Supersonic two-phase flow around a wedge

Supersonic two-phase flow around bodies is encountered in calculating the flow around the last stages of blades of condensing turbines, in studying the motion of airplanes under cloudy conditions, etc. In the latter case, there is, along with erosion of the forward edges of the wing profiles, a change in the wave structure and interference situation in the flow about the airplane, leading to off-design regimes of motion. Supersonic flow of a two-phase mixture around a wedge, without taking account of the influence of the particles on the flow, was investigated in [1–3]. In [4], also in this kind of simplified setting, a study was made of the interaction of particles with the surface of a wedge in which reflection of the particles from the wall was taken into account. Morganthaler [5] made an experimental study of the flow of a mixture of air and aluminum oxide particles around a wedge. In [6] a theoretical study was made of a supersonic two-phase flow around thin flat axially-symmetric bodies. In particular, for the flow around a wedge, closed form solutions were obtained for the form of the shock wave, the gas streamlines and particle paths, and the distribution of all the parameters along the surface of the wedge. On the basis of the equations given in [7] and the method of characteristics, which were developed for flows consisting of a mixture of a gas and heterogeneous particles in nozzles [8,9], we present below a study of a supersonic two-phase flow around a wedge.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 83–88, March–April, 1972.     

Droplet break-up through an oblique shock wave

The interaction of a two-phase flow with a wedge where a stationary shock wave is initially settled is studied in a two-dimensional configuration. Before the introduction of the dispersed phase, the flow around the wedge is a supersonic one phase flow such as an attached stationary shock wave is present. Then, the dispersed phase is introduced upstream the initial position of the stationary shock wave. The purpose of this study is to point out two-phase and droplets break-up effects on the oblique shock wave. The two-dimensional equations are solved by a TVD scheme where fluxes are computed by using Riemann solver for the gas phase equations and also for the dispersed phase equations wich is an original approach due to the authors (Saurel et al. 1994). In addition to drag forces and heat and mass transfers, the process of droplets fragmentation based on the particle oscillation is considered. Accepted April 28, 1995     

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.     

Supersonic flow of a combustible gas mixture past a sphere

In recent years considerable interest has developed in the problems of steady-state supersonic flow of a mixture of gases about bodies with the formation of detonation waves and slow combustion fronts. This is due in particular to the problem of fuel combustion in a supersonic air stream.In [1] the problem of supersonic flow past a wedge with a detonation wave attached to the wedge apex is solved. This solution is based on using the equation of the detonation polar obtained in [2]-the analog of the shock polar for the case of an exothermic discontinuity. In [3] a solution is given of the problem of cone flow with an attached detonation wave, and [4] presents solutions of the problems of supersonic flow past the wedge and cone with the formation of attached adiabatic shocks with subsequent combustion of the mixture in slow combustion fronts. In the two latter studies two different solutions were also found for the problem of flow past a point ignition source, one solution with gas combustion in the detonation wave, the other with gas combustion in the slow combustion front following the adiabatic shock. These solutions describe two different asymptotic pictures of flow of a combustible gas mixture past bodies.In an experimental study of the motion of a sphere in a combustible gas mixture [5] it was found that the detonation wave formed ahead of the sphere splits at some distance from the body into an ordinary (adiabatic) shock and a slow combustion front. Arguments are presented in [6] which make it possible to explain this phenomenon and in certain cases to predict its occurrence.The present paper presents examples of the calculation of flow of a combustible gas mixture past a sphere with a detonation wave in the case when the wave does not split. In addition, the flow near the point at which the detonation wave splits is analyzed for the case when splitting occurs where the gas velocity behind the wave is greater than the speed of sound. This analysis shows that in the given case the flow calculation may be carried out without any particular difficulties. On the other hand, the calculation of the flow for the case when the point of splitting is located in the subsonic portion of the flow behind the wave (or in the region of influence of the subsonic portion of the flow) presents difficulties. This flow case is similar to the problem of the supersonic jet of finite width impacting on an obstacle.     

Arbitrary body,close to a wedge,in a supersonic flow

The problem of supersonic flow around bodies close to a wedge was first discussed in the two-dimensional case in [1]. The shock wave was assumed to be attached, and the flow behind it to be supersonic; taking this into account, the angle of the wedge was assumed to be arbitrary. The surface of the body was also arbitrary, provided that it was close to the surface of the wedge. In solution of the three-dimensional problem, there was first considered flow around two supporting surfaces with only slightly different angles of attack [2], and then around a delta wing [3, 4]. In all these articles, the Lighthill method was used to solve the Hilbert boundary-value problem [5, 6]. A whole class of surfaces of bodies with arbitrary edges, under the assumption that the surface of the body was cylindrical, with generatrices directed along the flow lines of the unperturbed flow behind an oblique shock wave, was discussed in [7]. In the present work, the problem is regarded for a broad class of surfaces of bodies, using a new method which generalizes the results of [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 109–117, July–August, 1974.The author thanks G. G. Chernyi for his direction of the work.     

Study of flow in the vicinity of V-shaped wings formed by the stream surfaces behind a plane shock

In calculating the flow about bodies with plane surfaces and sharp edges it is assumed that in the flow regimes with attached shock the latter may be defined in a section normal to the edge from the corresponding relations for the wedge [1, 2], The solution is taken corresponding to a weak shock on a wedge with supersonic velocity behind it. While in the plane case (wedge) this solution will be the only physically realizable solution, in the case of three-dimensional bodies, when there is a slip velocity along the leading edge, the realization of a second wedge solution with a strong shock is conceivable in the section normal to the leading edge if the total velocity behind the shock (with account for the slip velocity along the edge) is supersonic [3].Relative to the undisturbed stream velocity both of these solutions correspond to a weak shock. We present an example when the solution with a strong shock in the section normal to the edge is possible.     

激波点燃粉尘的数值模拟研究

建立了用于模拟入射激波后可燃粉尘颗粒点火的一维非定常两相化学反应流模型，该模型考虑了气固两相间的相互作用、粉尘颗粒的加速、加热和化学反应。粉尘颗粒着火前的化学反应用发生在颗粒外表面和内孔表面的非均相反应描述，颗粒内部的温度变化用一含有化学反应源项的非稳态热传导方程来描述，以颗粒外表面温度的突跃上升作为可燃粉尘颗粒点燃的着火条件。我们用该模型和ＰＳＩＣ方法，对由中等强度激波从纯气相传入煤粉－氧气混合物而引起的非定常两相流动现象，包括气固两相间的相互作用、粉尘颗粒的加速、加热以及点火过程进行了数值研究，计算了对应于不同载荷比、马赫数为４～５的入射激波后煤尘颗粒的点火延迟时间，分析了由于可燃粉尘颗粒的存在，入射激波及波后气固两相流动参数的变化规律。数值计算结果与实验数据符合较好。文中建立的模型和所用的基于ＰＳＩＣ算法的数值方法，用最自然的方式描述气固两相流动，即用连续流模型（欧拉方程）描述输运相（气相）的流动，用轨道颗粒模型（拉格朗日方程）描述分散相（颗粒相）的运动。用这种方法模拟含尘介质中激波后颗粒的点火是很有效的，它可以清楚地确定哪一个颗粒群最先着火，它的初始位置以及在整个点火延迟时间内     

Supersonic motion of bodies in a gas with shock waves

The authors consider the problem of supersonic unsteady flow of an inviscid stream containing shock waves round blunt shaped bodies. Various approaches are possible for solving this problem. The parameters in the shock layer on the axis of symmetry have been determined in [1, 2] by using one-dimensional theory. The authors of [3, 4] studied shock wave diffraction on a moving end plane and wedge, respectively, by the through calculation method. This method for studying flow around a wedge with attached shock was also used in [5]. But that study, unlike [4], used self-similar variables, and so was able to obtain a clearer picture of the interaction. The present study gives results of research into the diffraction of a plane shock wave on a body in supersonic motion with the separation of a bow shock. The solution to the problem was based on the grid characteristic method [6], which has been used successfully to solve steady and unsteady problems [7–10]. However a modification of the method was developed in order to improve the calculation of flows with internal discontinuities; this consisted of adopting the velocity of sound and entropy in place of enthalpy and pressure as the unknown thermodynamic parameters. Numerical calculations have shown how effective this procedure is in solving the present problem. The results are given for flow round bodies with spherical and flat (end plane) ends for various different values of the velocities of the bodies and the shock waves intersected by them. The collision and overtaking interactions are considered, and there is a comparison with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–147, September–October, 1984.     

Entrainment of fine particles from surfaces by impinging shock waves

 When a shock wave impinges on a surface, it reflects and propagates across the surface at supersonic velocity. The gas is impulsively accelerated by the passing shock wave. The resulting high-speed flow imparts sufficiently strong forces to particles on the surface to overcome strong adhesive forces and entrain the surface-bound particles into the gas. This paper describes an experimental study of the removal of fine particles from a surface by impinging shock waves. The surfaces examined in this study were glass slides on which uniformly sized (8.3 μm diameter), spherical polystyrene particles had been deposited. Shock waves were generated in a small, open-ended shock tube at various heights above and impingement angles to the surface. Particle detachment from the carefully prepared substrates was determined from images of the surfaces recorded before and after shock impingement. A single shock wave effectively cleaned a large surface area. The centerline length of the cleared region was used to characterize the efficacy of shock cleaning. A model based upon the far field solution for a point source surface shock provides a good fit to the clearance length data and yields an estimate to the threshold shock strength for particle removal. Received: 13 November 1997/Accepted: 23 April 1998     

A numerical study of oblique shock wave reflections over wedges in an ideal quantum gas

The various oblique shock wave reflection patterns generated by a moving incident shock on a planar wedge using an ideal quantum gas model are numerically studied using a novel high resolution quantum kinetic flux splitting scheme. With different incident shock Mach numbers and wedge angles as flow parameters, four different types of reflection patterns, namely, the regular reflection, simple Mach reflection, complex Mach reflection and the double Mach reflection as in the classical gas can be classified and observed. Both Bose–Einstein and Fermi–Dirac gases are considered.      

Some unique features of supersonic unsteady gas flow around bodies

We consider unsteady supersonic gas flow about bodies for small Strouhal numbers. The amplitude of the angles of attack is assumed arbitrary under the condition that the bow shock remains attached, and the flow behind the shock is supersonic. A criterion is formulated which permits the comparison of the damping characteristic for small and large amplitudes of the disturbed motion. A comparison is made, using the wedge as an example, of the damping characteristics calculated by various theories, and the hypersonic similarity law is verified. Simple asymptotic equations are presented for the rotary derivatives of the thin wedge in a hypersonic gas stream.     

The shock–vortex interaction patterns affected by vortex flow regime and vortex models

We have used a third-order essentially non-oscillatory method to obtain numerical shadowgraphs for investigation of shock–vortex interaction patterns. To search different interaction patterns, we have tested two vortex models (the composite vortex model and the Taylor vortex model) and as many as 47 parametric data sets. By shock–vortex interaction, the impinging shock is deformed to a S-shape with leading and lagging parts of the shock. The vortex flow is locally accelerated by the leading shock and locally decelerated by the lagging shock, having a severely elongated vortex core with two vertices. When the leading shock escapes the vortex, implosion effect creates a high pressure in the vertex area where the flow had been most expanded. This compressed region spreads in time with two frontal waves, an induced expansion wave and an induced compression wave. They are subsonic waves when the shock–vortex interaction is weak but become supersonic waves for strong interactions. Under a intermediate interaction, however, an induced shock wave is first developed where flow speed is supersonic but is dissipated where the incoming flow is subsonic. We have identified three different interaction patterns that depend on the vortex flow regime characterized by the shock–vortex interaction.      

A soap film shock tube to study two-dimensional compressible flows

 A new experimental approach to the study of the two-dimensional compressible flow phenomena is presented. In this technique, a variety of compressible flows were generated by bursting plane vertical soap films. An aureole and a “shock wave” preceding the rim of the expanding hole were clearly observed using traditional high-speed flash photography and a fast line-scan charge coupled device (CCD) camera. The moving shock wave images obtained from the line-scan CCD camera were similar to the x–t diagrams in gas dynamics. The moving shock waves cause thickness jumps and induce supersonic flows. Photographs of the supersonic flows over a cylinder and a wedge are presented. The results suggest clearly the feasibility of the “soap film shock tube”. Received: 11 May 2000/Accepted: 2 November 2000     

Flow around a sphere by a two-phase supersonic jet

Results are presented of a calculation of the flow around a sphere of a two-phase supersonic jet, discharging into a vacuum. Calculations were performed by the determination method with use of a difference grid constructed on the basis of characteristic ratios [1], The parameters of the unperturbed jet were determined with the two-velocity and two-temperature model of mutually penetrating flows of continuous media (gas and particles) [2, 3] by the network method [4]. In calculating the flow around the sphere, as in [5–7], it was assumed that the particles do not affect the gas flow in the shock layer. An analysis of the effect of particles on gasdynamic parameters in a shock layer was performed in [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 171–176, November–December, 1978.The authors are grateful to A. N. Nikulin for providing the program for calculation of flow about a blunt body by a uniform supersonic flow.     

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.     

Shock wave standoff distance for a sphere slightly above Mach one

This paper describes a numerical simulation of bow shock formation ahead of a sphere at steady supersonic flow in the Mach number range of 1.025–1.20. Turbulent viscous flow results are presented using the Spalart–Allmaras turbulence model. The purpose of this study is to determine the shock standoff distance for a spherical projectile at slightly supersonic free flight speeds. Results are compared to experimental data, including double exposure holographic interferograms obtained from a 40 mm polycarbonate sphere launched by a light gas gun. The shock standoff distance was determined from the interferograms. The present numerical simulations were found to agree with previously published data, and reached down to M = 1.025—a range where almost no previously published data exists. The computed flow structure and shock wave locations agree well with recently obtained free-flight interferograms.     

Three-dimensional steady supersonic duct flow using Lagrangian formulation

In this paper, numerical simulation of three-dimensional supersonic flow in a duct is presented. The flow field in the duct is complex and can find its applications in the inlet of air-breathing engines. A unique streamwise marching Lagrangian method is employed for solving the steady Euler equations. The method was first initiated by Loh and Hui (1990) for 2-D steady supersonic flow computations and then extended to 3-D computation by the present authors Loh and Liou (1992). The new scheme is shown to be capable of accurately resolving complicated shock or contact discontinuities and their interactions. In all the computations, a free stream of Mach numberM=4 is considered.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

The formation of a secondary shock wave behind a shock wave diffracting at a convex corner

This paper deals with the formation of a secondary shock wave behind the shock wave diffracting at a two-dimensional convex corner for incident shock Mach numbers ranging from 1.03 to 1.74 in air. Experiments were carried out using a 60 mm 150 mm shock tube equipped with holographic interferometry. The threshold incident shock wave Mach number () at which a secondary shock wave appeared was found to be = 1.32 at an 81° corner and = 1.33 at a 120° corner. These secondary shock waves are formed due to the existence of a locally supersonic flow behind the diffracting shock wave. Behind the diffracting shock wave, the subsonic flow is accelerated and eventually becomes locally supersonic. A simple unsteady flow analysis revealed that for gases with specific heats ratio the threshold shock wave Mach number was = 1.346. When the value of is less than this, the vortex is formed at the corner without any discontinuous waves accompanying above the slip line. The viscosity was found to be less effective on the threshold of the secondary shock wave, although it attenuated the pressure jump at the secondary shock wave. This is well understood by the consideration of the effect of the wall friction in one-dimensional duct flows. In order to interpret the experimental results a numerical simulation using a shock adaptive unstructured grid Eulerian solver was also carried out. Received 1 May 1996 / Accepted 12 September 1996     

Explosive dispersal of solid particles

Abstract. The rapid dispersal of inert solid particles due to the detonation of a heterogeneous explosive, consisting of a packed bed of steel beads saturated with a liquid explosive, has been investigated experimentally and numerically. Detonation of the spherical charge generates a blast wave followed by a complex supersonic gas-solid flow in which, in some cases, the beads catch up to and penetrate the leading shock front. The interplay between the particle dynamics and the blast wave propagation was investigated experimentally as a function of the particle size (100–925 m) and charge diameter (8.9–21.2 cm) with flash X-ray radiography and blast wave instrumentation. The flow topology during the dispersal process ranges from a dense granular flow to a dilute gas-solid flow. Difficulties in the modeling of the high-speed gas-solid flow are discussed, and a heuristic model for the equation of state for the solid flow is developed. This model is incorporated into the Eulerian two-phase fluid model of Baer and Nunziato (1986) and simulations are carried out. The results of this investigation indicate that the crossing of the particles through the shock front strongly depends on the charge geometry, the charge size and the material density of the particles. Moreover, there exists a particle size limit below which the particles cannot penetrate the shock for the range of charge sizes considered. Above this limit, the distance required for the particles to overtake the shock is not very sensitive to the particle size but remains sensitive to the particle material density. Overall, excellent agreement was observed between the experimental and computational results. Received 16 August 1999 / Accepted 26 June 2000     

Experiments on free and impinging supersonic microjets

The fluid dynamics of microflows has recently commanded considerable attention because of their potential applications. Until now, with a few exceptions, most of the studies have been limited to low speed flows. This experimental study examines supersonic microjets of 100–1,000 μm in size with exit velocities in the range of 300–500 m/s. Such microjets are presently being used to actively control larger supersonic impinging jets, which occur in STOVL (short takeoff and vertical landing) aircraft, cavity flows, and flow separation. Flow properties of free as well as impinging supersonic microjets have been experimentally investigated over a range of geometric and flow parameters. The flowfield is visualized using a micro-schlieren system with a high magnification. These schlieren images clearly show the characteristic shock cell structure typically observed in larger supersonic jets. Quantitative measurements of the jet decay and spreading rates as well as shock cell spacing are obtained using micro-pitot probe surveys. In general, the mean flow features of free microjets are similar to larger supersonic jets operating at higher Reynolds numbers. However, some differences are also observed, most likely due to pronounced viscous effects associated with jets at these small scales. Limited studies of impinging microjets were also conducted. They reveal that, similar to the behavior of free microjets, the flow structure of impinging microjets strongly resembles that of larger supersonic impinging jets.