We studied the transport and deposition behaviour of point particles in Rayleigh–Bénard convection cells subjected to Couette-type wall shear. Direct numerical simulations (DNSs) are performed for Rayleigh number ($Ra$) in the range $10^{7} \leq Ra \leq ~10^9$ with a fixed Prandtl number $Pr = 0.71$, while the wall-shear Reynolds number ($Re_w$) is in the range $0 \leq Re_w \leq ~12\,000$. With the increase of $Re_w$, the large-scale rolls expanded horizontally, evolving into zonal flow in two-dimensional simulations or streamwise-oriented rolls in three-dimensional simulations. We observed that, for particles with a small Stokes number ($St$), they either circulated within the large-scale rolls when buoyancy dominated or drifted near the walls when shear dominated. For medium $St$ particles, pronounced spatial inhomogeneity and preferential concentration were observed regardless of the prevailing flow state. For large $St$ particles, the turbulent flow structure had a minor influence on the particles’ motion; although clustering still occurred, wall shear had a negligible influence compared with that for medium $St$ particles. We then presented the settling curves to quantify the particle deposition ratio on the walls. Our DNS results aligned well with previous theoretical predictions, which state that small $St$ particles settle with an exponential deposition ratio and large $St$ particles settle with a linear deposition ratio. For medium $St$ particles, where complex particle–turbulence interaction emerges, we developed a new model describing the settling process with an initial linear stage followed by a nonlinear stage. Unknown parameters in our model can be determined either by fitting the settling curves or using empirical relations. Compared with DNS results, our model also accurately predicts the average residence time across a wide range of $St$ for various $Re_w$.

We investigate the dynamics of heavy inertial particles in a flow field due to an isolated, non-axisymmetric vortex. For our study, we consider a canonical elliptical vortex – the Kirchhoff vortex and its strained variant, the Kida vortex. Contrary to the anticipated centrifugal dispersion of inertial particles, which is typical in open vortical flows, we observe the clustering of particles around co-rotating attractors near the Kirchhoff vortex due to its non-axisymmetric nature. We analyse the inertia-modified stability characteristics of the fixed points, highlighting how some of the fixed points migrate in physical space, collide and then annihilate with increasing particle inertia. The introduction of external straining, the Kida vortex being an example, introduces chaotic tracer transport. Using a Melnikov analysis, we show that particle inertia and external straining can compete, where chaotic transport can be suppressed beyond a critical value of particle inertia.

The orientation dynamics of inertialess prolate and oblate spheroidal particles in a directly simulated spanwise-rotating turbulent channel flow has been investigated by means of an Eulerian–Lagrangian point-particle approach. The channel rotation and the particle shape were parameterized using a rotation number Ro and the aspect ratio λ, respectively. Eleven particle shapes 0.05 ≤ λ ≤ 20 and four rotation rates 0 ≤ Ro ≤ 10 have been examined. The spheroidal particles retained their almost isotropic orientation in the core region of the channel, despite the significant mean shear rate set up by the Coriolis force. Irrespective of channel rotation rate Ro, rod-like spheroids tend to align in the streamwise direction, while disk-like particles are oriented in the wall-normal direction. These trends were accentuated with increasing departure from sphericity λ = 1. The changeover from the isotropic orientation mode in the centre to the highly anisotropic near-wall orientation mode commenced further away from the suction-side wall with increasing Ro, whereas the particle orientations on the pressure side of the rotating channel remained essentially unaffected by Ro. We observed that the alignments of the fluid rotation vector with the Lagrangian stretching direction were similarly unaffected by the imposed system rotation, except that the de-alignment set in deeper into the core at high Ro. This contrasts with the well-known substantial impact of system rotation on the velocity and vorticity fields. Similarly, slender rods and flatter disks were aligned with the Lagrangian stretching and compression directions, respectively, for all Ro considered, except in the vicinity of the walls. The typical near-wall de-alignment extended considerably further away from the suction-side wall at high Ro. We conjecture that this phenomenon reflects a change in the relative importance of mean shear and small-scale turbulence caused by the Coriolis force. Preferential particle alignment with Lagrangian stretching and compression directions are known from isotropic and anisotropic turbulence in inertial reference systems. The present results demonstrate the validity of this principle also in a non-inertial system.

The transient shear-induced particle migration of frictional non-Brownian suspensions is studied using particle-resolved simulations. The numerical method – the fictitious domain method – is well suited to heterogeneous flows thanks to a frame-invariant formulation of the subgrid (lubrication) corrections that does not involve the ambient flow (Orsi et al., J. Comput. Phys., vol. 474, 2023, 111823). The paper aims to give an accurate quantitative picture of the mass and momentum balance during the flow. The various assumptions and local constitutive laws that together form the suspension balance model (SBM) are thoroughly examined. To this purpose, the various quantities of interest are locally averaged in space and time, and their profile across the channel is extensively studied, with specific attention to the time evolution of the different contributions, either hydrodynamic in nature or from contact interactions, to the shear and normal stresses. The latter, together with the velocity gradient in the wall-normal direction and the volume fraction profile, yield the local constitutive laws, which are compared with their counterpart obtained in homogeneous shear flow. A fair agreement is observed except in a layering area at the boundaries and at the very centre of the channel. In addition, the main assumption of the SBM, i.e. the local relation between the hydrodynamic force on the particles and the particle flux, is meticulously investigated. The hydrodynamic force is found to be mainly a drag, except in the lower range of the probed volume fractions, where a non-drag contribution is observed.

The present work focuses on a specific bouncing behaviour as a spherical particle settles through a density interface in the absence of a neutral buoyant position. This behaviour was initially discovered by Abaid et al. (Phys. Fluids, vol. 16, issue 5, 2004, pp. 1567–1580) in salinity-induced stratification. Both experimental and numerical investigations are conducted to understand this phenomenon. In our experiments, we employ particle image velocimetry (PIV) to measure the velocity distribution around the particle and to capture the transient wake structure. Our findings reveal that the bouncing process begins after the wake detaches from the particle. The PIV results indicate that an upward jet forms at the central axis behind the particle following wake detachment. By performing a force decomposition procedure, we quantify the contributions from the buoyancy of the wake ($F_{sb}$) and the flow structure ($F_{sj}$) to the enhanced drag. It is observed that $F_{sb}$ contributes primarily to the enhanced drag at the early stage, whereas $F_{sj}$ plays a critical role in reversing the particle's motion. Furthermore, our results indicate that the jet is a necessary condition for the occurrence of the bouncing motion. We also explore the minimum velocities (where negative values denote the occurrence of bouncing) of the particle, while varying the lower Reynolds number $Re_l$, the Froude number $Fr$, and the upper Reynolds number $Re_u$, within the ranges $1 \leqslant Re_l\leqslant 125$, $115 \leqslant Re_u\leqslant 356$ and $2 \leqslant Fr\leqslant 7$. Our findings suggest that the bouncing behaviour is influenced primarily by $Re_l$. Specifically, we observe that the bouncing motion occurs below a critical lower Reynolds number $Re^\ast _{l}=30$ in our experiments. In the numerical simulations, the highest value for this critical number is $Re^\ast _{l}=46.2$, which is limited to the parametric ranges studied in this work.

Internal solitary waves are a widely observed phenomenon in natural waters. Mathematically, they are fundamentally a nonlinear phenomenon that differs from the paradigm of turbulence, in that energy does not move across scales. Internal solitary waves may be computed from the Dubreil–Jacotin Long equation, which is a scalar partial differential equation that is equivalent to the stratified Euler equations. When a background shear current is present the algebraic complexity of the problem increases substantially. We present an alternative point of view for characterizing the situation with a shear current using Lagrangian (particle-like) models analysed with graph theoretic methods. We find that this yields a novel, data-centric framework for analysis that could prove useful well beyond the study of internal solitary waves.

Research on the settling dynamics of snow particles, considering their complex morphologies and real atmospheric conditions, remains scarce despite extensive simulations and laboratory studies. Our study bridges this gap through a comprehensive field investigation into the three-dimensional (3-D) snow settling dynamics under weak atmospheric turbulence, enabled by a 3-D particle tracking velocimetry (PTV) system to record over a million trajectories, coupled with a snow particle analyser for simultaneous aerodynamic property characterization of four distinct snow types (aggregates, graupels, dendrites, needles). Our findings indicate that while the terminal velocity predicted by the aerodynamic model aligns well with the PTV-measured settling velocity for graupels, significant discrepancies arise for non-spherical particles, particularly dendrites, which exhibit higher drag coefficients than predicted. Qualitative observations of the 3-D settling trajectories highlight pronounced meandering in aggregates and dendrites, in contrast to the subtler meandering observed in needles and graupels, attributable to their smaller frontal areas. This meandering in aggregates and dendrites occurs at lower frequencies compared with that of graupels. Further quantification of trajectory acceleration and curvature suggests that the meandering frequencies in aggregates and dendrites are smaller than that of morphology-induced vortex shedding of disks, likely due to their rotational inertia, and those of graupels align with the small-scale atmospheric turbulence. Moreover, our analysis of vertical acceleration along trajectories elucidates that the orientation changes in dendrites and aggregates enhance their settling velocity. Such insights into settling dynamics refine models of snow settling velocity under weak atmospheric turbulence, with broader implications for more accurately predicting ground snow accumulation.

Small heavy particles cannot be attracted into a region of closed streamlines in a non-accelerating frame (Sapsis & Haller, Chaos, vol. 20, issue 1, 2010, 017515). In a rotating system of vortices, however, particles can get trapped (Angilella, Physica D, vol. 239, issue 18, 2010, pp. 1789–1797) in the vicinity of vortices. We perform numerical simulations to examine trapping of inertial particles in a prototypical rotating flow described by a rotating pair of Lamb–Oseen vortices of identical strength, in the absence of gravity. Our parameter space includes the particle Stokes number $St$, which is a measure of the particle's inertia, and a density parameter $R$, which measures the particle's density relative to the fluid. In particular, we study the regime $0< R<1$ and $0< St<1$, which corresponds to an inertial particle that is finitely denser than the fluid. We show that in this regime, a significant fraction of particles can be trapped indefinitely close to the vortices, and display extreme clustering into objects of smaller dimension: attracting fixed points and limit cycles of different periods including chaotic attractors. As $St$ increases for a given $R$, we may have an incomplete or complete period-doubling route to chaos, as well as an unusual period-halving route back to a fixed point attractor. The fraction of trapped particles can be a non-monotonic function of $St$, and we may even have windows in $St$ for which no particle trapping occurs. At $St$ larger than a critical value, beyond which trapping ceases to exist, significant fractions of particles can spend long but finite times in the vortex vicinity. The inclusion of the Basset–Boussinesq history (BBH) force is imperative in our study due to the finite density of the particle. We observe that the BBH force significantly increases the basin of attraction over which trapping occurs, and also widens the range of $St$ for which trapping can be realised. Extreme clustering can be of significance in a host of physical applications, including planetesimal formation by aggregation of dust in protoplanetary discs, and aggregation of phytoplankton in the ocean. Our findings in the prototypical model provide impetus to conduct experiments and further numerical investigations to understand clustering of inertial particles.

Laboratory experiments and particle-resolved simulations are employed to investigate the settling dynamics of a pair of rigidly connected spherical particles of unequal size. They yield a detailed picture of the transient evolution and the terminal values of the aggregate's orientation angle and its settling and drift velocities as functions of the aspect ratio and the Galileo number $Ga$, which denotes the ratio of buoyancy and viscous forces acting on the aggregate. At low to moderate values of $Ga$, the aggregate's orientation and velocity converge to their terminal values monotonically, whereas for higher $Ga$-values the aggregate tends to undergo a more complex motion. If the aggregate assumes an asymmetric terminal orientation, it displays a non-zero terminal drift velocity. For diameter ratios much larger than one and small $Ga$, the terminal orientation of the aggregate becomes approximately vertical, whereas when $Ga$ is sufficiently large for flow separation to occur, the aggregate orients itself such that the smaller sphere is located at the separation line. Empirical scaling laws are obtained for the terminal settling velocity and orientation angle as functions of the aspect ratio and $Ga$ for diameter ratios from 1 to 4 and particle-to-fluid density ratios from 1.3 to 5. An analysis of the accompanying flow field shows the formation of vortical structures exhibiting complex topologies in the aggregate's wake, and indicates the formation of a horizontal pressure gradient across the larger sphere, which represents the main reason for the emergence of the drift velocity.

This study conducts particle-resolved direct numerical simulations to analyse how finite-size spherical particles affect the decay rate of turbulent kinetic energy in non-sustained homogeneous isotropic turbulence. The decaying particle-laden homogeneous isotropic turbulence is generated with two set-ups, i.e. (1) releasing particles into a single-phase decaying homogeneous isotropic turbulence and (2) switching off the driving force of a sustained particle-laden homogeneous isotropic turbulence. With both set-ups, the decay of turbulent kinetic energy follows a power-law when the flow is fully relaxed, similar to their single-phase counterparts. The dependence of the power-law decay exponent $n$ on the particle-to-fluid density ratio, particle size and volume fraction is also investigated, and a predictive model is developed. We find that the presence of heavier particles slows down the long-time power-law decay exponent.

We study the behaviour of a particle bed immersed in water when a flow generated by an oscillating plate is induced above it. We first consider a rigid plate submerged and oscillated over a particle bed. During upward motion of the plate, a portion of the bed fails, allowing particle displacement, and the bed surface to deform into a heap. We have already determined the flow of the fluid above and within the bed. This work describes the particle motion within the failed region of the bed: when the particles are mobile, they follow the fluid. We depth average the balance of mass and obtain an evolution equation for the displacement of the bed surface. We solve this equation and compare the predictions with the measurements of surface displacement in earlier experiments on rigid square plates. We carry out new experiments to measure the surface displacements under elongated plates. Elongated rigid plates behave similarly to the rigid square ones. Flexible plates produce multiple heaps. We determine that the peaks of these heaps are correlated with the flexural modes of the plates and occur at points along the bed at which the fluid pressure has its extreme values. Different plate flexural modes, resulting in different numbers of heaps, are produced by driving the plate at different frequencies. The particle motion within the bed and heap evolution under a flexible plate can be roughly described by regarding it as two or more rigid plates. We test the predictions of the theory against experiments.

Turbulence can have a strong effect on the fall speed of snowflakes and ice crystals. In this experimental study, the behaviour of thin disks falling in homogeneous turbulence is investigated, in a range of parameters relevant to plate crystals. Disks ranging in diameter from 0.3 to 3 mm, and in Reynolds number $Re = 10\unicode{x2013}435$, are dispersed in two air turbulence levels, with velocity fluctuations comparable to the terminal velocity. For each case, thousands of trajectories are captured and reconstructed by high-speed laser imaging, allowing for statistical analysis of the translational and rotational dynamics. Air turbulence reduces the disk terminal velocities by up to 35 %, with the largest diameters influenced most significantly, which is primarily attributed to drag nonlinearity. This is evidenced by large lateral excursions of the trajectories, which correlate with cross-flow-induced drag enhancement as reported previously for falling spheres and rising bubbles. As the turbulence intensity is increased, flat-falling behaviour is progressively eliminated and tumbling becomes prevalent. The rotation rates of the tumbling disks, however, remain similar to those displayed in still air. This is due to their large moment of inertia compared to the surrounding fluid, in stark contrast with studies conducted in water. In fact, the observed reduction of settling velocity is opposite to previous findings on disks falling in turbulent water. This emphasizes the importance of the solid-to-fluid density ratio in analogous experiments that aim to mimic the behaviour of frozen hydrometeors.

We employ the methods of statistical mechanics to obtain closures for the balance equations of momentum and fluctuation kinetic energy that govern the ballistic motion of grains rebounding at a rigid, bumpy bed that are driven by turbulent or non-turbulent shearing fluids, in the absence of mid-trajectory collisions and fluid velocity fluctuations. We obtain semi-analytical solutions for steady and fully developed saltation over horizontal beds for the vertical profiles of particle concentration and stresses and fluid and particle velocities. These compare favourably with measurements in discrete-element numerical simulations in the wide range of conditions of Earth and other planetary environments. The predictions of the particle horizontal mass flux and its scaling with the amount of particles in the system, the properties of the carrier fluid and the intensity of the shearing also agree with numerical simulations and wind-tunnel experiments.

Small finite-size particles suspended in fluid flow through an enclosed curved duct can focus to points or periodic orbits in the two-dimensional duct cross-section. This particle focusing is due to a balance between inertial lift forces arising from axial flow and drag forces arising from cross-sectional vortices. The inertial particle focusing phenomenon has been exploited in various industrial and medical applications to passively separate particles by size using purely hydrodynamic effects. A fixed size particle in a circular duct with a uniform rectangular cross-section can have a variety of particle attractors, such as stable nodes/spirals or limit cycles, depending on the radius of curvature of the duct. Bifurcations occur at different radii of curvature, such as pitchfork, saddle-node and saddle-node infinite period (SNIPER), which result in variations in the location, number and nature of these particle attractors. By using a quasi-steady approximation, we extend the theoretical model of Harding et al. (J. Fluid Mech., vol. 875, 2019, pp. 1–43) developed for the particle dynamics in circular ducts to spiral duct geometries with slowly varying curvature, and numerically explore the particle dynamics within. Bifurcations of particle attractors with respect to radius of curvature can be traversed within spiral ducts and give rise to a rich nonlinear particle dynamics and various types of tipping phenomena, such as bifurcation-induced tipping (B-tipping), rate-induced tipping (R-tipping) and a combination of both, which we explore in detail. We discuss implications of these unsteady dynamical behaviours for particle separation and propose novel mechanisms to separate particles by size in a non-equilibrium manner.

In the present study, we perform direct numerical simulations of compressible turbulent boundary layers at free stream Mach numbers $2\unicode{x2013}6$ laden with dilute phase of spherical particles to investigate the Mach number effects on particle transport and dynamics. Most of the phenomena observed and well-recognized for inertia particles in incompressible wall-bounded turbulent flows – such as near-wall preferential accumulation and clustering beneath low-speed streaks, flatter mean velocity profiles, and trend variation of the particle velocity fluctuations – are identified in the compressible turbulent boundary layer as well. However, we find that the compressibility effects are significant for large inertia particles. As the Mach number increases, the near-wall accumulation and the small-scale clustering are alleviated, which is probably caused by the variations of the fluid density and viscosity that are crucial to particle dynamics. This can be affected by the fact that the forces acting on the particles with viscous Stokes number greater than 500 are modulated by the comparatively high particle Mach numbers in the near-wall region. This is also the reason for the abatement of the streamwise particle velocity fluctuation intensities with the Mach numbers.

Large amounts of small inertial particles embedded in a turbulent flow are known to modify the turbulent statistics and structures, a phenomenon referred to as turbulence modulation. While particle electrification is ubiquitous in particle-laden turbulence and significantly alters particle behaviour, the effects of inter-particle electrostatic forces on turbulence modulation and the underlying physical mechanisms remain unclear. To fill this gap, we perform a series of point-particle direct numerical simulations of turbulent channel flows at a friction Reynolds number of approximately 540, laden with uncharged and charged bidisperse particles. The results demonstrate that, compared to flows laden with uncharged particles, the presence of inter-particle electrostatic forces leads to substantial changes in both turbulent intensities and structures. In particular, the inner-scaled mean streamwise fluid velocity is found to shift towards lower values, indicating a noticeable increase in fluid friction velocity. Turbulent intensities appear to be further suppressed through facilitating the particles to extract momentum from the fluid phase and increasing extra turbulent kinetic dissipation by particles. Importantly, the overall drag is enhanced by indirectly strengthening the contribution of particle stress, even though the contribution of the total fluid stress is decreased. On the other hand, the magnitude of the large-scale motions is weakened by simultaneously reducing turbulent production and increasing particle feedback around the scales of the large-scale motions. Meanwhile, the average streaky fluid structures in the streamwise–spanwise planes and inclined fluid structures in the streamwise–wall-normal planes become expanded and flattened, respectively.

A simulation method has been developed to efficiently evaluate the motion of colloidal particles in a low-Reynolds-number confined microchannel flow using a Lagrangian-based approach. In this method, the background velocity within the channel, in the absence of suspended particles, is obtained from a fluid dynamics solver and is used to update the velocity at the particle centres using the Stokesian dynamics (SD) method, which incorporates multi-body hydrodynamic interactions. As a result, instead of computing the momentum of both the fluid and particles throughout the entire computational domain, the microscopic balance equation is solved only at the particle centres, increasing the computational efficiency. To accommodate complex boundary conditions within the SD framework, imaginary particles are placed on the channel walls, allowing the mobility relation to be reformulated to apply velocity constraints to immobilized wall particles. By employing this constrained SD approach, global mobility interactions that need to be computed at each time step are limited to the interior particles, resulting in a significant reduction in computational cost. The efficiency of this study is demonstrated through case studies on particulate flows in contraction and cross-flow microchannels. By using colloidal particles that incorporate Brownian motion and inter-particle attraction, observations through the entire stages of fouling dynamics are possible, from particle inflow to channel blockage. The fouling patterns observed in the simulations are consistent with experiments conducted under the same flow conditions. This study provides an efficient approach for analysing the effect of hydrodynamic interactions on particle dynamics in microfluidics and materials processing fields while allowing for predictions about structural changes over long-time scales, including complex phenomena such as clogging.

The clustering of debris floating on liquid interfaces such as water surfaces is a complex phenomenon that finds its applications in numerous examples from industrial processing and environmental systems. The recent paper by Shin & Coletti (J. Fluid Mech., vol. 984, 2024, R7) presents an experimental campaign investigating the three effects of turbulence, particle interactions and interfacial effects, to elucidate how the three force scales drag, capillary forces and lubrication give rise to three distinct regimes of clustering in dense suspensions. The study, hence, provides a useful systematic to categorize the clustering mechanisms. As an important finding, it is shown that, depending on volume fraction and non-dimensional turbulent shear, particles either tend to cluster into aggregate sizes larger than the Kolmogorov scale or can break into pieces that are as small as the primary particle size.

Low Stokes number particles at dilute concentrations in turbulent flows can reasonably be approximated as passive scalars. The added presence of a drift velocity due to buoyancy or gravity when considering the transport of such passive scalars can reduce the turbulent dispersion of the scalar via a diminution of the eddy diffusivity. In this work, we propose a model to describe this decay and use a recently developed technique to accurately and efficiently measure the eddy diffusivity using Eulerian fields and quantities. We then show a correspondence between this method and standard Lagrangian definitions of diffusivity and collect data across a range of drift velocities and Reynolds numbers. The proposed model agrees with data from these direct numerical simulations, offers some improvement to previous models in describing other computational and experimental data and satisfies theoretical constraints that are independent of Reynolds number.

We explored the settling dynamics of vertically aligned particles in a quiescent, stratified two-layer fluid using particle tracking velocimetry. Glass spheres of $d=4\,{\rm mm}$ diameter were released at frequencies of 4, 6 and 8 Hz near the free surface, traversing through an upper ethanol layer ($H_1$), where H is height or layer thickess, varying from $10d$ to $40d$ and a lower oil layer. Results reveal pronounced lateral particle motion in the ethanol layer, attributed to a higher Galileo number ($Ga = 976$, ratio of buoyancy–gravity to viscous effects), compared with the less active behaviour in the oil layer ($Ga = 16$). The ensemble vertical velocity of particles exhibited a minimum just past the density interface, becoming more pronounced with increasing $H_1$, and suggesting that enhanced entrainment from ethanol to oil resulted in an additional buoyancy force. This produced distinct patterns of particle acceleration near the density interface, which were marked by significant deceleration, indicating substantial resistance to particle motion. An increased drag coefficient occurred for $H_1/d = 40$ compared with a single particle settling in oil; drag reduced as the particle-release frequency ($\,f_p$) increased, likely due to enhanced particle interactions at closer proximity. Particle pair dispersions, lateral ($R^2_L$) and vertical ($R^2_z$), were modulated by $H_1$, initial separation $r_0$ and $f_p$. The $R^2_L$ dispersion displayed ballistic scaling initially, Taylor scaling for $r_0 < H_1$ and Richardson scaling for $r_0 > H_1$. In contrast, $R^2_z$ followed a $R^2_z \sim t^{5.5}$ scaling under $r_0 < H_1$. Both $R^2_L$ and $R^2_z$ plateaued at a distance from the interface, depending on $H_1$ and $f_p$.