Investigation of nonequilibrium two-phase flows in axisymmetric laval nozzles

The singularities of two-phase flows in Laval nozzles were investigated within the framework of the model of a two-fluid continuous medium [1, 2] mainly in a quasi-one-dimensional approximation ([3] and the bibliography therein). Two-dimensional computations of such flows were performed only recently by using the method of buildup [4–7]. However, systematic computations to clarify the influence of the second phase on such fundamental nozzle characteristics as the magnitude of the specific impulse, its losses, and discharge coefficient were performed only in the quasi-one-dimensional approximation [8, 9] and only for the supersonic parts of the nozzle in the two-dimensional approximation under the assumption of uniform flow in the throat [10, 3]. Such an investigation is performed in this paper in the two-dimensional case for the nozzle as a whole, including the sub-, trans-, and supersonic flow domains, and a comparative analysis is given of the magnitudes of the loss of a unit pulse obtained in the quasi-one-dimensional approximation [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 86–91, November–December, 1977.     

Investigation of the flow structure in nozzles with a throat section of constant height

The results of a theoretical and experimental investigation of the flow structure in plane nozzles with a throat section of constant height are presented. The effect of longitudinal and transverse deviations in the nozzle geometry on the transonic and supersonic flow parameters is studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 160–166, November–December, 1990.     

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.     

Gas dynamic ozone generator

The formation of ozone when partially dissociated oxygen flows out of a supersonic nozzle has been investigated experimentally and theoretically. The supersonic flow of a chemically reacting gas mixture containing excess O atoms is calculated in the one-dimensional approximation for a class of plane wedge-shaped nozzles. It is shown that for initial gas pressures ahead of the nozzle inlet of about 10 atm and a temperatureT ₀=1000 K in nozzles with a total vertex angle of 30°C and a throat dimensionh.=1 mm it is possible to obtain an ozone concentration of about 1%, which is comparable with ordinary ozonizers, while the output of the device is two to three orders greater. Experiments on a shock tube fitted with a nozzle to measure the absorption of UV radiation by oxygen recombining in the nozzle under highly nonoptimal conditions revealed the presence in the flow of ozone molecules formed as a result of O+O₂ recombination.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 139–148, November–December, 1994.     

Experimental investigation of turbulence in a supersonic flow

The structure of disturbances carried by the flow into the working section of a supersonic wind tunnel has been investigated by means of a constant-current hot-wire anemometer. In order to generate the disturbances grids consisting of round rods were introduced upstream from the nozzle throat. It was found that in the working section the disturbances consist of non-correlating vortex, entropy and acoustic modes. The latter is generated by the boundary layer on the nozzle walls and the first two by the grids. The spectral compositions of the various modes are compared. Because of the presence of grid turbulence the point of laminar-turbulent transition in the boundary layer on a flat plate varied widely.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 119–124, May–June, 1990.     

Starting of an axisymmetric convergent-divergent nozzle in a hypersonic flow

The starting of an axisymmetric convergent-divergent nozzle, with the result that supersonic flow is formed within almost the entire channel, is modeled, as applied to the hypersonic aerodynamic setup of the Institute of Mechanics of Moscow State University. A successful starting is realized when the nozzle is thrown in a uniform supersonic air flow at a fairly high Mach number. The steady flow structure is studied. It is numerically shown that in the convergent section of the channel there arises an oblique shock wave whose interaction with the nozzle axis leads to the formation of a reflected shock and a curvilinear Mach disk with a region of unsteady subsonic flow in the vicinity of the throat. The mathematical model is based on the two-dimensional Euler equations for axisymmetric gas flows.     

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.     

Nozzle thrust increase by flow rotation

Until recently, in constructing optimum supersonic axisymmetric nozzles, gas motions without rotation about the axis have been considered. This question has been studied rather completely for equilibrium gas flows in [1–6]. Avoidance of flow rotation is a limitation which may reduce the nozzle thrust, other conditions being equal. It is easy to show that for zero nozzle length or for a length that permits obtaining uniform flow at the exist that maximum thrust is provided by uniform flow parallel to the nozzle axis. If the length limitation does not make it possible to do this, then the use of freedom in the gas rotation may increase the nozzle thrust. Using a very simple example it is shown that this possibility is realized.     

Flow of gas mixtures with vibrational energy relaxation in plane and axisymmetric nozzles

A method is described for the calculation of plane and axisymmetric flows of gas mixtures with vibrational energy relaxation in the subsonic, transonic, and supersonic regions of the nozzle. The method is based on numerical solution of the inverse problem of nozzle theory. Results are given for the flow of a C0₂-N₂-H₂O-He mixture with vibrational relaxation and compared with the results of one-dimensional calculations. It is found that vibrational-energy relaxation has a significant effect on the gasdynamic parameters of flow in nozzles with large, relative expansion and therefore in choosing a nozzle shape, especially in the supersonic region, it is necessary to calculate the nonequilibrium flow. It is shown that the geometry of the transonic and supersonic regions of the nozzle has a considerable effect on the distribution of the inverse population of the level and the amplification factor.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 125–131, September–October, 1977.     

Numerical investigation of gasdynamic features of control jets

Numerical methods, based on first order difference schemes are used to investigate features of three-dimensional subsonic and supersonic flows of an inviscid non-heat-conducting gas in control jets. Elements of the nozzle channels considered are axisymmetric, and flow symmetry arises from the nonaxial feature of the prenozzle volume and the subsonic part of the nozzle, or because of nonaxiality of elements of the supersonic part. In the first case the nozzle includes an asymmetric subsonic region in which reverse-circulatory flow is observed, and in the second case it includes a region of sudden expansion of the supersonic flow from the asymmetric stagnation zone.Translated from Izvestiya Akademii NaukSSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 126–133, November–December, 1978.The authors thank A. N. Kraiko for useful comments and M. Ya. Ivanov for his interest in this work.     

Influence of nozzle profile on the characteristics of a gas-dynamic laser

The results are given of numerical profiling and analysis of the influence of nozzle shape and the gas-dynamic parameters on the characteristics of gas-dynamic lasers. Investigation of the two-dimensional nonequilbrium flow in a family of similar nozzles and nozzles with different angles of inclination of the contracting part show that it is expedient to choose a shape of the subsonic part that ensures a straight sonic line. Relationships between the geometrical parameters of the subsonic and transonic part of the nozzle are recommended which ensure separationless flow and a shape of the sonic surface that is nearly flat. A parametric investigation was made of the supersonic section of two classes of planar gas-dynamic laser nozzles constructed on the basis of uniform and symmetric characteristics at the exit. The parametric investigations of the influence of the degree of expansion, the total pressure and the temperature, and also the gas composition show that the smallest losses of useful vibrational energy in the cavity are achieved for nozzles constructed on the basis of uniform characteristics.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 163–167, November–December, 1982.     

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.     

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C ^1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C ^1,α multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine–Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.     

Interaction of a blown gas jet with a supersonic oncoming stream without formation of a three-dimensional region of boundary layer separation

A study is made of the flow resulting from the interaction of a supersonic stream with a transverse sonic or supersonic jet blown at right angles to the direction of the main stream through a nozzle whose exit section is situated on a flat wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 137–139, May–June, 1981.     

On the possibility of blowing a gas jet into a supersonic flow without formation of a three-dimensional boundary-layer separation zone

We consider the flow formed by the interaction of a supersonic flow and a transverse sonic or supersonic jet blown at right angles to the direction of the main flow through a nozzle whose exit section is in a flat wall. When a gas jet is blown through a circular opening [1] the pressure rises in front of the jet because of the stagnation of the oncoming flow. This leads to separation of the boundary layer formed on the wall in front of the blowing nozzle. The resulting three-dimensional separation zone leads to a sharp increase in the pressure and the heat fluxes to the wall in front of the blowing nozzle, which is undesirable in many modern applications. The aim of the present investigation was to find a shape of the exit section of the blowing nozzle for which there is no three-dimensional separation zone of the boundary layer in front of the blowing nozzle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 162–165, May–June, 1979.     

Stabilization of detonation combustion in a high-velocity flow of a hydrogen-oxygen mixture

The flow of a hydrogen-oxygen mixture diluted with argon in a supersonic axisymmetric nozzle consisting of an inlet cylinder, a convergent region, a cylindrical throat, and a divergent region is considered. The supersonic flow enters the channel along the axis of symmetry. The flow structure is calculated with allowance for hydrogen ignition. A possibility of stabilizing the combustion zone is studied and the forces acting on the nozzle from the flow are determined. The problem is solved in the two-dimensional approximation with account for detailed combustion kinetics.     

Theory of two-component gasdynamic laser

In the present paper a numerical calculation is made of the vibrational relaxation of a binary mixture of molecular nitrogen and carbon dioxide gas. The calculation is performed for the entire range of variation of the concentrations of the components and over a wide range of mixture temperatures and pressures for various geometries of the supersonic part of the nozzle (throat dimensions, degree of expansion). It is shown that population inversion of the CO₂ molecules exists within a certain range of variation of the parameters of the mixture and the nozzle. The population inversion of the vibrational levels and the gain of the gaseous mixture are calculated as functions of these parameters and of distance measured from the critical cross section of the nozzle. The energy characteristics of the two-component gasdynamic laser are optimized.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 23–30, May–June, 1974.     

On profiling supersonic nozzles to achieve uniform flow in the annular exit section

The problem of profiling supersonic nozzles with a central body in order to achieve uniform supersonic flow with Mach number M _e >1 in the annular exit section is studied. Equal lengths of the profiled inner (central body) and outer nozzle walls can be achieved by displacing downstream the initial bend in the inner contour, i.e., in the central body wall. A displacement of this kind was proposed and tested in [1, 2] for annular nozzles with circular exit sections.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 204–206, March–April, 1995.     

Results of interferometer study of laval nozzle

Interferometric measurement of the air density in a supersonic nozzle of rectangular cross section is described. The flow structure is studied in a real Laval nozzle. It is shown that the core flow follows the laws of motion of an ideal gas and has a wave nature. The relation _Z=(3–4)_y is obtained for the boundary layer thickness on the nozzle walls for nozzle width-height ratio L/h=3.75–7.5.The flow structure in a real supersonic nozzle may differ significantly from the theoretical structure, both because of defects in nozzle fabrication and because of boundary layer growth on the nozzle walls. In many casesitis important to know the param'eters of the supersonic flow in the actual nozzle. The determination of these parameters (density , pressure , temperature T, velocity u, Mach number M) at any section of the nozzle in question is the objective of the present investigation.The authors wish to thank V. P. Koronkevich for his assistance in this study.     

Detonation combustion of hydrogen in a Laval nozzle under rarefied atmosphere conditions

Detonation combustion of a hydrogen-air mixture entering an axisymmetric convergent-divergent nozzle at a supersonic velocity is considered under atmospheric conditions at altitudes up to 24 km. The investigation is carried out on the basis of the two-dimensional gasdynamic Euler equations for a multicomponent reacting gas. The limiting altitude ensuring detonation combustion in a Laval nozzle of given geometry is numerically established for freestream Mach numbers 6 and 7. The possibility of the laser initiation of detonation in a supersonic flow of a stoichiometric, preliminarily heated hydrogen-air mixture is experimentally studied. The investigation is carried out in a shock tube under conditions simulating a supersonic flow in the nozzle throat region.