Investigation of the starting process in the shaped nozzle of a large-diameter shock tube

The characteristics of flow formation in the shock-wave starting process of a shaped nozzle with a developed subsonic section are analyzed on the basis of numerical calculations in the inviscid two-dimensional formulation. It is shown that in this case the use of steady-state boundary conditions in the throat section can lead to a significant error in determining the startup time. The results of the calculations are compared with experimental data obtained over a broad interval of the leading parameters. The results of the comparison help to explain the part played in the starting process by flow separation from the nozzle walls, which was not taken into account in the numerical investigation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 88–95, September–October, 1986.The authors wish to express their thanks to R. I. Serikov and V. M. Khailov for the wind tunnel nozzle blowdown data and to V. P. Stulov for useful advice and discussions.     

The maximum flow rate principle and vortex chamber aerodynamics

It is shown that for fixed kinetic energy density the maximum flow rate principle previously used for determining the radius of the free surface of an incompressible fluid flowing from a centrifugal nozzle admits an equivalent formulation within the context of the Lagrangian formalism, which means that the principle can easily be extended to the case of a compressible gas. The results of the calculations are compared with experiment.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 49–55, May–June, 1989.     

Calculation of the flow of a two-phase mixture in a Laval nozzle,allowing for the turbulent diffusion of the particles

The flow of a mixture of gas and condensed particles in an axisymmetric Laval nozzle is considered. The motion of the particles is calculated in a specified field of gas flow, with due allowance for their turbulent diffusion. The results of calculations indicating the necessity of allowing for this phenomenon when considering the motion of particles toward the wall of a profiled nozzle are presented.Translated from Izvestiya Akademii Nauk SSSR, No. 2, pp. 161–165, March–April, 1973.     

Determination of optimum nozzle parameters in a gasdynamic laser

In connection with the use of supersonic nozzles to create lasers, the question arises of the optimum parameters of the nozzle and the gas mixture from the aspect of obtaining the greatest population inversion of the energy levels of internal degrees of freedom of molecules of the working gas and the greatest output power of the lasers. A rather complete concept of the kinetic processes taking place during the escape of a relaxing gas mixture containing carbon dioxide through a supersonic nozzle has now been developed on the basis of calculated and experimental data. In [1–4] the problems of optimization of the parameters of a CO₂-N₂-H₂O-He mixture and of the shape of the nozzle were set up and solved in a one-dimensional steady-state formulation. The influence of the two-dimensionality of the stream in an optimum nozzle on the laser characteristics is studied in the present report. The method of through calculation suggested in [5] is used to calculate the two-dimensional flow of a relaxing gas.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 23–26, September–October, 1978.     

Calculation of the swirling flow of an ideal gas in a laval nozzle

The numerical solution of the problem of the motion of a swirling flow of an ideal gas in a Laval nozzle in axisymmetric formulation is obtained by the method of stabilization. As a result, a number of effects appear that are essentially not one-dimensional, in particular, the drawing-in of the sonic line into the nozzle, an effect that leads to a decrease in the nozzle's expansion coefficient. The dependence of this coefficient on the intensity of the swirling is obtained. A number of problems connected with the control of the expansion of a gas through a Laval nozzle and with variation of the thrust of a nozzle can be solved successfully in cases where a rotary motion is imparted to the flow of gas exhausted from the nozzle. Investigation of such a swirling flow in [1, 2] and a number of other papers are based on a one-dimensional model of gas flow, which makes it possible in principle to obtain integrated characteristics of the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 72–76, September–October, 1971.     

Turbulent mixing of relaxing gases in a supersonic nozzle

Processes of turbulent mixing of hot nitrogen and carbon dioxide gas (CO₂) in a supersonic nozzle are investigated in connection with gasdynamic lasers with selective thermal excitation. The actual flow pattern in the nozzle and cavity is simulated by the system of equations of a turbulent boundary layer. The results of calculations of the gain profile, the CO₂ concentration, and the laser power agree satisfactorily with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 156–160, March–April, 1981.We are very grateful to B. S. Aleksandrov and V. K. Pozdyshev for numerous helpful discussions.     

Calculation of axially symmetrical free expansion of a no ne quilibrium hydrogen plasma

A scheme and the results of a calculation by the method of characteristics are presented for free expansion of a nonviscous, thermally nonconducting, nonequilibrium, optically thick hydrogen plasma from a round supersonic nozzle. The elementary process determined is considered to be collision-radiative recombination. A strong disturbance in the thermal and ionization equilibrium are observed in the flow field. The effect of relaxation processes on the geometry of flow and the field of gas-dynamic parameters is examined. The results of the calculations are compared with analogous data for an ideal perfect gas.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 27–30, July–August, 1972.     

Numerical simulation of the dynamics of a liquid in a moving axisymmetric cavity

The problem of the motion of an ideal liquid with a free surface in a cavity within a rigid body has been most fully studied in the linear formulation [1, 2]. In the nonlinear formulation, the problem has been solved by the small-parameter method [3] and numerically [4–7]. However, the limitations inherent in these methods make it impossible to take into account simultaneously the large magnitude and the threedimensional nature of the displacements of the liquid in the moving cavity. In the present paper, a numerical method is proposed for calculating such liquid motions. The results of numerical calculations for spherical and cylindrical cavities are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–177, March–April, 1984.     

Theory of vortical helical ideal gas flows in laval nozzles

An asymptotic solution is found for the direct problem of the motion of an arbitrarily vortical helical ideal gas flow in a nozzle. The solution is constructed in the form of double series in powers of parameters characterizing the curvature of the nozzle wall at the critical section and the intensity of stream vorticity. The solution obtained is compared with available theoretical results of other authors. In particular, it is shown that it permits extension of the known Hall result for the untwisted flow in the transonic domain [1]. The behavior of the sonic line as a function of the vorticity distribution and the radius of curvature of the nozzle wall is analyzed. Spiral flows in nozzles have been investigated by analytic methods in [2–5] in a one-dimensional formulation and under the assumption of weak vorticity. Such flows have been studied by numerical methods in a quasi-one-dimensional approximation in [6, 7]. An analogous problem has recently been solved in an exact formulation by the relaxation method [8, 9]. A number of important nonuniform effects for practice have consequently been clarified and the boundedness of the analytical approach used in [2–7] is shown.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 126–137, March–April, 1978.The authors are grateful to A. N. Kraiko for discussing the research and for valuable remarks.     

Three-dimensional interaction of a freely expanding jet with obstacles

A numerical scheme is proposed for calculating the steady three-dimensional flow in the subsonic and transonic regions of interaction of a freely expanding jet and an infinite flat obstacle at angles of attack up to 20 ° and the end of a cylinder placed relative to the jet axis at a distance up to ten nozzle radii. Results of calculations are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 22–27, January–February, 1981.We thank A. P. Zyuzin for making a number of calculations.     

Draft of a nozzle with the circulating outflow of a vortical flow of gas

The presence of circulation in an outflowing gas leads to a change in the working parameters of a nozzle. The question of the mass flow rate and the draft of a nozzle without a diffusor (a point) for twisted flows has been studied theoretically and experimentally [1–6]. The use of nozzles with a supersonic part introduces a considerable degree of complication into the method for the analytical calculation of the draft characteristics and the program for their experimental investigation. In [2, 7], a theory of a nozzle is formulated for a model of a potential circulating flow of gas; in [5, 8], an electronic computer was used to solve the complete system of the equations of gasdynamics for the motion of a rotating flow along a nozzle; in [7, 9], an investigation was made of a variational problem of the shaping of a diffusor for a circulation flow. The calculation of the draft, carried out in the above-mentioned communications (with the exception of [2], in which a study was made of a partial model of an eddyless rotational motion), is bound up with labor-consuming computer calculations. In the present article, in a development of [3, 6], a quasi-one-dimensional theory of a supersonic nozzle for a vortical flow of gas is formulated and verified experimentally.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–149, September–October, 1975.     

Experimental investigation of the internal ballistics of a pulsed water cannon

Measurements were made of the piston velocity and water pressure in the tube of a pulsed water cannon. The results are compared with calculations, which show that the quasistationary formulation of the problem gives sufficient accuracy.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 150–152, May–June, 1979.     

Calculation of three-dimensional weakly expanding flow in jets and channels

The results are given of a calculation of laminar flow in a channel of square section and the motion of a turbulent jet from a cruciform nozzle in an ambient flow. To calculate the secondary flows, the field of the transverse velocity is decomposed into irrotational and solenoidal components. The results of the calculation of the flow in the channel are compared with the calculations of other authors and experimental data. To calculate the flow in the turbulent jet, a one-parameter turbulence model is used, and the influence of the inhomogeneity of the distribution of the longitudinal component of the velocity on the components of the Reynolds stress tensor is taken into account. The results of calculation of the flow in the jet behind a cruciform nozzle are compared with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 36–44, July–August, 1984.     

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 study of flows of a viscous compressible gas and of an equilibrium gas mixture in a flat nozzle in the presence of strong blowing

The problem of the interaction of a viscous supersonic stream in a flat nozzle with a transverse gas jet of the same composition blown through a slot in one wall of the nozzle is examined. The complete Navier-Stokes equations are used as the initial equations. The statement of the problem in the case of the absence of blowing coincides with [1]. The conditions at the blowing cut are obtained on the assumption that the flow of the blown jet up to the blowing cut is described by one-dimensional equations of ideal gasdynamics. The proposed model of the interaction is generalized to the case of flow of a multicomponent gas mixture in chemical equilibrium. The exact solutions found in [2] are used as the boundary conditions at the entrance to the section of the nozzle under consideration. The results of numerical calculations of the flows of a homogeneous nonreacting gas and of an equilibrium mixture of gases consisting of four components (H₂, H₂O, CO, CO₂) are given for different values of the parameters of the main stream and of the blown jet. In the latter case it is assumed that the effect of thermo- and barodiffusion can be neglected.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 55–63, July–August, 1974.     

Ventilated entry of a slender profile into a compressible liquid at subsonic velocity

The problem of vertical entry at subsonic velocity into an ideal compressible fluid is solved in the linear formulation for a slender profile with open attached cavity. An integral equation is obtained for the potential of the accelerations. Expressions are given for the calculation of the drag of a thin wedge and also some results of calculations which show that in the limiting case of infinite depth of penetration, which corresponds to stationary flow past the thin wedge with separation of a jet, an analog of the Prandtl—Glauert theorem holds.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 9–17, July–August, 1980.     

Calculation of the aerodynamic characteristics of a helicopter lifting rotor subjected to air gusts

A numerical method of calculating the unsteady aerodynamic characteristics of a helicopter lifting rotor smoothly enveloped by an air gust arriving from an arbitrary direction is proposed. The problem is solved in the non-linear formulation for an incompressible medium. The flow over each blade is simulated by a system of discrete vortices. The deformation of the vortex trail from the blades under the influence of the gust is taken into account. The results of the calculations are reported.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 153–158, March–April, 1987.     

Numerical and experimental investigation of a laval nozzle with a cylindrical throat

The results are reported of experimental and numerical investigation of mixed flow and of the parameters of heat transfer in the transonic region of an axisymmetric Laval nozzle whose throat is formed by a cylindrical surface, i.e., the nozzle contour near the minimum cross section contains two bends.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 189–192, September–October, 1984.     

Variational problem of optimum side wall design for a narrow three-dimensional nozzle

The optimum design of the side walls of the supersonic section of a three-dimensional nozzle with two planes of symmetry is considered in the narrow channel model approximation, which reduces three-dimensional to two-dimensional flow. This nozzle realizes maximum thrust for given sonic or supersonic inlet flow, upper and lower walls, maximum permissible length and pressure outside the nozzle. In general, an approximate solution of the variational problem can be obtained by the indeterminate control contour method [1]. For nozzles with nonexpanding end sections of the upper and lower walls this is a rigorous solution. Numerical algorithms, based on the method of characteristics, for constructing the optimum, side walls and calculating the flow in narrow channels are developed in the formulation adopted using the optimality conditions found, which generalize the wellknown conditions for plane and axisymmetric configurations [1]. In addition, the three-dimensional supersonic flow in the nozzles thus designed has been calculated in accordance with a shock-capturing marching scheme [2], which for the uniform grids employed in the calculations gives a second-order approximation. A rather complex relation is established between the thrust of the optimum configurations constructed and the shape of their inlet cross sections.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 102–112, March–April, 1992.The authors are grateful to L. E. Sternin for drawing their attention to the problem and to V. A. Vostretsova for assisting with the work.     

Nonequilibrium homogeneous condensation in rarefaction waves

Results of numerical calculations of condensing vapor expanding into a vacuum and in a supersonic conical nozzle are presented. Nonequilibrium homogeneous condensation is compared with the theory of determining parameters. It is proved that equality of the determining parameters ensures the unity of the course of such processes in different devices. The influence of unsteady nucleation on the maximum supercooling was also investigated. All the calculations were made for water and nitrogen vapors.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 152–156, March–April, 1986.