Parameters of flow near the axis in the shock layer in the interaction of a supersonic jet with a barrier

The axisymmetric interaction between a supersonic jet with a finite expansion ratio and a barrier is accompanied by the formation of complex sub- and supersonic flow in a shock layer whose thickness depends on the parameters of the jet and the position of the barrier. The main relationships of the interaction process have been established experimentally ([1–3] and others) and individual results of numerical calculations of such flows are known [4]. An analytical investigation of the parameters in the shock layer formed ahead of a plane barrier when an underexpanded jet impinges on it is presented below. The results of [5], where the region near the axis of a shock layer of arbitrary thickness is analyzed within the framework of a model of flow with a constant density, is placed at the basis of the analysis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 63–70, September–October, 1978.The author thanks Yu. M. Tsirkunov for useful discussions.     

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.     

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.     

Characteristics of unsteady type IV shock/shock interaction

Characteristics of the unsteady type IV shock/shock interaction of hypersonic blunt body flows are investigated by solving the Navier–Stokes equations with high-order numerical methods. The intrinsic relations of flow structures to shear, compression, and heating processes are studied and the physical mechanisms of the unsteady flow evolution are revealed. It is found that the instantaneous surface-heating peak is caused by the fluid in the “hot spot” generated by an oscillating and deforming jet bow shock (JBS) just ahead of the body surface. The features of local shock/boundary layer interaction and vortex/boundary layer interaction are clarified. Based on the analysis of flow evolution, it is identified that the upstream-propagating compression waves are associated with the interaction of the JBS and the shear layers formed by a supersonic impinging jet, and then the interaction of the freestream bow shocks and the compression waves results in entropy and vortical waves propagating to the body surface. Further, the feedback mechanism of the inherent unsteadiness of the flow field is revealed to be related to the impinging jet. A feedback model is proposed to reliably predict the dominant frequency of flow evolution. The results obtained in this study provide physical insight into the understanding of the mechanisms relevant to this complex flow.     

Calculation of interaction of a turbulent near-wake behind a step with a supersonic jet

Existing computational methods [1–5] do not enable one to calculate complex flows behind steps, accounting for nonuniformity of the incident supersonic flow and the effect of compression and expansion waves arriving in the near-wake region. For example, computational methods based on the methods of [1] or [2] are used mainly in uniform supersonic flow ahead of the base edge and, for the most part, cannot be used to calculate flow in annular nozzles with irregular conditions. An exception is reference [6], which investigated flow in an annular nozzle behind a cylindrical center-body. The present paper suggests a method, based on references [7, 8] for calculating the base pressure behind two-dimensional and three-dimensional steps, washed by a supersonic jet.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 43–51, November– December 1977.     

Supersonic motion of bodies in a gas with shock waves

The authors consider the problem of supersonic unsteady flow of an inviscid stream containing shock waves round blunt shaped bodies. Various approaches are possible for solving this problem. The parameters in the shock layer on the axis of symmetry have been determined in [1, 2] by using one-dimensional theory. The authors of [3, 4] studied shock wave diffraction on a moving end plane and wedge, respectively, by the through calculation method. This method for studying flow around a wedge with attached shock was also used in [5]. But that study, unlike [4], used self-similar variables, and so was able to obtain a clearer picture of the interaction. The present study gives results of research into the diffraction of a plane shock wave on a body in supersonic motion with the separation of a bow shock. The solution to the problem was based on the grid characteristic method [6], which has been used successfully to solve steady and unsteady problems [7–10]. However a modification of the method was developed in order to improve the calculation of flows with internal discontinuities; this consisted of adopting the velocity of sound and entropy in place of enthalpy and pressure as the unknown thermodynamic parameters. Numerical calculations have shown how effective this procedure is in solving the present problem. The results are given for flow round bodies with spherical and flat (end plane) ends for various different values of the velocities of the bodies and the shock waves intersected by them. The collision and overtaking interactions are considered, and there is a comparison with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–147, September–October, 1984.     

“Lateral” interaction of a supersonic underexpanded ideal-gas jet with surfaces of different shape

A numerical investigation is made of the interaction of an underexpanded jet of an inviscid and nonheat-conducting gas issuing from an axisymmetric conical nozzle with plane, cylindrical, and spherical surfaces. It is assumed that the flow turning angle for flow about a barrier is smaller than the critical angle, and subsonic regions are absent in the flow field studied. The effect of the characteristic parameters (Mach number at the nozzle exit, jet underexpansion) on the flow pattern and jet forces is analyzed. The results of numerical calculations are compared to the results of approximate theories and experimental data. A theoretical solution of the problem of the effect of a supersonic jet on a surface of given shape, even in the approximation of an inviscid, nonheat-conducting gas, is quite difficult. A reason for this is that the flow region contains shock waves interacting with each other, contact discontinuities, and zones of mixed sub-and supersonic flow. As far as is known to the authors, the results obtained for three-dimensional problems for the interaction of supersonic jets with each other or with barriers are primarily experimental (for example, [1–6]). A numerical analysis of the interaction of axisymmetric ideal-gas jets was carried out in [7–10]. In [7] a three-dimensional form of the method of characteristics was used to calculate the initial interaction region for two supersonic cylindrical jets (with Mach number M=10) intersecting at an angle of 60. The interaction of several jets has been considered in [8, 9], where the solution was obtained according to the Lax—Wendroff method without elimination of the discontinuity lines of flow parameters. In [10] the lateral interaction of axisymmetric supersonic jets with each other and with a plate is investigated by means of a straight-through calculationTranslated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–8, November–December, 1974.The authors thank A. N. Kraiko for useful discussions of the results, and A. L. Isakov and É. N. Gasparyan for kindly providing the experimental data.     

Strong mass injection at the surface of a blunt body in a supersonic flow

The case of supersonic flow over a blunt body when another gas is injected through the surface of the body in accordance with a given law is theoretically investigated. If molecular transport processes are neglected, the flow between the shock wave and the surface of the body should be regarded as two-layer, that is, as consisting of the flow in the shock layer between the shock wave and the contact surface and the flow in the layer of injected gas. A numerical solution of the problem is obtained near the front of the body and its accuracy is estimated. Approximate analytic solutions are obtained in the injected-gas layer: a constant-density solution and a solution of the boundary-layer type in the local similarity approximation. Near the flow axis the numerical and analytic solutions are fairly close, but at a distance from the axis the assumptions made reduce the accuracy of the approximate solutions. The flow in question can serve as a gas-dynamic model of a series of problems describing the radiant heating of blunt bodies in a hypersonic flow. In the presence of intense radiative heat transfer, vaporization is so great that the thickness of the vapor layer is comparable with the thickness of the shock layer. Moreover, the thermal shielding of various kinds of obstacles in channels through which a radiating plasma flows can be organized by means of the forced injection of a strong absorber. The formulation of a similar problem was reported in [1], but the results of the solution were not given. A two-layer model of the flow of an ideal gas over a blunt body was used in [2, 3] for the analysis of radiative heat transfer. In [2] the neighborhood of the stagnation point is considered. In [3] preliminary results relating to two-layer flow over blunt cones are presented. The solution is obtained by Maslen's approximate method.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 89–97, March–April, 1972.     

Numerical investigation of “lateral” interaction with an obstacle of an axisymmetric jet exhausting into vacuum

The problem of the interaction of a strongly underexpanded axisymmetric jet with an obstacle for which the normal to the surface makes an angle near /2 with the jet axis is rather laborious for numerical solution due to the high disequilibrium of the gas-dynamic parameters in the peripheral part of the jet and the three-dimensional nature of the flow in the interaction region. Therefore, the results at present available have mainly been obtained experimentally [1, 2]. Among the theoretical studies made in this direction, it is necessary to mention Ivanov and Nazarov's [3], which gives the results of numerical investigation of lateral interaction of a jet with obstacles of various shapes in the case of weakly underexpanded jets when the flow in the interaction region is everywhere supersonic. In the present paper, a study is made of the case when a jet exhausts into vacuum and in front of the obstacle there is a detached shock wave, behind which there is mixed subsonic and supersonic three-dimensional flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 49–54, November–December, 1982.We thank V. I. Uskov for assistance in the present work.     

NUMERICAL SIMULATION OF THREE-DIMENSIONAL SHOCK-SHOCK INTERACTIONS ON A BLUNT BODY

A numerical study is conducted to simulate the effects of extraneous shock impingement on a blunt body in viscous hypersonic flow. The interaction of extraneous shock with the leading-edge shock results in a very complex flow field that contains local regions of high pressure and intense heating. The heating and pressure can be orders of magnitude higher than the peak values in the absence of shock impingement. The flow field is calculated by solving thin-layer Navier-Stokes equations with a finite-volume flux splitting technique developed by van Leer. For a zero or small sweep of the body, a type IV interaction occurs, which produces a lambda shock structure with a supersonic jet embedded in the otherwise subsonic flow; for a moderate sweep of about 25^°, a type V interaction occurs in which a subsonic shear layer sandwiched in supersonic flow is produced with a transmitted shock. In the present study, both type IV and type V interactions are investigated. Results of the present numerical investigation are compared with available experimental results. For the present conditions, the peak pressure is 2.2 times the unimpinged stagnation point pressure and the peak heating is 3 times the unimpinged stagnation point heating. The flow for a type IV interaction is found to be unsteady.     

Numerical investigation of the steady-state conditions of interaction of a supersonic underexpanded jet with a plane obstacle located perpendicular to its axis

The results are presented of the numerical investigation of the interaction of a supersonic axisymmetrical jet of a nonviscous and nonthermally conductive gas, flowing from a conical nozzle into a space with reduced pressure, with a plane obstacle. The presence of a triple point of intersection of the shock wave issuing from the obstacle with the trailing and reflected oblique compression shock is characteristic for the conditions considered in the paper. The solution of the problem is obtained by numerical integration of the gasdynamic equations by means of monotonic difference schemes of a straight-through calculation with first-order accuracy. The interaction of supersonic gas jets with surfaces is a vast problem and is one of the trends being developed intensively in the theory of jet streams. Of the whole multiplicity of problems of practical interest, the two-dimensional case of the normal collision between a supersonic axisymmetrical jet and a plane obstacle has been studied in most detail. As a result of the investigations carried out, many characteristic mechanisms of these flows have been revealed. Together with the numerous experimental papers, several reports have been published (for example, [1–4]) in which various numerical methods are employed to solve this problem. In addition to the method of integral relations used in [1], an implicit difference scheme [2] and explicit schemes of straight-through calculation [3, 4] have been used to calculate the subsonic zone of increased pressure in front of the obstacle. However, an extensive investigation of the special features of the action of a supersonic underexpanded jet on a plane obstacle, at a very small distance from the nozzle exit, still has not been carried out up to the present. In this paper, a solution of this problem is undertaken by the numerical method described in [4] using difference schemes [5, 8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 49–56, September–October, 1976.The authors express sincere thanks to A. N. Kraiko, é. A. Ashratov, and U. G. Pirumov for constant interest and support in carrying out this project.     

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.     

Compression shock in the transonic flow around a convex corner

The interaction between mixed sub-and supersonic flow near a convex breakpoint of a profile with a rectilinear wall downstream of this breakpoint is investigated. If we start from the fact that the initial flow has the character of a singularity in the domain ahead of the last characteristic of the rarefaction node [1], then the solution in the interaction domain, obtained in the hodograph plane under the assumption of its continuity in the physical plane, is not realizable because of the presence of limit lines. This governs the hypotheses of the formation of the compression shock emerging from the corner point and having zero intensity there.     

Supersonic flow around the trailing edge of a thin profile

The method of mergeable asymptotic expansions has recently been used effectively in investigations devoted to the study of boundary layer interaction with an external inviscid flow at high subcritical Reynolds numbers Re. The asymptotic analysis permits obtaining a limit pattern of the flow around a solid as Re þ, and determining the similarity and quantitative regularity laws which are in good agreement with experimental results. Thus by using the method of mergeable asymptotic expansions it is shown in [1–4] that near sites with high local curvature of the body contour and flow separation and attachment points, an interaction domain appears that has a small length on the order of Re^-3/8. In this flow domain, which has a three-layer structure, the pressure distribution in a first approximation already depends on the change in boundary-layer displacement thickness, while the induced pressure gradient, in turn, influences the flow in the boundary layer. An analogous situation occurs in the neighborhood of the trailing edge of a flat plate where an interaction domain also appears [5, 6]. The flow in the neighborhood of the trailing edge of a flat plate around which a supersonic viscous gas flows was examined in [7]. Numerical results in this paper show that the friction stress on the plate surface remains positive everywhere in the interaction domain, and grows on approaching the trailing edge. The supersonic flow around the trailing edge of a flat plate at a small angle of attack was investigated in [8, 9], Supersonic flow of a viscous gas in the neighborhood of the trailing edge of a flat plate at zero angle of attack is examined in [10], but with different velocity values in the inviscid part of the flow on the upper and lower sides of the plate. The more general problem of the flow around the trailing edge of a profile with small relative thickness is investigated in this paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 36–42, May–June, 1981.     

The blunt-body problem in a nonuniform hypersonic viscous gas flow

The laws of heat transfer associated with the interaction of underexpanded supersonic gas jets and obstacles or blunt bodies have been investigated, for example, in [1–3]. Similar problems of nonuniform flow occur when bodies move in the wake behind other bodies; however, in this case the laws of heat transfer have so far received little attention [4–8]. It has been established that for a certain Reynolds number and flow nonuniformity parameters a zone of reverse-circulatory flow develops near the front of the blunt body. However, the conditions of transition to separated flow have not been determined. This paper presents a self-similar solution of the equations of the viscous shock layer near the stagnation line in supersonic flow past an axisymmetric blunt body located behind another body. On the basis of this solution a separationless flow criterion is proposed. The effect of the nonuniformity and the Reynolds number on the shock standoff distance, the convective heat flux and the friction drag of the blunt body is investigated. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–125, November–December, 1986. In conclusion the authors wish to thank I. G. Eremeitsev for useful suggestions and G. A. Tirskii for discussing their work.     

Plane-parallel supersonic flow with a discontinuity

This article considers a plant-parallel supersonic flow, with a shock wave terminating within the flow; the shock wave is regarded as a distortion. A line of discontinuity is located ahead of the shock wave in the supersonic zone. The problem is solved by the method of indeterminate coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 95–100, July–August, 1970.The authors thank S. V. Fal'kovich for his valuable advice and for his evaluation of the results obtained.     

Investigation of the properties of a shock wave arising in a supersonic flow of plasma,passing through a transverse magnetic field

In a flow of plasma, set up by an ionizing shock wave and moving through a transverse magnetic field, under definite conditions there arises a gasdynamic shock wave. The appearance of such shock waves has been observed in experimental [1–4] and theoretical [5–7] work, where an investigation was made of the interaction between a plasma and electrical and magnetic fields. The aim of the present work was a determination of the effect of the intensity of the interaction between the plasma and the magnetic field on the velocity of the motion of this shock wave. The investigation was carried out in a magnetohydrogasdynamic unit, described in [8]. The process was recorded by the Töpler method (IAB-451 instrument) through a slit along the axis of the channel, on a film moving in a direction perpendicular to the slit. The calculation of the flow is based on the one-dimensional unsteady-state equations of magnetic gasdynamics. Using a model of the process described in [9], calculations were made for conditions close to those realized experimentally. In addition, a simplified calculation is made of the velocity of the motion of the above shock wave, under the assumption that its front moves at a constant velocity ahead of the region of interaction, while in the region of interaction itself the flow is steady-state.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 86–91, January–February, 1975.     

Circulation flow at the front surface of a sphere in a supersonic wake

Several theoretical and experimental studies of supersonic flow past a blunt body located in the wake behind another body have been made [1–7]. It has been shown that a reverse-circulation flow can occur in the shock layer at the front surface. The possibility of such a flow forming depends on the nonuniformity of the freestream flow and the Reynolds number. This paper presents new results of the theoretical study of the structure of the shock wave at the front surface of such a sphere, obtained on the basis of numerical solution of Navier-Stokes equations. It is shown that for a fixed nonuniformity of the freestream flow, an increase in the Reynolds number and cooling of the surface of the body lead to the formation of a secondary vortex in the region where the contour of the body intersects the axis of symmetry. A study is made of the variations of the drag and heat transfer parameters over the front surface of a cooled and thermally insulated sphere. The possibility of numerical simulation of the flow on the basis of the Euler equations is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–148, May–June, 1985.     

Combined method of calculating supersonic ideal-gas flow over a wing with a supersonic blunt leading edge

A combined numerical method, based on the successive calculation of the flow regions near the blunt leading edge and center of a wing, is proposed on the assumption that the angle of attack and the relative thickness and bluntness radius of the leading edge are small. The flow in the neighborhood of the leading edge of the wing is assumed to be identical to that on the windward surface of a slender body coinciding in shape with the surface of the blunt nose of the wing and is numerically determined in accordance with [1]. The flow parameters near the center of the wing are calculated within the framework of the law of plane sections [2]. In both regions the equations of motion of the gas are integrated by the Godunov method. The flow fields around elliptic cones are obtained within the framework of the combined method and the method of [3], A comparative analysis of the results of the calculations makes it possible to estimate the region of applicability of the method proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 159–164, January–February, 1989.The authors wish to express their gratitude to A. A. Gladkov for discussing their work, and to G. P. Voskresenskii, O. V. Ivanov, and V. A. Stebunov for making available a program for calculating supersonic flow over a wing with a detached shock.     

Effect of sound on a supersonic jet

It is known that under the influence of sound from an external source or the sound emitted by the supersonic jet itself at discrete frequencies in nonoptimal flow regimes the supersonic jet expands more rapidly and its range is reduced [1, 2], However, the mechanism of action of the sound on the supersonic jet has not been adequately investigated and, in particular, no one has determined the intensity of the external source capable of producing a marked change in the gas dynamic parameters of the jet, its characteristics or how the interaction process develops. These questions are examined below. By means of shadow photography with a pulsed light source it is shown that a significant change in the gas dynamic characteristics of the supersonic jet can be achieved by directing at its base along the normal to the jet boundary sound with an intensity corresponding to 0.1–0.2% of the total pressure in the jet. The appearance of large-scale disturbances on the irradiated side of jet and the directional emission of sound by the jet at the frequency of the external source are noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 170–174, November–December, 1989.The author is grateful to A. A. Kochetkov for assisting with the work.