Hypersonic flow past a supersonic two-phase source

The interaction of a uniform hypersonic gas flow with a supersonic two-phase gas-particle source is considered. In the symmetry-axis neighborhood between the bow and termination shock waves, an approximate analytical solution for the carrier-phase parameters is found. On the basis of parametric numerical calculations, the behavior of the particle trajectories and the concentration distribution in the shock layers are studied for both continuum and free-molecule flow regimes around the particles. The appearance of regions with multiple intersections of the particle trajectories and the formation of "layer structures" in the particle concentration distributions (particle accumulation regions near the envelopes of the particle trajectories) are indicated. The dependence of the number of the high concentration layers on the governing parameters is studied. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 134–147, May–June, 1998. The work received financial support from the Russian Foundation for Basic Research (project No. 96-01-00313) and the National Foundation for Natural Sciences of China (joint RFBR-NFNS grant No.96-01-00017c).     

Determination of the particle velocity,size, and distribution function along the diameters in a two-phase jet

A method for simultaneous determination of the particle size and velocity in a supersonic, two-phase luminescent jet consisting of a gas phase and the particles is considered. The jet is subjected to external transverse blowing by a hypersonic gas stream in two modes under whose effect it separates into the gas phase and a particle stream. The particle trajectories were determined by photography. A comparison between the theoretically computed and the experimental trajectories permits determination of the particle diameter and velocity. The particle distribution function over the diameters was found by means of the light intensity emitted by the stream of particles.     

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.     

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.     

Particle grouping in oscillating flows

An equation describing the dynamics of spherical particles in an oscillating Stokesian flow in the frame of reference moving with the phase velocity of the wave, and only taking into account the contribution of the drag force, is simplified in two limiting cases. Firstly, the case when Stokes numbers are small is considered. Secondly, the analysis focuses on the case when the initial location of the particles is close to the location where the particles are grouped (their velocities and accelerations in the wave frame of reference are equal to zero), $x_{\lim }$. This is followed by an analysis of the dynamics of non-Stokesian particles. In all cases, the analytical results are validated against the results of numerical solution of the equation of particle motion. Three types of trajectories are predicted when particles approach $x_{\lim }$: the trajectories describing the monotonic approach to $x_{\lim }$, the trajectories describing the approach to $x_{\lim }$ with oscillations and trajectories repelled from $x_{\lim }$. These are identified with stable nodes, stable foci and saddles. The trajectories in the zone between stable nodes and foci are identified as stable stars. Using Dulac's criterion, it is pointed out that none of the particle trajectories in the position–velocity plane can be closed. This result is illustrated by the trajectories calculated using the numerical solution of the equation for particle dynamics for various parameter values.     

Measurements of single spherical particle trajectories with lateral migration in a slit with one porous wall under laminar flow conditions

Lateral migration of spherical, neutrally buoyant particles moving in Poiseuille flow was measured in a slit with a porous membrane at one wall. Neutrally buoyant particles with diameters of 226 μm were injected into 22% glycerol-water solution flowing vertically in a slit channel (1.6 × 50 × 152 mm). The particles were illuminated with a strobe at 25 flashes/sec and photographed with a 4 × 5 camera under dark field conditions. Measured particle trajectories are compared with theoretically predicted trajectories based on Altena and Belfort's (1984) model. The theoretical trajectories are within the limits of error in the measured trajectories. By confirming the Altena and Belfort (1984) model within the range of experimental conditions tested here, inertial interactions should be included when modeling membrane fouling of dilute suspensions (Belfort et al. 1982).     

Particle tracking in living cells: a review of the mean square displacement method and beyond

The focus of many particle tracking experiments in the last decade has been active systems, such as living cells. In active systems, the particles undergo simultaneous active and thermally driven transport. In contrast to thermally driven transport, particle motion driven by active processes cannot directly be correlated to the rheology of the probed region. The rheology in particle tracking experiments is typically obtained through the mean square displacements (MSD) of the trajectories. Hence, the MSD and its functional form remain the only basic tools to evaluate and compare living cells or other active systems. However, the mechano-structural characteristics of the intracellular environment and the mechanisms driving particle transport cannot be revealed by the MSD alone. Hence, approaches for advanced analysis of particle trajectories have been introduced recently. Here, we present a broad review of the extensive intracellular particle tracking experiments that have been carried out on a wide variety of cell types. Those works utilize the MSD, revealing similarities and differences relating to cell type and experimental setup. We also highlight several advanced trajectory-and displacement-based analysis methods and illustrate their capabilities using particle tracking data obtained from two cancer cell lines. We show that combining these analysis methods with the MSD can reveal additional information on intracellular structure and the existence and nature of active processes driving particle motion in cells.     

Turbulent transport of particles in a straight square duct

Turbulent flow through a duct of square cross-section gives rise to off-axis secondary flows, which are known to transfer momentum between fluid layers thereby flattening the velocity profile. The aim of this study is to investigate the role of the secondary flows in the transport and dispersion of particles suspended in a turbulent square duct flow. We have numerically simulated a flow through a square duct having a Reynolds number of Re_τ = 300 through discretization of the Navier–Stokes equations, and followed the trajectories of a large number of passive tracers and finite-inertia particles under a one-way coupling assumption. Snapshots of particle locations and statistics of single-particle and particle pair dispersion were analyzed. It was found that lateral mixing is enhanced for passive tracers and low-inertia particles due to the lateral advective transport that is absent in straight pipe and channels flows. Higher inertia particles accumulate close to the wall, and thus tend to mix more efficiently in the streamwise direction since a large number of the particles spend more time in a region where the mean fluid velocity is small compared to the bulk. Passive tracers tend to remain within the secondary swirling flows, circulating between the core and boundary of the duct.     

Inertial migration of sedimenting particles in a suspension flow through a Hele-Shaw cell

Within the framework of the model of two interpenetrating continua, a horizontal laminar dilute-suspension flow in a vertical Hele-Shaw cell is investigated. Using the method of matched asymptotic expansions, an asymptotic model of the transverse migration of sedimenting particles is constructed. The particle migration in the horizontal section of the cell is caused by an inertial lateral force induced by the particle sedimentation and the shear flow of the carrier phase. A characteristic longitudinal length scale is determined, on which the particles migrate across the slot through a distance of the order of the slot half-width. The evolution of the particle number concentration and velocity fields along the channel is studied using the full Lagrangian method. Depending on the particle inertia parameter, different particle migration regimes (with and without crossing of the channel central plane by the particles) are detected. A critical value of the particle inertia parameter corresponding to the change in migration regime is found analytically. The possibility of intersection of the particle trajectories and the formation of singularities in the particle number concentration is demonstrated.     

Numerical study of bubble and particle motion in a turbulent boundary layer using proper orthogonal decomposition

We have studied the motion of bubbles and particles in the near-wall region of a turbulent boundary layer, to investigate the influence of the unsteady turbulent structure. The velocity field was computed using Proper Orthogonal Decomposition (POD), and the trajectories of bubbles and particle have been computed by integrating their equation of motion. We have used this to investigate the roles, and the relative importance, of the different forces acting on bubbles and particles, We find that the unsteady turbulent structure plays an important role in the preferential accumulation of bubbles and particles. The accumulation of bubbles depends on a rather complicated interaction between the pressure gradient and the lift force; neither is sufficient, acting on its own, to explain the strong accumulation observed when they act together.     

3D scanning particle tracking velocimetry

In this article, we present an experimental setup and data processing schemes for 3D scanning particle tracking velocimetry (SPTV), which expands on the classical 3D particle tracking velocimetry (PTV) through changes in the illumination, image acquisition and analysis. 3D PTV is a flexible flow measurement technique based on the processing of stereoscopic images of flow tracer particles. The technique allows obtaining Lagrangian flow information directly from measured 3D trajectories of individual particles. While for a classical PTV the entire region of interest is simultaneously illuminated and recorded, in SPTV the flow field is recorded by sequential tomographic high-speed imaging of the region of interest. The advantage of the presented method is a considerable increase in maximum feasible seeding density. Results are shown for an experiment in homogenous turbulence and compared with PTV. SPTV yielded an average 3,500 tracked particles per time step, which implies a significant enhancement of the spatial resolution for Lagrangian flow measurements.     

Structure of inertial-admixture accumulation zones in a tornado-like flow

The motion of a dispersed inertial admixture in a steady-state axisymmetric 3D viscous incompressible flow formed by a semi-infinite vortex filament interacting with an orthogonally located substrate surface is considered. The carrier-phase parameters are found from the numerical solution of the Navier-Stokes equations under the assumption of flow self-similarity of a known type [1]. Different phase force interaction schemes corresponding to different ratios of the phase densities are considered. For calculating the dispersed-phase continuum parameters, a full Lagrangian approach is used, which makes it possible to calculate the dispersed-phase concentration in particle accumulation zones and regions of intersecting particle trajectories. On the basis of parametric calculations, it is found that in the case of heavy particles (whose density is greater than that of the carrier phase) a “cup-shaped” particle accumulation surface visualizing a high-vorticity region is formed. The dependence of this surface shape on the governing parameters is investigated. It is shown that for different phase density ratios the dispersed-phase concentration fields are qualitatively different.     

The Kinetic Limit of a System of Coagulating Brownian Particles

We consider a random model of diffusion and coagulation. A large number of small particles is randomly scattered in at an initial time. Each particle has some integer mass and moves as a Brownian motion whose rate of diffusion is determined by that mass. When any two particles are close, they are liable to combine into a single particle that bears the mass of each of them. The range of interaction between pairs of particles is chosen so that a typical particle is liable to interact with a unit order of other particles in a unit of time. We determine the macroscopic evolution of the system, in any dimension d ≧ 3. The density of particles evolves according to the Smoluchowski system of partial differential equations, indexed by the mass parameter, in which the interaction term is a sum of products of densities. Central to the proof is the task of establishing the so-called Stosszahlansatz, which asserts that, at any given time, the presence of particles of two given masses at any given point in macroscopic space is asymptotically independent, as the initial number of particles is taken to be high. Nonetheless, there is, in a microscopic region about each particle, a reduced probability of finding another particle. Determining this deficit precisely is necessary in computing the coefficients appearing in the interaction terms of the Smoluchowski partial differential equation.     

Particle tracking velocimetry in three-dimensional flows

The photogrammetric determination of three-dimensional particle coordinates from a 3-camera system is described in Part I. In Part II we describe a fully automated tracking scheme for the determination of a sequence of velocity vectors within a three-dimensional observation volume of a fluid flow. From this sequence long-time particle trajectories are reconstructed.The tracking scheme is tested on trajectories obtained using the Kinematic Simulation Inertial Model (KSIM). Estimates of the yield of links between adjacent data sets of particle positions and of the yield of long-time particle trajectories are obtained. The limits of efficient tracking as a function of the spacing-displacement ratio p = _o/ut are also obtained. The effect of noise, in the form of the apparent appearance and disappearance of particles between one image and the next, and of jitter, which is the error in the determination of particle coordinates, is examined. It is shown that noise reduces the number of links per frame, but does not increase the number of erroneous links which is always small. However, the yield of long trajectories drops sharply with increasing noise. A small level of jitter, on the other hand, does not significantly influence any of the results.The tracking scheme is used on two sets of particle coordinate data obtained from real flows: a non-turbulent flow in a small water tank and a turbulent open channel flow.     

Three-dimensional particle imaging with a single camera

A new approach to the instantaneous three-dimensional mapping of flow fields is introduced. A single camera system uses defocusing in conjunction with a mask (three pin holes) embedded in the camera lens to decode three-dimensional point sources of light (i.e., illuminated particles) on a single image. The sizes and locations of the particle image patterns on the image plane relate directly to the three-dimensional positions of the individual particles. Using sequential images, particles may be tracked in space and time, yielding whole-field velocity information. Calibration of the system is straightforward, whereas the self-similarity of the particle image patterns can be used in automating the data-extraction process. The described technique was used to obtain particle trajectories in the flow field of a vortex ring impinging on a wall.     

垂直旋转流场中单颗粒运动状态判别及受力分析

为研究单颗粒在旋转流场中的运动状态及受力情况,以毫米级球形颗粒为例,利用旋转流场颗粒运动装置,通过使用摄像机记录颗粒在流场中的运动轨迹以获取其运动参数,分析了不同转速和颗粒直径条件下颗粒的运动轨迹,拟合得到了颗粒运动状态判别公式以及颗粒运动轨迹公式,分析了颗粒在旋转流场中的受力情况。结果表明,颗粒在旋转流场平衡状态下运动状态主要分为两类,一类是未离开壁面保持静止,另一类是离开壁面保持稳定周向运动;颗粒进行周向运动的轨迹为椭圆形,并且圆心随着转速的增大靠近旋转中心,而随着粒径的增大靠近壁面;颗粒在旋转流场的运动过程中主要受到离心力和旋转科式力作用。     

Particle trajectories under interactions between solitary waves and a linear shear current

This paper is concerned with particle trajectories beneath solitary waves when a linear shear current exists. The fluid is assumed to be incompressible and inviscid, lying on a flat bed. Classical asymptotic expansion is used to obtain a Korteweg-de Vries(Kd V) equation, then a forth-order Runge-Kutta method is applied to get the approximate particle trajectories. On the other hand, our particular attention is paid to the direct numerical simulation(DNS) to the original Euler equations. A conformal map is used to solve the nonlinear boundary value problem. Highaccuracy numerical solutions are then obtained through the fast Fourier transform(FFT) and compared with the asymptotic solutions, which shows a good agreement when wave amplitude is small. Further, it also yields that there are different types of particle trajectories. Most surprisingly,periodic motion of particles could exist under solitary waves, which is due to the wave-current interaction.     

Numerical simulation of the hydrodynamic interaction between a sedimenting particle and a neutrally buoyant particle

A new method for the simulation of the translational and rotational motions of a system containing a sedimenting particle interacting with a neutrally buoyant particle has been developed. The method is based on coupling the quasi-static Stokes equations for the fluid with the rigid body equations of motion for the particles. The Stokes equations are solved at each time step with the boundary element method. The stresses are then integrated over the surface of each particle to determine the resultant forces and moments. These forces and moments are inserted into the rigid body equations of motion to determine the translational and rotational motions of the particles. Unlike many other simulation techniques, no restrictions are placed on the shape of the particles. Superparametric boundary elements are employed to achieve accurate geometric representations of the particles. The simulation method is able to predict the local fluid velocity, resolve the forces and moments exerted on the particles, and track the particle trajectories and orientations.     

Investigation of trajectories of inviscid fluid particles in two-dimensional rotating boxes

The particle trajectories of inviscid fluid flow within two-dimensional rotating (elliptic, triangular, and square) boxes are numerically investigated. The source panel method is employed to represent the instantaneous potential interior flow field, and the Runge–Kutta method is used to track the fluid particles. The analytic solutions for the fluid trajectories for the elliptic box are used to verify the numerical accuracy of the method. The numerical error can be reduced to the level of the round-off error if the panels are properly configured and an appropriate number of panels is used. The stagnation of the particles at the corners of the triangular box is successfully predicted with this method. The corner of the square box is found to be a singularity. A logarithmic complex potential is proposed to account for the singularity, using which the stagnation of the particles at the corner in the square box is also captured. The natural frequency of the particles in the rotating elliptic box is constant throughout the flow domain, and the fluid trajectories are epitrochoidal curves. In the triangular box and the square box, the natural frequency strongly depends on the particle position, and the particle trajectories are similar to epitrochoidal curves. In general, the trajectory patterns depend only on the box rotating frequency and the natural frequency of the fluid particle motion.      

Two-way coupling model for shock-induced laminar boundary-layer flows of a dusty gas

The present paper describes a numerical two-way coupling model for shock-induced laminar boundary-layer flows of a dust-laden gas and studies the transverse migration of fine particles under the action of Saffman lift force. The governing equations are formulated in the dilute two-phase continuum framework with consideration of the finiteness of the particle Reynolds and Knudsen numbers. The full Lagrangian method is explored for calculating the dispersed-phase flow fields (including the number density of particles) in the regions of intersecting particle trajectories. The computation results show a significant reaction of the particles on the two-phase boundary-layer structure when the mass loading ratio of particles takes finite values.