Effect Of Particle Inertia Research Articles

Relative particle velocity in a turbulent flow

On the basis of a statistical approach using a probability density function for the coordinates of two particles in a turbulent flow, the parameters of the relative particle motion are investigated. For the functions describing particle entrainment in the turbulence, rigorous results are obtained using a 3D turbulence spectrum. A method of calculating the particle relative-velocity rate with account for particle trajectory correlation is presented. The effects of particle inertia and velocity slip on the parameters of the relative particle motion are studied. Simple approximating formulas for calculating the relative particle motion in a turbulent flow are proposed. The calculation results are compared with the data of direct numerical simulation of stochastic particle trajectories in an isotropic turbulent field.

Acceleration of heavy particles in isotropic turbulence

The paper concerns the effect of particle inertia on acceleration statistics. A simple analytical model for predicting the acceleration of heavy particles suspended in an isotropic homogeneous turbulent flow field is developed. This model is capable of describing the influence of both Stokes and Reynolds numbers on the particle acceleration variance. Comparisons of model predictions with numerical simulations are presented.

Effect of particle inertia on the dynamics of depositional particulate density currents

In this work we address the effect of particle inertia on depositional particulate density currents. We employ a two-fluid model based on the equilibrium Eulerian approach and present highly resolved two-dimensional (2D) simulations of the flow for a Reynolds number of 3450 (this particular choice corresponds to values of Grashof number Gr of 1.5 × 10 6 ).The simulations capture physical aspects of two-phase flows, such as particle preferential concentration and particle migration down turbulence gradients (turbophoresis), which modify substantially the structure and dynamics of the flow. In the simulations we observe the migration of particles from the core of Kelvin–Helmholtz vortices shed from the front of the current as well as their accumulation in the current head. This redistribution of particles in the current affects the propagation velocity of the front, bottom shear stress patterns, as well as deposition rates and patterns.

An Eulerian–Eulerian model for gravity currents driven by inertial particles

In this work we introduce an Eulerian–Eulerian formulation for gravity currents driven by inertial particles. The model is based on the equilibrium Eulerian approach and on an asymptotic expansion of the two-phase flow equations. The final model consists of conservation equations for the continuum phase (carrier fluid), an algebraic equation for the disperse phase (particles) velocity that accounts for settling and inertial effects, and a transport equation for the disperse phase volume fraction. We present highly resolved two-dimensional (2D) simulations of the flow for a Reynolds number of Re = 3450 (this particular choice corresponds to a value of Grashof number of Gr = Re 2 / 8 = 1.5 × 10 6 ) in order to address the effect of particle inertia on flow features. The simulations capture physical aspects of two-phase flows, such as particle preferential concentration and particle migration down turbulence gradients (turbophoresis), which modify substantially the structure and dynamics of the flow. We observe the migration of particles from the core of Kelvin–Helmholtz vortices shed from the front of the current as well as their accumulation in the current head. This redistribution of particles in the current affects the propagation speed of the front, bottom shear stress distribution, deposition rate and sedimentation. This knowledge is helpful for the interpretation of the geologic record.

The magnetron instability in a pulsar's cylindrical electrosphere

(abridged) The physics of the pulsar magnetosphere remains poorly constrained by observations. Little is known about their emission mechanism. Large vacuum gaps probably exist, and a non-neutral plasma partially fills the neutron star surroundings to form an electrosphere. We showed that the differentially rotating equatorial disk in the pulsar's electrosphere is diocotron unstable and that it tends to stabilise when relativistic effects are included. However, when approaching the light cylinder, particle inertia becomes significant and the electric drift approximation is violated. In this paper, we study the most general instability, i.e. by including particle inertia effects, as well as relativistic motions. This general non-neutral plasma instability is called the magnetron instability. We linearise the coupled relativistic cold-fluid and Maxwell equations. The non-linear eigenvalue problem for the perturbed azimuthal electric field component is solved numerically. The spectrum of the magnetron instability in a non-neutral plasma column confined between two cylindrically conducting walls is computed for several cylindrical configurations. For a pulsar electrosphere, no outer wall exists. In this case, we allow for electromagnetic wave emission propagating to infinity. When the self-field induced by the plasma becomes significant, it can first increase the growth rate of the magnetron instability. However, equilibrium solutions are only possible when the self-electric field, measured by the parameter $s_{\rm e}$ and tending to disrupt the plasma configuration, is bounded to an upper limit, $s_{\rm e,max}$. For $s_{\rm e}$ close to but smaller than this value $s_{\rm e,max}$, the instability becomes weaker or can be suppressed as was the case in the diocotron regime.

Influence of added mass on anomalous high rise velocity of light particles in cellular flow field: A note on the paper by Maxey (1987)

In this Brief Communication, we examine in detail the motion of light particles—or dense gas bubbles—rising under gravity in a cellular flow. We follow the methodology of Maxey [Phys. Fluids 30, 1915 (1987)], and we examine a range of parameters not fully discussed in the past, corresponding to particles lighter and yet more inertial than the surrounding fluid if the added mass effect is considered. We observe a nonmonotonic behavior of the particle rise velocity, which exhibits a maximum for density of the particles only slightly smaller than the density of the fluid. The maximum value of the average rise velocity corresponding to the “optimum” density is several times (more than 40) higher than those obtained for small density changes around the optimum. The occurrence of this maximum is due to the combined effects of particle inertia and fluid added mass, which segregate the rising particles into specific, fast pathlines where the local fluid velocity adds to the particle gravitational velocity. A plot covering particle rise velocity for this range—not covered by the analysis of Maxey—is provided.

Asymptotic analysis of chaotic particle sedimentation and trapping in the vicinity of a vertical upward streamline

The sedimentation of a heavy Stokes particle in a laminar plane or axisymmetric flow is investigated by means of asymptotic methods. We focus on the occurrence of Stommel’s retention zones and on the splitting of their separatrices. The goal of this paper is to analyze under which conditions these retention zones can form, and under which conditions they can break and induce chaotic particle settling. The terminal velocity of the particle in still fluid is of the order of the typical velocity of the flow, and the particle response time is much smaller than the typical flow time scale. It is observed that if the flow is steady and has an upward streamline where the vertical velocity has a strict local maximum, then inertialess particle trajectories can take locally the form of elliptic Stommel cells, provided the particle terminal velocity is close enough to the local peak flow velocity. These structures only depend on the local flow topology and do not require the flow to have closed streamlines or stagnation points. If, in addition, the flow is submitted to a weak time-periodic perturbation, classical expansions enable one to write the particle dynamics as a Hamiltonian system with one degree of freedom, plus a perturbation containing both the dissipative terms of the particle motion equation and the flow unsteadiness. Melnikov’s method therefore provides accurate criteria to predict the splitting of the separatrices of the elliptic cell mentioned above, leading to chaotic particle trapping and chaotic settling. The effect of particle inertia and flow unsteadiness on the occurrence of simple zeros in Melnikov’s function is discussed. Particle motion in a plane cellular flow and in a vertical pipe is then investigated to illustrate these results.

Influence of rotary inertia on the fiber dynamics in homogeneous creeping flows

AbstractThe rotation of an inertialess ellipsoidal particle in a Newtonian fluid has been firstly analyzed by Jeffery [1]. He found that in the shear flow the particle rotates such that the end of its axis of symmetry describes a closed periodic orbit. The aim of this paper is to discuss the effects of particle inertia on the rotary motion of a fiber in uniform flow fields. We recall the equations of motion and the constitutive equation for the hydrodynamic moment in the case of a slender fiber. These equations are solved numerically for several flow fields. We demonstrate that for the plane flow fields with dominant vorticity (elliptic and rotational flows) the effect of inertia is the slow particle drift toward the flow plane.

Computational investigation of particle inertia effects on submicron aerosol deposition in the respiratory tract

Current models of submicron particle transport and deposition often ignore particle inertia for aerosols smaller than 200 nm. In the absence of inertial effects, a highly efficient Eulerian transport model can be applied that treats the particle phase as a dilute chemical species. However, the effects of inertia have not been fully quantified for aerosols in the fine and ultrafine ranges. The objective of this study is to evaluate conditions for which current chemical species Eulerian and Lagrangian particle transport models can be applied in order to predict submicron particle deposition characteristics on a regional and local basis in upper and central respiratory models. Differences between the chemical species Eulerian and Lagrangian model results have been used to evaluate conditions for which particle inertia becomes important relative to diffusional effects. The deposition characteristics of particles ranging from 5 nm to 1 μ m have been evaluated in a tubular entrance flow geometry, a double bifurcation model of upper respiratory generations G3–G5 and a double bifurcation model of central respiratory generations G7–G9. Considering the regional area-averaged deposition of submicron aerosols, the minimum particle diameters (and Stokes numbers) for which particle inertia became significant were approximately 20 nm ( St = 6.1 × 10 - 6 ) for the tubular entrance flow geometry, 70 nm ( St = 5.1 × 10 - 5 ) for the upper bifurcation model, and 140 nm ( St = 4.4 × 10 - 5 ) for the central bifurcation model. Below these critical particle diameters, numerical estimates of regional deposition were shown to be consistent with currently available analytic correlations of diffusional deposition efficiencies. In comparison to regional-averaged values, the effects of particle inertia on localized deposition characteristics were found to be much more dramatic. For the upper airway bifurcation model, inclusion of particle inertia increased the maximum local microdosimetry factor by one order of magnitude for 40 nm particles at an inhalation flow rate of 30 L/min. Results of this study indicate that particle inertia may be more significant in regional and local depositions of fine and ultrafine aerosol than previous considered. Therefore, effective models of particle transport are necessary that can maintain the efficiency of the chemical species Eulerian approach while accounting for local finite particle inertia.

Stochastic Stokes' drift with inertia

We consider both the effect of particle inertia on stochastic Stokes' drift, and also a related process which could be considered as a crude model of stochastic Stokes' drift driven by an eddy diffusivity. In the latter, the stochastic forcing is a stable Ornstein–Uhlenbeck process rather than Brownian motion. We show that the eddy Stokes' drift velocity has a peak at a non-zero value of the correlation time-scale for particles that have the same (limiting) diffusivity. For both of the models considered, this study shows that not only can stochastic Stokes' drift be used to sort particles with different diffusivities, but also it can be used to sort particles of the same diffusivities, but with different particle masses or correlation time-scales. This effect may be important in particle sorting applications.

Collisions of liquid coated solid spherical particles in a viscous fluid

An analytical description is presented for the head-on collision of two spherical rigid particles that are coated with a thin layer of one liquid and immersed in another. Lubrication theory is used to resolve the spatio-temporal evolution of the coating surfaces, in conjunction with the fluid flow in the gap region between the particles. The analysis is carried out up to the point where the gap region has almost completely been drained; intermolecular forces are neglected. The effects of particle inertia, the ratio of particle radii, surface tension, and the viscosity ratio of the coating and carrier fluids are studied; these are parameterised by St, β, Ca and m, respectively. The results of the present work elucidate the effect of the above-mentioned factors on the conditions under which particles rebound (assumed to occur if the distance between the particles becomes very short while the relative velocity does not vanish) or stick. In particular, summarizing flowmaps show that the likelihood of particles rebounding increases with increasing St and decreasing β, Ca and m. On the other hand, it is shown that the force on approaching particles depends on all of these parameters in a non-monotonic manner.

Time scales for predicting dispersion of arbitrary-density particles in isotropic turbulence

The effects of particle inertia and particle-to-fluid density on the time scales that measure the dispersion of particles by isotropic turbulence are investigated. Numerical calculations are performed by means of a kinematic simulation method coupled with Lagrangian tracking of discrete particles. Moreover, a simple semi-empirical model for predicting the time scales of the fluid velocity seen by a particle as well as of the fluid–particle velocity covariance and the particle velocity variance is proposed. Comparisons between simulations and model predictions are presented.

Effect of particle inertia on temperature statistics in particle-laden homogeneous isotropic turbulence

The fluid temperature statistics along particle trajectories is crucial to understand the mechanisms of turbulent non-isothermal or reactive fluid-particle flow, especially for the Lagrangian model of non-isothermal particle-laden turbulent flow. In the present study, direct numerical simulations were utilized to generate temperature field statistics in particle-laden incompressible stationary homogeneous isotropic turbulent flows, which is focused on the effect of particle response time on the Lagrangian statistics of the particle and the fluid temperature seen by particles. It shows that, for the particles with τ p / τ k τ p / τ k increased; while for larger particles ( τ p / τ k > 1), the trend is inversed. For small particles ( τ p / τ k R p T decreases as the particle inertia ( τ p / τ k ) increases. The trend is reversed for larger particles. The autocorrelation of fluid temperature along the particle path, R pf T , decreased as the particle inertia increased. And as the particle inertia increased, the autocorrelation coefficient of the fluid temperature seen by particle decreased more rapidly than that of the particle temperature. The mean temperature gradient contributes to the correlation between the particles velocity component and temperature fluctuations in the direction of the gradient. For the particles with τ p / τ k <1, the magnitude of the correlation coefficient increases as the particle inertia increases, while this value is independent of the particle time constant for larger particles.

Effect of particle inertia and gravity on the turbulence in a suspension

A theoretical model is presented for the effect of particle inertia and gravity on the turbulence in a homogeneous suspension. It is an extension of the one-fluid model developed by L’vov, Ooms, and Pomyalov [Phys. Rev. E 67, 046314 (2003)], in which the effect of gravity was not considered. In the extended model the particles are assumed to settle in the fluid under the influence of gravity due to the fact that their density is larger than the fluid density. The generation of turbulence by the settling particles is described, with special attention being paid to the turbulence intensity and spectra. A comparison is made with direct numerical simulation calculations and experimental data. Also a sensitivity study is carried out to investigate at which conditions the gravity effect becomes important.

Particle behavior in homogeneous isotropic turbulence

Direct numerical simulations were conducted to investigate the behavior of heavy particles in homogeneous isotropic turbulence. The present study focused on the effect of particle inertia and drift on the autocorrelations of the particle velocity and the fluid seen by particles and the dispersion characteristics of particles. The Lagrangian integral time scale of particles monotonically increased as the magnitude of the particle response time increased, while that of the fluid seen by particles remained relatively constant; it reached a maximum when the particle response time was close to the Kolmolgorov time scale of the flow. Particle dispersion increased as the particle inertia increased for small particles, while for larger particles, it decreased as particle inertia increased; particle eddy diffusion coefficient was maximal, and greater than that of the fluid by about 30%, at the preferential concentration. The concentration field of the particles with τp/τk≈1.0 showed that particles tend to collect in regions of low vorticity (high strain) due to preferential concentration. As the drift velocity of a particle is increased it crosses the paths of fluid elements more rapidly and will tend to lose correlation with its previous velocity faster than a fluid element will. And the correlation of particle velocities along the drift direction is more persistent than that perpendicular to the direction of drift. Simulations also showed that the continuity effect and the crossing-trajectory effect are weakened for particles with infinite inertia.

Brownian motion of finite-inertia particles in a simple shear flow

Simultaneous diffusive and inertial motion of Brownian particles in laminar Couette flow is investigated via Lagrangian and Eulerian descriptions to determine the effect of particle inertia on diffusive transport in the long-time limit. The classical fluctuation dissipation theorem is used to calculate the amplitude of random-force correlations, thereby neglecting corrections of the order of the molecular relaxation time to the inverse shear rate. In the diffusive limit (time much greater than the particle relaxation time) the fluctuating particle-velocity autocorrelations functions are found to be stationary in time, the correlation in the streamwise direction being an exponential multiplied by an algebraic function and the cross correlation nonsymmetric in the time difference. The analytic, nonperturbative, evaluation of the particle-phase total pressure, which is calculated to be second order in the Stokes number (a dimensionless measure of particle inertia), shows that the particle phase behaves as a non-Newtonian fluid. The generalized Smoluchowski convective-diffusion equation, determined analytically from a combination of the particle-phase pressure tensor and the inertial acceleration term, contains a shear-dependent cross derivative term and an additional term along the streamwise direction, quadratic in the particle Stokes number. The long-time diffusion coefficients associated with the particle flux relative to the carrier flow are found to depend on particle inertia such that the streamwise diffusion coefficient becomes negative with increasing Stokes number, whereas one of the cross coefficients is always negative. The total diffusion coefficients measuring the rate of change of particle mean-square displacement are always positive as expected from general stability arguments.

Simulation of particle deposition on filter fiber in an external electric field

Particle deposition onto a filter fiber was numerically simulated when a uniform external electric field was applied. The effects of electric field strength, particle inertia, and electrical conductivity of particles on p article deposition characteristics such as particle loading patterns and collection efficiency were qualitatively investigated. As a result, the electrostatic forces between a newly introdu ced particle and the already captured particles on the fiber were found to have a great influence on th e particle deposition patterns compared with the results where the electrostatic forces were neglected. Conductive particle s and filter fibers lead to higher collection efficiency and more linear structure of particle deposits than those of dielectrics, and the particle inertia could also be more important to the collection effi- ciency of a fibrous filter when electric fields are present. The simulated particle deposits obtained from this work agreed well with the existing experimental results, in which the photogra phs of particle loaded fibers , within an external electric field, were reported.

Coagulation of monodisperse aerosol particles by isotropic turbulence

The rate of coagulation of initially monodisperse aerosols due to isotropic turbulence is studied with particular emphasis on the effects of noncontinuum hydrodynamics and particle inertia. The prevalence of these two factors distinguishes aerosol coagulation from the coagulation of colloidal particles. The turbulent flow seen by an interacting pair of particles is modelled as a stochastically varying flow field that is a linear function of position. This approximation is valid because the 1–10 micron diameter particles for which turbulence dominates coagulation are much smaller than the smallest eddies of a typical turbulent flow field. It is shown that the finite mean-free path of the gas enhances the rate of coagulation and leads to a finite coagulation rate even in the absence of van der Waals attractions. The coupled effects of turbulent shear and Brownian motion are treated. As in the case of laminar shear flows, it is found that Brownian motion plays an important role in the coagulation process even when the Peclet number is moderately large. It is shown that particle inertia increases the coagulation rate in two ways. First, preferential concentration increases the radial distribution function on length scales intermediate between the Kolmogorov length scale and the particle diameter. Second, the greater persistence of particles’ relative motion during their local interaction leads to an increase in coagulation rate with increasing particle Stokes number.

Towards a general semi-empirical model for the aspiration efficiencies of aerosol samplers in perfectly calm air

In previous papers we have described a body of experimental work, backed up by numerical studies, to acquire a large set of new data for the aspiration efficiencies of aerosol samplers of simple geometries in perfectly calm air and for a range of orientations. The samplers studied were idealized thin-walled and spherical blunt probes, and experiments were conducted for upwards, downwards and horizontal sampling. Taken as a whole, this is believed to be the most comprehensive data set yet obtained for aerosol sampling in calm air. In this paper, we have examined this complete data set—including both experimental and numerical data—and used it to construct a set of semi-empirical models for aspiration efficiency. The starting point for these is the basic physical model of Levin (1957) for the entry of particles into a point sink in calm air. Additional terms are introduced that account for the role of the sampler having finite dimensions and defined shape. The models are expressed in terms of groups of dimensionless combinations of variables that embody the effects of particle inertia, gravitational settling, sampler bluntness and sampler orientation with respect to the vertical. They are semi-empirical in that, although they thus embody physical mechanisms and ideas, the added terms (beyond the Levin-model starting point) containing those dimensionless quantities are incorporated into mathematical constructs that are entirely empirical. The constructs are such that they may be combined together in a single universal form that reverts back to the individual specific cases when those conditions are inserted. Agreement is generally good across the whole range of conditions studied. The small number of areas where agreement is less good may be explained in terms of experimental error or technical limitations in acquiring the experimental data.

A stochastic description of wall sources in a turbulent field: part 2. Calculation for a simplified model of horizontal annular flows

A stochastic approach that uses a modified Langevin equation to calculate fluid velocities seen by particles dispersing in a turbulent flow was verified at Re τ =150, V T +=0.11, τ p +=20 in Part 1 of this paper. It is now used to examine the effects of particle inertia and gravitational settling on the deposition rate at Re τ =590. The system considered is an idealized annular flow in a horizontal channel. Drops are admitted from the top and the bottom walls at relative rates such that a fully-developed field is realized. The walls are considered to be arrays of point sources. The theoretical problem is to model the behavior of an instantaneous wall source. Particles are ejected from the source with a velocity V i 0 at time zero. They are mixed by the fluid turbulence and eventually deposit at the top and bottom walls. Of particular interest are the strong effect of very small gravitational settling on the rate of deposition and the definition of a critical condition for particles (ejected from the bottom wall) to reach the top wall to create an annular flow.