Investigation of a Gas Flow with Solid Particles in a Supersonic Nozzle

A two-phase flow with high Reynolds numbers in the subsonic, transonic, and supersonic parts of the nozzle is considered within the framework of the Prandtl model, i.e., the flow is divided into an inviscid core and a thin boundary layer. Mutual influence of the gas and solid particles is taken into account. The Euler equations are solved for the gas in the flow core, and the boundary-layer equations are used in the near-wall region. The particle motion in the inviscid region is described by the Lagrangian approach, and trajectories and temperatures of particle packets are tracked. The behavior of particles in the boundary layer is described by the Euler equations for volume-averaged parameters of particles. The computed particle-velocity distributions are compared with experiments in a plane nozzle. It is noted that particles inserted in the subsonic part of the nozzle are focused at the nozzle centerline, which leads to substantial flow deceleration in the supersonic part of the nozzle. The effect of various boundary conditions for the flow of particles in the inviscid region is considered. For an axisymmetric nozzle, the influence of the contour of the subsonic part of the nozzle, the loading ratio, and the particle diameter on the particle-flow parameters in the inviscid region and in the boundary layer is studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 65–77, November–December, 2005.     

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.     

Two-layer quasi-one-dimensional model for calculating gas-particle mixture flow in nozzles

The one-dimensional approximation is widely used at the present time to calculate gas-particle (solid or liquid) mixture flows in nozzles within the framework of the two-velocity (or multi-velocity) continuum model. Other studies have been made [1–6] in which the calculations of the two-phase flow in the supersonic part of the nozzle was made by the method of characteristics, and, within the limits of the model adopted, these results may be considered exact. Comparison of the exact and approximate results [6] has shown that even for nozzles of quite simple form (nearly conical) the accuracy of the one-dimensional approximation in the case of mixture flow is considerably lower than for the pure gas, and the computation error increases with increase in the relative particle flow rate. This deterioration of the accuracy is to a considerable degree caused by flow stratification, which arises because of particle lag and leads to the formation of a wall region of pure gas. For high particle content, the wall layer, in which the gas is not subjected to thermal and dynamic input from the particles, has the nature of a low-entropy, low-temperature, high-velocity layer with parameters which differ significantly from the gas parameters in the region occupied by the particles.Therefore, in the present study a modification was made in the one-dimensional theory, based on separate averaging of the flow in the wall layer and in the core, where the gas flows together with the foreign particles. Comparison of the exact results with those obtained with the aid of conventional one-dimensional theory and the proposed two-layer model showed that this modification of one-dimensional theory led to a considerable reduction in the errors of calculation for the flow parameters.In conclusion, the authors wish to thank S. Yu. Krasheninnikov for suggesting this study and also N. S. Galyun, A. M. Konkin, and L. P. Frolov for assistance in the investigation.     

Solution of the direct problem of a laval nozzle working with a two-phase medium

A study is made of a method of numerical solution of the system of ordinary differential equations describing the flow of a two-phase medium in a Laval nozzle in the one-dimensional approximation. The paper is concerned with the direct problem, in which the law of variation of the area of the transverse section of the nozzle and functional relationships between the entrance parameters are given and the unknown parameters of the two-phase flow are found along the length of the nozzle.     

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.     

Calculation of the parameters of two-phase flow in an axisymmetrical laval nozzle allowing for the coagulation and atomization of the particles

The laws governing changes taking place in the parameters of the heterogeneous flow in a Laval nozzle were studied on the one-dimensional approximation in [1, 2]; a flow containing particles of uniform size was considered in [3–5]. In this paper we shall consider a method of calculating the parameters of a two-phase flow in the sub- and supersonic parts of an axisymmetrical Laval nozzle with due allowance for the coagulation and atomization of the particles, and shall present the results of some corresponding calculations.     

Extremal nozzle contours for gas flows with particle lag

The determination of the extremal nozzle contour for gas flow without foreign particles has been carried out in several studies [1–6], based on the calculation of the flow field using the method of characteristics.In [7, 8] the equations are derived for the characteristics and the relations along the streamlines which are required for calculating two-dimensional gas flow with foreign particles. The variational problem for two-phase flow in the two-dimensional formulation may be solved by the method of Guderley and Armitage [9] with the use of equations given in [7] or [8]; however this method is very tedious, even with the use of high-speed computers.In [10, 11] studies are made of two-phase one-dimensional flows by expanding the unknown functions in series in a small parameter, defined by the particle dimensions. In [12] a solution is given for the variational problem (in the one-dimensional formulation) of designing the contour of a nozzle with maximal impulse. However that study does not take account of the static term appearing in the impulse and the solution is obtained in relative cumbersome form. Moreover, the question of account for the losses due to nonparallelism and nonuniformity of the discharge was not considered.The present paper considers in the one-dimensional formulation the flow of a two-phase medium in a Laval nozzle with small particle lags (in velocity and temperature). The variational problem of determining the maximal nozzle impulse is formulated along the nozzle contour for fixed geometric expansion ratio. The impulse losses due to nonparallelism of the discharge are simulated by a function which depends on the ordinates which are variable along the contour and on the slope of the tangent to the contour.The author wishes to thank Yu. D. Shmyglevskii and A. N. Kraiko for helpful discussions and V. K. Starkov for carrying out the calculations on the computer.     

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.     

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.     

Hyperbolic Approximation of the Navier-Stokes Equations for Viscous Mixed Flows

Simplified two-dimensional Navier-Stokes equations of the hyperbolic type are derived for viscous mixed (with transition through the sonic velocity) internal and external flows as a result of a special splitting of the pressure gradient in the predominant flow direction into hyperbolic and elliptic components. The application of these equations is illustrated with reference to the calculation of Laval nozzle flows and the problem of supersonic flow past blunt bodies. The hyperbolic approximation obtained adequately describes the interaction between the stream and surfaces for internal and external flows and can be used over a wide Mach number range at moderate and high Reynolds numbers. Examples of the calculation of viscous mixed flows in a Laval nozzle with large longitudinal throat curvature and in a shock layer in the neighborhood of a sphere and a large-aspect-ratio hemisphere-cylinder are given. The problem of determining the drag coefficient of cold and hot spheres is solved in a new formulation for supersonic air flow over a wide range of Reynolds numbers. In the case of low and moderate Reynolds numbers a drag reduction effect is detected when the surface of the sphere is cooled.     

Investigation of isothermal flow of a mixture of gas and particles in a channel of variable section

Isothermal flow of a gas with particles is investigated analytically, which makes it possible to analyze all possible flow regimes in channels of different shapes. It is shown that in a channel of constant section there are two possibilities: either an equilibrium regime is established with constant flow parameters, or the gas reaches the velocity of sound, and then further flow in the channel is impossible (blocking of the channel). In a contracting nozzle, blocking also occurs if the channel is sufficiently long. In an expanding nozzle when there are particles in the gas with a velocity lower than the gas velocity, it is possible to have flow regimes with transition through the velocity of sound: a subsonic flow goes over into a supersonic flow and, conversely, it is also possible to have a flow in which there is blocking of the channel, which is quite different from the flow of a pure gas in an expanding nozzle and is due to the influence of interphase friction on the flow. The variation of the pressure along the flow can be nonmonotonic with points of local maximum or minimum which do not coincide with the singular point at which the gas velocity reaches the velocity of sound. In the case of nonequilibrium gas flows with particles in a Laval nozzle, the velocity of the gas may become equal to the isothermal velocity of sound not only in the exit section of the nozzle or in its expanding part, as noted in [4–6], but also at the minimal section, since it is possible to have flows for which the velocities of the phases are equalized at this section.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 61–68, October–December, 1981.     

Flow of a two-phase medium in a laval nozzle

The back reaction of particles on a gas flow in Laval nozzles was investigated experimentally. Experimental data were obtained that characterize the change produced by the particles of a solid phase in the shape of the sonic line, the pressure distribution on the nozzle profile, and the configuration of the shock waves in the jet. Flow rate coefficients are given for different nozzle profiles and mass fraction and sizes of the particles in the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 107–111, January–February, 1981.     

Stagnation region of a blunt body in two-phase hypersonic flow

The fluid-mechanics equations of a two-velocity, two-temperature medium are used to investigate flow near the stagnation point of a blunt body washed by a hypersonic stream of gas containing solid or liquid deformed particles. The effect of particles of the gasdynamic flow parameters is analyzed. A relaxation layer was found to occur near the body, with marked changes in the gas parameters. It is shown that the presence of particles in the flow reduces the shock stand-off distance. The results of computations on the dynamics and heating of particles in the shock layer are discussed. A solution in finite form is obtained in the limiting case of fine particles by the method of asymptotic expansions. The motion of solid or liquid particles in hypersonic shock layers has been the subject of several papers [1–6], in which particle dynamics was examined, assuming that the particles have a negligible influence on the gasdynamic flow parameters. The solutions obtained are therefore limited to the case of low mass particle concentration in the incident flow. A numerical solution not subject to this limitation was obtained in [7] for supersonic two-phase flow over a wedge.     

Solution of the direct problem of the mixed flow of a conducting gas in a laval nozzle with a meridional magnetic field

We consider the direct problem in the theory of the axisymmetric Laval nozzle (including sonic transition) for the steady flow of an inviscid and nonheat-conducting gas of finite electrical conductivity. The problem is solved by numerical integration of the equations of unsteady gas flow using an explicit difference scheme that was proposed by Godunov [1,2], and was used to calculate steady and unsteady flows of a nonconducting gas in nozzles by Ivanov and Kraiko [3]. The subsonic and the supersonic flows of a conducting gas in an axisymmetric channel when there is no external electric field, the magnetic field is meridional, and the magnetic Reynolds numbers are small have previously been completely investigated. Thus, Kheins, Ioller and Élers [4] investigated experimentally and theoretically the flow of a conducting gas in a cylindrical pipe when there is interaction between the flow and the magnetic field of a loop current that is coaxial with the pipe. Two different approaches were used in the theoretical analysis in [4]: linearization with respect to the parameter S of the magnetogasdynamic interaction and numerical calculation by the method of characteristics. The first approach was used for weakly perturbed subsonic and supersonic flows and the solutions obtained in analytic form hold only for small S. This is the approach used by Bam-Zelikovich [5] to investigate subsonic and supersonic jet flows through a current loop. The numerical calculations of supersonic flows in a cylindrical pipe in [4] were restricted to comparatively small values of S since, as S increases, shock waves and subsonic waves appear in the flow. Katskova and Chushkin [6] used the method of characteristics to calculate the flow of the type in the supersonic part of an axisymmetric nozzle with a point of inflection. The flow at the entrance to the section of the nozzle under consideration was supersonic and uniform, while the magnetic field was assumed to be constant and parallel to the axis of symmetry. The plane case was also studied in [6]. The solution of the direct problem is the subject of a paper by Brushlinskii, Gerlakh, and Morozov [7], who considered the flow of an electrically conducting gas between two coaxial electrodes of given shape. There was no applied magnetic field, and the induced magnetic field was in the direction perpendicular to the meridional plane. The problem was solved numerically in [7] using a standard process. However, the boundary conditions adopted, which were chosen largely to simplify the calculations, and the accuracy achieved only allowed the authors [7] to make reliable judgments about the qualitative features of the flow. Recently, in addition to [7], several papers have been published [8–10] in which the authors used a similar approach to solve the direct problem in the theory of the Laval nozzle (in the case of a nonconducting gas).Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 5, pp. 14–20, September–October, 1971.In conclusion the author wishes to thank M. Ya. Ivanov, who kindly made available his program for calculating the flow of a conducting gas, and also A. B. Vatazhin and A. N. Kraiko for useful advice.     

Numerical solution of an inverse problem of nozzle theory for a two-phase gas-particle mixture

In the present paper gas flows with monodisperse and polydisperse particles in plane and axisymmetric nozzles are calculated by the inverse method [1, 2]. The gas velocity distribution is specified on the axis of symmetry of the nozzle, while the gas and particle parameters are specified in the entrance section. As a result of the numerical integration of a system of equations describing a flow of gas with condensate particles in it we determine the gas and particle parameters, the gas streamlines, and the particle trajectories with allowance for the mutual influence of the gas and particles. One of the gas streamlines is taken as the nozzle contour and the limiting trajectories and pure gas zone are found. A difference method is described which makes it possible to calculate the subsonic, transonic, and supersonic flow regions using a single algorithm, its features are noted, and the results of the calculation for monodisperse mixtures with particle diameters 1 and 5 m and fractions by weight 0.3 are given. A comparison is made with the results of calculations by other methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 106–114, July–August, 1986.The authors express their gratitude to N. B. Ponomarev and G. E. Dumnov for their useful discussions and help in carrying out the calculations.     

An exact solutin for incompressible flow through a two-dimensional Laval nozzle

A careful examination of the variation of the velocity along the centerline and the contour of a Laval nozzle in the physical plane shows that either the upper or the lower half of the Laval nozzle assumes the same form of a slitted thick airfoil with tandem trailing edges. These two airfoils lie on different Riemann sheets in the hodograph plane. The interior of the airfoil is then mapped onto an infinite strip in the complex potential plane. Making use of these results, we obtained an exact solution for the incompressible potential flow through a two-dimensional Laval nozzle. The solution is applicable for nozzles with any given contraction ratio mexpansion rations, and throat wall radius R^*. As examples of the method, various nozzle contours, the velocity distribution of the flow, and the locations of the fluid particles at different time intervals are presented.     

Possibility of separationless flow in a nozzle with strongly curved walls

An example is given of calculation of the flow in a two-dimensional Laval nozzle whose profile in the subsonic part is concave with respect to the direction of the oncoming flow. Under the hypothesis of a separationless flow of ideal gas on the walls of the nozzle, regions of deceleration of the flow are absent. Then the well-known criteria suggest the existence of a separationless boundary layer, which must ensure that the flow as a whole is separationless.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 188–189, September–October, 1980.     

Laval nozzle flow calculation

The inverse problem of the theory of the Laval nozzle is considered, which leads to the Cauchy problem for the gasdynamic equations; the streamlines and the flow parameters are found from the known velocity distribution on the axis of symmetry.The inverse problem of Laval nozzle theory was considered in 1908 by Meyer [1], who expanded the velocity potential into a series in powers of the Cartesian coordinates and constructed the subsonic and supersonic solutions in the vicinity of the center of the nozzle. Taylor [2] used a similar method to construct a flowfield which is subsonic but has local supersonic zones in the vicinity of the minimal section. Frankl [3] and Fal'kovich [4] studied the flow in the vicinity of the nozzle center in the hodograph plane. Their solution, just as the Meyer solution, made it possible to obtain an idea of the structure of the transonic flow in the vicinity of the center of the nozzle.A large number of studies on transonic flow in the vicinity of the center of the nozzle have been made using the method of small perturbations. The approximate equation for the transonic velocity potential in the physical plane, obtained in [3–6], has been studied in detail for the plane and axisymmetric cases. In [7] Ryzhov used this equation to study the question of the formation of shock waves in the vicinity of the center of the nozzle, and conditions were formulated for the plane and axisymmetric cases under which the flow will not contain shock waves. However, none of the solutions listed above for the inverse problem of Laval nozzle theory makes it possible to calculate the flow in the subsonic and transonic parts of the nozzles with large gradients of the gasdynamic parameters along the normal to the axis of symmetry.Among the studies devoted to the numerical calculation of the flow in the subsonic portion of the Laval nozzle we should note the study of Alikhashkin et al., and the work of Favorskii [9], in which the method of integral relations was used to solve the direct problem for the plane and axisymmetric cases.The present paper provides a numerical solution of the inverse problem of Laval nozzle theory. A stable difference scheme is presented which permits analysis with a high degree of accuracy of the subsonic, transonic, and supersonic flow regions. The result of the calculations is a series of nozzles with rectilinear and curvilinear transition surfaces in which the flow is significantly different from the one-dimensional flow. The flowfield in the subsonic and transonic portions of the nozzles is studied. Several asymptotic solutions are obtained and a comparison is made of these solutions with the numerical solution.The author wishes to thank G. D. Vladimirov for compiling the large number of programs and carrying out the calculations on the M-20 computer.     

Influence of the mixing of two flows with different total parameters in a laval nozzle on its integral characteristics

A calculation is made of the turbulent zone of mixing of two flows of viscous and heat conducting gas in a Laval nozzle. For such a nozzle of given geometry, a comparison is made of calculations of the integrated characteristics of flows that are nonuniform with respect to the total parameters in the framework of various models: laminar hydraulics, viscous laminar hydraulics, and total mixing without hydraulic losses. The calculations are made for a stationary, nonswlrling flow of a viscous heat conducting gas with nearly discontinuous step distribution of the total parameters at the entrance to an axisymmetric Laval nozzle of given geometry. In this situation, the gas flows with different total parameters at the entrance to the nozzle are separated by a surface near which the profiles of the flow parameters are specified on the basis of boundary-layer theory. In the blocked regime investigated here, the flow in the part where the nozzle becomes narrower and at least at the beginning of the expanding part does not depend on the pressure of the surrounding medium. The integrated characteristics of the nozzle (gas flow rate G, impulse I, specific impulse i = I/G, etc.) depend on the parameter distributions at the entrance to the nozzle, and also on the turbulent mixing of the flows in the mixing zone. To analyze the dependence of the integrated characteristics on the turbulent mixing, the values of these characteristics calculated in the framework of the three models are compared. The model of mixing without hydraulic losses presupposes complete equalization of the parameters of the original inhomogeneous flow in the constant-area chamber in front of the nozzle with conservation of the mass, energy, and momentum fluxes. The model of laminar hydraulics is described in detail in [1, 2]. The model of viscous laminar hydraulics will be described in Sec. 1.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 114–119, July–August, 1979.I thank A. N. Kraiko for supervising the work, A. N. Sekundov for helpful discussions, and I. P. Smirnova and A. B. Lebedev for making available the computer program.