We investigate the mechanisms governing particle settling in turbidity currents using Eulerian-Lagrangian direct numerical simulations. The Eulerian carrier flow is driven either by solutal buoyancy or particle feedback, with the Lagrangian phase comprising passive tracers or inertial particles, respectively. The effective particle settling velocity is decomposed into a fluid velocity sampled at particle positions and a particle-fluid slip velocity. The Eulerian mean profiles of these velocities are obtained using a concentration-weighted average of the coarse-grained fields. The mean sampled fluid velocity is shown to be approximately equal to the ratio of the vertical turbulent flux of particles to their mean concentration and reflects biased sampling of upward turbulent fluctuations at particle positions, despite the zero Eulerian mean vertical fluid velocity. The passive-tracer cases show that the upward bias is inertia-independent and arises from turbulent transport acting on concentration gradients, as it persists for inhomogeneous tracer seeding but disappears under uniform seeding. For the weakly inertial regime considered here, the upward bias dominates downward-directed biases associated with particle inertia. The mean slip velocity is well approximated by the terminal settling velocity predicted for a quiescent fluid. This is consistent with a leading-order balance between buoyancy and drag in the slope-normal direction. Modelling the sampled fluid velocity from the turbulent flux and using the slip-velocity approximation yield an Eulerian prediction for the settling velocity, in good agreement with the simulation data for the dilute, weakly inertial particles considered here.

The dynamics of buoyant droplets rising freely in a large body of an immiscible liquid is investigated numerically for a moderate drop-to-fluid viscosity ratio μ^∗. We focus on toluene droplets rising in clean water, for which μ^∗=0.62, and vary the radius over 0.5 mm≤ R≤3.0 mm. Direct numerical simulations are performed in imposed axisymmetric and fully three-dimensional configurations. As R increases, the system displays a rich sequence of rise regimes. Starting from steady vertical rise with an axisymmetric disturbance flow, it first undergoes an internal flow instability associated with an azimuthal mode m=2, leading to a biplanar-symmetric wake and reduced terminal speed. This state is followed by a steady oblique regime, in which the m=1 mode also becomes unstable and coexists with the m=2 mode. At larger radii, the path becomes nearly vertical again before the flow enters an m=2 rotating-wave regime, where the wake drifts azimuthally at an approximately constant angular velocity. For still larger droplets, persistent shape oscillations and vortex shedding lead to fully three-dimensional chaotic paths. Simulations initialised from finite-amplitude asymmetric states further reveal several multistable size ranges, in which distinct terminal states coexist depending on the initial condition. Taken together, these findings show that the path instability of moderate-viscosity-ratio droplets differs fundamentally from that of bubbles and solid particles: in most regimes encountered here, axisymmetry breaking is initiated within the droplet, highlighting the central role of the internal flow instability in shaping the subsequent wake structure, rise speed and droplet dynamics.

Drops play a pivotal role in both environmental systems and industrial processes, where their physical properties, dynamics, and interactions have far-reaching implications across diverse domains. In environmental contexts, water droplets are central to critical phenomena such as carbon dioxide exchange at the ocean-atmosphere interface, cloud formation, and precipitation-all of which are essential to agriculture, climate regulation, and ecosystem sustainability. In industrial applications, drops underpin technologies such as spray systems, coatings, inkjet printing, and fuel injection systems. Their behavior governs key processes like heat transfer, material deposition, and combustion efficiency, offering opportunities for innovation and enhanced performance across various sectors. The multiscale nature of drop-laden flows - ranging from molecular-scale interfacial dynamics to macroscale phenomena - poses significant challenges for numerical or experimental approaches. This necessitates the development of models capable of bridging these disparate scales effectively. This paper provides a concise overview of the current state of the art in drop-laden flow research, reviewing experimental and computational techniques. Additionally, it outlines key challenges and proposes strategic guidelines to shape future investigations, emphasizing the need for interdisciplinary efforts to address unresolved questions and drive progress in this multifaceted field.

Direct numerical simulations of a uniform flow past a fixed spherical droplet are performed to determine the parameter range within which the axisymmetric flow becomes unstable. The problem is governed by three dimensionless parameters: the drop-to-fluid dynamic viscosity ratio, $\mu <^>\ast$ , and the external and internal Reynolds numbers, ${\textit{Re}}<^>e$ and ${\textit{Re}}<^>i$ , which are defined using the kinematic viscosities of the external and internal fluids, respectively. The present study confirms the existence of a regime at low-to-moderate viscosity ratio where the axisymmetric flow breaks down due to an internal flow instability. In the initial stages of this bifurcation, the external flow remains axisymmetric, while the asymmetry is generated and grows only inside the droplet. As the disturbance propagates outward, the entire flow first transits to a biplanar-symmetric flow, characterised by two pairs of counter-rotating streamwise vortices in the wake. A detailed examination of the flow field reveals that the vorticity on the internal side of the droplet interface is driving the flow instability. Specifically, the bifurcation sets in once the maximum internal vorticity exceeds a critical value that decreases with increasing ${\textit{Re}}<^>i$ . For sufficiently large ${\textit{Re}}<^>i$ , internal flow bifurcation may occur at viscosity ratios of $\mu <^>\ast = {\mathcal{O}}(10)$ , an order of magnitude higher than previously reported values. Finally, we demonstrate that the internal flow bifurcation in the configuration of a fixed droplet in a uniform fluid stream is closely related to the first path instability experienced by a buoyant, deformable droplet of low-to-moderate $\mu <^>\ast$ freely rising in a stagnant liquid.

Direct numerical simulations of a uniform flow past a fixed spherical droplet are performed to investigate the parameter range within which the axisymmetric flow becomes unstable due to an external flow bifurcation. The hydrodynamics is governed by three dimensionless parameters: the viscosity ratio μ*, the external Reynolds numbers Ree, and internal Reynolds numbers Rei, respectively. The drop-to-fluid density ratio is related to these parameters as ρ* = μ*Rei/Ree. This study focuses on highly viscous droplets with μ* ≥ 5, where wake instability is driven by the vorticity flux transferred from the droplet surface into the surrounding fluid. By analysing the wake structure, we confirm that the onset of the external bifurcation is linked to the tilting of the azimuthal vorticity ωϕ, in the wake and that the bifurcation occurs once the isocontours of ωϕ align nearly perpendicular to the symmetry axis. We propose an empirical criterion for predicting the onset of the external bifurcation, formulated in terms of the maximum vorticity on the external side of the droplet surface. This criterion is applicable for sufficiently high Rei and holds over a wide range of μ* and Ree. Additionally, we examine the bifurcation sequence for two specific external Reynolds numbers, Ree = 300 and Ree = 500, and show that, beyond a critical viscosity ratio, the axisymmetric wake first transitions to a steady planar-symmetric state before undergoing a secondary Hopf bifurcation. Finally, we highlight the influence of Rei on external bifurcation and show that, at moderate Rei, wake instability may set in at a lower vorticity threshold than predicted by our criterion. These findings provide new insights into the external flow bifurcation of viscous droplets.

Chemical reactions can have a significant impact on the overall rate of mass transfer in fluid-fluid applications (such as gas absorption in a liquid or liquid-liquid extraction). However, as for non reacting systems, the rate at which a solute is exchanged between the two fluids is difficult to predict accurately. Yet from a theoretical point of view, most studies address uniform diffusion transport or creeping flow, i.e. conditions enabling the derivation of analytical or simple numerical solutions of the solute transport equation, therefore neglecting the effect of convection in the case of coupled mass transfer problems. On the other hand, experimental investigations available in the litterature provided correlations corresponding to flow conditions, shape and contamination of the interface which are extremely difficult to control precisely.Direct numerical simulation (DNS) is therefore a powerful tool to investigate the coupling of hydrodynamics with transfer and reaction. It was recently applied to model the internal and external resistances (or Sherwood numbers, Shi and She) prevailing during the transfer of a non reacting solute, in the general case of conjugate mass transfer from a moving spherical droplet (Godé et al., 2023; 2024). This study is complemented by considering a first order reaction that consumes the transferred solute in the continuous phase. The results indicate that the double film model, based on these last correlations for Shi and She, with She corrected by an enhancement factor allows obtaining accurate predictions of the overall mass transfer coefficient in the case of a reactive conjugate problem, thus extending the work of Juncu (2002) to flows at moderate or even high Re, as long as the droplet remains spherical and the flow axisymmetric.

We study numerically the flow around a spherical droplet set fixed in a linear shear flow with moderate shear rates (Sr <= 0.5, Sr being the ratio between the velocity difference across the drop and the relative velocity) over a wide range of external Reynolds numbers (0.1 < Re <= 250, Re based on the slip velocity and the viscosity of the external fluid) and drop-to-fluid viscosity ratios (0.01 <= mu(& lowast; )<= 100). The flow structure, the vorticity field and their intrinsic connection with the lift force are analysed. Specifically, the results on lift force are compared with the low-Re solution derived for droplets of arbitrary mu(& lowast;), as well as prior data at finite Re available in both the clean-bubble limit (mu(& lowast; )-> 0) and the solid-sphere limit (mu(& lowast; )-> infinity). Notably, at Re = O(100), the lift force exhibits a non-monotonic transition from mu(& lowast; )-> 0 to mu(& lowast; )-> infinity, peaking at mu(& lowast; )approximate to 1. This behaviour is related to an internal three-dimensional flow bifurcation also occurring under uniform-flow conditions, which makes the flow to evolve from axisymmetric to biplanar symmetric. This flow bifurcation occurs at low-but-finite mu(& lowast;) when the internal Reynolds number (Re-i, based on the viscosity of the internal fluid) exceeds approximately 300. In the presence of shear, the corresponding imperfect bifurcation enhances the extensional rate of the flow in the wake. Consequently, the streamwise vortices generated behind the droplet can be more intense compared with those behind a clean bubble. Given the close relation between the lift and these vortices, a droplet with Re = O(100) and mu(& lowast; )approximate to 1 typically experiences a greater lift force than that in the inviscid limit.

Many industrial processes involve grinding and milling operations to produce powders of well-controlled particle size distribution. Improving these processes requires researchers to consider particle-particle interactions, breakage, aggregation, and in the case of wet milling, hydrodynamics. The purpose of this study is to develop a numerical model, accounting for these mechanisms, with the final goal to guide the design of an optimal process. Computational fluid dynamics is coupled with a population balance model. The hydrodynamics of the multiphase flow is predicted using the multifluid Euler approach, whereas the class method is considered to solve the population balance equation. A theoretical definition of the breakage functions, based on hydrodynamics inside the tank, is proposed. The results show that breakage is highly heterogeneous inside the tank and that particle breakage frequency allows researchers to evaluate milling efficiency.

Clustering of plankton plays a vital role in several biological activities, including feeding, predation, and mating. Gyrotaxis is one of the mechanisms that induces clustering. A recent study (Candelier et al., 2022) reported a fluid inertial torque acting on a spherical microswimmer, which has the same effect as a gyrotactic torque. In this study, we model plankton cells as microswimmers that are subject to gravitational sedimentation as well as a fluid inertial torque. We use direct numerical simulations to obtain the trajectories of swimmers in homogeneous isotropic turbulence. We also investigate swimmers' clustering using Voronoï analysis. Our findings indicate that fluid inertial torque leads to notable clustering, with its intensity depending on the swimming and settling speeds of swimmers. Using Voronoï analysis, we demonstrate that swimmers preferentially sample downwelling regions where clustering is more prevalent.

In this paper, the physics of external flow past a superhydrophobic SD7003 hydrofoil is studied at Reynolds number of 105. We used Navier’s slip length to model the superhydrophobic surfaces, and investigated the effect of different slip lengths on hydrofoil performance by means of RANS (Reynolds-averaged Navier–Stokes) simulations. The slip length boundary condition is imposed on upper, or/and lower side(s), to study the effect of each superhydrophobic region on lift coefficient. By increasing the slip length (ls), it was expected to see lift enhancement and drag reduction. However, the lift enhancement trend had some exceptions. The unexpected trends were explained by studying the location of laminar separation and reattachment for different angles of attack (AOA) and slip lengths. For example, the lift coefficient of a full superhydrophobic hydrofoil at AOA = 2° does not change monotonically against slip length. By increasing the slip length, separation and reattachment points are shifted towards the trailing edge and at slip length 200 μm, there is separation with no reattachment which results in lift reduction. By further increasing the slip length, the separation is shifted further down and eventually there would be no separation n\and flow becomes fully attached.

Recently superhydrophobic surfaces are widely used to reduce drag, but there has been less focus on their in-fluence on lift to drag ratio. In this paper, it is shown that using a superhydrophobic coating on the whole surface of a hydrofoil is not always beneficial to hydrodynamic performance (i.e. lift to drag ratio). Depending on angle of attack (AOA), Reynolds number (Re) or slip length (l(s)), some regions on the surface of hydrofoil should be left uncoated (optimum superhydrophobic pattern). To find the optimum superhydrophobic pattern, we developed a genetic algorithm (GA) combined with two dimensional RANS (Reynolds Averaged Navier Stokes) equations solver. To demonstrate the methodology a SD7003 foil surface is divided into 10 equal regions. Every region can either be superhydrophobic or uncoated, i.e. the problem is a binary optimization one. In the first step the initial population is evaluated by fluid solver and the individuals with higher fitness value are selected as parents by roulette wheel selection method. Combined (multi random points) cross-over is performed on the chromosome of two parents to generate the offspring. Parents and offspring are merged in a set and the individuals with higher fitness value are selected as the next generation. The results show that depending on angle of attack (0 < AOA <10 degrees), Reynolds number (0.5 x 10(5) < Re < 2 x 10(5)) and slip length (100 mu m < l(s) < 400 mu m), the optimum superhydrophobic pattern is different. It was found that the idea of piecewise superhydrophobic pattern works for the cases with detached boundary layers and in transition regime. According to results, at higher Re, AOA and l(s), the optimum superhydrophobic pattern is the full superhydrophobic one.

Most studies of heat or mass transfer in dispersed, two-phase flows are either analytical or empirical, and therefore limited to simplified or specific operating conditions. In this frame, Direct Numerical Simulations (DNS) can be a powerful tool to address the needs arising from the development of new liquid–liquid solvent extraction processes, involving e.g., more viscous solvents or different hydrodynamic conditions. In this study, a large parametric study, in the range: Reynolds number Re∈[1,500], Péclet number Pee∈[10,1000], viscosity ratio μ∗∈[0.05,20] and a thermodynamic equilibrium coefficient k∈[0.1,10] , of mass transfer from an isolated droplet is achieved by DNS to propose a physically based correlation of the apparent Sherwood number, Sh, that is sufficiently generic to apply to emerging and future applications. The updated correlations derive first from a solid analysis of the features, in terms of concentration fields inside and outside the droplet and interface properties, to challenge the validity of the assumptions of mass transfer correlations. Then, we propose proper modifications in the formulation of existing models. In a second step, the accuracy and the robustness of the new models have been assessed by comparison with Sh obtained from DNS. The proposed correlations are applicable for mass transfer as well as for heat transfer problems, in most configurations encountered in chemical plants.

The total force experienced by a droplet embedded in a uniform oscillating flow can be divided into four contributions: steady drag force, inertial or pressure gradient force, added-mass force, and Basset-Boussinesq history force. The effect of the history force is often neglected, due to numerical difficulties for its time integration, but also because no analytic expression is available in the time domain for droplets. The contribution of the history force acting on a spherical droplet in an oscillatory flow is determined using direct numerical simulation. Variation of the viscosity ratio makes the analysis relevant to bubbles, droplets, and solid particles. By changing the flow oscillation frequency, we can determine the range of physical parameters for which the contribution of the history force is significant. Additionally, the relevance of the kernel expression recently proposed by Legendre et al. [Phys. Rev. Fluids 4, 073603 (2019)] is discussed and compared to theoretical predictions in the frequency domain. We demonstrate that this kernel can be used to model history effects for droplets, especially since the contribution of the history force to the total force is important. The proposed history force expression provides a significant improvement for droplet trajectory prediction when the history force contribution cannot be neglected.

The flow around a circular cylinder is studied with the presence of solid finite-size spherical particles. The simulations of suspension flows are based on highly efficient simulation algorithms to generate scalable Lattice Boltzmann Method calculations. Simulations of particle resolved dynamics with up to 4800 million grid cells and 1.8 million spherical particles are carried out for different flow regimes and particle concentrations (from dilute, 5% volume concentration to semi-dilute regime, 20%). We investigated a range of parameters that spans from suspensions of iso-dense particles to highly inertial particles in a granular flow, while considering particle to cylinder diameter ratios of 5 and 10. The flow response is described by characteristic properties such as vortex shedding frequency or wake recirculation length. Flow regimes with iso-dense particles can be rationalized by means of mixture material properties while we observe a progressive transition to granular flow for inertial particles. Drag and lift forces acting onto the cylinder are presented in time and frequency domains and the physical analysis is based on the respective hydrodynamic and particle collision contributions.

A hybrid IBM-LES method is presented with the objective to simulate high-Reynolds number pipe flows on coarse Cartesian meshes. The IBM method is first used to simulate a laminar pipe flow and results have shown to converge with second order accuracy to the exact solution. A new forcing scheme inside the IBM wall thickness improves significantly numerical accuracy and provides an interesting way to control the fluid–solid interaction. Based on this new modeling of the IBM wall boundary condition, turbulent pipe flows for Reynolds numbers in the range 50,000 to 500,000 are then considered. The IBM wall forcing under these conditions is developed based on the classical turbulent wall laws, namely the log-law and the power-law, able to reproduce the mean velocity profile. We show that adjusting the control parameters of these two models makes possible to recover the correct bulk velocity and mean velocity profile. In order to improve the fluctuations level and spatial distribution of turbulent structures inside the pipe, we propose to extend the log-law modeling using local and unsteady value of the wall shear stress obtained from a stochastic model. The latter preserves spatiotemporal correlations of the wall friction and enhances the reliability of the simulations in terms of both mean bulk flow and fluctuations. The effects of both the Reynolds number and the grid resolution are also discussed and empiric correlations for the model parameters are proposed.

Turbulence can cause particles to accumulate within specific regions of the flow. One mechanism responsible for this phenomenon, called preferential concentration, consists in particle–fluid interactions yielding inhomogeneous spatial distribution of particles into clusters or depleted regions due to density difference or finite-size effects. In the case of living particles such as plankton, clustering may also originate from their motility or from their behavioral response to turbulent forcing. Preferential concentration of plankton has attracted much attention, because it is a key determinant of encounter rates and therefore relevant for a wide range of ecological processes. However, most studies have focused on microscopic cells, and consequently the case of larger organisms remains poorly studied. Here, we use high-performance particle tracking and three-dimensional Voronoï analysis to test for the emergence of clustering in the spatial distribution of calanoid copepods, the most important metazoans in the oceans in terms of biomass. We found that neither inertia nor motility resulted in significant departure from a random Poisson process over a range of turbulence intensity from very strong to moderate. However, we observed weak clustering in calm water, which may originate from hydrodynamic and olfactory interactions between organisms. Our results improve our understanding of fluid–particle interactions in the zooplankton and have important implications for the modeling of their encounter rates in turbulence.

A detailed understanding of the physical mechanisms driving gyrotactic species to migrate vertically towards the surface allows better quantification of biogeochemical fluxes across the ocean. We focus on marine phytoplankton cells that are motile under gyrotactic forcing. Some species spontaneously swim in the direction opposite to gravity [1]. Gyrotaxis is originating either from morphological aspects (elongated shape, density heterogeneity) or the coupled effect of swimming and settling which results in an inertial torque. Indeed, fluid inertial torque may have a potential impact on the gyrotaxis for elongated planktonic swimmers, especially for those forming long chains and thus having large swimming and settling speeds Based on numerical simulations of hundreds of thousands of micro-organisms swimming in homogeneous isotropic turbulence, we will comment on the different sources of gyrotactic induced spatial clustering [2, 3] and vertical migration [4]. Some specific configurations lead to the accumulation of elongated plankton cells in upwelling flow regions enhancing their ability to move across turbulence through the water column. [1] Kessler J.O. (1985), Nature - 313, 218–220. [2] Durham W. M., et al. (2013) Nat. Commun. - 4, 2148. [3] De Lillo F., et al. (2014) Phys. Rev. Lett. – 112, 044502 [4] Lovecchio S., et al. (2019) Sci. Adv. - 5: eaaw7879

Hydrofoils are among the main components of underwater vehicles and play an important role in power generation of ocean current turbines. In this paper, the physics of external flow past a superhydrophobic SD7003 hydrofoil is studied. We used Navier’s slip length to model the superhydrophobic surfaces, and investigated the effect of different slip lengths on hydrofoil performance by means of RANS (Reynolds-averaged Navier-Stokes) simulations. In addition to drag and lift forces, laminar separation bubble and reattachment are investigated at Reynolds number of 105 before stall angle of attack (AOA). The slip length boundary condition is imposed on upper, and/or lower side(s), in order to study the effect of each superhydrophobic region on the flow field. By increasing the slip length ( [[EQUATION]] ), it was expected to see lift enhancement and drag reduction. However, the lift enhancement trend had some exceptions. For example, lift of full superhydrophobic hydrofoil at [[EQUATION]] and [[EQUATION]] presents an unexpected trend. The unexpected trends were explained by studying the location of laminar separation and reattachment. By increasing the slip length, the laminar separation and reattachment shift towards the trailing edge, and at some slip length, flow separates with no reattachment (i.e. no laminar separation bubble). For such a case, a strong lift reduction is observed. By further increasing the slip length, the laminar separation eventually vanishes and the lift is recovered again. It was shown that in the range of studied Reynolds number, larger slip length does not necessarily lead to better performance of hydrofoil lift. In fact, for all the studied cases and AOA<8ᵒ slip length had negligible influence on lift coefficient, but for AOA>8ᵒ the influence of slip becomes significant by delaying the stall. The lift enhancements of superhydrophobic foil at AOA=10ᵒ for [[EQUATION]] =50 µm, [[EQUATION]] =100 µm and [[EQUATION]] =200 µm are 9%, 13.9% and 19.4%, respectively.