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.     

Boundary layer on a blunt body in a flow of dusty gas

The problem of interaction of gas-dust flows with solid surfaces arose in connection with the study of the motion of aircraft in a dusty atmosphere [1–2], the motion of a gas suspension in power generators, and in a number of other applications [3]. The presence of a disperse admixture may lead to a significant increase in the heat fluxes [4] and to erosion of the surface [5]. These phenomena are due to the joint influence of several factors — the change in the structure of the carrier-phase boundary layer due to the presence of the particles, collisions of the particles with the surface, roughness of the ablating surface, and so forth. This paper continues an investigation begun earlier [6–7] into the influence of particles on the structure of the dynamical and thermal two-phase boundary layer formed around a blunt body in a flow. The model of the dusty gas [8] has an incompressible carrier phase. The method of matched asymptotic expansions [9] is used to obtain the equations of the two-phase boundary layer. In the frame-work of the refined classification made by Stulov [6], it is shown that the form of the boundary layer equations is different in the presence and absence of inertial precipitation of the particles. The equations are solved numerically in the neighborhood of the stagnation point of the blunt body. The temperature and phase velocity distributions in the boundary layer, and also the friction coefficients and the heat transfer of the carrier phase are found for a wide range of the determining parameters. In the case of an admixture of low-inertia particles that are not precipitated on the body, it is shown that even when the mass concentration of the particles in the undisturbed flow is small their accumulation in the boundary layer can lead to a sharp increase in the thermal fluxes at the stagnation point.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–107, September–October, 1985.I thank V. P. Strulov for a discussion.     

Stress–strain relation for water-driven particle erosion of quasi-brittle materials

This paper is concerned with the behaviour of quasi-brittle materials eroded by water-driven particles (WDP). Determined are the compressive stress–strain curves for the material. The measured relations are approximated by parabolic regression. The areas under these curves are then used to characterise the specific energy absorption capability of the investigated materials. Erosion is then investigated by WDP with varying water pressures and exposure time. The depth of penetration, the material volume loss, and the kerf geometry are measured. Based on these measurements, the specific energy of the erosion process is calculated. A characteristic efficiency number Φ is then defined and estimated for all materials at different erosion conditions. It is shown that this parameter effectively characterises the erodability of the materials. Moreover, relations are found between the characteristic erosion parameters, such as threshold pressure, machinability number, depth of penetration, material removal rate, and the specific energy absorption capability. A preliminary phenomenological model based on the relationship between crack propagation processes and energy absorption capability is developed.     

Rheological equations of state of weakly concentrated suspensions of rigid ellipsoidal particles

Rheological equations of state of dilute suspensions of rigid ellipsoidal particles (ellipsoids of revolution) are derived [1–4] from the vantage point of the structural-continuum approach, with attention given both to rotational Brownian motion of particles and to their inertia and the outer force fields. Interaction between particles is ignored in those treatments given the low concentration of the suspended particles. In this paper, the earlier findings [1–4] are generalized to higher concentrations. The effect of hydrodynamical interaction between particles on the rheological behavior of the suspension is treated in the light of the Simha approach [5].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 141–145, January–February, 1973.     

Thermal stresses and the shattering of solid particles in a shock layer

The motion of blunt bodies through two-phase media at high supersonic velocities is accompanied by strong heating of particles when they enter the shock layer. Because the ratio of the heating time of nonmetallic particles to the time of their thermal relaxation with the gas exceeds unity, large temperature gradients are developed in the particles, which are stressed and deformed and under the influence of the force and inertial loads they can then shatter, which significantly changes their force and thermal effect on the supersonic body. A special case of this problem — the shattering of ice particles in a shock layer under the influence of pressure forces — was investigated in [1]. In the present paper, the results of numerical calculations and known analytic solutions are used in the development of an approximate method for estimating the stresses that arise in spherical particles. Simple criteria are established for determining when the tensile stresses in the particles reach critical values above which the particles may shatter. As an example, the distribution of the temperature and stresses in silicon dioxide particles is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 66–73, January–February, 1981.We thank V. G. Pchelkina for assistance in calculating the temperature fields.     

Manipulation of particles using dielectrophoresis

A numerical scheme based on the distributed Lagrange multiplier method (DLM) is used to study the motion of particles of electrorheological suspensions subjected to non-uniform electric fields. At small Reynolds number, the time taken by the particles to collect at the minimums or maximums of the electric field is primarily determined by a parameter defined to be the ratio of the dielectrophoretic and viscous forces. Simulations show that in non-uniform electric fields the collection time is also influenced by a parameter defined by the ratio of the electrostatic particle–particle interaction and dielectrophoretic forces. The collection time decreases as this parameter decreases because when this parameter is less than one, particles move to the regions of high or low electric field regions individually. However, when this parameter is greater than one, particles regroup into chains which then move toward the electric field maximums or minimums without breaking. It is also shown that when the real part of the Clausius–Mosotti factor (β) is negative the positions of the local minimums of the electric field, and thus also the locations where particles collect, can be modified by changing the electric potential boundary conditions.     

Motion of inhomogeneities of a developed fluidized bed at small reynolds numbers

The instability of a fluidized system in which the particles are uniformly distributed in space [1–3] leads to the development of local inhomogeneities in the internal structure, these taking the form of more or less stable formations of packets of particles [4]. In accordance with the existing ideas based on experimental data [5–8, 13], the particle concentration within a packet may vary in a wide range from very small values (10^–2–10^–3 [8]) for bubbles to the concentration of the unfluidized bed for bunches of particles in a nearly closely packed state. The paper considers the steady disturbed motion of the fluid and solid phases near an ascending or descending packet of particles in a developed fluidized bed. It is assumed that the motion of the solid phase corresponds to a creeping flow of viscous fluid, and the viscosity of the fluidizing agent is taken into account only in the terms that describe the interphase interaction. The velocity fields and pressure distributions of the phases inside and outside a packet are determined. If the particle concentration within a packet tends to zero, the solution describes the slow motion of a bubble in a fluidized bed. The results of the paper are compared with results obtained earlier for the model of ideal fluids [9] and Batchelor's model [10], in which the fluidized bed is treated in a simplified form as a viscous quasihomogeneous continuum.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 57–65, July–August, 1984.     

Experimental study of the velocity field of parametrically excited waves in a two-layer liquid

The results of measuring the velocities of the particles of a two-layer liquid in standing parametrically excited waves are presented. The horizon on which the vertical harmonic components of the velocity are equal to zero is determined. It is established that the oscillatory motion of the particles is accompanied by a system of slow circulating flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–166, September–October, 1991.     

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.     

Calculation of the motion of two-component media

The one-dimensional motion of a viscous incompressible liquid in which particles are suspended is described by the mathematical model used in [1], Two examples are discussed: the precipitation of particles from the suspension, and a boiling layer. The results are presented in the form of graphs.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 102–108, November–December, 1973.     

Numerical investigation of nonequilibrium flows of mixtures of fusible metal particles and gases in a laval nozzle

There are numerous papers [1–11] on the determination of the parameters of condensed oxide particles which are formed during combustion of metallized fuels. The ambiguity, and sometimes the contradictoriness, of the test results obtained [3–5, 9–11] indicate the difficulties in conducting correct experimental investigations. In this connection, numerical studies using mixtures of calibrated liquid-metal particles and different gases are of practical interest. Different probes can be calibrated by using calibrated two-phase flows, the two-phase flow around models and probes can be studied, as can the interaction between liquid-metal particles and the front of an aerodynamic compression shock, their intrusion in different entraining media, the interaction between fine particles (particle-projectiles) and large size particles (particle-targets), etc. In many cases, the prehistory of the flow and the parameters of the gas mixture with the particles in the area of the nozzle exit section must be known to investigate the above-mentioned phenomena. The parameters of different nonequilibrium flows of mixtures of gallium particles and gases in a Laval nozzle are investigated numerically in this paper; the maximum diameter (upper boundary of the spectrum) of the particles (d_s = 30 ) which are not destroyed in the nozzle under the effect of the aerodynamic forces and are suitable for use in a calibrated two-phase stream is determined. The computations were carried out in a one-dimensional approximation according to [12–14].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 86–91, March–April, 1976.The authors are grateful to V. K. Starkov and U. G. Pirumov for discussing the results of the research and to N. M. Alekseev for aid in constructing the graphs.     

Particle suspension in a simple shear flow

A hydrodynamic model describing the particle distribution over the cross-section of a finely dispersed flow is proposed. The model is constructed on the basis of notions concerning the diffusion of particles induced by their random displacements in the process of relative motion of neighboring layers at constant shear velocity. It is shown that the suspension capacity of the flow is large for small particles due to thermal fluctuations and for relatively large particles due to shear-induced particle pulsations. There are critical particle sizes for which the particles are suspended and transported by the flow less effectively than larger or smaller particles.Ekaterinburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 112–121, January–February, 1995.     

Influence of the viscous boundary layer on the deposition of particles from a gas suspension flowing past a sphere

The flow of incompressible gas containing particles past bodies of simple shapes at moderate and high velocities is investigated in [1–5], in which the flow of the carrier medium is assumed to be irrotational. The estimates made in [3] for the neighborhood of the stagnation point show that it is necessary to take into account the viscous boundary layer in the case of fine particles. In the present paper, the viscous flow of a gas suspension over the front surface of a sphere at Reynolds numbers R = 10³–10⁷ is considered. It is assumed that the carrier gas is incompressible and the particle concentra ion negligibly small. The influence of the boundary layer on the particle trajectories and the deposition of the disperse phase on the surface of the sphere is investigated. It is shown that there is a wide range of flow parameters for the gas suspension in which the influence of the boundary layer is important. The limits of this range are established.Translated from Izvestiya Akademli Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–66, January–February, 1982.I thank Yu. P. Savel'ev for a helpful discussion of the work.     

Self-consistent bunches of oscillating heavy particles in an accelerator

The possibility of the existence of self-consistent spheroidal bunches having a constant density which consists of heavy particles or bodies is demonstrated. In a projection onto the transverse plane the particles describe circles or ellipses, while along the spheroid axis the particles oscillate sinusoidally. It is possible for an infinite set of particle distributions to appear with respect to the half-axes of the transverse ellipses and with respect to the amplitudes of the longitudinal oscillations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 140–142, July–August, 1970.     

Ablation under the combined erosive and thermal influence of a two-phase flow

A study is made of the combined effect of a high-velocity two-phase (gas—solid particles) flow to the surface of a body. Different ablation models are proposed for simultaneous thermal (or thermochemical) and impact influence of the flow on the material. A numerical investigation is made of the unsteady thermal state of a body on which solid particles impinge, and the ranges of the determining parameters in which the thermal effect is negligibly small compared with the erosive effect are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 120–126, January–February, 1984.     

Rotation of dispersed particles in a vortex field

An expression is obtained for the angular velocity of a spherical dispersed particle in a viscous fluid in an external vortex field with an harmonic time dependence. This expression is then used for investigating a system of two rotating dispersed particles whose rotation is the result of the interaction of the particles in the field of an incident sound wave. It is found that such a system possesses a rather interesting nontrivial property: under certain conditions it has a resonant frequency at which the rotation of the particles relative to the fluid is most intense.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 186–188, July–August, 1992.     

The necessary conditions for stable combustion of a powder in a half-closed chamber

The low-frequency stability of the steady-state combustion cycle is investigated in linear approximation for a powder in a half-closed chamber, taking account of incompleteness of combustion, heat losses at the walls, and dynamic erosion. The necessary conditions for stable combustion are obtained. Qualitative conclusions are drawn of the incompleteness of the chemical reactions taking place and of the destabilizing effect of thermal losses and erosion on the combustion process. Only two out of three possible steady-state mechanisms are stable. The existence of limits of combustion with respect to pressure is shown.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 90–98, November–December, 1972.     

Locally uniform no-collision suspensions in euler approximation

Pseudoturbulent motions of phases are considered as well as the internal structure of monodispersional locally uniform suspension of particles in a liquid when the derivatives of the dynamic variables which describe the mean flow of the suspension are ignored. One assumes that the number of collision dissipations, which expresses the ratio of the dissipation force due to acceleration of the fluid phase in the case of stepwise change in the velocities of the colliding particles to the forces of viscous interaction, is small. Dynamic equations which determine the motion of the suspension in its continuous approximation and the balance equation of the pseudoturbulent energy of the particles are obtained with an approximation which is similar to the Euler approximation in the hydrodynamics of a single-phase medium.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 91–101, July–August, 1970.     

Peessure intensification on the front of a shock wave propagating in a heterogeneous system

An increase in pressure in the wave front as compared to the pulse initiating the wave has been observed experimentally in a study of shock-wave propagation in aqueous suspensions of bentonite [1]. In suspensions in which the solid phase is in the form of colloidal size particles =10^–7–10^–8 m of the mineral montmorillonite with mass content c=6%, with multishock loading an intensification of this effect from experiment to experiment was observed [2]. In order to study the principles involved in this anomalous intensification of pressure on the shock wave front, as well as to clarify the effect of the nature of the material in the dispersed phase, experiments were performed with particles of another broad class of clay-like minerals — kaolinite.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 86–92, September–October, 1986.     

Direct simulation of the motion of neutrally buoyant balls in a three-dimensional Poiseuille flow

In a previous article the authors introduced a Lagrange multiplier based fictitious domain method. Their goal in the present article is to apply a generalization of the above method to: (i) the numerical simulation of the motion of neutrally buoyant particles in a three-dimensional Poiseuille flow; (ii) study – via direct numerical simulations – the migration of neutrally buoyant balls in the tube Poiseuille flow of an incompressible Newtonian viscous fluid. Simulations made with one and several particles show that, as expected, the Segré–Silberberg effect takes place. To cite this article: T.-W. Pan, R. Glowinski, C. R. Mecanique 333 (2005).