Aspects of self-similar diffraction gas flows and their numerical simulation

A large number of papers has been devoted to the investigation of the interaction of a plane shock wave with bodies of various geometric shapes, and they have been generalized and classified for a stationary body in [1, 2]. Separate results of experimental and theoretical investigations of the interaction of a shock wave with a wedge, cone, sphere, and cylinder moving with supersonic velocities are contained in [3–9]. Analysis of the available results shows that the features of the unsteady gas flows formed in this case largely depend on the nature of the boundary-value problem that arises for the system of differential gas dynamic equations. The question of the wave structure of the unsteady gas flow and the accuracy of the obtained solution is central to the numerical investigation of the present class of problems. The most characteristic types of unsteady self-similar gas flows that arise on the interaction of a plane shock wave with bodies of a wedge or convex corner type are calculated on the basis of an explicit numerical continuous calculation method of the second order of accuracy. The accuracy of the numerical solutions is discussed on the basis of a comparison with the experimental data. The case of the interaction of a shock wave with the rarefaction wave that arises in a supersonic flow past a convex corner is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 146–152, July–August, 1986.     

Arbitrary body,close to a wedge,in a supersonic flow

The problem of supersonic flow around bodies close to a wedge was first discussed in the two-dimensional case in [1]. The shock wave was assumed to be attached, and the flow behind it to be supersonic; taking this into account, the angle of the wedge was assumed to be arbitrary. The surface of the body was also arbitrary, provided that it was close to the surface of the wedge. In solution of the three-dimensional problem, there was first considered flow around two supporting surfaces with only slightly different angles of attack [2], and then around a delta wing [3, 4]. In all these articles, the Lighthill method was used to solve the Hilbert boundary-value problem [5, 6]. A whole class of surfaces of bodies with arbitrary edges, under the assumption that the surface of the body was cylindrical, with generatrices directed along the flow lines of the unperturbed flow behind an oblique shock wave, was discussed in [7]. In the present work, the problem is regarded for a broad class of surfaces of bodies, using a new method which generalizes the results of [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 109–117, July–August, 1974.The author thanks G. G. Chernyi for his direction of the work.     

Numerical simulation of shock wave intersections

A large number of papers, generalized and classified in [1, 2], have been devoted to unsteady gas flows arising in shock wave interaction. Experimental results [3–5] and theoretical analysis [6–9] indicate that the most interesting and least studied types of interaction arise in cases when there are several shock waves. At the same time, nonlinear effects, which depend largely on the nature of the shock wave intersections, become appreciable. Regions of existence of different types, of plane shock wave intersections have been analyzed in [10–13]. It has been shown that in a number of cases the simultaneous existence of different types of intersections is possible. The aim of the present paper is to study unsteady shock wave intersections in the framework of a numerical solution of the axisymmetric boundary-value problem that arises in the diffraction of a plane shock wave on a cone in a supersonic gas flow. Flow regimes that augment the experimental data of [3–5] and the theoretical analysis of [9] are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 134–140, September–October, 1986.     

Unsteady reflection of a shock wave from a body with a cylindrical recess

In a number of cases of supersonic flow past bodies with recesses pulsations in the flow arise [1–3]. Experiments [4, 5] indicate that stabilization of the steady supersonic flow past the body with a recess on which a shock wave is incident takes place after a series of oscillations of the bow wave. Numerical calculation of the interaction of a supersonic jet with a cylindrical cavity [6] reveals that damped pressure pulsations arise inside the cavity if the jet is homogeneous, and undamped pulsations it is inhomogeneous. The authors explain the damping of the pulsations by the influence of artificial viscosity. This paper investigates experimentally and theoretically (by numerical methods) the oscillations of the bow shock wave and the parameters of the flow behind it in the case of unsteady reflection of a shock wave from a body with a cylindrical recess turned towards the flow. The problem is posed as follows. A plane shock wave with constant parameters impinges on a cylinder with a cavity. The unsteady flow originating from this interaction is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 199–202, September–October, 1984.     

Study of flow in the vicinity of V-shaped wings formed by the stream surfaces behind a plane shock

In calculating the flow about bodies with plane surfaces and sharp edges it is assumed that in the flow regimes with attached shock the latter may be defined in a section normal to the edge from the corresponding relations for the wedge [1, 2], The solution is taken corresponding to a weak shock on a wedge with supersonic velocity behind it. While in the plane case (wedge) this solution will be the only physically realizable solution, in the case of three-dimensional bodies, when there is a slip velocity along the leading edge, the realization of a second wedge solution with a strong shock is conceivable in the section normal to the leading edge if the total velocity behind the shock (with account for the slip velocity along the edge) is supersonic [3].Relative to the undisturbed stream velocity both of these solutions correspond to a weak shock. We present an example when the solution with a strong shock in the section normal to the edge is possible.     

Solution of the direct problem of the mixed flow of a conducting gas in a laval nozzle with a meridional magnetic field

We consider the direct problem in the theory of the axisymmetric Laval nozzle (including sonic transition) for the steady flow of an inviscid and nonheat-conducting gas of finite electrical conductivity. The problem is solved by numerical integration of the equations of unsteady gas flow using an explicit difference scheme that was proposed by Godunov [1,2], and was used to calculate steady and unsteady flows of a nonconducting gas in nozzles by Ivanov and Kraiko [3]. The subsonic and the supersonic flows of a conducting gas in an axisymmetric channel when there is no external electric field, the magnetic field is meridional, and the magnetic Reynolds numbers are small have previously been completely investigated. Thus, Kheins, Ioller and Élers [4] investigated experimentally and theoretically the flow of a conducting gas in a cylindrical pipe when there is interaction between the flow and the magnetic field of a loop current that is coaxial with the pipe. Two different approaches were used in the theoretical analysis in [4]: linearization with respect to the parameter S of the magnetogasdynamic interaction and numerical calculation by the method of characteristics. The first approach was used for weakly perturbed subsonic and supersonic flows and the solutions obtained in analytic form hold only for small S. This is the approach used by Bam-Zelikovich [5] to investigate subsonic and supersonic jet flows through a current loop. The numerical calculations of supersonic flows in a cylindrical pipe in [4] were restricted to comparatively small values of S since, as S increases, shock waves and subsonic waves appear in the flow. Katskova and Chushkin [6] used the method of characteristics to calculate the flow of the type in the supersonic part of an axisymmetric nozzle with a point of inflection. The flow at the entrance to the section of the nozzle under consideration was supersonic and uniform, while the magnetic field was assumed to be constant and parallel to the axis of symmetry. The plane case was also studied in [6]. The solution of the direct problem is the subject of a paper by Brushlinskii, Gerlakh, and Morozov [7], who considered the flow of an electrically conducting gas between two coaxial electrodes of given shape. There was no applied magnetic field, and the induced magnetic field was in the direction perpendicular to the meridional plane. The problem was solved numerically in [7] using a standard process. However, the boundary conditions adopted, which were chosen largely to simplify the calculations, and the accuracy achieved only allowed the authors [7] to make reliable judgments about the qualitative features of the flow. Recently, in addition to [7], several papers have been published [8–10] in which the authors used a similar approach to solve the direct problem in the theory of the Laval nozzle (in the case of a nonconducting gas).Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 5, pp. 14–20, September–October, 1971.In conclusion the author wishes to thank M. Ya. Ivanov, who kindly made available his program for calculating the flow of a conducting gas, and also A. B. Vatazhin and A. N. Kraiko for useful advice.     

Supersonic two-phase flow around a wedge

Supersonic two-phase flow around bodies is encountered in calculating the flow around the last stages of blades of condensing turbines, in studying the motion of airplanes under cloudy conditions, etc. In the latter case, there is, along with erosion of the forward edges of the wing profiles, a change in the wave structure and interference situation in the flow about the airplane, leading to off-design regimes of motion. Supersonic flow of a two-phase mixture around a wedge, without taking account of the influence of the particles on the flow, was investigated in [1–3]. In [4], also in this kind of simplified setting, a study was made of the interaction of particles with the surface of a wedge in which reflection of the particles from the wall was taken into account. Morganthaler [5] made an experimental study of the flow of a mixture of air and aluminum oxide particles around a wedge. In [6] a theoretical study was made of a supersonic two-phase flow around thin flat axially-symmetric bodies. In particular, for the flow around a wedge, closed form solutions were obtained for the form of the shock wave, the gas streamlines and particle paths, and the distribution of all the parameters along the surface of the wedge. On the basis of the equations given in [7] and the method of characteristics, which were developed for flows consisting of a mixture of a gas and heterogeneous particles in nozzles [8,9], we present below a study of a supersonic two-phase flow around a wedge.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 83–88, March–April, 1972.     

Interaction between an external compression shock and a blunt body in a hypersonic stream

A complex shock configuration with two triple points can occur during the interaction between an external oblique compression shock and the detached shock ahead of a blunt body (for instance, ahead of a wing or stabilizer edge). This results in the formation of a high-pressure, low-entropy supersonic gas jet [1–6]. Here two flow modes are possible [1], which differ substantially in the intensity of the thermal and dynamic effects of the stream on the blunt body: mode I corresponds to the impact of a supersonic jet [2–6], while the supersonic jet in mode II does not reach the body surface in the domain of shock interaction because of curvature under the effect of a pressure drop. Conditions for the realization of the above-mentioned flow modes are investigated experimentally and theoretically, and an approximate method is proposed to determine the magnitude of the compression shock standoff in the interaction domain. Blunt bodies with plane and cylindrical leading edges are examined. The results of a computation agree satisfactorily with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 97–103, January–February, 1976.The author is grateful to V. V. Lunev for discussing the research and for useful remarks.     

Study of unsteady flow past bodies behind shock waves

Several theoretical and experimental studies have been devoted to the problem of the nonstationary action of the stream behind a shock wave on bodies of varied shape. In particular, in [1], the pressure and density are calculated for flow about bodies of the more typical shapes in the initial stage of the process. The basic relations which accompany the interaction of shock waves are considered in [2, 3]. The analysis of the phenomena of diffraction of shock waves on the sphere, cylinder, and cone is presented in [4]. Problems of unsteady flow about a wing are examined in [5, 6]. A detailed review of the foreign studies on unsteady flow is given in [7]. Of great practical interest is the question of the time for flow formation and the magnitudes of the unsteady loads during this period. Experimental investigations have been made recently [8, 9] in which some criteria are presented for estimating the bow shock formation time for supersonic flow about the sphere and cylinder with flat blunting. However the question of the formation time of the stationary pressure on the body surface is not referred to in these studies and no relationship is shown between the transient position of the reflected wave and the corresponding unsteady pressure on the surface. Moreover, in [8] the dimensionless time criterion is determined very approximately, independently of the Mach number of the shock wave. The present study was undertaken with the object of determining the basic criteria which characterize unsteady flow about bodies behind a plane shock wave which has time-independent parameters, and clarification of the shock wave reflected from the body and the pressure on the surface of the body during the transient period. The most typical body shapes were studied: 1) a cylinder with flat face aligned with the stream; 2) a spherically-blunted cylinder; and 3) a cylinder transverse to the stream. The experiments were conducted in a conventional shock tube using the single-diaphragm scheme. The measurements of the pressure on the models and the velocity of the incident shock wave were made using the technique analogous to that of [10, 11]. A highspeed movie camera was used to record the pattern of the wave diffraction on the body. The Mach number of the incident shock wave varied in the range from M=1.5 to M≈6.0, which corresponded to a range of Mach numbers M_∞ of the stream behind the shock wave from 0.6 to 2.1. The calculations of the required gas dynamic parameters for high temperatures were made with account for equilibrium dissociation of the air on the basis of the data of [10, 12, 13]. The magnitude of the relative maximal shock wave standoff Δ at the stagnation point obtained in the present experiments was compared with the values of Δ from other studies. In the case of the flat-blunted cylinder it was in good agreement with the results of [8–14], and in the case of the spherically-blunted cylinder and the transverse cylinder it was in agreement with the results of [15].     

Flow past a sphere of two co-axial supersonic streams

We investigate the flow past a sphere of a parallel supersonic stream which is nonuniform in magnitude; such a flow can be considered as two co-axial streams of an ideal gas. The problem is solved numerically by the method of establishment [1]. Supersonic flow of nonuniform magnitude and direction past blunt bodies and a plane wall was investigated in [2–5],Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 89–94, September–October, 1970.The author wishes to thank G. F. Telenin for his attention to the paper.     

Supersonic flow past weak sources of radiation

In the cited works [1, 2], a study was made of supersonic flow round a source of x-ray radiation by stellar wind. It was found that if the energy release is sufficiently high, a bow shock wave forms in the flow, and a zone of low density and high temperature of the gas arises behind the source. If the influx of energy to the gas is small, the flow remains supersonic everywhere [2]. The question of the formation of a shock wave in the case of weak heating through of the gas by the radiation was not considered in [2]. However, on the basis of the qualitative analogy between supersonic flow past impermeable bodies and radiation sources, one would naturally expect that if the power of the energy release is low a shock wave which does not intersect the axis of symmetry could appear. The following article indicates the conditions under which this actually occurs, and a quantitative analogy is established between supersonic flow past thin bodies and sources of radiation, and the range of flow where such an analogy is absent is also determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 133–136, July–August, 1984.The author thanks V. P. Stulov and M. M. Gilinskii for discussing this work and for their extremely useful comments.     

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.     

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.     

Regular reflection of a magnetohydrodynamic shock wave from a conducting wall

In two-dimensional supersonic gasdynamics, one of the classical steady-state problems, which include shock waves and other discontinuities, is the problem concerning the oblique reflection of a shock wave from a plane wall. It is well known [1–3] that two types of reflection are possible: regular and Mach. The problem concerning the regular reflection of a magnetohydrodynamic shock wave from an infinitely conducting plane wall is considered here within the scope of ideal magnetohydrodynamics [4]. It is supposed that the magnetic field, normal to the wall, is not equal to zero. The solution of the problem is constructed for incident waves of different types (fast and slow). It is found that, depending on the initial data, the solution can have a qualitatively different nature. In contrast from gasdynamics, the incident wave is reflected in the form of two waves, which can be centered rarefaction waves. A similar problem for the special case of the magnetic field parallel to the flow was considered earlier in [5, 6]. The normal component of the magnetic field at the wall was equated to zero, the solution was constructed only for the case of incidence of a fast shock wave, and the flow pattern is similar in form to that of gasdynamics. The solution of the problem concerning the reflection of a shock wave constructed in this paper is necessary for the interpretation of experiments in shock tubes [7–10].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 102–109, May–June, 1977.The author thanks A. A. Barmin, A. G. Kulikovskii, and G. A. Lyubimov for useful discussion of the results obtained.     

Calculation of unsteady supersonic flow past a two-dimensional cascade of plates ween vorticity inhomogeneities of the flow act on it

In the framework of the linear theory of small perturbations the problem of unsteady subsonic flow past a two-dimensional cascade of plates has been considered in a number of papers. Thus, the unsteady aerodynamic characteristics of a cascade of vibrating plates were calculated in [1] by the method of integral equations, while the same method was used in [2, 3] to calculate the sound fields that are excited when sound waves Coming from outside or vorticity inhomogeneities of the oncoming flow act on the cascade. The problem of a two-dimensional cascade of vibrating plates in a supersonic flow was solved in [4, 5]. In [4] the solution was constructed on the basis of the well-known solution of the problem of vibrations of a single plate, while in [5] a variant of the method of integral equations was used which differed slightly from the usual formulation of this method [1–3]. The approach proposed in [5] is used below to calculate the unsteady flow past a two-dimensional cascade of plates in the case when vorticity inhomogeneities of a supersonic oncoming flow act on it. Equations are obtained for the strength of the unsteady pressure jumps arising in such a flow and the vortex wakes shed from the trailing edges of the plates. Examples of the calculations illustrating the accuracy of the method and its possibilities are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp, 152–160, May–June, 1986.     

Calculation of hydrogen-air combustion behind detached shock wave for supersonic flow past a sphere

Many of the published theoretical studies of quasi-one-dimensional flows with combustion have been devoted to combustion in a nozzle, wake, or streamtube behind a normal shock wave [1–6].Recently, considerable interest has developed in the study of two-dimensional problems, specifically, the effective combustion of fuel in a supersonic air stream.In connection with experimental studies of the motion of bodies in combustible gas mixtures using ballistic facilities [7–9], the requirement has arisen for computer calculations of two-dimensional supersonic gas flow past bodies in the presence of combustion.In preceding studies [10–12] the present author has solved the steady-state problem under very simple assumptions concerning the structure of the combustion zone in a detonation wave.In the present paper we obtain a numerical solution of the problem of supersonic hydrogen-air flow past a sphere with account for the nonequilibrium nature of eight chemical reactions. The computations encompass only the subsonic and transonic flow regions.The author thanks G. G. Chernyi for valuable comments during discussion of the article.     

Flow of a supersonic electrogasdynamic stream,with formation of an electroconducting gaseous shock layer

Surfaces of strong discontinuity in electrogasdynamics were considered in [1, 2]. An investigation was done for the case when a gas has the properties of a unipolar charged medium on both sides of a surface of discontinuity. However, with sufficiently high supersonic gas flow over bodies the gas becomes electroconducting and acquires the properties of a low-temperature plasma in the compressed layer between the shock wave and the body, because of the temperature increase. Therefore, there is great interest in investigating type S* Shockwaves dividing a unipolar charged medium and a low-temperature plasma. The S* waves separating the uncharged medium and a gas with high electrical activity in the presence of an electrical field were studied in [3]. Below we examine the general properties of S* waves (physicalmodel, relations at the wave, conditions for development, shock adiabats, and polars). We formulate the problem of flow of a supersonic electrogasdynamic stream over bodies, with formation of S* waves. A perturbation method is proposed for solution of the problem, using a small parameter to describe electrogasdynamic interaction. By way of example a complete solution for flow over a wedge is constructed.     

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.     

The reflection of a plane shock wave from a body with groove

When a plane shock wave impinges on bodies with grooves and when a supersonic stream of gas flows past such bodies a complicated flow pattern develops. In a number of cases oscillations of the bow wave [1–3] and an anomalous heating of the gas in the groove [4–6] have been observed. Unsteady reflection of shock waves from bodies with grooves and the processes occurring inside the grooves have been investigated comparatively little.Translated from Izvestiya Akademii Nauk SSSR, Hekhanika Zhidkosti 1 Gaza, No. 5, pp. 180–186, September–October, 1935.The authors wish to thank V. I. Ivanov for carrying out the calculations.     

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.