–11]. With the further development of numerical methods the next problem has become the analysis of supersonic flow over blunt bodies of large elongation having cross sections of sufficiently arbitrary shape. The effects of essentially three-dimensional flow (without planes of symmetry) over bodies whose cross sections represent ellipses with a constant or variable ratio of axes along the length of the body are discussed in the present paper.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 155–159, November–December, 1976.     

Dependence of the drag of a conical axisymmetric cavitation body on the cone angle and the cavitation number

Numerical investigations of flow past axisymmetric conical captation bodies have shown that the drag coefficient of the cavitation body, calculated from the maximum cross-sectional area of the cavity (midsection), depends on the cavitation number and the cone angle.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 170–173, May–June, 1995.In conclusion, the author is grateful to G. Yu. Stepanov whose critical remarks were helpful in making this paper more conclusive and more illustrative.     

Determination of the body shape for minimum radiative heating along a flight trajectory

Space vehicles are subject to intense aerodynamic heating in planetary entry. According to estimates in [1], the heat shield mass for entry of a probe into the atmospheres of the outer planets can make up 20–50% of its total mass; here the radiative component predominates in the aerodynamic heating. It is therefore interesting to investigate methods of reducing the heat flux to the nose region of a vehicle. Analysis shows [2–6] that, for a given atmospheric composition, the heat-shield weight is determined by the trajectory, the body shape, the heat-protection method, and the chemical composition and the thermophysical and optical properties of the heat shield material. In such a general statement of the problem, optimization of the heat-shield mass depends on many parameters, and has not been solved hitherto. A number of papers have examined simpler problems, associated with reducing spacevehicle heating: optimization of the trajectory from the condition that the total heat flux to the body stagnation point should be a minimum for given probe parameters [2, 3], optimization of the characteristic probe size for a given trajectory [2–4], and optimization of the probe shape in a class of conical bodies at a given trajectory point [3, 5, 6J. In [7] a variational problem was formulated to determine the shape of an axisymmetric body from the condition that the radiative heat flux to the body at a given trajectory point should be a minimum for the entire surface, and an analytical solution was found for this in limiting cases. The present paper investigates a more general variational problem: determination of the shape of an axisymmetric body from the condition that the total radiative influx of heat to the body along its atmospheric trajectory should be a minimum. A solution has been obtained for a class of slender bodies for different isoperimetric conditions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 94–100, March–April, 1978.     

On turbulent separated flow past a truncated body of revolution

Results of testing a series of truncated bodies of revolution with convergent afterbodies in a hydrodynamic tunnel are presented. It is shown that the base pressure can be substantially raised and hence the total drag reduced by varying the shape and convergence of the afterbodies. This effect is caused by intense reverse jets formed as a result of the collision of flow particles moving toward the axis of symmetry.The turbulent flow past the bodies is calculated using the method of viscous-inviscid interaction. The formulas derived for the base pressure and drag coefficients agree satisfactorily with the experimental data.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 50–55, November–December, 1996.     

Forces and moments acting on bodies rotating about a symmetry axis in a free molecular flow

Analytic expressions are obtained for the forces and moments acting on symmetrically rotating convex figures of revolution moving in a free molecular flow of rarefied gas under the following assumptions: the velocity distribution function of the molecules of the oncoming flow is Maxwellian and the incident molecules have a diffuse—specular interaction with the surface of the body. For bodies with arbitrary piecewise smooth generator, general expressions are found in terms of quadrature for the components of the aerodynamic forces and moments. For a disk, sphere, and cylindrical and conical surfaces, the integration of the forces and the moments, which depend on the rotation of the body, is carried out to the end. For the moments of the forces, graphs are plotted of the errors of the hypothermal approximation as a function of the velocity ratio.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp, 151–155, May–June, 1980.     

Optimal blunt-nosed figures of revolution in a gas of different rarefactions

Many studies have been made of the optimization of the shape of bodies in a gas stream. However, the majority of these have been made for supersonic and hypersonic flow in the limiting case of a continuum [1], and only a few studies have been made [1–3] for the case of flow over a body of a rarefied gas (mainly a free molecular stream). In the present paper, the problem of shape optimization is considered for hypersonic flow of a gas of different rarefactions over a body. Numerical methods are used to investigate the influence of the Reynolds number on the shape and drag of optimal figures of revolution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 158–161, May–June, 1980.     

Flow past blunt bodies with nose spikes and surface injection

Axisymmetric supersonic ideal-gas flow past a blunt body with a forward-projecting spike is numerically investigated with allowance for injection from the surface. The effect of the length and shape of the spike, the parameters of the injected gas and the position of the permeable zone on the flow pattern and drag is studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 128–133, July–August, 1987.     

Behavior of sonic lines in a shock layer behind a detached shock wave

A numerical investigation is made into the formation of local supersonic zones in the subsonic flow region between a detached shock wave and the surface of the body in the case of supersonic three-dimensional flow over conical bodies with opening angle _k = 120 ° of the cone in the range of Mach numbers M = 2.5–15.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No.4 pp. 143–145, July–August, 1979.We thank G. I. Petrov for suggesting the problem and for helpful advice and O. M. Belotserkovskii for constant interest in the work.     

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.     

Interference of oblique shock waves of one family in a hypersonic flow

A study is made of the irregular regime of interaction of two shock waves of the same direction when a hypersonic gas stream flows past bodies of complicated shape. It is shown that the rarefaction waves formed in the flow field significantly weaken the shock wave that approaches the body. This effect is confirmed by the results of an experiment and numerical calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 134–138, September–October, 1982.     

Investigation of gas blowing into a base cavity

The results of an experimental investigation of various methods of influencing the flow in the near wake by injecting a system of jets and varying the rear-end configuration for a Mach number M=3 are presented. The structure of the base flow and the pressure fluctuations behind an axisymmetric body with a base cavity when circular and annular gas jets are injected is investigated. The results are compared with the data for bodies with a flat base.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 170–172, July–August, 1989.     

Body shape with minimum total radiant energy flux toward its surface

During a space vehicle's entry into a planet's atmosphere at hypersonic speed one of the important problems is the aerodynamical surface heating due to convective and radiant heat fluxes from the gas after passing through a strong shock wave. Due to the high destructive action of this heating, an important problem is the selection of the aerodynamic shape allowing the minimum heat influx to its surface. The problem of determining the shapes of an axisymmetric body from the condition of minimum total convective heat flux along the lateral face of the body was considered under various assumptions in [1–7]. There are a number of entry conditions (for example, into the earth's atmosphere with a speed of 11 km/ sec at an altitude of about 60 km [12]) during which the radiative component becomes dominant in the total heat flux toward the body. A numerical solution of the problem of hypersonic flow of a nonviscous, non-heat-conducting radiating gas around a body is obtained at this time only for a limited class of bodies and primarily for certain entry conditions (for example, [8–12]). On the basis of these calculations it is impossible to make general conclusions concerning arbitrary body shapes. Therefore, approximate methods were proposed which permit the distribution of radiant heat flux to be obtained for an arbitrary axisymmetric body in a wide range of flight conditions [13–15]. In the present work an expression is derived for the total radiant heat flux over the entire body surface and similarity criteria are found. A variational problem is formulated to determine the shape of an axisymmetric body from the condition of minimum total radiant-heat flux over the entire body surface. It is solved analytically for the class of thin bodies and in the case of a strongly radiating gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 84–89, July–August, 1976.     

Parameters determining the shape of the subsonic section of a shock wave in the case of axisymmetric flow

Supersonic flow around a blunt body by an ideal gas with a constant specific heat ratio is considered. The dependence of the geometry of the subsonic section of the shock wave on the shape of the body and the freestream Mach number is studied. Analysis of the large quantity of numerical data has permitted simple approximate relations to be formulated for the principal geometrical parameters of the wave, which can be used for solving the problem of flow around a quite broad class of bodies. The question of the characteristic dimensions in such problems is also discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 77–83, July–August, 1976.In conclusion, the author expresses his thanks to V. N. Ivanova for carrying out the calculations and to é. é. Shnol' for for several useful comments.