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.     

On the center of pressure of a body

It is shown that under conditions of local interaction of a flow with the surface of a body the position of the center of pressure does not depend on the angle of attack or the form of the model of the flow past the body for a large class of configurations including conical, cylindrical, spherical, and flat elements. The problem of finding the shape of conical bodies with unchanged position of the center of pressure in a supersonic flow without slip was formulated by Ostapenko [1], who also found the shape of some such bodies.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 129–133, September–October, 1982.     

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.     

Slender axisymmetric cavities in a supersonic flow

Slender axisymmetric cavities in a subsonic flow of compressible fluid were investigated in [1–4]. In [5] a finite-difference method was used to calculate the drag coefficient of a circular cone, near which the shape of the cavity was determined for subsonic, transonic, and supersonic water flows; however, in the supersonic case the entire shape of the cavity was not determined. Here, on the basis of slender body theory an integrodifferential equation is obtained for the profile of the cavity in a supersonic flow. The dependence of the cavity elongation on the cavitation number and the Mach number is determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 179–181, January–February, 1989.     

On the formulation of the problem of the motion of a body in water

The selection of solutions describing steady irrotational flow of an ideal incompressible fluid over bodies is considered. The selection is based on restrictions that follow from the physical properties of a real fluid and from the presence of a boundary layer on the body. In particular, for any body one can specify a minimal Euler number below which flow without cavitation becomes physically impossible. In the limiting case of an Euler number equal to zero, only the Kirchhoff scheme is physical admissible, and the cavity section tends to a circle. An equation is derived for the limiting shapes of cavities at small cavitation numbers, and a comparison is made with known results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–26, September–October, 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.     

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.     

Area rule for heat transfer coefficient of three-dimensional ablating bodies in heat flows that depend locally on the angle of attack

The area rule, which is well known for wave resistance [1, 2], is generalized to the heating of three-dimensional bodies by flows which depend locally on the angle of attack. Calculations are made for triaxial ellipsoids with different ratios of the semiaxes, and the limits of applicability of the rule are found. The problem of determining the ablation of a three-dimensional body that changes its shape in a heat flow is solved. It is shown that the area rule also holds for a change of mass of three-dimensional bodies, and expressions are given for calculating the ablation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 71–76, March–April, 1979.We are grateful to G. G. Chernyi and G. A. Tirskii for discussing the results.     

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.     

Singularities of supersonic flows around pointed bodies with a detached shock

Results are presented of a numerical investigation of the axisymmetric flow around a family of bodies with a pointing angle for which the shock is detached. It is shown that the supersonic part of the stream remains the same for all bodies of the family for a fixed value of M despite the fact that the shaper of the subsonic zone is related quite strongly to the pointing angle _*. The dependence of the shock standoff and its radius of curvature on the spreading line on the body shape is studied. Effects inherent in flows around sharply pointing bodies are discussed. A dimensionless parameter characterizing each body of the family under consideration is introduced and used to establish general flow regularities. Data illustrating the possibility of applying such parametrization are analyzed for a wider class of pointing bodies.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 111–115, July–August, 1975.     

Optimal shapes of bodies with attached shock waves

We consider the problem of finding the shape of two-dimensional and axisymmetric bodies having minimal wave drag in a supersonic perfect gas flow. The solution is sought among bodies having attached shock waves. The limitations on the body contour are arbitrary: these constraints may be body dimensions, volume, area, etc. Such problems with arbitrary isoperimetric conditions may be solved by the method suggested in [1, 2]. This method involves the use of the exact equations of gasdynamics which describe the flow as additional constraints. This method was developed further in [3–6] in the solution of several problems.The author wishes to thank V. M. Borisov, A. N. Kraiko and Yu. D. Shmyglevskii for their interest in this study.     

Unsteady three-dimensional viscous shock layer in nonuniform gas flow in the neighborhood of a stagnation point

The combined influence of unsteady effects and free-stream nonuniformity on the variation of the flow structure near the stagnation line and the mechanical and thermal surface loads is investigated within the framework of the thin viscous shock layer model with reference to the example of the motion of blunt bodies at constant velocity through a plane temperature inhomogeneity. The dependence of the friction and heat transfer coefficients on the Reynolds number, the shape of the body and the parameters of the temperature inhomogeneity is analyzed. A number of properties of the flow are established on the basis of numerical solutions obtained over a broad range of variation of the governing parameters. By comparing the solutions obtained in the exact formulation with the calculations made in the quasisteady approximation the region of applicability of the latter is determined. In a number of cases of the motion of a body at supersonic speed in nonuniform media it is necessary to take into account the effect of the nonstationarity of the problem on the flow parameters. In particular, as the results of experiments [1] show, at Strouhal numbers of the order of unity the unsteady effects are important in the problem of the motion of a body through a temperature inhomogeneity. In a number of studies the nonstationary effect associated with supersonic motion in nonuniform media has already been investigated theoretically. In [2] the Euler equations were used, while in [3–5] the equations of a viscous shock layer were used; moreover, whereas in [3–4] the solution was limited to the neighborhood of the stagnation line, in [5] it was obtained for the entire forward surface of a sphere. The effect of free-stream nonuniformity on the structure of the viscous shock layer in steady flow past axisymmetric bodies was studied in [6, 7] and for certain particular cases of three-dimensional flow in [8–11].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 175–180, May–June, 1990.     

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.     

Aerodynamic equivalence of asymmetric bodies with allowance for nonlinear body shape influence factors and angle of attack

In order to determine the aerodynamic coefficients of asymmetric bodies the principle of aerodynamic equivalence is used. In the gas dynamic function expansions in the asymmetry parameter and the angle of attack both linear and second-order terms are taken into account. This makes it possible to establish theoretically the principle of aerodynamic equivalence with allowance for nonlinear body shape influence factors and angle of attack. A universal structure of the aerodynamic characteristics of asymmetric bodies with an arbitrary cross-sectional shape is obtained. The theoretical conclusions are confirmed by the results of a numerical solution of the three-dimensional gas dynamic problem.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 116–122, November–December, 1993.     

Conical wing of maximum lift-drag ratio in a supersonic gas flow

The calculation of supersonic flow past three-dimensional bodies and wings presents an extremely complicated problem, whose solution is made still more difficult in the case of a search for optimum aerodynamic shapes. These difficulties made it necessary to simplify the variational problems and to use the simplest dependences, such as, for example, the Newton formula [1–3]. But even in such a formulation it is only possible to obtain an analytic solution if there are stringent constraints on the thickness of the body, and this reduces the three-dimensional problem for the shape of a wing to a two-dimensional problem for the shape of a longitudinal profile. The use of more complicated flow models requires the restriction of the class of considered configurations. In particular, paper [4] shows that at hypersonic flight velocities a wing whose windward surface is concave can have the maximum lift-drag ratio. The problem of a V-shaped wing of maximum lift-drag ratio is also of interest in the supersonic velocity range, where the results of the linear theory of [5] or the approximate dependences of the type of [6] can be used.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 128–133, May–June, 1986.We note in conclusion that this analysis is valid for those flow regimes for which there are no internal shock waves in the shock layer near the windward side of the wing.     

Flow of electrically charged aerosols past bodies

An experimental investigation was made of the flow of electrically charged aerosols with solid or liquid disperse phase past bodies. Air flows with solid particles (sand, iron) and water drops were produced by special apparatus. The regimes in which the disperse particles and the bodies were charged or neutral were studied. A system of dimensionless numbers and their values are indicated for the different gas-dynamic and electric flow regimes. The main features of the flows are explained and the integrated electric characteristics of the bodies determined. The possibility of controlling the flow of a disperse medium by means of electric fields is demonstrated. It is shown that the inductive charging of liquid films near sharp edges of a body has a strong influence on the electric characteristics of the body (when air with water drops flows past it).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 71–79, July–August, 1982.     

On the propulsive efficiency of vibrating bodies in a subsonic gas flow

In the cited work [1], a variational problem of the minimum power required to obtain a given thrust for a wing with a surface able to vibrate in a supersonic flow is considered. In the present work the theoretical maximum efficiency in a subsonic flow of bodies whose shape can be simulated by unsteady sources is found. For axisymmetric bodies a shape is found giving an efficiency as close as required to the theoretical limit. The results obtained make it possible to evaluate possible improvements not only of artificial devices but also of living creatures, since they move at subsonic speeds.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 128–132, July–August, 1984.     

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.     

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.     

Heat transfer at the surface of a triangular body in a hypersonic viscous gas flow

The boundary layer on a semi-infinite triangular body of power-law shape is calculated for viscous interaction with an external hypersonic flow. The results of calculating the characteristics of the three-dimensional boundary layer are presented. The formation of secondary return flows and zones of intensified heat transfer on the surface of the body in the neighborhood of lines of flow divergence is noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 77–82, January–February, 1988.