Effect Of Particle Inertia Research Articles

Dispersion and particle pressure in sedimenting suspensions with hydrodynamic interactions and particle inertia

Particle inertia is a feature of suspension dynamics that is often neglected. However, its neglect changes the order of the governing equations of particle motion. In this work, we examine the impact of the inertia of particles over their velocity fluctuations and the resulting stresses. In order to observe effects of particle inertia, we consider dusty-gas suspensions of spherical weakly Brownian microparticles sedimenting in a Newtonian fluid at low Reynolds numbers. We first investigate the motion of a single spherical particle. The simulations for this simple case allow us to define the appropriate physical parameters of the flow and to validate the numerical calculation of velocity statistics over a range of Péclet and Stokes numbers. Next, we perform Langevin dynamics simulations with periodic boundary conditions to integrate the equations governing the translational and rotational motions of N hydrodynamically interacting particles suspended in the viscous fluid. The first aim of this paper is to examine the behavior of the velocity variance of inertial particles, with small fluctuations produced by Brownian motion at the moderate Péclet numbers. The interplay between mild Brownian forces and hydrodynamic interactions decreases the anisotropy of velocity fluctuations observed in non-Brownian suspensions (Pe≫1) of sedimenting particles. The simulation results suggest that particle inertia attenuates the magnitude of velocity fluctuations produced by both Brownian motion and viscous hydrodynamic interactions. Additionally, we study the long-time behavior of the velocity fluctuations by calculating their autocorrelation functions and the particle diffusivities. The numerical simulations show clear evidence of particle pressure arising from the flow disturbance produced by hydrodynamic interactions in a suspension. From the particle velocity variance obtained in the present numerical simulations, we propose a simple model for particle-phase pressure as a linear function of the particle volume fraction ϕ⩽5%, which is usually a closure quantity required in models of more complex particulate systems such as fluidized beds.

The merger of co-rotating vortices in dusty flows

We investigate the effect of particle inertia on the merger of co-rotating dusty vortex pairs at semi-dilute concentrations. In a particle-free flow, the merger is triggered once the ratio of vortex core size to vortex separation reaches a critical value. The vortex pair separation then decreases monotonically until the two cores merge together. Using Eulerian–Lagrangian simulations of co-rotating particle-laden vortices, we show substantial departure from the vortex dynamics previously established in particle-free flows. Most strikingly, we find that disperse particles with moderate inertia cause the vortex pair to push apart to a separation nearly twice as large as the initial separation. During this stage, the drag force exerted by particles ejected out of the vortex cores on the fluid results in a net repulsive force that pushes the two cores apart. Eventually, the two dusty vortices merge into a single vortex with most particles accumulating outside the core, similar to the dusty Lamb–Oseen vortex described in Shuai & Kasbaoui (J. Fluid Mech., vol 936, 2022, p. A8). For weakly inertial particles, we find that the merger dynamics follows the same mechanics as that of a single-phase flow, albeit with a density that must be adjusted to match the mixture density. For highly inertial particles, the feedback force exerted by the particles on the fluid may stretch the two cores during the merger to a point where each core splits into two, resulting in inner and outer vortex pairs. In this case, the merger occurs in two stages where the inner vortices merge first, followed by the outer ones.

The Role of Particle Inertia and Thermal Inertia in Heat Transfer in a Non-Isothermal Particle-Laden Turbulent Flow

We present an analysis of the effect of particle inertia and thermal inertia on the heat transfer in a turbulent shearless flow, where an inhomogeneous passive temperature field is advected along with inertial point particles by a homogeneous isotropic velocity field. Eulerian–Lagrangian direct numerical simulations are carried out in both one- and two-way coupling regimes and analyzed through single-point statistics. The role of particle inertia and thermal inertia is discussed by introducing a new decomposition of particle second-order moments in terms of correlations involving Lagrangian acceleration and time derivative of particles. We present how particle relaxation times mediate the level of particle velocity–temperature correlation, which gives particle contribution to the overall heat transfer. For each thermal Stokes number, a critical Stokes number is individuated. The effect of particle feedback on the attenuation or enhancement of fluid temperature variance is presented. We show that particle feedback enhances fluid temperature variance for Stokes numbers less than one and damps is for larger than one Stokes number, regardless of the thermal Stokes number, even if this effect is amplified by an increasing thermal inertia.

Mixed convection in turbulent particle-laden channel flow at Re τ = 180

A numerical investigation of mixed convection in a turbulent particle-laden channel flow is presented. Using Eulerian-Lagrangian Direct Numerical Simulations (DNSs) within the point-particle approach, the interaction between wall turbulence, particle inertia and thermal inertia, and buoyancy in the two-way coupling regime has been studied. The flow dynamics is controlled by the friction Reynolds number, Richardson number, particle Stokes and thermal Stokes numbers, and the Prandtl number. The effects of particle inertia and buoyancy on fluid and particle statistics are shown for a friction Reynolds number of 180, Stokes numbers ranging from 0.6 to 120, Richardson numbers ranging from 2.72 × 10−4 to 27.2 at a single Prandtl number, 0.71. The results indicate that the effect of particle inertia is significant on particle statistics, but not as pronounced on fluid statistics at the relatively low volume fraction considered. Particle inertia modulates the overall heat flux in a non-monotonic way.

Statistical Models for the Dynamics of Heavy Particles in Turbulence

When very small particles are suspended in a fluid in motion, they tend to follow the flow. How such tracer particles are mixed, transported, and dispersed by turbulent flow has been successfully described by statistical models. Heavy particles, with mass densities larger than that of the carrying fluid, can detach from the flow. This results in preferential sampling, small-scale fractal clustering, and large relative velocities. To describe these effects of particle inertia, one must consider both particle positions and velocities in phase space. In recent years, statistical phase-space models have significantly contributed to our understanding of inertial-particle dynamics in turbulence. These models help to identify the key mechanisms and nondimensional parameters governing the particle dynamics and have made qualitative and, in some cases, quantitative predictions. This article reviews statistical phase-space models for the dynamics of small, yet heavy, spherical particles in turbulence. We evaluate their effectiveness by comparing their predictions with results from numerical simulations and laboratory experiments, and we summarize their successes and failures.

Particle inertial effects on radar Doppler spectra simulation

Abstract. Radar Doppler spectra observations provide a wealth of information about cloud and precipitation microphysics and dynamics. The interpretation of these measurements depends on our ability to simulate these observations accurately using a forward model. The effect of small-scale turbulence on the radar Doppler spectra shape has been traditionally treated by implementing the convolution process on the hydrometeor reflectivity spectrum and environmental turbulence. This approach assumes that all the particles in the radar sampling volume respond the same to turbulent-scale velocity fluctuations and neglects the particle inertial effect. Here, we investigate the inertial effects of liquid-phase particles on the forward modeled radar Doppler spectra. A physics-based simulation (PBS) is developed to demonstrate that big droplets, with large inertia, are unable to follow the rapid change of the velocity field in a turbulent environment. These findings are incorporated into a new radar Doppler spectra simulator. Comparison between the traditional and newly formulated radar Doppler spectra simulators indicates that the conventional simulator leads to an unrealistic broadening of the spectrum, especially in a strong turbulent environment. This study provides clear evidence to illustrate the droplet inertial effect on radar Doppler spectrum and develops a physics-based simulator framework to accurately emulate the Doppler spectrum for a given droplet size distribution (DSD) in a turbulence field. The proposed simulator has various potential applications for the cloud and precipitation studies, and it provides a valuable tool to decode the cloud microphysical and dynamical properties from Doppler radar observation.

Particle dynamics in compressible turbulent vertical channel flows

In this work, we carry out direct numerical simulations of particle suspensions in the compressible turbulent vertical channel (TVC) flows with Mach number Ma = 1.5 and particle Stokes number St = 1–100. The compressibility effect is considered in the particle dynamic model for the first time in the study of compressible particle-laden wall turbulence. We find that in both incompressible and compressible flow, gravity weakens the wall-normal and spanwise fluctuations of particle velocities as the Stokes number increases. However, compared to the incompressible flow case, the compressible effect amplifies the mean velocity, fluctuations of velocity, and slip velocity of particle in the streamwise direction. The wall-normal and spanwise fluctuations of particle velocities are augmented by the compressible effect in the channel core region. Moreover, in the core region, the effect of fluid compressibility on the wall-normal and spanwise fluctuations of particle velocities attenuates as the Stokes number increases, indicating a competition between the compressible effect and the particle inertia effect. We, furthermore, conduct the quadrant analysis of the local fluctuation velocities of fluid at particle positions and observe preferential distributions in the second and the fourth quadrants at y+ = 12.5–13.5. For compressible TVC flows, the pattern of probability distributions is more elongated, and the percentage is slightly higher in the second and fourth quadrants than that of incompressible flows. This observation implies that more particles locate in the ejection and sweep events in compressible flows than that in incompressible flows, which is anticipated to influence the particle wall-normal transport.

Inertial effects in just-saturated axisymmetric column collapses

This work introduces a scaling analysis of sub-aerial axisymmetric column collapses of glass beads and Newtonian glycerol-water solutions mimicking some of the behaviours of debris flows. The beads were in a size range where their inertia partly decouples their collapse behaviour from the water column. Experiments explored a range of viscous, surface tension and particle inertia effects through systematic variation of particle size and fluid viscosity. Crucially a geotechnical centrifuge was used to access elevated effective gravitational accelerations driving the collapse, allowing field-scale viscous and surface tension effects to be replicated. Temporal pore pressure and run out front position evolution data was extracted using a pressure transducer and high speed imaging, respectively. A least-squares fitted scale analysis demonstrated that all characteristic dimensionless quantities of the acceleration phase could be described as a function of the column-scale Bond number text{ Bo }, the Capillary number text{ Ca }, the system size r^*, and the grain-fluid density ratio rho ^*. This analysis demonstrated that collapses as characterised by pore pressure evolution and front positions were controlled by the ratio of text{ Bo}/text{Ca}. This indicates that grain-scale surface tension effects are negligible in such inertial column collapses where they may dominate laboratory-scale granular-fluid flow behaviour where geometric similarity between grain and system size is preserved.Graphical abstract

Investigating the magnitude and temporal localization of inertial particle mixing in turbulent channel flows

Mixing of inertial point particles in a turbulent channel flow at Reτ=950 is investigated by means of direct numerical simulations. We consider inertial particles, at varying Stokes number, released from pairs of sources located at different positions inside the channel and analyze the rate at which particles come into close proximity to each other. To do so, we employ a Lagrangian framework, which is suitable for the analysis of trajectories and in general for the study of mixing and dispersion problems. By varying the release position of particles along the wall-normal direction we obtain a thorough description of mixing in an anisotropic turbulent flow. Moreover, we analyze the effects of particle inertia and show that these are not univocal but also depend on the position and alignment of the sources, owing in particular to the dependence of the flow timescales on the distance from the wall.

Deposition velocity of inertial particles driven by wall-normal external force in turbulent channel flow

The effect of wall-normal external force and particle inertia on the deposition of particles in wall-bounded turbulence has been investigated by a point-particle Lagrange simulation. The preferential distribution of particles in the near-wall regions and the insufficient deceleration in the viscous sublayer jointly cause the enhancement of deposition velocity. The external forces strongly affect the clustering of particles in coherent flow structures. The increasing of the wall-normal force inhibits the clustering of particles, which further affects the deposition velocity.

Drag model from interface-resolved simulations of particle sedimentation in a periodic domain and vertical turbulent channel flows

A drag correlation is established for laminar particle-laden flows, based on data from the interfaced-resolved direct numerical simulations (IR-DNS) of particle sedimentation in a periodic domain at density ratio ranging from 2 to 1000, particle concentration ranging from 0.59 % to 14.16 %, and particle Reynolds number below 132. Our drag decreases slightly with increasing density ratio when the other parameters are fixed. The drag correlation is then corrected to account for the turbulence effect by introducing the relative turbulent kinetic energy, from the IR-DNS data of the upward turbulent channel flows laden with the particles larger than the Kolmogorov length scale at relatively low particle volume fractions. A drift velocity model is developed to obtain the effective slip velocity from the interphase mean velocity difference for the vertical turbulent channel flow by considering the effects of particle inertia, particle concentration distribution and large-scale streamwise vortices.

Simulation of bubble–particle attachment process and estimation of attachment probability using a coupled smoothed particle hydrodynamics–discrete element method model

The bubble–particle attachment is an important step in determining the success of recovery in froth flotation. In this study, the bubble–particle attachment process was investigated using a coupled smoothed particle hydrodynamics (SPH)–discrete element method (DEM) model. The effects of particle inertia and the drag of fluid flow on the particle motion were examined. The sliding time was determined for different particle sizes, densities, and bubble sizes, and the simulated sliding time was compared with the analytical solution and our own experimental results using a high-speed camera. As the particle size and density increased, the sliding time decreased. However, this value was independent of the bubble size. The induction time was calculated as a function of the particle size and contact angle from an empirical model, and it was used to determine the attachment and net probabilities in a dynamic simulation. Based on the SPH–DEM simulation results, novel probability models were developed for bubble–particle interactions. Both attachment and net probabilities could be decoupled into hydrodynamic and thermodynamic effects, given that the collision probability was also derived as a single function of the particle Stokes number. The developed probability models embodying the mutual influence of bubbles and particles can be extended to macroscopically describe flotation cells and evaluate cell performance including particle recovery.

Scale-dependent particle clustering in transitional wake flow

Dispersion and mixing of inertial point particles in the unsteady and three-dimensional wake of a circular cylinder at$Re=200$are investigated via one-way coupled simulations. The presence of streamwise-oriented vortical braids originating from the mode A instability in the transition regime has a profound impact on the particle dynamics and preferential concentration in the near wake. Particles trapped between the Kármán rollers and the streamwise braids form an inner layer of densely concentrated particles, while discrete particle clumps on the otherwise thin ribbon-like clusters between two consecutive Kármán rollers are another manifestation of the streamwise braids. The effect of particle inertia on the size of clusters and voids ascribed to centrifugal ejection is examined by volume-averaged Voronoï analysis. A distinct dependence on Stokes number ($Sk$) is seen at the cluster scale, whereas the void scale exhibits self-similarity. New physics-based threshold values of clusters and voids are distinctly different from the probability-distribution-based threshold at cluster scale. Increasing inertia is found to suppress particle acceleration more than deceleration. The particle velocity is further suppressed by the presence of the streamwise vortical braids. The effects of particle inertia vary non-monotonically in$Sk$with the strongest effect at$Sk=1$when most particles tend to reside in high-strain/low-vorticity regions.

Lane dynamics in pair-ion plasmas: effect of obstacle and geometric aspect ratio

Lane formation dynamics of driven two-dimensional pair-ion plasmas is investigated in under-damped cases where the effect of particle inertia cannot be neglected. Extensive Langevin dynamics simulations using an OpenMP parallel program are carried out to analyse the effect of obstacle and geometric aspect ratio on lane formation dynamics previously reported in Sarma et al. (Phys. Plasmas, vol. 27, 2020, 012106) and Baruah et al. (J. Plasma Phys., vol. 87, 2021, 905870202). Lanes are found to form when like particles move along or opposite to the applied field direction. Lane order parameter, cumulative order parameter and distribution of the order parameter have been implemented to detect phase transition. The effect of geometric aspect ratio on the stability of lanes is systematically determined in both the presence and absence of an obstacle. Here, a specular reflective boundary condition is implemented to mimic an obstacle. We demonstrate that an obstacle promotes the merging of lanes, and the system gradually transitions to a partially mixed phase with higher value of aspect ratio. The occurrence of lane mixing phenomena at the separation boundary of two oppositely flowing lanes at higher value of aspect ratio is observed. In the presence of an oscillatory electric field, the lane merging tendency is reduced to a large extent as compared to the system where a constant electric field is applied. Furthermore, in the presence of both space- and time-varying electric fields, an appearance of a void is observed on either side of the obstacle. The study finds that the presence of an external magnetic field promotes acceleration of the phase transition process towards the lane mixing phase; it also reveals the existence of electric field drift in the system. Our findings may prove to be useful in understanding the nature of lane dynamics in naturally occurring pair-ion plasma systems as well as their relevance to technological applications that exploit or mitigate self-organization.

Effect of Particle Inertia on the Alignment of Small Ice Crystals in Turbulent Clouds

AbstractSmall nonspherical particles settling in a quiescent fluid tend to orient so that their broad side faces down because this is a stable fixed point of their angular dynamics at small particle Reynolds number. Turbulence randomizes the orientations to some extent, and this affects the reflection patterns of polarized light from turbulent clouds containing ice crystals. An overdamped theory predicts that turbulence-induced fluctuations of the orientation are very small when the settling number Sv (a dimensionless measure of the settling speed) is large. At small Sv, by contrast, the overdamped theory predicts that turbulence randomizes the orientations. This overdamped theory neglects the effect of particle inertia. Therefore, we consider here how particle inertia affects the orientation of small crystals settling in turbulent air. We find that it can significantly increase the orientation variance, even when the Stokes number St (a dimensionless measure of particle inertia) is quite small. We identify different asymptotic parameter regimes where the tilt-angle variance is proportional to different inverse powers of Sv. We estimate parameter values for ice crystals in turbulent clouds and show that they cover several of the identified regimes. The theory predicts how the degree of alignment depends on particle size, shape, and turbulence intensity, and that the strong horizontal alignment of small crystals is only possible when the turbulent energy dissipation is weak, on the order of 1 cm2 s−3 or less.

Mechanisms governing the settling velocities and spatial distributions of inertial particles in wall-bounded turbulence

We use theory and Direct Numerical Simulations (DNS) to explore the average vertical velocities and spatial distributions of inertial particles settling in a wall-bounded turbulent flow. The theory is based on the exact phase-space equation for the Probability Density Function describing particle positions and velocities. This allowed us to identify the distinct physical mechanisms governing the particle transport. We then examined the asymptotic behavior of the particle motion near the wall, revealing the fundamental differences to the near wall behavior that is produced when incorporating gravitational settling. When the average vertical particle mass flux is zero, the averaged vertical particle velocity is zero away from the wall due to the particles preferentially sampling regions where the fluid velocity is positive, which balances with the downward Stokes settling velocity. When the average mass flux is negative, the combined effects of turbulence and particle inertia lead to average vertical particle velocities that can significantly exceed the Stokes settling velocity, by as much as ten times. Sufficiently far from the wall, the enhanced vertical velocities are due to the preferential sweeping mechanism. However, as the particles approach the wall, the contribution from the preferential sweeping mechanism becomes small, and a downward contribution from the turbophoretic velocity dominates the behavior. Close to the wall, the particle concentration grows as a power-law, but the nature of this power law depends on the particle Stokes number. Finally, our results highlight how the Rouse model of particle concentration is to be modified for particles with finite inertia.

Effect of the local wind reduction zone on seed dispersal from a single shrub element on sparsely vegetated land.

Accurate predictions of seed dispersal kernels are crucial for understanding both vegetation communities and landscape dynamics. The influences of many factors, including the physical properties of seeds, the time-averaged wind speed and the wind turbulence, on seed dispersal have been studied. However, the influence of local wind speed reduction around a single shrub element (e.g. a small patch of scrub) on seed dispersal is still not well understood. Here, the spatial distribution of the wind intensity (represented by the wind friction speed u*) around a single shrub element is described, with an emphasis on the variation in the streamwise direction, and assuming that the time-averaged lateral and vertical speeds are equal to zero. The trajectories of the seeds were numerically simulated using a Lagrangian stochastic model that includes the effects of wind turbulence and particle inertia. The patterns of seed deposition with and without the effect of local wind reduction were compared. The variation in seed deposition with changing wind intensity, release height and shrub porosity were also simulated. The simulation results revealed that the local wind reduction increased seed deposition in nearby regions and therefore decreased seed deposition in the regions farther away. Local wind reduction had a greater impact on short-distance dispersal than on long-distance dispersal. Moreover, the dispersal in the circumferential direction decreased once the motion of a seed moving in the streamwise direction was reduced due to the local wind reduction. As the wind intensity and release height increased, the effect of local wind reduction on seed dispersal weakened. Seed dispersal was both wider and farther as the shrub porosity increased. These results may help explain the disagreement between the mechanistic models and the fitting curves in real cases. In addition, the results of this study may improve the currently used mechanistic models by either increasing their flexibility in case studies or by helping explain the variations in the observed distributions.

Study of Colliding Particle-Pair Velocity Correlation in Homogeneous Isotropic Turbulence

This paper deals with the numerical analysis of the particle inertia and volume fraction effects on colliding particle-pair velocity correlation immersed in an unsteady isotropic homogeneous turbulent flow. Such correlation function is required to build reliable statistical models for inter-particle collisions, in the frame of the Euler–Lagrange approach, to be used in a broad range of two-phase flow applications. Computations of the turbulent flow have been carried out by means of Direct Numerical Simulation (DNS) by the Lattice Boltzmann Method (LBM). Moreover, the dependence of statistical properties of collisions on particle inertia and volumetric fraction is evaluated and quantified. It has been found that collision locations of particles of intermediate inertia, StK~1, occurs in regions where the fluid strain rate and dissipation are higher than the corresponding averaged values at particle positions. Connected with this fact, the average kinetic energy of colliding particles of intermediate inertia (i.e., Stokes number around 1) is lower than the value averaged over all particles. From the study of the particle-pair velocity correlation, it has been demonstrated that the colliding particle-pair velocity correlation function cannot be approximated by the Eulerian particle-pair correlation, obtained by theoretical approaches, as particle separation tends to zero, a fact related with the larger values of the relative radial velocity between colliding particles.

Direct numerical simulation of turbulent liquid–solid flow in a small-scale stirred tank

Particle-resolved direct numerical simulation of a 4-baffled cylindrical stirred tank with a 6-pitched blade-turbine-downflow (PBTD) impeller was carried out for the first time to understand the hydrodynamic characteristics of its turbulent solid–liquid flow. The fluid-particle and fluid-wall interactions are numerically described by the immersed boundary method (IBM) with smeared and sharp interfaces jointly on a staggered Cartesian grid. The particle diameter is 1 mm, and the impeller rotation speed changes between 23.88 and 15.88 rev/s, resulting in the impeller Reynolds number at 7031 and 4676, respectively, belonging to medium turbulent regime. The average solid volume fraction ranged from 0 to 0.1. The effect of particle inertia was also investigated by changing the particle–fluid density ratio from 1.5 to 3.0. Analysis of the simulation results indicates turbulence suppression in the bulk region and particle-induced strengthening of the axial liquid circulation. The quasi-steady and unsteady drag is correlated to the relative turbulence intensity and the turbulence dissipation rate obtained from the simulation data, respectively.

Transport of inertial particles in high-Reynolds-number turbulent boundary layers

Abstract