A two-phase stream containing liquid particles in the sub- and transonic regions of plane nozzles

The results of experimental studies of the influence of the entrance conditions, the particle size, the profiles of the sub- and transonic parts of the nozzle, and the initial concentration on the distribution of the solid discrete phase in the exit cross sections of axisymmetric nozzles were analyzed in [1]. The results of a study of the influence of the profiling of the nozzle and the size of the particles at the nozzle entrance on the formation of the distribution fields of the discrete liquid phase and its size at the cut of a plane nozzle are presented in the present report, which is a continuation of [1]. The experimental data presented permit a deeper understanding of the mechanism of flow of a two-phase medium in a nozzle and are required for an evaluation of efficiency of the calculation methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 167–170, March–April, 1978.     

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.     

Transonic Shock Problem for the Euler System in a Nozzle

In this paper, we study the well-posedness problem on transonic shocks for steady ideal compressible flows through a two-dimensional slowly varying nozzle with an appropriately given pressure at the exit of the nozzle. This is motivated by the following transonic phenomena in a de Laval nozzle. Given an appropriately large receiver pressure P _r , if the upstream flow remains supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the flow is compressed and slowed down to subsonic speed, and the position and the strength of the shock front are automatically adjusted so that the end pressure at exit becomes P _r , as clearly stated by Courant and Friedrichs [Supersonic flow and shock waves, Interscience Publishers, New York, 1948 (see section 143 and 147)]. The transonic shock front is a free boundary dividing two regions of C ^2,α flow in the nozzle. The full Euler system is hyperbolic upstream where the flow is supersonic, and coupled hyperbolic-elliptic in the downstream region Ω₊ of the nozzle where the flow is subsonic. Based on Bernoulli’s law, we can reformulate the problem by decomposing the 3 × 3 Euler system into a weakly coupled second order elliptic equation for the density ρ with mixed boundary conditions, a 2 × 2 first order system on u ₂ with a value given at a point, and an algebraic equation on (ρ, u ₁, u ₂) along a streamline. In terms of this reformulation, we can show the uniqueness of such a transonic shock solution if it exists and the shock front goes through a fixed point. Furthermore, we prove that there is no such transonic shock solution for a class of nozzles with some large pressure given at the exit. This research was supported in part by the Zheng Ge Ru Foundation when Yin Huicheng was visiting The Institute of Mathematical Sciences, The Chinese University of Hong Kong. Xin is supported in part by Hong Kong RGC Earmarked Research Grants CUHK-4028/04P, CUHK-4040/06P, and Central Allocation Grant CA05-06.SC01. Yin is supported in part by NNSF of China and Doctoral Program of NEM of China.     

Adaptive mesh refinement and coarsening for aerodynamic flow simulations

A new mesh refinement technique for unstructured grids is discussed. The new technique presents the important advantage of maintaining the original grid skewness, thanks to the capability of handling hanging nodes. The paper also presents an interpretation of MacCormack's method in an unstructured grid context. Results for a transonic convergent–divergent nozzle, for a convergent nozzle with a supersonic entrance and for transonic flow over a NACA 0012 airfoil are presented and discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Cavitation: a contributory factor in the transition from symmetric to asymmetric jets in cross-flow nozzles

The structure and evolution of cavitation and its influence on jet patterns from two transparent cross-flow nozzles with holes inclined at 90 degrees (nozzle A) and 80 degrees (nozzle B) to the nozzle axis have been investigated using high-speed motion pictures, flash photography and stroboscopic visualization. At the onset, cavitation inception was in the form of travelling bubbles, which were transported along the flow and clearly detached from the wall. As the flow was increased the bubbles grew and merged into a dense group of bubbles (cloud cavitation), partly unsteady and shedding. Further increasing the flow caused the cavitation at the entrance to transform mainly into a glassy appearance and at this stage the cavitation was well inside the hole and the spray appeared symmetric. When the flow was increased beyond this stage, cavitation extended to the exit of the hole, occupying a significant part of the hole on one side, resulting in a jet that atomized on the side where cavitation was most extensive and a non-atomizing jet on the side with less cavitation. The distribution of cavitation in the hole is very sensitive to the nozzle geometry and it substantially influences the spray dispersion.     

SiC/(W,Ti)C梯度陶瓷喷嘴材料的制备及其冲蚀磨损机理研究

针对陶瓷喷嘴磨损特点,在喷嘴材料的设计和制造中提出运用梯度功能材料理论,通过控制陶瓷喷嘴材料的成分分布以实现其力学性能的合理梯度变化,将梯度陶瓷喷嘴材料制备过程中所产生的残余压应力引入喷嘴入口以提高喷嘴入口的力学性能,从而缓解喷嘴入口的高应力,提高其抗冲蚀磨损能力.采用热压烧结工艺制备SiC/(W,Ti)C梯度陶瓷喷嘴材料并分析其冲蚀磨损机理.结果表明,在相同冲蚀磨损条件下,梯度陶瓷喷嘴材料的抗冲蚀磨损性能较非梯度陶瓷喷嘴材料显著提高,这是由于梯度陶瓷喷嘴应力状态的改善及其力学性能提高的缘故.梯度陶瓷喷嘴材料的磨损机制为入口处呈现疲劳断裂、中间呈现微切削、出口处呈现疲劳断裂和脆性断裂特征.     

Formation of multiple shocklets in a transonic diffuser flow

Multiple shocklets are frequently generated in transonic diffuser flows. The present paper investigates the formation of these shocklets with a high-speed CCD camera combined with the schlieren method. It is observed that compression waves steepen while propagating upstream, and eventually become new shock waves. The ordinary shock wave is found to move upstream beyond the nozzle throat or to disappear while moving downstream depending on the pressure ratio across the nozzle. This phenomenon is also analyzed with the one-dimensional Euler equations by assuming a pressure disturbance given by the sine function at the channel exit. The calculated results are found to reproduce quite well the experimental behavior of the shocklets. The effect of the frequency of disturbance is also studied numerically, and it is shown that the multiple shocklet pattern appears when the amplitude of disturbance is not large and the diverging part of the channel downstream of the ordinary shock wave is long. Received 26 June 1998 / Accepted 15 March 1999     

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.     

Transonic Shocks for the Full Compressible Euler System in a General Two-Dimensional De Laval Nozzle

In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure p _e(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p _e(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Hölder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.     

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.     

Gas and Particle Dynamics of a Contoured Shock Tube for Pre-clinical Microparticle Drug Delivery

We investigate the gas-particle dynamics of a device designed for biological pre-clinical experiments. The device uses transonic/supersonic gas flow to accelerate microparticles such that they penetrate the outer skin layers. By using a shock tube coupled to a correctly expanded nozzle, a quasi-one-dimensional, quasi-steady flow (QSF) is produced to uniformly accelerate the microparticles. The system utilises a microparticle “cassette” (a diaphragm sealed container) that incorporates a jet mixing mechanism to stir the particles prior to diaphragm rupture. Pressure measurements reveal that a QSF exit period – suitable for uniformly accelerating microparticles – exists between 155 and 220 mus after diaphragm rupture. Immediately preceding the QSF period, a starting process secondary shock was shown to form with its (x,t) trajectory comparing well to theoretical estimates. To characterise the microparticle, flow particle image velocimetry experiments were conducted at the nozzle exit, using particle payloads with varying diameter (2.7–48 μm), density (600–16,800 kg/m³) and mass (0.25–10 mg). The resultant microparticle velocities were temporally uniform. The experiments also show that the starting process does not significantly influence the microparticle nozzle exit velocities. The velocity distribution across the nozzle exit was also uniform for the majority of microparticle types tested. For payload masses typically used in pre-clinical drug and vaccine applications (≤ 1 mg), it was demonstrated that payload scaling does not affect the microparticle exit velocities. These characteristics show that the microparticle exit conditions are well controlled and are in agreement with ideal theory. These features combined with an attention to the practical requirements of a pre-clinical system make the device suitable for investigating microparticle penetration into the skin for drug delivery.     

Flow in a viscous jet escaping through a supersonic nozzle into a semi-infinite ambient space

The problem of an axisymmetric gas flow in a supersonic nozzle and in the jet escaping from the nozzle to a quiescent gas is solved within the framework of Navier-Stokes equations. The calculated pressure distribution is compared with that measured in the jet by a Pitot tube. The influence of the jet pressure ratio, Reynolds number, and half-angle of the supersonic part of the nozzle on nozzle flow and jet flow parameters is studied. It is shown that the distributions of gas-dynamic parameters at the nozzle exit are nonuniform, which affects the jet flow. The flow pattern for an overexpanded jet shows that jet formation begins inside the nozzle because of boundary-layer displacement from the nozzle walls. This result cannot be obtained with the inviscid formulation of the problem.     

Transonic flow around a solid of revolution during the outflow of a reactive jet from its stern section

A numerical method is developed for calculating the motion of a subsonic or transonic flow around a solid of revolution with due allowance for the effects of the reactive jet, the bottom projection, possible noncoincidence between the planes of the nozzle tip and the stern section, nonuniformity of the flow at the nozzle outlet, the ejecting action of the supersonic jet, and also the displacement of the boundary layer. Examples of such calculations are given for solids of revolution of different shapes and nozzles of different types; comparison is also made with experimental data.     

Transonic Shocks in Compressible Flow Passing a Duct for Three-Dimensional Euler Systems

In this paper we study the transonic shock in steady compressible flow passing a duct. The flow is a given supersonic one at the entrance of the duct and becomes subsonic across a shock front, which passes through a given point on the wall of the duct. The flow is governed by the three-dimensional steady full Euler system, which is purely hyperbolic ahead of the shock and is of elliptic–hyperbolic composed type behind the shock. The upstream flow is a uniform supersonic one with the addition of a three-dimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem of determining the transonic shock and the flow behind the shock is reduced to a free-boundary value problem. In order to solve the free-boundary problem of the elliptic–hyperbolic system one crucial point is to decompose the whole system to a canonical form, in which the elliptic part and the hyperbolic part are separated at the level of the principal part. Due to the complexity of the characteristic varieties for the three-dimensional Euler system the calculus of symbols is employed to complete the decomposition. The new ingredient of our analysis also contains the process of determining the shock front governed by a pair of partial differential equations, which are coupled with the three-dimensional Euler system. The paper is partially supported by National Natural Science Foundation of China 10531020, the National Basic Research Program of China 2006CB805902, and the Doctorial Foundation of National Educational Ministry 20050246001.     

Density distribution in a supersonic jet exhausting into vacuum from a nozzle with an inclined exit

Fairly effective methods have been developed to calculate axisymmetric jets of an inviscid perfect gas exhausting into vacuum, and approximations have also been obtained for the density field in such jets at large distances from the nozzle exit (see, for example, [1—3] and the bibliography in them). If the plane of the exit of the nozzle is not perpendicular to its axis, the flow in the jet is three dimensional. In a number of cases one can take into account the influence of the inclined exit with sufficient accuracy by a correction in the density distribution. An expression for such correction is given in the present paper. It is obtained on the basis of earlier calculations of the author [4], in which flow from a source was specified at the nozzle exit.     

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.     

Experimental analysis of liquid–gas interface at low Weber number: interface length and fractal dimension

The present paper reports an experimental investigation on atomizing liquid flows produced by simplified cavity nozzles. The Weber number being kept low, the sprays produced by these injectors depend on the liquid flow characteristics only, and more precisely, on the non-axial kinetic energy and of the turbulent kinetic energy at the nozzle exit. The investigation reported here concentrates on the characterization of liquid flows during atomization by measuring the spatial variation of the local interface length and of the local interface fractal dimension. Both parameters were found representative of the physics of atomization process: they depend on the characteristics of the flow issuing from the nozzle and they are related to the subsequent drop size distribution. The local interface length is representative of the amount of liquid–gas interface surface area, and is a function of both the non-axial and the turbulent kinetic energies at the nozzle exit. The fractal dimension is representative of the tortuosity of the liquid–gas interface and, as expected, is mainly related to the turbulent kinetic energy at the nozzle exit. As far as the drop size distribution is concerned, it is found that the local interface length at the instant of break-up determines a representative drop diameter of some kind, whereas the fractal dimension at the same instant controls the dispersion of the distribution.     

On nitrogen condensation in hypersonic nozzle flows: numerical method and parametric study

A numerical method for calculating two-dimensional planar and axisymmetric hypersonic nozzle flows with nitrogen condensation is developed. The classical nucleation theory with an empirical correction function and the modified Gyarmathy model are used to describe the nucleation rate and the droplet growth, respectively. The conservation of the liquid phase is described by a finite number of moments of the size distribution function. The moment equations are then combined with the Euler equations and are solved by the finite-volume method. The numerical method is first validated by comparing its prediction with experimental results from the literature. The effects of nitrogen condensation on hypersonic nozzle flows are then numerically examined. The parameters at the nozzle exit under the conditions of condensation and no-condensation are evaluated. For the condensation case, the static pressure, the static temperature, and the amount of condensed fluid at the nozzle exit decrease with the increase of the total temperature. Compared with the no-condensation case, both the static pressure and temperature at the nozzle exit increase, and the Mach number decreases due to the nitrogen condensation. It is also indicated that preheating the nitrogen gas is necessary to avoid the nitrogen condensation even for a hypersonic nozzle with a Mach number of 5 operating at room temperatures.     

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.