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.     

Flow around a sphere by a two-phase supersonic jet

Results are presented of a calculation of the flow around a sphere of a two-phase supersonic jet, discharging into a vacuum. Calculations were performed by the determination method with use of a difference grid constructed on the basis of characteristic ratios [1], The parameters of the unperturbed jet were determined with the two-velocity and two-temperature model of mutually penetrating flows of continuous media (gas and particles) [2, 3] by the network method [4]. In calculating the flow around the sphere, as in [5–7], it was assumed that the particles do not affect the gas flow in the shock layer. An analysis of the effect of particles on gasdynamic parameters in a shock layer was performed in [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 171–176, November–December, 1978.The authors are grateful to A. N. Nikulin for providing the program for calculation of flow about a blunt body by a uniform supersonic flow.     

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.     

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.     

Three-dimensional nonequilibrium reacting airflow around a body penetrating into an equilibrium heated zone

The problem of the supersonic penetration of a spherically blunt body at an angle of attack into a medium containing a temperature and chemical inhomogeneity is considered. In order to determine the flow parameters the unsteady Euler equations of gas dynamics, supplemented by the continuity equations for the chemical components that compose the dissociated air, are solved numerically. The variation with time of the shape of the bow shock, the flow characteristics and the component concentrations is determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 130–137, November–December, 1990.The author wishes to thank V. B. Baranov for discussing his results.     

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.     

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.     

Experimental investigation of three-dimensional supersonic flow of a gas in the interference region of intersecting surfaces

A contemporary high-speed aircraft represents a complex three-dimensional configuration, where supersonic gas flow is accompanied by numerous local flow interaction zones, in particular, near the intersection of different surfaces. Such a flow is characterized by three-dimensional systems of shock and expansion waves, and close to the surfaces one finds interaction of boundary layers and, above all, interaction of shock waves with the boundary layer. In general, the angular configurations are formed by intersection or contact of nonplanar surfaces with swept-back or blunted leading edges. This makes it practically impossible to obtain a rigorous theoretiical solution to the problem of gas flow over these surfaces, and presents considerable difficulty in an experimental investigation. It is therefore of interest to study the physical features of gas flow in corner configurations of very simple form [1–3]. The present paper examines the results of an experimental investigation of typical features of symmetric and asymmetric interaction of compressive, expansive, and mixed flows in the interference region of planar surfaces intersecting at an angle of less than 180?.     

Solution of a problem of supersonic gas flow past solids of revolution in the theory of small perturbations

A small-parameter method is widely used today for solving many problems of aerodynamics. It has made it possible to obtain results which are interesting from the practical point of view. In particular, it has helped to solve a problem dealing with a substantially three-dimensional field of flow around a wing with finite wingspread past which a gas is flowing at supersonic speed [1]. However, in the problem of flow past solids of revolution, certain difficulties have been encountered in the application of the method. In the present paper, within the framework of the theory of small perturbations, we consider a method for obtaining a solution in which these difficulties can be avoided. We obtain simple analytic expressions for the aerodynamic characteristics of solids of revolution in a supersonic stream of gas. We give a comparison of the results with experimental data and with calculations carried out by the method of characteristics.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 99–106, November–December, 1977.In conclusion, the authors wish to thank V. V. Sychev for his comments on the results of the study.     

Computation of base flow characteristics with uniform blowing

Blowing at bluff body base was considered under different conditions and for small amount of blowing this problem was solved using dividing streamline model [1]. The effect of supersonic blowing on the flow characteristics of the external supersonic stream was studied in [2–4]. The procedure and results of the solution to the problem of subsonic blowing of a homogeneous fluid at the base of a body in supersonic flow are discussed in this paper. Analysis of experimental results (see, e.g., [5]) shows that within a certain range of blowing rate the pressure distribution along the viscous region differs very little from the pressure in the free stream ahead of the base section. In this range the flow in the blown subsonic jet and in the mixing zones can be described approximately by slender channel flow. This approximation is used in the computation of nozzle flows with smooth wall inclination [6, 7]. On the other hand, boundary layer equations are used to compute separated stationary flows with developed recirculation regions [8] in order to describe the flow at the throat of the wake. The presence of blowing has significant effect on the flow structure in the base region. An increasing blowing rate reduces the size of the recirculation region [9] and increases base pressure. This leads to a widening of the flow region at the throat, usually described by boundary-layer approximations. At a certain blowing rate the recirculation region completely disappears which makes it possible to use boundary-layer equations to describe the flow in the entire viscous region in the immediate neighborhood of the base section.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 76–81, January–February, 1984.     

Numerical investigation of axisymmetric flows in the wake of blunt bodies in a supersonic stream of viscous gas

The axisymmetric flow in the near wake of spherically blunted cones exposed to a supersonic stream of viscous perfect heat-conducting gas is numerically investigated on the basis of the complete Navier-Stokes equations. The free-stream Mach numbers considered M = 2.3 and 4 were such that the gas can be assumed to be perfect, and the Reynolds numbers such that for these Mach numbers the flow in the wake is laminar but close to laminar-turbulent transition [1–4]. The flow structure in the near wake is described in detail and the effect of the Mach and Reynolds numbers on the base pressure, the total drag and the wake geometry is investigated. The results of calculating the flow in the wake of spherically blunted cones are compared with the experimental data [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–47, July–August, 1988.     

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.     

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.     

Stationary flow past the lower surface of a piecewise planar delta wing with supersonic leading edges

The considered wing has any finite number of inflections in its plane with lines of inflection intersecting at the point of inflection of the leading edge. In the present paper, this generalizes the author's earlier work [1] on flow past the undersurface of a flat wing at unite angle of attack with finite angle of slip and supersonic leading edges. In [1], calculations were not given. The special case of flow without slip in the same situation was considered later in [2], However, this paper contains errors, indicated at the end of the present paper. The calculations given in [2] are not correct. In the quoted papers, the gas flow is assumed to be a perturbation of a homogeneous flow behind a plane oblique shock wave. Such flows are treated systematically in [3]. Here and in [1], we use and generalize the representation of the linearized conservation laws across the shock front as the conditions of a boundary-value problem for an analytic function of a complex variable as obtained in [4, 5]. Calculations are given of the pressure distribution over the span for a number of different flow regimes and the pressure coefficients in the middle of the wing are compared with a numerical solution presented partly in [6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 80–90, September–October, 1979.I am very grateful to V. I. Lapygin for making available a large number of variants of his numerical solution, and to L. E. Pekurovskii for assistance in the calculations.     

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.     

Planar gas flows with weak energy supply

Perfect gas flows in an unlimited space, which occur during rectilinear motion of a system of distributed heat sources, are investigated. The next modes in order of growth of the number M are examined: the heat conductive, convective, subsonic, transonic, supersonic, hypersonic. Examples of computations are presented. Flows with distributed heat sources attract ever-increasing attention. Such flows are important, e.g., in the problem of radiation propagation [1–5], in the analysis of a gasdynamic laser resonator and the optical characteristics of a ray [6]. Changes in the density because of absorption of the ray energy, which can result in an essential redistribution of the radiation intensity, are of great interest in these problems. Theoretical investigations of a general nature with distributed heat supply [7–10] are also important for the development of further applications. Gas flows for a given distribution of relatively weak heat sources switched on at a certain time are examined in this paper.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 95–102, September–October, 1978.     

Numerical investigation of supersonic flow past blunt bodies with protruding spikes

A procedure for the calculation of a supersonic flow of ideal gas near axisymmetric blunt bodies with protruding spikes is developed. The flow past a frustum of a cone with a protruding spherically blunt cylindrical spike as a dependence on the ratio K of the spike length1 to the diameter D of the flat end of the body and the Mach number M of the oncoming flow is studied. Several steady flow regimes are obtained, including the formation of circulation zones and internal shock waves in the shock layer. It is shown that mounting a spike in front of the frustum of a cone can lead to a 40–50% reduction in its drag. A full investigation of the variation of the drag coefficient as a dependence on K is carried out for M = 3.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 119–127, May–June, 1986.The authors express their gratitude to V. A. Levin for the formulation of the problem and his constant attention to the work.     

Motion of a gas shock layer with a free boundary under nonadiabatic conditions

This paper is a study of the effect of heat input (removal) on the characteristics of a shock layer produced by a gas at high supersonic velocity encountering a mobile boundary, which for generality is assumed to be free. We will use the Chernyi method, which was employed previously to solve the problem of a shock layer in an adiabatic flow [1, 2]. The results obtained can be useful for analysis of the effect of radiation (absorption) and processes involving the relaxation of internal degrees of freedom of molecules, condensation, chemical reactions, etc., whose effect on the gasdynamics of the flow in a shock layer may be similar to heat input or removal [3–5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 152–154, May–June, 1976.The author thanks A. K. Rebrov for discussion of the results.     

Blowing into a supersonic stream through a permeable surface

Distributed blowing of gas into a supersonic stream from flat surfaces using an inviscid flow model was studied in [1–9]. A characteristic feature of flows of this type is the influence of the conditions specified on the trailing edge of the body on the complete upstream flow field [3–5]. This occurs because the pressure gradient that arises on the flat surface is induced by a blowing layer whose thickness in turn depends on the pressure distribution on the surface. The assumption of a thin blowing layer makes it possible to ignore the transverse pressure gradient in the layer and describe the flow of the blown gas by the approximate thin-layer equations [1–5]. In addition, at moderate Mach numbers of the exterior stream the flow in the blowing layer can be assumed to be incompressible [3]. In [7, 8] a solution was found to the problem of strong blowing of gas into a supersonic stream from the surface of a flat plate when the blowing velocity is constant along the length of the plate. In the present paper, a different blowing law is considered, in accordance with which the flow rate of the blown gas depends on the difference between the pressures on the surface over which the flow occurs and in the reservoir from which the gas is supplied. As in [8, 9], the solution is obtained analytically in the form of universal formulas applicable for any pressure specified on the trailing edge of the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 108–114, September–October, 1980.I thank V. A. Levin for suggesting the problem and assistance in the work.     

Normal interaction of shock waves in the presence of vibrational relaxation

The effect of nonequilibrium physicochemical processes on the flow resulting from the normal collision and reflection of shock waves is studied by the example of nonequilibrium excitation of molecular oscillations in nitrogen. It is shown that the thermal effect of vibrational relaxation is small and the problem can be linearized around a known solution [1]. A similar approach to the solution of the problem of flow around a wedge and certain one-dimensional non-steady-state problems was used earlier in [2–4]. The solution of these problems was constructed in an angular domain, bounded by the shock wave and a solid wall (or the contact surface) and was reduced to a well-known functional equation [6]. The solution of this problem, because of the presence of two angular domains divided by a tangential discontinuity, reduces to a functional equation of more general form than in [6]. The results are obtained in finite form. In the special case of shocks of equal intensity, the normal reflection parameters are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 90–96, July–August, 1976.