Numerical investigation of two-phase flow with coagulation and fragmentation of particles in axisymmetric laval nozzles

The method and results of a calculation of the parameters of a polydis-perse two-phase flow with coagulation and fragmentation of particles in collisions are described. The problem is considered in a two-dimensional formulation. The secondary particles (fragments) formed by fragmentation are assumed to have arbitrary mass and velocity distributions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 145–153, May–June, 1982.We thank F. G. Gaponich for assistance in the computer calculations.     

Subsonic and transonic flow field in two-dimensional de laval nozzles

This paper presents solutions of subsonic and transonic flow fields in two-dimensional De Laval nozzles with preassinged contraction ration ₁, expansion ration ₂, and throat wall radiusR _*. The effects of the contraction and the expansion angle on nozzle flow, the transformation of flow pattern of a De Laval nozzle in the throat region, and the conditions of occurrence and the governing parameters of the supersonic bubbles are discussed.     

Some peculiarities of two-phase flow in nozzles

Calculations are conducted for unidimensional two-phase flow in nozzles for a wide range of particle concentrations and dimensions. It is established that there exists a maximum in loss of specific momentum due to a lag in particle velocity and temperature relative to the gas. The results obtained are compared with calculations using linearized theory as well as with experimental data. The agreement between calculation and experiment is noted. Equilibrium flow of a two-phase mixture with solidification of liquid particles is considered. The presence of an anomalous flow region is established, where in the model of an ideal unidimensional equilibrium flow in a nozzle with discharge into a vacuum the presence of two successively located minimum sections is necessary.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 50–57, May–June, 1973.     

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.     

Numerical investigation of two-phase flows in axisymmetric laval nozzles with allowance for processes of coagulation and division of condensed particles

In the framework of a model in which a polydisperse two-phase medium is represented as a multivelocity continuum [1, 2] a numerical investigation is made into the features of two-phase flows in axisymmetric Laval nozzles with allowance for processes of coagulation and division of particles. Calculations were made for a step distribution of the condensed particles with respect to their diameters (Lagrange's method [3, 4]). The flow pattern as a whole was determined, and also the most important integrated characteristics of the nozzles, including the unit impulse loss. Some qualitative results of calculations of such two-dimensional flows were obtained earlier in [6], but systematic investigations have been made only in the quasione-dimensional approximation (see [3, 4] and the bibliography there).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 82–90, January–February, 1980.I thank A. A. Glazunov for great assistance in the calculations and am indebted to I. M. Vasenin for his interest in the work and also for a number of comments of a critical nature.     

Dynamics of two-phase downwards flows in submerged entry nozzles and its influence on the two-phase flow in the mold

Gas–liquid flows inside the submerged entry nozzle (SEN) of a slab mold and its influence in the flow field in the mold were studied using video recording, mathematical simulations and particle image velocimetry (PIV) approaches. Bubbly and annular flows in the SEN yield structurally-uncoupled and structurally coupled flows in mold, respectively. High gas loads (ratio of mass flux of gas and mass flux of liquid) at high casting rates lead to increases of bubble population and bubbles sizes due to coalescence processes whose rate exceeds that of their breakup. The presence of gas bubbles or gas layers inside the SEN lead to periodical twisting of the liquid flow that induces biased flows through both ports yielding uneven flows in the mold. A multiphase mathematical model predicts acceptably well the flow dynamics of two-phase flows inside the SEN.     

Flow of a stream containing charged inertial particles past a conducting sphere

The problem of the flow of a hydrodynamic stream containing electrically charged particles past a conducting sphere is solved. The influence of the volume density of the electric charge and the potential of the sphere on the capture coefficient is determined for different values of the inertia parameter of the particles and different drag laws for them. It is shown that for an earthed sphere can appreciably exceed unity. In the formulation of the problem, it is assumed that the region of electrogasdynamic flow is bounded by two electrode grids, which simulate the exit of the source of charged particles and the surrounding electric conditions. The velocity field near the sphere is assumed irrotational. The problem is solved numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 21–26, July–August, 1982.I thank A. B. Vatazhin for assistance in the work.     

Theory of a laval nozzle for a two-phase mixture containing particles of small lag

A two-velocity and two-temperature model is considered for a continuous medium in relation to the flow of a mixture of gas and particles in the subsonic, transsonic, and supersonic parts of a Laval nozzle. It is assumed that the particles are small, and hence that the coefficients f and ^q, which define the interaction with the gas, are large (these coefficients are inversely proportional to the square of the particle radius for a Stokes mode of flow). This means that the velocity or thermal lag of the particles relative to the gas is small. The solution is sought as expansions with respect to the small parameters ₁=1/f and ₂=1/^q.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 89–100, July–August, 1973.     

Flow around a flexible cylindrical shell by a plane stream of an ideal liquid

The flow by a plane stream of an ideal liquid around a cylindrical shell of zero flexural stiffness (a soft cylindrical shell), or a gas bubble on the boundary of which forces of tension act, was studied in [1–6]. The flow around an elastic plate in a linear formulation was considered in [7, 8]. We consider the flow, around a flexible cylindrical shell which possesses a flexural stiffness and at the same time admits large displacements, by a plane system of an ideal incompressible liquid. An application of methods of the theory of functions of a complex variable leads to an effective solution of the problem. The shape of the shell, the forces in it, the forces acting on the shell, and the field of velocities of the flow of the liquid are determined.     

Calculation of mixed sub- and supersonic gas flow in a plane pressure nozzle

Plane vortex-free sub- and supersonic gas flow is considered without taking account of viscosity and heat conduction. For the system of equations of motion of the mixed elliptic-hyperbolic type in the potential plane, a particular solution is found which corresponds to gas motion in a nozzle with a curved sonic line. The system of equations used to construct the solution is neatly homogeneous, which made it possible to separate the principal part of the solution in the transonic region. The validity of the simplifications made is well confirmed by comparison with calculation, using the method of characteristics, and with experiment.The author wishes to thank S. V. Fal'kovich for valuable comment on this study.     

Asymptotic representation of the solution in the hodograph plane in transonic flow problems around profiles

The flow around a slender profile by an ideal gas flow at a constant, almost sonic, velocity at infinity is considered. The behavior of the perturbed stream in the domain upstream of the compression shocks sufficiently remote from the streamlined body is studied. The question is investigated of what conditions the solution in the hodograph plane satisfies when it corresponds to a flow without singularities on the limit characteristic in the physical flow plane. It is known that cases are possible when a regular solution in the hodograph plane loses its regularity property upon being mapped into the physical plane [1]. A regular flow on the limit characteristic can be continued analytically downstream into the supersonic domain between the limit characteristic and the shock. The requirement of analyticity of the streamlined profile is essential for realizability of the flow under consideration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 84–88, January–February, 1976.In conclusion, the author is grateful to O. S. Ryzhov for discussing the research.