“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.     

Boundary-layer separation on a cooled body and interaction with a hypersonic flow

In previous papers, e.g., [1, 2], boundary-layer separation was investigated by analyzing solutions of the boundary-layer equations with a given external pressure distribution. In general, this kind of solution cannot be continued after the separation point. Study of the asymptotic behavior of solutions of the Navier-Stokes equations [3–5] shows that, in boundarylayer separation in supersonic flow over a smooth surface, the main effect on the flow in the immediate vicinity of the separation point is a large local pressure gradient induced by interaction with the external flow. The solution can be continued beyond the separation point and linked to the solutions in the other regions, located downstream [5]. Similar results for incompressible flow were recently obtained in [6]. We can assume that in general there is always a small region near the separation point in which separation is self-induced, and where the limiting solution of the Navier-Stokes equations does not contain unattainable singular points. However, this limiting slope picture can be more complex and can contain more regions where the behavior of the functions differed from that found in [3–6]. The present paper investigates separation on a body moving at hypersonic speed, where the ratio of the stagnation temperature to the body temperature is large. It is shown that both. for hypersonic and supersonic speeds the flow near the separation point is appreciably affected by the distribution of parameters over the entire unperturbed boundary layer, and not only in a narrow layer near the body, as was true in the flows studied earlier [3–6]. Regions may appear with appreciable transverse pressure drops within the zone occupied by layers of the unperturbed boundary layer. Similarity parameters are given, the boundary problems are formulated, and the results of computer calculation are presented. The concept of subcritical and supercritical boundary layers is refined, and the dependence of pressure coefficients responsible for separation on the temperature factor is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 99–109, November–December 1973.     

Iterative method for supersonic flow laminar boundary layer interaction in the presence of separation

The classical two-dimensional compressible boundary-layer equations supplemented by a relation describing the interaction of boundary layer with external inviscid flow (see, e.g., [1]) are treated as the governing equations in one of the methods to study the viscous-inviscid interaction. It is then necessary in the case of supersonic flow to specify certain downstream boundary conditions for the closure of the governing system, i.e., it is a boundary-value problem (e.g., [2]). The shooting technique for parameters at the beginning of the computational region to obtain the solution satisfying such a condition usually requires large computer time since the integral curves are highly sensitive to small changes in upstream boundary conditions. A more effective method is the algorithm of global relaxations of pressure distribution along the entire computational region [1]. A numerical method to compute supersonic interacting boundary layer in the presence of separation is presented in this paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 89–93, January–February, 1984.     

Flow behind the boundary layer separation point in a supersonic stream

The flow structure behind the separation point of a laminar boundary layer in a supersonic stream has been investigated. Analytic and numerical solutions are obtained for simple semiinfinite separation zones starting from the leading edge or a point on the smooth surface. The question of the pressure plateau in a separation zone of finite length is discussed and its value is calculated on the basis of asymptotic theory. The asymptotic theory of flow [1, 2] in the neighborhood of the separation point of the laminar boundary layer in a supersonic gas stream (region of free interaction) is employed. The local solution obtained is subsequently used to construct the flow pattern in the separation zone [3]. An analysis is made of the behavior of the solution for the free-interaction region on transition to the region of reverse flows. The results make it possible actually to compute (in the first approximation) the pressure in the plateau region after establishing the mathematical significance of this concept, previously introduced on the basis of the experimental results. At the same time relatively simple solutions are obtained for semiinfinite separation zones.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 19–25, May–June, 1971.     

The flow structure near the windward side of V-shaped wings with an attached shock wave on the leading edges

The results of a numerical calculation of a symmetric flow of supersonic gas with the Mach number M=3 past the windward side of V-shaped wings with an opening angle =40° and apex angles =30, 45, and 90° are given. The possibility of the ascent of one or two Ferri points from the break point of the transverse contour of the wing is discovered and explained. It is shown that conical flow near wings of finite length need not exist in flow regimes corresponding to angles of attack at which a Ferri point ascends, while at angles of attack smaller and larger than a certain interval, conical flow will exist. The investigation is conducted by means of a numerical method of stabilization with an artificial viscosity. The longitudinal coordinate, relative to which the steady system of equations is hyperbolic, played the part of the time variable, usual for methods of stabilization. The numerical method constructed using the scheme of [1] is described in [2] and was successfully applied to the calculation of different regimes of supersonic flow past conical wings with supersonic leading edges [2–6]. In. the present investigation the calculation algorithm of [2] is modified and makes it possible to realize motion with respect to the parameter a, this being particularly important for the stabilization of the solution in the calculation of flow regimes for which regions with a total velocity Mach number close to unity arise in the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–131, January–February, 1986.     

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.     

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.     

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.     

Supersonic flow past blunt bodies involving fast nonequilibrium processes

In the numerical integration of the system of equations of relaxation gasdynamics the solution may become unstable. Instability arises in those cases when the characteristic time for the nonequilibrium process becomes less than the characteristic flow time. To ensure stability it is necessary to reduce the integration step. With approach to equilibrium conditions, when the process rate increases, the step reduction may lead to excessive computational time. Preceding studies have overcome the difficulty in solving the one-dimensional [1–3] and two dimensional [4] problems by various techniques, the basic idea being the use of implicit difference schemes for approximating the relaxation equations.In the present paper analogous considerations are used to develop a scheme for calculating supersonic flow past blunt bodies with fast non-equilibrium processes within the framework of [5]. The basic coordinate system , is used to approximate the equations, just as in [5]. However the relaxation equation is solved along a streamline element. Calculations are presented for the air flow past a sphere with account for the oxygen dissociation reaction. The validity of the binary similarity law for this model is verified. As an example of the applicability of the technique, a calculation is made of the flow of a chemically reacting mixture with heat release about a sphere.     

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.     

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.     

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 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.     

Steady flow around a plane cascade by an ideal gas

The steady separation-free flow around a flat cascade by an ideal gas is discussed. Most of the attention is devoted to blocking regimes with a supersonic velocity in the entire flow and its subsonic component normal to the front of the cascade. A directing action of the cascade (the direction of the velocity and the Mach number of the advancing flow turn out to be related) is exhibited in these regimes which is a consequence of an independence of the flow in front of the cascade of the conditions behind it [1–5]. The most widespread method of their calculation [3, 4, 6] is based on the method of characteristics with establishment of the flow outside the cascade in a timelike coordinate. Although the integrated conservation laws also permit finding the parameters at infinity, the numerical construction of as long-range fields as desired with periodic sequences of attenuating discontinuities is practically impossible. The approximation of nonlinear acoustics (ANA) [7, 8] is justified here, as it is very effective in such problems [8–12]. A combination of ANA, the integrated conservation laws, and establishment in a calculation according to [13, 14] with isolation of the discontinuities has been realized in [5] for the construction of a solution on the entrance section of a cascade and everywhere in front of it. Below the method of [5] is extended to the entire flow and simplified even more. The flow on the entrance section of the cascade is, just as in [3], found in the approximation of a simple wave, in the rest of it and in a finite strip behind it-the flow is found with the help of the straight-through version of the scheme of [13, 14], and in the long-range field-in the ANA. A simpler version is proposed. In it ANA is applied outside the cascade and the linear theory is applied inside the cascade. Examples of the calculations are given. Similarity laws are formulated for all the regimes of streamline flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 35–43, November–December, 1984.     

Calculation of viscous compressible gas flow past a corner

We consider the problem of calculating the parameters for supersonic viscous compressible gas flow past a corner (angle greater than ). The complete system of Navier-Stokes equations for the viscous compressible gas is solved in the small vicinity Q₁. (characteristic dimensionl~1/R) of the corner point. The conditions for smooth matching of the solution of the Navier-Stokes equations and the solution of the ideal gas or boundary layer equations are specified on the boundary of Q₁. All these solutions are a priori unknown, and the conditions for smooth matching reduce to certain differential equations on the boundary of Q₁. Here account is taken of the interaction of the flows near the wall surface and in the so-called outer region [1].We note that no a priori assumptions are made in Q₁ concerning the qualitative behavior of the solution, in contrast with other studies on viscous flow past a corner (for example, [2–4]).The Navier-Stokes system in Q₁ is solved numerically, using the difference scheme suggested in [5]. This scheme permits obtaining the steady-state solution by the asymptotic method for large Reynolds numbers R, and also has an approximation accuracy adequate to account for the effects of low viscosity and thermal conductivity.     

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.     

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.     

Oblique interaction of alfvén and contact discontinuities in magnetohydrodynamics

A general method of solving problems of the interaction of stationary discontinuities is proposed. The problem of the oblique incidence of an Alfvén plane-polarized discontinuity on a contact discontinuity is examined in the general formulation. A solution is constructed numerically over the entire range of variation of the governing parameters. A number of effects associated with the magnetohydrodynamic nature of the interaction are explored. For example, the formation in space of sectors in which the density falls by several orders (almost to a vacuum) is detected. The solutions obtained are of interest, for example, for investigating the interaction between Alfvén discontinuities in the solar wind and the magnetopause, plasmopause and other inhomogeneities whose boundary can be approximated by a contact discontinuity [13–15].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 131–142, January–February, 1990.     

Calculation of hysteresis and flow-rate oscillations of base pressure in supersonic annular nozzles

The interaction of the turbulent axisymmetric near wake behind the face of the central body of an annular nozzle with the supersonic annular jet discharging from this nozzle is analyzed. The flow in the monoparametric near wake is calculated by the integral method [1] while the flow in the nonviscous jet is calculated by the method of through calculation using a monotonic explicit difference system of the first order of accuracy [2]. The interaction between the nonviscous and turbulent streams is determined by the displacement thickness of the wake. The initial conditions of the wake are determined from the integral conditions of attachment with the mixing flow in the isobaric base region. The interaction flow is described by the particular solution of the equations which passes through the singular saddle point — the throat of the wake. The near wake and base pressure in different modes of discharge from an annular nozzle at the exit cross section of which the ratio of outer and inner radii is y₂/y₁ = 1.3 and the Mach number is M = 2.54 are calculated as an example. The region of hysteresis of the base pressure, connected with the ambiguity of the interaction flow owing to the formation of the throat of the wake within the first or second barrel of the jet, and the parameters of the low-frequency flow-rate oscillations of base pressure in this region are determined. The results of the calculations are in satisfactory agreement with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 125–130, January–February, 1977.     

Numerical Simulation of a Supersonic Flow with Transverse Injection of Jets

A plane supersonic flow with symmetric perpendicular injection of jets through slots in the walls is numerically simulated with the use of Navier–Stokes equations. The effect of the jet pressure ratio and Mach number on the flow structure is considered. The angle of inclination of the shock wave and the separationregion length are found as functions of the jet pressure ratio. The influence of the jet pressure ratio on the increase in the lift force arising owing to interaction of the flow with the injected jet is found.