Diffraction of a plane shock wave by a disk that enters a compressible liquid at an inclination

A study is made of the three-dimensional problem of the interaction between a disk that enters water asymmetrically and a shock wave that is moving toward the disk. The water is assumed to be a perfect compressible liquid and the flow adiabatic. The changes in the flow parameters and the state are determined by numerical integration of the equations that describe the flow. A three-dimensional version of the finite-difference scheme of [1] is used in accordance with the method of [2]. The influence of the intensity of the shock wave on the drag coefficient of the disk and the shape of the free surface is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 185–187, January–February, 1990.     

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.     

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.     

Transonic separation flow of water past a circular cone

The problem of subsonic, transonic, and supersonic separation flow of water past a circular cone of finite length is solved. The water is assumed to be an ideal compressible fluid. A steady flow picture is obtained in a process of stabilization with respect to the time by means of a two-dimensional finite-difference scheme [1]. The dependence of the drag coefficient on the Mach number of the oncoming flow, the distribution of the pressure over the conical surface, and the shape of the free surface formed behind the cone are investigated.     

Separated inviscid gas flow past a disk and a body with maximum critical Mach numbers

The problem of the separated axisymmetric subsonic flow of an inviscid perfect gas with the specific heat ratio 1.4 past a disk in accordance with the Riabouchinsky scheme is solved using the method developed in [1]. Formulas relating the main parameters with the base pressure coefficient and the Mach number at the free boundary are presented. Formulas which make it possible to determine the shape of the body of revolution giving the maximum critical Mach numbers are also derived.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 166–172, May–June, 1996.     

Transition flow of a fibrous suspension in a pipe as the flow of an anisotropic fluid

The transition flow is considered of a fibrous suspension in a pipe. The flow region consists of two subregions: at the center of the flow a plug formed by interwoven fibers and fluid moves as a rigid body; between the solid wall and the plug is a boundary layer in which the suspension is a mixture of the liquid phase and fibers separated from the plug [1–3]. In the boundary region the suspension is simulated as an anisotropic Ericksen—Leslie fluid [4, 5] which satisfies certain additional conditions. Equations are obtained for the velocity profile and drag coefficient of the pipe, which are both qualitatively and quantitatively in good agreement with the experimental results [6–8]. Within the framework of the model, a mechanism is found for reducing the drag in the flow of a fibrous suspension as compared to the drag of its liquid phase.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 91–98, September–October, 1985.     

Subsonic gas-liquid cavitation flow past a disk

The problem of axisymmetric subsonic gas-liquid cavitation flow past a disk in accordance with the Riabouchinsky scheme is solved using the method of [1]. Formulas relating the main flow parameters with the cavitation number, the Mach number on the free boundary and the gas/liquid volume ratio under stagnation conditions are presented.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 202–206, March–April, 1996.     

Influence of the atmosphere on the magnitude of the hydrodynamic forces in the case of a disk in a flat encounter with the surface of a compressible liquid

If the velocities with which bodies enter liquids are small, and the bodies are not too blunt, the magnitudes of the hydrodynamic forces can be satisfactorily determined in the framework of the approximation of an incompressible liquid and depend on the density of the liquid, the velocity of entry, and the geometrical parameters (shape of the body, angles of entrance and attack). If the velocity is increased or the encounter with the surface becomes nearly flat, the compressibility of the liquid and the presence of an atmosphere begin to influence the hydrodynamic forces significantly. The influence of the compressibility on the magnitude of the impact loads has been investigated theoretically and experimentally [1–8]. The influence of the atmosphere in the case of an incompressible liquid has also been taken into account [9–11]. In the case of a flat encounter the two factors (compressibility of the liquid and presence of the atmosphere) simultaneously influence the development of the impact process. The present paper reports experimental results and computer calculations of the impact loads in the case of a flat encounter of a disk and the surface of a compressible liquid in atmospheres of helium, air, and freon.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 15–20, May–June, 1984.     

Entry of a disk into a compressible fluid at an angle to the free surface

The available experimental results are used to find the dependence on the Mach number of the maximal impact drag coefficient and the growth time of the impact load when a disk enters a compressible fluid at an angle to the free surface. The elastic properties of the disk are not taken into account, i.e., the disk is assumed to be absolutely rigid.     

Investigation of Aerodynamic Drag of Two Bodies in Trans‐ and Supersonic Flows

Results of experimental investigations of trans or supersonic flow around two bodies (cone–disk or sphere–disk) connected by a cylindrical rod along the axis of symmetry are presented. The special features of the flow are analyzed. It is found that the dependence of the drag coefficient C_x of a pair of bodies on the Mach number within the range 0.6 M 1.7 is nonmonotonic. The reasons for the hysteresis in the dependences C_x(M) for two bodies at the stages of flow acceleration and deceleration and discrete variation of the Mach number are clarified. The influence of cone angles and sizes of both bodies on the drag coefficient is estimated.     

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 study of the entry of a plate and a disk into a compressible fluid

The dependences of the drag force on the time and the Mach number are found, as also are pressure distribution, and the shape of the free surface. It is shown that with the passage of time the drag force rapidly approaches its asymptotic value, which corresponds to flow around a body by a compressible fluid in accordance with Kirchhoff's scheme. It is also shown that with increasing Mach number the dimensions of the cavity decrease, the unsteady cavity always being narrower than the Kirchhoff cavity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–107, March–April, 1985.     

Hydrodynamic drag of an ellipsoidal droplet at small Reynolds numbers

At present, there is an absence of the accurate data on the influence of the shape of a droplet on its hydrodynamic drag and mass transfer without which the design of mass transfer apparatus is impossible [1–3]. Most often it is assumed that the drag of an ellipsoidal liquid droplet as it moves along the axis of symmetry is determined by the product of the drag of a spherical liquid droplet and a coefficient which takes into account the shape and is determined from the drag of a solid ellipsoid for which the exact solutions are known. It is shown below that this assumption is not always valid.Translated from Tzvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 4–8, May–June, 1987.     

Transverse subsonic flow past a cylinder

At around the critical Reynolds number Re = (1.5–4.0)·10⁵ there is an abrupt change in the pattern of transverse subsonic flow past a circular cylinder, and the drag coefficient C_x decreases sharply [1]. A large body of both experimental and computational investigations has now been made into subsonic flow past a cylinder [1–4]. A significant contribution to a deeper understanding of the phenomenon was made by [4], which gives a physical interpretation of a number of theoretical and experimental results obtained in a wide range of Re. Nevertheless, the complicated nonstationary nature of flow past a cylinder with separation and the occurrence of three-dimensional flows when two-dimensional flow is simulated in wind tunnels do not permit one to regard the problem as fully studied. The aim of the present work was to make additional experimental investigations into transverse subsonic flow past a cylinder and, in particular, to study the possible asymmetric stable flow regimes near the critical Reynolds number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 154–157, March–April, 1980.     

Theory of regular and mach reflection of shock waves in a two-phase gas-liquid medium

An equilibrium model of a two-phase gas-liquid medium, with allowance for the proportion, density, and compressibility of the components, and with a difference from [1] in that the adiabatic velocity of sound is introduced, has been used in order to study the regular and Mach (elementary theory) reflection of a shock wave of moderate intensity from a solid wall throughout the whole range of gas proportions. A complicated nonmonotonic variation has been found for the pressure on the wall behind the reflected wave, the angle of reflection, and the angle of departure of the triple point as functions of the gas proportion, the angle of incidence, and the intensity of the incident wave. In particular, it is shown that oblique reflection for moderate and low gas contents leads to the formation of a stronger reflected shock wave than does normal reflection. The effect of the gas proportion on the position of the boundary between the regions of regular and Mach reflection has already been studied in [2]. The results are described of serial calculations of the parameters of reflection for an air-water mixture, and these results agree fairly well for normal reflection with the known experimental data [3] for low and moderate gas contents. In the limiting case, the results agree with the known results for single-phase media [4, 5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 188–190, March–April, 1985.     

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.     

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.     

Existence of an optimal cavitation body

The results are given of an exact numerical calculation of Riabouchinsky flow past a cavitator of optimal shape determined in accordance with the approximate theory of Gonor and Zabutnaya [5]. It is shown that the drag C_xk of the cavitator is less than for a disk.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 63–68, March–April, 1991.We are most grateful to G. Yu. Stepanov for valuable comments made during preparation of this paper.     

Optimal lifting surfaces of complicated-geometry wings at supersonic flight velocities

Infinitely thin wings weakly perturbing a supersonic flow of perfect gas are investigated. The flow problem is solved in a linear formulation [1]. The shape of the wing in plan and the Mach number M of the oncoming flow are specified. The optimal wing surface is determined as a result of finding the function of the local angles of attack _M(x, z) which ensures a minimum of the drag coefficient c_x when there are limitations in the form of equalities on the lift coefficient c_y and the pitching moment m_z. A separationless flow regime is realized on the optimal wing for the given number M, and its subsonic leading edge does not experience a load [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 154–160, November–December, 1985.     

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.