Interaction of a supersonic flow with a transverse jet injected through a circular aperture in a plate

A flow pattern created by the interaction of a supersonic flow with a transverse sonic or supersonic jet injected normally to the direction of the main flow through a circular aperture in a plate is considered. The pressure rises in front of the jet owing to the retarding action of the incident flow. The boundary layer building up on the wall in front of the injection nozzle is accordingly detached. The flow pattern in the region of interaction between the jet and the external flow is illustrated in Fig. 1. The three-dimensional zone of detachment thus formed deflects the incident flow from the wall, and in front of the jet a complicated system of sharp jumps in contraction develops. A three-dimensional system of jumps also develops in the jet itself.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 5, pp. 193–197, September–October, 1970.     

Investigation of interaction between a plane transverse gas jet and a supersonic stream

The problem of interaction between a two-dimensional transverse gas jet emerging from a slot on a flat plate and a supersonic stream is considered. Several theoretical methods based on various approaches and physical models have been proposed to determine the characteristics of such a flow. The following fundamental directions can be isolated: a quasi-one-dimensional method [1], use of blast theory [2], and methods based on the equivalence of the effect of the jet and some solid on the external flow [3, 4]. However, the listed computational schemes [1–4] do not permit any clarification of the flow configuration in the jet and in the outer stream (the shock configuration, the jet boundaries, the distribution of the gasdynamic parameters in the flow field, etc.). Extensive experimental investigations of this phenomenon have been carried out simultaneously with the development of the theoretical methods, wherein the flow picture was determined, the pressure distribution was measured in the interaction domain, etc. [5, 6]. A computation method is proposed in this paper which will permit a detailed investigation of the flow structure in the jet and in the outer flow outside the separation region. Underlying the method is a hypothesis verified experimentally: The separating streamline in the mixing layer of the separated boundary layer (the “constant mass∝ line) intersects the jet boundary at the point of maximum jet standoff from the nozzle exit towards the incoming stream.     

Determination of the shape of the shocks ahead of a transverse jet injected into a supersonic flow

The three-dimensional shape of the shock wave formed ahead of a sonic jet flowing out into a supersonic flow through the surface of a sharp cone is determined. The shape of the wave in the longitudinal and transverse cross-sections of the model is constructed using schlieren photographs taken for various angles of rotation and freestream Mach numbers M=1.75–3. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 41–44, March–April, 1998. This research was carried out with financial support from the Russian Foundation for Basic Research (project No. 95-01-00709a).     

Interaction of a transverse gas jet with supersonic flow in a dihedral

The results of an experimental investigation of supersonic flow at the Mach number M₁=3 past a transverse gas jet flowing from an orifice in the edge of a dihedral with a linear angle of 90° are presented. The Mach number of the jet was varied from 1 to 3, and the ratio of the total pressure in the jet to the free stream pressure from 90 to 760. Visualization of the flow near the faces of the dihedral revealed the existence of internal lines of flow convergence and divergence in the region of three-dimensional separated flow, which indicates the presence of complex vortex structures. The dependence of the dimensions of the separated flow zone and the characteristic pressures in it on the jet parameters is explored.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 161–165, November–December, 1994.     

Penetration depth of a jet injected into an oncoming supersonic flow

The geometrical characteristics of jets injected through an opening in a flat plate into an oncoming supersonic flow have been studied on a number of occasions [1, 3]. However, the results were analyzed under different suppositions about the important dimensionless parameters. In [1], the degree of underexpansion of the jet, characterized by n = p _a /p, was regarded as decisive; in [3], the experimental points were plotted against the relative dynamic head _a u² _a /(u² ) of the jet. In the present paper, dimensional considerations are used to determine the dimensionless parameters which influence the flow field when an injected jet interacts with an oncoming supersonic gas flow. The influence of these determining dimensionless parameters on the depth of penetration of injected jets into a flow was investigated experimentally. It is shown that the relative depth of penetration is determined basically by the relative specific impulse of the jet, the injection angle, and the shape of the blowing nozzle section.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 151–154, July–August, 1980.     

Sound emission by a vortex ring passing through a circular hole in a large flat plate

Acoustic emission has been studied experimentally when a vortex ring passes through a circular hole in a large flat plate, along its normal axis. The speed of the vortex ring is made high enough for the sound emission to be detectable, but can be regarded as sufficiently low in comparison to the sound speed. Substantial monopole and quadrupole components are observed in the detected wave profiles. Translational motion of the vortex ring in the presence of the flat plate with a hole has been observed optically, and its relation with the sound emission is determined. In this case, the power law of the acoustic pressure amplitude of monopolar vortex sound versus the translation speed U of the vortex ring is first measured in detail and is found to be U^2.1–U^2.4. This means that experimentally determined powers for the monopole components in the two half-spaces also agree approximately with the corresponding values predicted by the theory of vortex sound.     

Extension of a viscoplastic plate with a circular hole

International Applied Mechanics -     

Numerical study of unsteady compressible flow driven by supersonic jet injected into elliptical cell with small exit hole

The unsteady behavior of flow driven by a jet suddenly injected into an elliptical cell is numerically studied by solving the axisymmetric two-dimensional compressible Navier–Stokes equations. This system is a model of laser ablation of a certain duration followed by a discharging process through the exit hole at the downstream end of the cell. The parameters for the calculations are the exit diameter of the cell, the Mach number and duration of the injected jet. The injected jet becomes a traveling plume approaching the downstream end of the elliptical cell and discharges from the cell through an exit hole. The plume generates and interacts with a shock wave in the elliptical cell. The unsteady flow properties downstream of the cell are found to be attenuated by the combination of the phenomena occurring in the cell and at the exit. Monitoring the velocity at the exit hole is used to clarify the characteristics of the flow and apply them to applications in pulse laser ablation. The results show that the vortex in the plume head with the same radius as the exit diameter (i.e., D_e/D_j = 2.7, where D_e is the exit diameter and D_j is the injected jet diameter) causes a relatively constant velocity at the exit for about 10 μs. In the downstream flow characteristics, the suddenly injected jet makes a single or double peak in the velocity variation outside the cell depending on the combination of parameters. This suggests that a single laser pulse might generate two beams through the exit hole of a cell, which could increase the efficiency of beam generation with the combination of an elliptical cell and the laser ablation. It is also found that the wave form of the variations and their level are roughly determined by the durations of the jet and the exit diameters of the cell exit, respectively. PACS 51.35.+a; 47.40.Nm     

Calculation of a supersonic jet emerging from a hole with planar walls

The coordinates of the sonic line are derived by means of Frankl's solution [1], while the supersonic part of the jet is considered by the method of characteristics. The numerical solution has been used to calculate the family of rarefaction waves and a family of nozzles having a corner point and a curvilinear transition surface. These calculations show that, when there is a corner point, the shape of the sonic line has hardly any effect on the velocity distribution along the symmetry axis. It is also shown that a positive pressure gradient arises on the surface of the nozzle beyond the corner point if that point lies upstream from the limiting characteristic of the first family.     

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.     

Displacement field around a circular hole in a viscoelastic plate

Moiré experimental techniques are used to measure displacement fields in viscoelastic plates undergoing large deformations at elevated temperatures. These experimental procedures are applicable to determining displacement fields in nonlinear materials. As preliminary information, the material properties are determined from creep studies. The moiré method is used to determine the strains under constant load and isothermal conditions. Tests are conducted for several combinations of load and temperature for 2.5 decades of time. Assuming thermorheologically simple behavior, the data are shifted to establish the creep extensional compliance over ten decades in time. The constitutive equations are formulated as integral equations, the kernels of which are the functions that were measured in this work. These equations are solved exactly for the infinitesimal case. The finite case is then approximated by an incremental superposition of a series of successiye infinitesimal solutions. The results are applied to a plate initially containing a circular hole, and are shown to agree closely with the experimental measurements.     

Initiation and development of separated flow behind a circular cylinder in a supersonic stream

On the basis of the two-dimensional Navier-Stokes equations, the initiation and development of the separated flow behind a thermally insulated circular cylinder in a supersonic perfect-gas stream is investigated in relation to the Reynolds number. It is shown that the entire Re-range can be subdivided into a number of intervals with their own characteristic features. In particular, the conditions for the generation and development of global separation are established. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 27–36, November–December, 1998. The study was carried out with the support of the Russian Foundation for Basic Research (project No. 95-01-01129a).     

Hypersonic jet issuing into resting medium or a co-flowing supersonic stream

The calculation of supersonic ideal thermodynamically perfect gas jets discharging from a nozzle has been carried out previously in many studies (for example, [1, 2]) on the basis of the numerical method of characteristics. Here we give an approximate analytic solution of the problem of a supersonic jet discharging into a medium at rest or into a supersonic co-flowing stream.The author wishes to express his gratitude to G. G. Chernyi for his continued interest in this study.     

Vortical structures behind a transverse jet in a supersonic flow at high jet to crossflow pressure ratios

A three-dimensional supersonic turbulent flow with symmetric normal injection of circular jets from the channel walls is numerically simulated. The initial Favre-averaged Navier–Stokes equations closed by the k–ω turbulence model are solved by an algorithm based on an ENO scheme. The mechanism of the formation of vortical structures due to the interaction of the jet with the free stream is studied for jet to crossflow total pressure ratios ranging from 3 to 50. It is known from experiments reported in the literature that, for n ? 10, mixing of the jet with the high-velocity flow leads to the formation of a pair of vortices and of an additional separation zone near the wall behind the jet. It is demonstrated that the present numerical results are consistent with such findings and that the pressure distribution on the wall ahead of the jet in the plane of symmetry is also in reasonable agreement with available experimental data.     

Reinforcement of a plate with a circular hole by an eccentric circular patch

We study the reinforcement of an infinite elastic plate with a circular hole by a larger eccentric circular patch completely covering the hole and rigidly adjusted to the plate along the entire boundary of itself. We assume that the plate and the patch are in a generalized plane stress state generated by the action of some given loads applied to the plate at infinity and on the boundary of the hole. We use the power series method combined with the conformal mapping method to find the Muskhelishvili complex potentials and study the stress state on the hole boundary and on the adhesion line. We consider several examples, study how the stresses depend on the geometric and elastic parameters, and compare the problem under study with the case of a plate with a circular hole without a patch. In scientific literature, numerous methods for reinforcing plates with holes, in particular, with circular holes, have been studied. In the monographs [1, 2], the problem of reinforcing the hole edges by stiffening ribs is solved. Methods for reinforcing a circular hole by using two-dimensional patches pasted to the entire plate surface are studied in [3, 4]. The case of a plate with a circular cut reinforced by a concentric circular patch adjusted to the plate along the boundary of itself or along some other circle was studied in [5, 6]. The reinforcement of an elliptic hole by a confocal elliptic patch was considered in [7].