Developing reduced order models for the transport of solid particles in turbulence typically requires a statistical description of the particle-turbulence interactions. In this work, we utilize a statistical framework to derive continuum equations for the moments of the slip velocity of inertial settling Lagrangian particles in a turbulent boundary layer. Using coupled Eulerian-Lagrangian direct numerical simulations, we then identify the dominant mechanisms controlling the slip velocity variance, and find that for a range of St+, Sv+, and Re, the slip variance is primarily controlled by local differences between the "seen" variance and the particle velocity variance, while terms appearing due to the inhomogeneity of the turbulence are sub-leading until Sv+ becomes large. We also consider several comparative metrics to assess the relative magnitudes of the fluctuating slip velocity and the mean slip velocity, and we find that the vertical mean slip increases rapidly with Sv+, rendering the variance relatively small -- an effect found to be most substantial for Sv+>1. Finally, we compare the results to a model of the acceleration variance Berk and Coletti (2021) based the concept of a response function described in Csanady (1963), highlighting the role of the crossing trajectories mechanism. We find that while there is good agreement for low Sv+, systematic errors remain, possibly due to implicit non-local effects arising from rapid particle settling and inhomogeneous turbulence. We conclude with a discussion of the implications of this work for modeling the transport of coarse dust grains in the atmospheric surface layer.
We present an experimental study on the settling velocity of dense sub-Kolmogorov particles in active-grid-generated turbulence in a wind tunnel. Using phase Doppler interferometry, we observe that the modifications of the settling velocity of inertial particles, under homogeneous isotropic turbulence and dilute conditions $ϕ_v\leq O(10)^{-5}$, is controlled by the Taylor-based Reynolds number $Re_λ$ of the carrier flow. On the contrary, we did not find a strong influence of the ratio between the fluid and gravity accelerations (i.e., $γ\sim(η/τ_η^2)/g$) on the particle settling behavior. Rema…
We investigate the sedimentation properties of quasi-neutrally buoyant inertial particles carried by incompressible zero-mean fluid flows. We obtain generic formulae for the terminal velocity in generic space-and-time periodic (or steady) flows, along with further information for flows endowed with some degree of spatial symmetry such as odd parity in the vertical direction. These expressions consist in space-time integrals of auxiliary quantities which satisfy partial differential equations of the advection--diffusion--reaction type, that can be solved at least numerically since our scheme im…
Marine plankton are usually modeled as settling elongated micro-swimmers. For the first time, we consider the torque induced by fluid inertia on such swimmers, and we discover that they spontaneously swim in the direction opposite to gravity. We analyze the equilibrium orientation of swimmers in quiescent fluid and the mean orientation in turbulent flows using direct numerical simulations. Similar to well-known gyrotaxis mechanisms, the effect of fluid inertial torque can be quantified by an effective reorientation time scale. We show that the orientation of swimmers strongly depends on the re…
Using Direct Numerical Simulations (DNS), we examine the effects of Taylor Reynolds number, $R_λ$, and Froude number, $Fr$, on the motion of settling, bidisperse inertial particles in isotropic turbulence. Particle accelerations play a key role in the relative motion of bidisperse particles, and we find that reducing $Fr$ leads to an enhancement of the accelerations, but a suppression of their intermittency. For Stokes numbers $St>1$, the effect of $R_λ$ on the accelerations is enhanced by gravity, since settling causes the particle accelerations to be affected by a larger range of flow scales…
The settling behaviour of small inertial particles in turbulent convection is a fundamental problem across several disciplines, from geophysics to metallurgy. In a geophysical context, the settling of dense crystals controls the mode of solidification of magma chambers and planetary-scale magma oceans, while rising of light bubbles of volatiles drives volcanic outgassing and the formation of primordial atmospheres. Motivated by these geophysical systems, we perform a systematic numerical study on the settling rate of particles in a rectangular two-dimensional Rayleigh-Benard system with Raylei…
We propose an experimental study on the gravitational settling velocity of dense, sub-Kolmogorov inertial particles under different background turbulent flows. We report Phase Doppler Particle Analyzer measurements in a low-speed wind tunnel uniformly seeded with micrometer scale water droplets. Turbulence is generated with three different grids (two consisting on different active-grid protocols while the third is a regular static grid), allowing us to cover a very wide range of turbulence conditions in terms of Taylor-scale based Reynolds numbers ($Re_λ\in [30-520]$), Rouse numbers ($Ro \in […
The rate at which small inertial particles collide with a moderate-Reynolds-number spherical body is found to be strongly affected when the formers are also settling under the effect of gravity. The sedimentation of small particles indeed changes the critical Stokes number above which collisions occur. This is explained by the presence of a shielding effect caused by the unstable manifolds of a stagnation-saddle point of an effective velocity field perceived by the small particles. It is also found that there exists a secondary critical Stokes number above which no collisions occur. This is du…
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 gr…
In this article, we investigate the inertial settling of an arbitrarily oriented cylinder settling under gravity. We focus on two regimes: the very short-time and long-time dynamic. By using the generalized Kirchhoff equations to describe the particle motion, we demonstrate that during the very short dynamic regime, a cylinder starting from rest behaves with sedimenting velocities and angular velocity proportional to $t$ and $t^3$, respectively. We then explore the long-time behaviour and evaluate the validity of the quasi-steady assumption under which the fluid unsteady term can be neglected.…
Inertial microfluidic is able to focus and separate particles in microchannels based on the characteristic geometry and intrinsic hydrodynamic effect. Yet, the vertical position of suspended particles in the microchannel cannot be manipulated in real time. In this study, we utilize the boundary slip effect to regulate the parabolic velocity distribution of fluid in the microchannel and present a scheme to active control the vertical position of particles in inertial microfluidics. The flow field of a microchannel with a unilateral slip boundary is equivalent to that of the microchannel widened…
We study experimentally the spatial distribution, settling, and interaction of sub-Kolmogorov inertial particles with homogeneous turbulence. Utilizing a zero-mean-flow air turbulence chamber, we drop size-selected solid particles and study their dynamics with particle imaging and tracking velocimetry at multiple resolutions. The carrier flow is simultaneously measured by particle image velocimetry of suspended tracers, allowing the characterization of the interplay between both the dispersed and continuous phases. The turbulence Reynolds number based on the Taylor microscale ranges from $Re_λ…
We explore the role of gravitational settling on inertial particle concentrations in a wall-bounded turbulent flow. While it may be thought that settling can be ignored when the settling parameter $Sv\equiv v_s/u_τ$ is small ($v_s$ - Stokes settling velocity, $u_τ$ - fluid friction velocity), we show that even in this regime the settling may make a leading order contribution to the concentration profiles. This is because the importance of settling is determined, not by the size of $v_s$ compared with $u_τ$ or any other fluid velocity scale, but by the size of $v_s$ relative to the other mechan…
One of the most enigmatic science question concerning inertial particle transport by a turbulent boundary layer flow is the value of the turbulent Schmidt number as the ratio of particle diffusivity and turbulent eddy viscosity. Using direct acoustic measurement of turbulent particle flux profile, and two-phase flow turbulence-resolving numerical simulation, it is demonstrated that turbulent dispersion of particles is reduced rather than enhanced when predicted with existing literature model. The explanation lies in the misleading assumption of settling velocity in quiescent water to estimate …
We examine the dynamics of small anisotropic particles (spheroids) sedimenting through homogeneous isotropic turbulence using direct numerical simulations and theory. The gravity-induced inertial torque acting on sub-Kolmogorov spheroids leads to pronouncedly non-Gaussian orientation distributions localized about the broadside-on(to gravity) orientation. Orientation distributions and average settling velocities are obtained over a wide range of spheroid aspect ratios, Stokes and Froude numbers. Orientational moments from the simulations compare well with analytical predictions in the inertiale…
Using 3D Vorono\text{ï} analysis, we explore the local dynamics of small, settling, inertial particles in isotropic turbulence using Direct Numerical Simulations (DNS). We independently vary the Taylor Reynolds number $R_λ\in[90,398]$, Froude number $Fr\equiv a_η/g\in[0.052,\infty]$ (where $a_η$ is the Kolmogorov acceleration, and $g$ is the acceleration due to gravity), and Kolmogorov scale Stokes number $St\equivτ_p/τ_η\in[0,3]$. In agreement with previous results using global measures of particle clustering, such as the Radial Distribution Function (RDF), we find that for small Vorono\text{…
Using Direct Numerical Simulations (DNS), we investigate how gravity modifies the multiscale dispersion of bidisperse inertial particles in isotropic turbulence. The DNS has a Taylor Reynolds number $R_λ=398$, and we simulate Stokes numbers (based on the Kolmogorov timescale) in the range $St\leq 3$ , and consider Froude numbers $Fr = 0.052$ and $\infty$, corresponding to strong gravity and no gravity, respectively. The degree of bidispersity is quantified by the difference in the Stokes number of the particles $|ΔSt|$. We first consider the mean-square separation of bidisperse particle-pairs …
In this work, we analyze the inertial migration of an electrophoretic particle in a 2-D Poiseuille flow with an electric field applied parallel to the walls. For a thin electrical double layer, the particle exhibits a slip-driven electrokinetic motion along the direction of the applied electric field, which causes the particle to lead or lag the flow (depending on its surface charge). The fluid disturbance caused by this slip-driven motion is characterized by a rapidly decaying source-dipole field which alters the inertial lift on the particle. We determine this inertial lift using the recipro…
We present a numerical study of settling and clustering of small inertial particles in homogeneous and isotropic turbulence. Particles are denser than the fluid, but not in the limit of being much heavier than the displaced fluid. At fixed Reynolds and Stokes numbers we vary the fluid-to-particle mass ratio and the gravitational acceleration. The effect of varying one or the other is similar but not quite the same. We report non-monotonic behavior of the particles' velocity skewness and kurtosis with the second parameter, and an associated anomalous behavior of the settling velocity when compa…
The dynamic behavior of crystals in convecting fluids determines how magma bodies solidify. In particular, it is often important to estimate how long crystals stay in suspension in the host liquid before being deposited at its bottom (or top, for light particles). We perform a systematic 3D numerical study of particle-laden Rayleigh-Benard convection, and derive a robust model for the particle residence time. For Rayleigh numbers higher than 10^7, inertial particles' trajectories exhibit a monotonic transition from fluid tracer-like to free-fall dynamics, the control parameter being the ratio …
In a seminal article, \citet[J. Fluid Mech., 174:441-465]{maxey87} presented a theoretical analysis showing that enhanced particle settling speeds in turbulence occur through the preferential sweeping mechanism, which depends on the preferential sampling of the fluid velocity gradient field by the inertial particles. However, recent Direct Numerical Simulation (DNS) results in \citet[J. Fluid Mech., 796:659--711]{ireland16b} show that even in a portion of the parameter space where this preferential sampling is absent, the particles nevertheless exhibit enhanced settling velocities. Further, th…
We derive the torque on a spheroid of an arbitrary aspect ratio $κ$ sedimenting in a linearly stratified ambient. The analysis demarcates regions in parameter space corresponding to broadside-on and edgewise (longside-on) settling in the limit $Re, Ri_v \ll 1$, where $Re = ρ_0UL/μ$ and $Ri_v =γL^3g/μU$, the Reynolds and viscous Richardson numbers, respectively, are dimensionless measures of the importance of inertial and buoyancy forces relative to viscous ones. Here, $L$ is the spheroid semi-major axis, $U$ an appropriate settling velocity scale, $μ$ the fluid viscosity, and $γ\,(>0)$ the (co…
We analyze the effect of the Basset history force on the sedimentation of nearly neutrally buoyant particles, exemplified by marine snow, in a three-dimensional turbulent flow. Particles are characterized by Stokes numbers much smaller than unity, and still water settling velocities, measured in units of the Kolmogorov velocity, of order one. The presence of the history force in the Maxey-Riley equation leads to individual trajectories which differ strongly from the dynamics of both inertial particles without this force, and ideal settling tracers. When considering, however, a large ensemble o…
How non-spherical particles orient as they settle in a flow has important practical implications in a number of scientific and engineering problems. In a quiescent fluid, a slowly settling particle orients so that it settles with its broad side first. This is an effect of the torque due to convective inertia of the fluid set in motion by the settling particle, which maximises the drag experienced by the particle. Turbulent flows tend to randomise the particle orientation. Recently the settling of non-spherical particles in turbulence was analysed neglecting the effect of convective fluid inert…
For one-way coupled (1WC) flows, Tom & Bragg (J. Fluid Mech., 871, pp. 244-270, 2019) advanced the analysis of Maxey (J. Fluid Mech., 174, pp. 441-465, 1987), which applied to weakly inertial particles, to particles of arbitrary inertia, and the new theoretical result revealed the role that different scales play in the preferential sweeping mechanism that leads to enhanced particle settling in turbulent flows. Monchaux & Dejoan (Phys. Rev. Fluids, 2, 104302, 2017) showed using direct numerical simulations (DNS) that while for low particle loading the effect of two-way coupling (2WC) on the glo…
We present experimental and computational results for the orientation distributions of slender fibers and ramified particles settling in an isotropic turbulent flow. The rotational dynamics of the particles is modeled using a slender-body theory that includes the inertial torque due to sedimentation that tends to rotate the particles toward a broadside orientation. The particles are assumed to rotate due to viscous forces associated with the turbulent velocity gradients occurring on the particle length scale. In the simulations, the turbulence is obtained from a stochastic model of the velocit…
On the testing of grain shape corrections to bedload transport equations with grain-resolved numerical simulations
Institute of Port, Coastal and Offshore Engineering, Ocean College, Zhejiang University（浙江大学海洋学院港口海岸与近海工程学院）；Department of Ocean Engineering, Texas A&M University（德克萨斯农工大学海洋工程系）
The history force is one of the hydrodynamic forces which act on a particle moving through a fluid. It is an integral over the full time history of the particle's motion and significantly complicates the equations of motion (accordingly it is often neglected). We present here a study of the influence of this force on particles moving in a turbulent flow, for a wide range of particle parameters. It is shown that the magnitude of history force can be significant and that it can have a considerable effect on the particles' slip velocity, acceleration, preferential concentration and collision rate…
At finite Reynolds numbers particles migrate across flow streamlines to their equilibrium positions in microchannels. Such a migration is attributed to an inertial lift force, and it is well-known that the equilibrium location of neutrally-buoyant particles is determined only by their size and the Reynolds number. Here we demonstrate that the decoration of a bottom wall of the channel by superhydrophobic grooves provides additional possibilities for manipulation of neutrally-buoyant particles. It is shown that the effective anisotropic hydrodynamic slip of such a bottom wall can be readily use…
How anisotropic particles rotate and orient in a flow depends on the hydrodynamic torque they experience. Here we compute the torque acting on a small spheroid in a uniform flow by numerically solving the Navier-Stokes equations. Particle shape is varied from oblate (aspect ratio $λ= 1/6$) to prolate ($λ= 6$), and we consider low and moderate particle Reynolds numbers (${\rm Re} \le 50$). We demonstrate that the angular dependence of the torque, predicted theoretically for small particle Reynolds numbers remains qualitatively correct for Reynolds numbers up to ${\rm Re} \sim 10$. The amplitude…
We experimentally investigate the behaviour of instantaneous localised releases of heavy particles falling as turbulent clouds in quiescent water, both with and without background rotation. We present the results of 514 systematic experiments for no rotation and for three rotation rates $Ω= 5,10,20$rpm, and for the size of particles in the range $5μm$ to $1mm$, exploring four decades of the Rouse number $\mathcal{R} \in [6\times 10^{-4},4]$ which quantifies the inertia of particles. In the canonical framework of turbulent thermals described by Morton \textit{et al.}, [Proc. R. Soc. A: Math. Ph…
At finite Reynolds numbers, Re, particles migrate across laminar flow streamlines to their equilibrium positions in microchannels. This migration is attributed to a lift force, and the balance between this lift and gravity determines the location of particles in channels. Here we demonstrate that velocity of finite-size particles located near a channel wall differs significantly from that of an undisturbed flow, and that their equilibrium position depends on this, referred to as slip velocity, difference. We then present theoretical arguments, which allow us to generalize expressions for a lif…
We present an experimental study of the preferential concentration of sub-Kolmogorov inertial particles in active-grid-generated homogeneous and isotropic turbulence, characterized via Voronoï tessellations. We show that the detection and quantification of clusters and voids is influenced by the intensity of the laser and high values of particles volume fraction $ϕ_v$. Different biases on the statistics of Voronoï cells are analyzed to improve the reliability of the detection and the robustness in the characterization of clusters and voids. We do this by adapting Big-Data techniques that allow…
