Lubrication Forces Research Articles

A Review of the Homogenized Lattice Boltzmann Method for Particulate Flow Simulations: From Fundamentals to Applications

The homogenized lattice Boltzmann method (HLBM) has emerged as a flexible computational framework for studying particulate flows, providing a monolithic approach to modeling pure fluid flows and flows through porous media, including moving solid and porous particles, within a unified framework. This paper presents a thorough review of HLBM, elucidating its underlying principles and highlighting its diverse applications to particle-laden flows in various fields as reported in literature. These include studies leading to new fundamental knowledge on the settling of single arbitrarily shaped particles as well as application-oriented research on wall-flow filters, hindered settling, and evaluation of the damage potential during particle transport. Among the strengths of HLBM are its monolithic approach, which allows seamless simulation of different fluid-solid interactions, and its ability to handle arbitrary particle shapes, including irregular and concave geometries, while resolving surface interactions to capture local forces. In addition, its parallel scheme based on the lattice Boltzmann method (LBM) results in high computational efficiency, making it suitable for large-scale simulations, even though LBM requires small time steps. Important future development needs are identified, including the addition of a lubrication force correction model, performance enhancements, such as support for hybrid parallelization and GPU, and the extension of compatible contact models to accommodate concave shapes. These advances promise expanded capabilities for HLBM and broader applicability for solving complex real-world problems.

The method of fundamental solutions for multi-particle Stokes flows: Application to a ring-like array of spheres

A method is presented for calculating Stokes flow around multiple particles of arbitrary shape. It uses the Method of Fundamental Solutions (MFS) applied to single particles, combined with an iterative scheme to resolve the many particle-particle hydrodynamic interactions; an approach that is reminiscent of the Method of Reflections. The attractive features of the proposed method are inherited from the MFS — simplicity and accuracy — while providing orders of magnitude computational speed-up for large particle systems. The method is verified through a series of test cases, including those involving strong lubrication forces and non-spherical particles. Unlike applications of the Method of Reflections reported in the literature, the iterative scheme we propose (a block Gauss-Seidel approach to solving a particle-particle interaction matrix) converges for all the cases we consider, for both resistance and mobility problems. The scheme is applied to the study of Stokes flow around ring-like arrays of spheres. We show that the relationship between globally applied velocity or force to the response of individual spheres can be described by just 5 coefficients (or 9 in total) for any given configuration. The results indicate that for 7–10 spheres in sedimentation, there exists a certain spacing that produces steady-state translation of the ring, independent of its orientation. For large numbers of spheres, slender-body theory can be applied to the problem, providing remarkably close agreement to the numerical results over a wide range of parameters.

Short-time self-diffusion in binary colloidal suspensions

The Brownian dynamics of a colloidal particle is consistently modified by the presence in the solvent of other particles of comparable size, whose effects on the particle diffusion coefficient cannot be attributed to a change of the effective solvent viscosity. So far, despite their impact on subjects ranging from microrheology to phoretic transport in crowded environments, a detailed experimental survey of these effects is still lacking. By exploiting the peculiar properties of fluorinated colloidal particle, we have performed an extensive dynamic light scattering (DLS) investigation of short-time self-diffusion in binary colloidal mixtures, focusing on systems where one of the two species (the “tracer” particles) is very diluted compared to the other one (the “host” particles). From the dependence on the host concentration of the DLS correlation function, we have obtained the first-order correction hs1s to the tracer single-particle diffusion coefficient, varying the tracer-to-host size ratio q in the range 0.2 ≤ q ≤ 2. Our results support the functional relation of hs1s on q proposed to account for the theoretical and numerical results for hard-sphere mixtures. However, hs1s seems to have a weaker dependence on the size ratio than theoretically predicted, possibly because of an imperfect matching of the suspensions we used for an ideal hard-sphere mixture. This may be due to the presence of a stabilizing surfactant layer on the particle surface that, although very thin, has significant effects on hydrodynamic lubrication forces.

Coupled CFD-DEM simulation of pin-type wet stirred media mills using immersed boundary approach and hydrodynamic lubrication force

Qualitative description of stressing energies between grinding beads/media in wet stirred media mills at different operation settings could provide a comprehensive understanding of the process parameter influence. In the current work, a modeling approach for wet stirred media mills is described, employing a coupled CFD-DEM framework. This study employs an immersed boundary methodology to emulate the pin-type stirrer rotation in CFD and utilizes a hydrodynamic lubrication force to model the squeezing out and sucking in of fluid between two grinding bead surfaces. The results indicate the influence of process parameters such as the stirrer rotation speed, bead diameter, and bead filling degree. Key findings include the necessity of coupled CFD-DEM with lubrication (i.e., comparison of results from DEM, and coupled CFD-DEM simulations). Dependence of system dynamics on the fluid properties is numerically analyzed using the properties of water–glycerol mixtures.

Solid–fluid force modeling: Insights from comparing a reduced order model for a pair of particles with resolved CFD-DEM

Solid–fluid force models are essential to efficiently model multiple industrial apparatuses such as fluidized beds, spouted beds, and slurry transport. They are generally built using strong hypotheses (e.g. fully developed flow and no relative motion between particles) that affect their accuracy. We study the effect of these hypotheses on particle dynamics using the sedimentation of a pair of particles. We develop new induced drag, lift and torque models for pairs of particles based on an artificial neural network (ANN) regression. The fluid force model covers a range of Reynolds numbers of 0.1 to 100 and particle centroid distance of up to 9 particle diameters. The ANN model uses 3475 computational fluid dynamics (CFD) simulation results as the training data set. Using this fluid force model, we develop a reduced-order model (ROM), which includes the virtual mass force, the Meshchersky force, the history force, the lubrication force, and the Magnus force. Using the results of a resolved computational fluid dynamics coupled with a discrete element method (CFD-DEM) model as a reference, we analyze the discrepancies between the ROM and CFD-DEM results for a series of sedimentation cases that cover particle Archimedes number from 20 to 2930 and particle to fluid density ratio of 1.5 to 1000. The errors primarily stem from particle history interactions that are not accounted for by the fully developed flow hypothesis. The importance of this effect on the dynamic of two particles is isolated and it is shown that it is more pronounced in cases with a lower particle-to-fluid density ratio (such as solid–liquid cases). This work underscores the need for more research on these effects to increase the precision of solid–fluid force models for small particle-to-fluid density ratios (1.5).

Gap in drop collision rate between diffusive and inertial regimes explains the stability of fogs and non-precipitating clouds

Rain drops form in clouds by collision of submillimetric droplets falling under gravity: larger drops fall faster than smaller ones and collect them on their path. The puzzling stability of fogs and non-precipitating warm clouds with respect to this avalanche mechanism has been a longstanding problem. How can droplets of diameter around 10 $\mathrm {\mu }$ m have a low collision probability, inhibiting the cascade towards larger and larger drops? Here we review the dynamical mechanisms that have been proposed in the literature and quantitatively investigate the frequency of drop collisions induced by Brownian diffusion, electrostatics and gravity, using an open-source Monte Carlo code taking all of them into account. Inertia dominates over aerodynamic forces for large drops, when the Stokes number is larger than $1$ . Thermal diffusion dominates over aerodynamic forces for small drops, when the Péclet number is smaller than $1$ . We show that there exists a range of size (typically 3–30 $\mathrm {\mu }$ m for water drops in air) where neither inertia nor Brownian diffusion are significant, leading to a gap in the collision rate. The effect is particularly important, due to the lubrication film forming between the drops immediately before collision, and secondarily to the long-range aerodynamic interaction. Two different mechanisms regularise the divergence of the lubrication force at vanishing separation: the transition to a non-continuum regime in the lubrication film, when the separation is comparable to the mean free path of air, and the induction of a flow inside the drops due to shear at their surfaces. In the gap between inertia-dominated and diffusion-dominated regimes, dipole–dipole electrostatic interactions becomes the major effect controlling the efficiency of drop collisions.

Effects of confinement-induced non-Newtonian lubrication forces on the rheology of a dense suspension

In this work, we propose a functionalised bi-viscous lubrication model to study the material properties of concentrated non-Brownian suspensions and explore the possible confinement-induced non-Newtonian effects of the lubricant in the rheological response of this type of suspensions. From tribological studies, it is well-known that even macroscopically Newtonian liquids under strong confinement might exhibit properties which deviate significantly from their bulk behaviour. When two surfaces separated by an extremely small gap (still large compared to the molecular size) are sheared, strong shear-thinning of the lubricant viscosity at low shear-rates is observed, in spite of its Newtonian-like bulk response. This is connected to a significant increase of the zero-shear-rate viscosity under extreme confinement. We start from an effective lubrication algorithm recently proposed and develop a new gap-size-dependent interparticle bi-viscous lubrication model, able to capture qualitatively the main phenomenology of confined lubricants. We solve the lubrication interaction between particles iteratively via a semi-implicit splitting scheme. Since the handling of lubrication is made implicitly here, the method copes efficiently with large increases of the inter-particle effective viscosities, which would otherwise lead to simulation blow-up or the use of vanishing time-steps in standard explicit schemes. We analyse the rheological response of the suspension systematically in terms of model parameters. In contrast to pure Newtonian lubrication interactions, distinct shear-thinning and shear-thickening regimes in the relative suspension viscosity are observed, which are discussed in terms of particle microstructure coupled with the complex rheology of the confined lubricant. In addition, normal-stress response is negative in both N1 and N2, which is difficult to achieve with standard contact frictional models.

Simulating the rheology of dense suspensions using pairwise formulation of contact, lubrication and Brownian forces

Dense suspensions of solid particles in viscous liquid are ubiquitous in both industry and nature, and there is a clear need for efficient numerical routines to simulate their rheology and microstructure. Particles of micron size present a particular challenge: at low shear rates, colloidal interactions control their dynamics while at high rates, granular-like contacts dominate. While there are established particle-based simulation schemes for large-scale non-Brownian suspensions using only pairwise lubrication and contact forces, common schemes for colloidal suspensions generally are more computationally costly and thus restricted to relatively small system sizes. Here, we present a minimal particle-based numerical model for dense colloidal suspensions that incorporates Brownian forces in pairwise form alongside contact and lubrication forces. We show that this scheme reproduces key features of dense suspension rheology near the colloidal-to-granular transition, including both shear thinning due to entropic forces at low rates and shear thickening at high rates due to contact formation. This scheme is implemented in LAMMPS, a widely used open source code for parallelised particle-based simulations, with a runtime that scales linearly with the number of particles, making it amenable for large-scale simulations.

Dynamics of meniscus-bound particle clusters in extensional flow

Capillary suspensions are three-phase mixtures containing a solid particulate phase, a continuous liquid phase, and a second immiscible liquid forming capillary bridges between particles. Capillary suspensions are encountered in a wide array of applications including 3D printing, porous materials, and food formulations, but despite recent progress, the micromechanics of particle clusters in flow is not fully understood. In this work, we study the dynamics of meniscus-bound particle clusters in planar extensional flow using a Stokes trap, which is an automated flow control technique that allows for precise manipulation of freely suspended particles or particle clusters in flow. Focusing on the case of a two-particle doublet, we use a combination of experiments and analytical modeling to understand how particle clusters rearrange, deform, and ultimately break up in extensional flow. The time required for cluster breakup is quantified as a function of capillary number Ca and meniscus volume V. Importantly, a critical capillary number Cacrit for cluster breakup is determined using a combination of experiments and modeling. Cluster relaxation experiments are also performed by deforming particle clusters in flow, followed by flow cessation prior to breakup and observing cluster relaxation dynamics under zero-flow conditions. In all cases, experiments are complemented by an analytical model that accounts for capillary forces, lubrication forces, hydrodynamic drag forces, and hydrodynamic interactions acting on the particles. Results from the analytical models are found to be in good agreement with experiments. Overall, this work provides a new quantitative understanding of the deformation dynamics of capillary clusters in extensional flow.

Flocculation rate of locally densely distributed cohesive particles in Taylor–Green vortex flow

We employ the three-way coupled numerical simulations to investigate the flocculation of primary cohesive particles which are locally densely distributed in the Taylor–Green cellular vortex flow. The hydrodynamic and inertial forces as well as the direct contact, lubrication, and cohesion forces between particles during the growth, deformation, and breakup of flocs are captured in detail. The flocculation rate of the primary particles decreases gradually from its maximum value at the initial moment, then levels off during flocculation, yielding the flocculation and equilibrium stages. The flocculation rate is determined by the equilibrium floc size and a flocculation coefficient. A larger equilibrium floc size and a smaller value of the flocculation coefficient yield faster flocculation. An initially dense distribution of cohesive particles accelerates the growth of flocs during flocculation but has minor effects on the equilibrium floc size, compared to an initially dilute distribution. A larger particle-to-fluid density ratio, a smaller size ratio between the particle diameter and the Kolmogorov length scale, and stronger cohesion yield a larger equilibrium floc size and a higher flocculation coefficient. Their influence on the flocculation coefficient becomes more evident when the initial particle distribution becomes more concentrated, while their impact on the maximum flocculation rate is very limited. A simple new model is proposed to describe the flocculation process of unevenly distributed cohesive particles in turbulence.

Hard congestion limit of the dissipative Aw–Rascle system

In this study, we analyse the famous Aw–Rascle system in which the difference between the actual and the desired velocities (the offset function) is a gradient of a singular function of the density. This leads to a dissipation in the momentum equation which vanishes when the density is zero. The resulting system of PDEs can be used to model traffic or suspension flows in one dimension with the maximal packing constraint taken into account. After proving the global existence of smooth solutions, we study the so-called ‘hard congestion limit’, and show the convergence of a subsequence of solutions towards a weak solution of a hybrid free-congested system. This is also illustrated numerically using a numerical scheme proposed for the model studied. In the context of suspension flows, this limit can be seen as the transition from a suspension regime, driven by lubrication forces, towards a granular regime, driven by the contacts between the grains.

Numerical Simulation of the Two-Phase Flow Dynamics in an Alkaline Water Electrolyzer Considering Difference in Bubble Diameter

For more efficient hydrogen production, predicting the characteristics of bubble flow inside a water electrolyzer is critical since bubbles affect ion transport, and electrode coverage by bubbles influence overpotentials and the entire electrolysis performance. Transport of bubble significantly depends on its size. In this study, a CFD model of an alkaline water electrolyzer which considers difference in sizes of generated bubbles was developed. The model treats bubbles of different sizes as different gas phases, which are independent of each other. The total bubble mass source for a given current density is divided into phases of each bubble size according to experimental visualizations. Drag, virtual mass, lift, wall lubrication, and turbulent dispersion forces are considered as interaction between gas and liquid phase.The following three sizes of bubbles were considered: diameters d = 36 μm (small), 75 μm (medium), and 208 μm (large). The entire calculation domain represents the anode side of a zero-gap monopolar alkaline water electrolyzer. Figure 1 shows the volume fraction fields of the three gas phases (i.e., bubble sizes) at a constant current density of 0.6 A/cm2. The bubbles are generated on the electrode surface at the z=0 plane. As the liquid phase forms a circulating flow inside the electrolyzer, small bubbles are transported along the circulating downstream flow due to smaller buoyancy force compared to their viscous drag. In contrast, large bubbles flow out directly to the upper outlet because buoyancy force dominates. This result points out the need to consider the difference in bubble diameter for designing flow field inside alkaline water electrolyzer.[Acknowledgements]This study was based on results obtained from the Development of Fundamental Technology for Advancement of Water Electrolysis Hydrogen Production in Advancement of Hydrogen Technologies and Utilization Project (P14021) commissioned by the New Energy and Industrial Technology Development Organization (NEDO).Fig. 1. Total gas phase volume fraction field obtained from the sum of the volume fraction field for each size bubble, and its experimental visualization. Figure 1

Intermediate temperature of supercritical water enhances the dispersion of cohesive particles

Supercritical water gasification represents a promising technology for the clean conversion of coal to hydrogen, so that it is expected to replace the traditional use of coal by burning. However, the flocculation of cohesive coal particles significantly reduces the contact area between particles and supercritical water, which limits the chemical reaction that produces hydrogen. Based on Lifshitz theory, we propose a cellular flow model for numerically reproducing the flocculation of cohesive particles in supercritical fluid turbulence. Cohesion, lubrication, and direct contact forces between particles are fully accounted for, so that the model allows us to conduct a detailed analysis of the translation, rotation, growth, and breakage of flocs in supercritical fluid turbulence. The results show a transient flocculation phase of cohesive particles characterized by an increasing average floc size, followed by a statistically steady equilibrium phase. Increasing the density ratio between particles and supercritical water significantly accelerates the flocculation of dispersed cohesive particles. Higher temperature of the fluid yields faster flocculation and larger flocs when the temperature is below the critical value of 660 K. Upon raising the temperature above 660 K, the flocculation rate and floc size first decrease, and then increase again, yielding an optimal temperature around 750 K where the primary particles are most effectively dispersed by the supercritical water turbulence.

A barrier method for contact avoiding particles in Stokes flow

Rigid particles in a Stokesian fluid experience an increasingly strong lubrication resistance as particle gaps narrow. Numerically, resolving these lubrication forces comes at an intractably large cost, even for moderate system sizes. Hence, it can typically not be guaranteed that artificial particle collisions and overlaps do not occur in a dynamic simulation, independently of the choice of method to solve the Stokes equations. In this work, the potentially large set of non-overlap constraints, in terms of the Euclidean distance between boundary points on disjoint particles, are efficiently represented via a barrier energy. We solve for the minimum magnitudes of repelling contact forces and torques between any particle pair in contact to correct for overlaps by enforcing a zero barrier energy at the next time level, given a contact-free configuration at a previous instance in time. Robustness for the method is illustrated using a multiblob method to solve the mobility problem in Stokes flow, applied to suspensions of spheres, rods and boomerang shaped particles. Collision free configurations are obtained at all instances in time, and considerably larger time-steps can be taken than without the technique. The effect of the contact forces on the collective order of a set of rods in a background flow that naturally promote particle interactions is also illustrated.

A two-dimensional void fraction distribution fast prediction model and application to a rod bundle

Accurate prediction of void fraction distribution is essential for system optimization, design, and safety analysis which involves gas-liquid two-phase flow. This paper introduces a novel conceptual two-dimensional Void Fraction Distribution Fast Prediction model (VDFP) designed to rapidly predict the void fraction distribution of fully developed two-phase pipe flow. By achieving this, the N-S equations were simplified into a readily solvable Laplacian form. The simplification was based on two assumptions: 1) For the steady simulation of fully developed pipe flow, the convective and axial diffusion terms in the time-averaged N-S equations are negligible; 2) The pressure field can be predefined as linearly decreasing along the axial direction while ignoring the radial pressure difference. After determining the axial velocity distribution, the void fraction distribution can be calculated through the balance of interfacial forces (lift force, wall lubrication force, and turbulent dispersion force) in the radial direction, which has a solid physical foundation and is applicable across a wide range of operating conditions. The feasibility of the VDFP model was validated by comparing the predicted results with experimental data of air-water two-phase flow in a rod bundle channel with a rod diameter of 25 mm and a rod pitch of 34 mm. Additionally, considering the constraints imposed by the rod bundle subchannel, the impact of bubble geometric restrictions on void fraction prediction was also evaluated. The validation results indicate that the VDFP model predicts the wall peak and core peak void fraction distribution results with satisfactory accuracy, and its solving time is <3 s, showing considerable potential for further reduction.

Iterative screening methodology for optimal modeling of bubble coalescence/breakup and interphase force in CMFD simulation of flow boiling

The Euler–Euler two fluid approach has been widely adopted in computational multi-fluid dynamic (CMFD) simulation of nucleate boiling flow. Accurate prediction of bubble size and interphase forces are desired to properly model interphase momentum transfer in the CMFD simulation. Bubble dynamics, including nucleation, coalescence/breakup and condensation, should be modeled to calculate bubble size variation. In the meantime, modeling of interphase forces is crucial to predict distribution of void fraction. Numerous closure models have been developed for bubble dynamics and interphase forces. Diverse model combinations were reported depending on the specific application. As a step forward to converge the diversity, efforts are dedicated to establish a methodology to iteratively screen the closure models for nucleate flow boiling at elevated pressures based on the DEBORA experiment. In the proposed methodology, the closure models for each physics are first evaluated and sorted based on the predicted value in fixed-point analysis and its effect on CMFD results observed in separate-effect analysis. The second part of the methodology is iterative screening. To achieve accurate bubble size prediction, the optimal coalescence model is identified for each breakup model. With the optimal coalescence counterpart, the optimal breakup model is then identified. For the interphase force models, the optimal lift force model is first determined based on the CMFD simulation with the same drag force model and without wall lubrication force model. The optimal wall lubrication force and drag force model are selected in sequence. Utilized one single case of DEBORA experiments, the optimal set of the coalescence/breakup models, i.e., Prince et al.’s and Luo et al.’s models, and the interphase models, i.e., Tomiyama et al.’s models for drag and lift force, Lubchenko et al.’s model for wall lubrication force and Burns et al.’s model for turbulence dispersion force has been selected. Validation of the model combination based on more DEBORA experiment datasets and Bartolomei et al.’s experiment shows promising prediction accuracy.

A front-tracking method for simulating interfacial flows with particles and soluble surfactants

In this research, a three-dimensional model is proposed to simulate the behavior of particle-covered droplets that contain soluble surfactant. The model includes a full two-way coupling between surfactant transport and particle motion, with surfactant concentrations at the interface and in the bulk linked to the flow and motion of the interface by a convection-diffusion equation. A front-tracking method is used to track the interfacial flow, while a semi-resolved particle method is used to model the particles, accounting for various forces acting on them, such as the excluded volume force, the direct contact force and the hydrodynamic lubrication force. The model also includes a surfactant-dependent surface tension that can affect the particle-interface interactions. The model is validated against previous numerical data and a parametric study is performed to examine the delicate interplay between flow fields and interfaces associated with surfactant concentration and particle motion. The study then investigates the role of surfactants in the dynamics of a particle-covered droplet, focusing on their coupled influence on droplet deformation. In addition, it is observed that particles bridge colliding droplet interfaces and suppress fluid drainage from the gap, preventing droplet coalescence along with the interparticle repulsion force.

The Influence of Gravity and Mass Loading on the Coalescence of Aerodynamically Interacting Droplets in Homogeneous Isotropic Turbulence

Abstract The collision–coalescence of cloud droplets in atmospheric turbulent flow is analyzed numerically using direct numerical simulation coupled to a Lagrangian particle tracking. The droplet aerodynamic interactions (AI) are represented for employing two complementary approaches. For large separations, the interaction forces are evaluated by the superposition of Stokes disturbance velocities generated by moving particles. When the distance between droplets is comparable to their mean radii, lubrication forces are additionally considered. Simulation results show that without gravitational acceleration, aerodynamic interactions decrease the kinetics of the coalescence process but do not significantly impact the size spectrum broadening. The influence of AI on the coalescence kinetics is more complex in the presence of gravity and depends on the mass loading and on droplet inertia. Long-range aerodynamic interactions reduce the coalescences in dilute suspensions but increase the collision rate in dense suspensions of high-inertia droplets. In contrast, lubrication forces decrease the collision rate regardless of the mass loading. The collision efficiency induced by aerodynamic interactions additionally is influenced by the radius ratio of colliding droplets and the mechanisms leading to raindrops formation and growth. In cloud-like conditions, both long- and short-range AI decrease the fraction of raindrops created by collisions between droplets (autoconversion) while promoting raindrops growth by accretion (collection by settling drops). In turn, aerodynamic interactions favor the growth of a limited number of droplets and promote the broadening of the droplet size spectrum. This effect is stronger in dilute suspensions of weakly inertial droplets, corresponding to the flow properties encountered in developing precipitation.

Experimental and numerical determination of the lubrication force between a spherical particle and a micro-structured surface

In many particulate processes suspensions need to be handled. Hydrodynamic forces in presence of a liquid as a surrounding continuum medium can significantly affect the particle collision behaviour. When particles approach a wall, lubrication force can become dominant with decreasing distance. This force was described analytically by different authors for a smooth flat wall. Roughness was found to be an important factor in this context, but the mechanisms are still not fully understood. In this work, the effects of topology on the lubrication force were studied using a regular prismatic micro-structured titanium surface produced by micro-milling. A nanoindentation setup was modified for the direct measurement of this force during the particle approach to polished and micro-structured surfaces in liquid. For a more detailed insight on the behaviour of the fluid in the decreasing gap between particle and surface microstructure, resolved computational fluid dynamics (CFD) simulations were performed using an overset mesh method. The comparison of simulation results with nanoindentation tests and analytical solution showed a good agreement. The effects of structure size and particle contact location at various approaching velocities on the lubrication force were investigated.

Collision Statistics of Droplets in Turbulence Considering Lubrication Interactions, Mobility of Interfaces, and Non-continuum Molecular Effects

Collision statistics of same-size aerodynamically interacting droplets in homogeneous isotropic turbulence is examined via hybrid direct numerical simulations combined with Lagrangian particle tracking under a point-particle assumption and a one-way momentum coupling. The simulations are performed both in the absence and presence of gravity for various droplet Stokes numbers (0.1–2) over a wide range of liquid water contents (LWCs = 1–30 g/m3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {g/m}^3$$\\end{document}). Also, the effects of using different representations of the lubrication forces, i.e. a continuum and a non-continuum one, have been investigated. The results have highlighted the importance of considering the aerodynamic interaction, especially in the presence of gravity for the systems that have high LWCs. Likewise, taking the lubrication force into account shows a notable change in statistics, but a non-continuum representation yields results that are close to the continuum one. In addition, the impact of modeling droplets as fluid drops instead of rigid particles has been examined. Obtaining quite close statistics under both cases demonstrates that a rigid particle assumption for water droplets in air is sufficiently accurate owing to the high water-to-air viscosity ratio. Moreover, the collision efficiency is shown to be in the range 60–100% in the absence of gravity, which approaches 100% as the LWC is enlarged. In the presence of gravity, however, the efficiency grows, with both the Stokes number and the LWC, to values as high as 300%. Finally, for settling droplet systems, the enhancement factor due to turbulence is quantified, exhibiting a growth with the LWC and a reduction with the Stokes number.