Particles In Isotropic Turbulence Research Articles

Particle motion under turbulent eddies: Inspiration for fine minerals flotation

Turbulent flow ubiquitous in flotation machines and contain multi-scale eddies. Compared to static environments, turbulence can increase particle kinetic energy and promote particle-bubble collision in mineral flotation. However, there have been few quantitative studies on turbulent eddies that affect particle motion, which limits the precise flow control of flotation processes. In this study, the motion characteristics of particles in isotropic turbulence were measured by particle tracking velocimetry, and the relationship between turbulent eddy and particle motion was quantitatively analyzed by 7-scale wavelet transform and fast Fourier transform. The measurement results show that the particle moves upward following a rotating turbulent eddy, and turbulent eddies with sizes of 35–119 µm and 127–288 µm drives the motion of dp = 74 µm and dp = 200 µm particles respectively. The turbulent acceleration of particles within a turbulent eddy can be calculated as aλ=Cε2/3λ-1/3. This study provides an eddy control basis for mineral flotation performance improvement.

Spectrum of passive scalar carried by particles in isotropic turbulence

Spectrum of passive scalar carried by particles in isotropic turbulence

Detailed characterization of extreme clustering at near-contact scales in isotropic turbulence

Recent measurements of inertial particles in isotropic turbulence (Hammond & Meng, J. Fluid Mech., vol. 921, 2021, A16) revealed surprising extreme clustering of particles at near-contact separations $(r)$ , whereby the radial distribution function, $g(r)$ , grows from $O(10)$ to $O(10^3)$ with a $(r/a)^{-6}$ scaling (where $a$ is the particle radius), and a surprising upturn of the mean inward particle-pair relative velocity (MIRV). Hydrodynamic interactions (HIs) were proposed to explain the extreme clustering, but despite predicting the correct scaling $(r/a)^{-6}$ , the HI theory underpredicted $g(r)$ by at least two orders of magnitude (Bragg et al., J. Fluid Mech., vol. 933, 2022, A31). To further understand the extreme clustering phenomenon and the relevance of HI, we characterize $g(r)$ and particle-pair kinematics for Stokes numbers $0.07 \leq St \leq 3.68$ in a homogeneous isotropic turbulence chamber using three-dimensional (3-D) particle tracking resolved to near–contact. A drift–diffusion equation governing $g(r)$ is presented to investigate the kinematic mechanisms of particle pairs. Measurements in all 24 conditions show that when $r/a\lessapprox 20$ , extreme clustering consistently occurs, scaling as $g(r) \sim (r/a)^{-k}$ with $4.5 \leq k \leq 7.6$ , which increases with $St$ . Here $g(r)$ varies with $St$ , particle size, density and polydispersity in ways that HI cannot explain. The extreme clustering region features an inward drift contributed by particle-pair turbophoresis and an inward radial relative acceleration. The latter indicates an interparticle attractive force at these separations that HI also cannot explain. The MIRV turns upward when approaching the extreme clustering region, opposite to direct numerical simulation predictions. These observations further support our previous assessment that extreme clustering arises from particle–particle interactions, but HI is not the main mechanism.

Large-eddy simulation of particle-laden isotropic turbulence using machine-learned subgrid-scale model

We apply a machine-learned subgrid-scale model to large-eddy simulations (LES) of heavy particles in isotropic turbulence with different Stokes numbers. The data-driven model, originally developed for high Reynolds number isotropic turbulent flows based on the gene expression programming (GEP) method, has explicit model equations and is for the first time tested in multiphase problems. The performance of the GEP model has been investigated in detail, focusing on the particle statistics including particle acceleration, velocity, and clustering. Compared with the commonly used dynamic Smagorinsky model, the GEP model provides significantly improved predictions on the particle statistics with Stokes numbers varying from 0.01 to 20, showing satisfactory agreement with the results from direct numerical simulations. The reasons for the enhanced predictions of the GEP model are further discussed. As the GEP model is less dissipative and it introduces high-order terms closely related to vorticity distribution, the fine-scale structures usually missing in LES simulations can be better recovered, which are believed to be closely related to the intermittency of particle motion and also particle clustering.

Particle radial velocity and concentration kernels estimation in isotropic grid turbulence experiments of inertialess particles at small separation distances

We report experimental measurements of kinematic and dynamic particle concentration kernels conditioned by the separation distances of solid inertialess particles in isotropic turbulence by three-dimensional particle tracking velocimetry with particle diameters smaller than the Kolmogorov length scale. Particle radial relative velocity statistics are measured from the dissipation to the integral length-scale range. The radial scaling of particle and fluid relative velocity variance $\langle w_r(r)^{2} \rangle \sim r^{2/3}$ in the inertial subrange, consistent with Kolmogorov's theory, is reported, while a new scaling is found for small distances due to finite-size effects between particles. The measured concentration kernels at small separation distances therefore deviate from those in the theory of Saffman & Turner (J. Fluid Mech., vol. 1, 1956, pp. 16–30) at small inter-particle distances due to hydrodynamic interactions. A real kernel taking into account the history of the particle tracks and excluding multiple events is also calculated, while the normalised particle concentration kernels are found to be essentially insensitive to the flow Reynolds number.

Hydrodynamic interactions and extreme particle clustering in turbulence

Expanding recent observations by Hammond & Meng (J. Fluid Mech., vol. 921, 2021, A16), we present a range of detailed experimental data of the radial distribution function (r.d.f.) of inertial particles in isotropic turbulence for different Stokes number,$St$, showing that the r.d.f. grows explosively with decreasing separationr, exhibiting$r^{-6}$scaling as the collision radius is approached, regardless of$St$or particle radius$a$. To understand such explosive clustering, we correct a number of errors in the theory by Yavuzet al.(Phys. Rev. Lett., vol. 120, 2018, 244504) based on hydrodynamic interactions between pairs of small, weakly inertial particles. A comparison between the corrected theory and the experiment shows that the theory by Yavuzet al.underpredicts the r.d.f. by orders of magnitude. To explain this discrepancy, we explore several alternative mechanisms for this discrepancy that were not included in the theory and show that none of them are likely the explanation. This suggests new, yet-to-be-identified physical mechanisms are at play, requiring further investigation and new theories.

Local analysis of the clustering, velocities, and accelerations of particles settling in turbulence

Using 3D Vorono\text{\"i} analysis, we explore the local dynamics of small, settling, inertial particles in isotropic turbulence using Direct Numerical Simulations (DNS). We independently vary the Taylor Reynolds number $R_\lambda \in[90,398]$, Froude number $Fr\equiv a_\eta/g\in[0.052,\infty]$ (where $a_\eta$ is the Kolmogorov acceleration, and $g$ is the acceleration due to gravity), and Kolmogorov scale Stokes number $St\equiv\tau_p/\tau_\eta\in[0,3]$. In agreement with previous results using global measures of particle clustering, such as the Radial Distribution Function (RDF), we find that for small Vorono\text{\"i} volumes (corresponding to the most clustered particles), the behavior is strongly dependent upon $St$ and $Fr$, but only weakly dependent upon $R_\lambda$, unless $St>1$. However, larger Vorono\text{\"i} volumes (void regions) exhibit a much stronger dependence on $R_\lambda$, even when $St\leq 1$, and we show that this, rather than the behavior at small volumes, is the cause of the sensitivity of the standard deviation of the Vorono\text{\"i} volumes that has been previously reported. We also show that the largest contribution to the particle settling velocities is associated with increasingly larger Vorono\text{\"i} volumes as the settling parameter $Sv\equiv St/Fr$ is increased. Our local analysis of the acceleration statistics of settling inertial particles shows that clustered particles experience a net acceleration in the direction of gravity, while particles in void regions experience the opposite. The particle acceleration variance, however, is a convex function of the Vorono\text{\"i} volumes, with or without gravity, which seems to indicate a non-trivial relationship between the Vorono\text{\"i} volumes and the sizes of the turbulent flow scales. Results for the variance of the fluid acceleration at the inertial particle "..."

Impact of turbulence forcing schemes on particle clustering

This letter investigates the clustering of particles in isotropic turbulence sustained by linear and spectral forcing. The Stokes numbers range from 0.1 to 100. The time series of particles are analyzed with autocorrelation functions of the particle velocity and local fluid velocity. The spatial distributions of particles are analyzed with probability density functions of Voronoï volumes. These statistics show significant differences between the clustering of particles in linearly and spectrally forced turbulence.

Influence of Reynolds number on the motion of settling, bidisperse inertial particles in turbulence

Using Direct Numerical Simulations (DNS), we examine the effects of Taylor Reynolds number, $R_\lambda$, and Froude number, $Fr$, on the motion of settling, bidisperse inertial particles in isotropic turbulence. Particle accelerations play a key role in the relative motion of bidisperse particles, and we find that reducing $Fr$ leads to an enhancement of the accelerations, but a suppression of their intermittency. For Stokes numbers $St>1$, the effect of $R_\lambda$ on the accelerations is enhanced by gravity, since settling causes the particle accelerations to be affected by a larger range of flow scales. The results for the Probability Density Function (PDF) of the particle relative velocities show that for bidisperse particles, decreasing $Fr$ leads to an enhancement of their relative velocities in both the vertical (parallel to gravity) and horizontal directions. Importantly, our results show that even when the particles are settling very fast, turbulence continues to play a key role in their vertical relative velocities, and increasingly so as $R_\lambda$ is increased. This occurs because although the settling velocity may be much larger than typical velocities of the turbulence, due to intermittency, there are significant regions of the flow where the turbulence contribution to the particle motion is of the same order as that from gravitational settling...

Enhanced and suppressed multiscale dispersion of bidisperse inertial particles due to gravity

Using Direct Numerical Simulations (DNS), we investigate how gravity modifies the multiscale dispersion of bidisperse inertial particles in isotropic turbulence. The DNS has a Taylor Reynolds number $R_\lambda=398$, and we simulate Stokes numbers (based on the Kolmogorov timescale) in the range $St\leq 3$ , and consider Froude numbers $Fr = 0.052$ and $\infty$, corresponding to strong gravity and no gravity, respectively. The degree of bidispersity is quantified by the difference in the Stokes number of the particles $|\Delta St|$. We first consider the mean-square separation of bidisperse particle-pairs and find that without gravity (i.e. $Fr = \infty$), bidispersity leads to an enhancement of the the mean-square separation over a significant range of scales. When $|\Delta St|\geq O(1)$, the relative dispersion is further enhanced by gravity due to the large difference in the settling velocities of the two particles. However, when $|\Delta St|\ll1$, gravity suppresses the relative dispersion as the settling velocity contribution is small, and gravity suppresses the non-local contribution to the particle dynamics. In order to gain further insights, we consider separately the relative dispersion in the vertical (parallel to gravity) and horizontal directions. As expected, the vertical relative dispersion can be strongly enhanced by gravity due to differences in the settling velocities of the two particles. However, a key finding of our study is that gravity can also significantly enhance the horizontal relative dispersion. This non-trivial effect occurs because fast settling particles experience rapid fluctuations in the fluid velocity field along their trajectory, leading to enhanced particle accelerations and relative velocities. For sufficiently large initial particle separations...

Wavelet multiresolution analysis of particle-laden turbulence

The preferential concentration of inertial particles in isotropic turbulence is studied using wavelets, enabling the joint scale-space analysis of the flow field of the carrier phase, the concentration field of the dispersed phase, and interphase conditioned statistics.

Effects of deterministic and stochastic forcing schemes on the relative motion of inertial particles in DNS of isotropic turbulence

In direct numerical simulations (DNS) of homogeneous isotropic turbulence, statistical stationarity is achieved by artificially forcing the low-wavenumber scales of turbulence. In this study, the effects of two such forcing schemes on the relative positions and velocities of heavy, monodisperse, “point” particles in isotropic turbulence are studied. The first forcing scheme considered is a deterministic method that maintains the turbulent kinetic energy constant by resupplying the energy dissipated during a time step to the velocity components in a low-wavenumber band. The second is the stochastic forcing scheme of Eswaran and Pope [Comput. Fluids 16, 257 (1988)] that involves adding a forcing acceleration to the fluid momentum equation at the low wavenumbers. The forcing acceleration is based on a combination of six independent Uhlenbeck-Ornstein (UO) random processes. An input parameter to the stochastic scheme is the correlation time scale Tf of the UO processes. Among our objectives is to assess the effects of varying Tf on the relative-motion statistics of inertial particle pairs.Direct numerical simulations of homogeneous, isotropic turbulence containing disperse particles were undertaken using both the deterministic and stochastic forcing schemes for three grids sizes (1283, 2563, and 5123). For each grid size, DNS runs based on the stochastic forcing were performed for five values of the UO correlation time scale Tf = TE/4, TE/2, TE, 2TE, and 4TE, where TE is the large-eddy time scale obtained from the DNS run with deterministic forcing at the same grid size. Thus, six DNS runs (one with deterministic forcing and five with stochastic forcing) were performed for each grid resolution, with Reλ held nearly constant (varying by 2–3%) among these runs. The nominal values of Reλ were ≈ 80, 131, and 210 for the 1283, 2563, and 5123 grids, respectively. In each DNS run, heavy, monodisperse particles were tracked corresponding to twelve Stokes numbers ranging from Stη = 0.05 to 40, where Stη is the Stokes number based on the Kolmogorov time-scale. The motivation was to understand how the applied forcing impacted particle-pair relative motion in the three main Stokes number regimes, namely Stη < 1, Stη ∼ 1, and Stη > 1. We focus our attention on three statistics quantifying the relative positions and velocities of particles: the radial distribution function (RDF), the variance 〈Ur2〉 of the component of pair relative velocity along the separation vector (Ur), and the probability density function (PDF), P(Ur). Using the RDF and the PDF P(Ur), we also computed the particle collision kernel. The pair statistics obtained from the various forcing cases are compared among each other, as well as with those from the DNS study of Ireland et al. that employed only the deterministic forcing [J. Fluid Mech. 796, 617 (2016)].At all three Reynolds numbers, we find that the forcing method and the time scale Tf have a noticeable effect on the RDFs for both Stη < 1 and Stη > 1. For Stη < 1 (at a given Reλ), the differences between the RDFs for the various forcing cases increased with Stokes number, reaching a maximum around Stη = 0.4. However, for Stη ∼ 1 (Stη = 0.7 and 1), the RDFs seem to be relatively unaffected by the forcing schemes. When Stη > 1, we find that the RDFs for the various forcings differed the most at Stη = 2, with the differences decreasing thereafter for higher Stokes numbers. When considering the effects of Reλ, it is seen that the RDFs computed from the DNS with deterministic forcing were more sensitive to Reλ variation than those obtained from DNS with stochastic forcing.For Stη < 1, the variance 〈Ur2〉 showed only a weak sensitivity to the forcing scheme at the two lower Reynolds numbers, but a much clearer dependence is seen for Reλ = 210. When Stη ≥ 1, the forcing has a relatively significant impact on the variances at all Reynolds numbers, and these effects are amplified as Reλ is increased. Using the RDF and the PDF P(Ur), we also computed the collision kernels. At the two lower Reλ, collision kernels were found to be weakly dependent on Reλ for Stη < 1, but showed significant increase with Reλ for Stη ≳ 1. However, when Reλ is increased to 210, the collision kernel is seen to increase at all Stokes numbers.

Rotations of large inertial cubes, cuboids, cones, and cylinders in turbulence

Rotations of large, neutrally buoyant, anisotropic particles in isotropic turbulence are studied. The rotation rate probability density function and its moments are found to be simple functions of particle shape and size. These are compared with structure functions in the flow to understand the effects of finite particle size.

Small-scale dynamics of settling, bidisperse particles in turbulence

Mixing and collisions of inertial particles at the small scales of turbulence can be investigated by considering how pairs of particles move relative to each other. In real problems the two particles will have different sizes, i.e. they are bidisperse, and the effect of gravity on their motion is often important. However, how turbulence and gravity compete to control the motion of bidisperse inertial particles is poorly understood. Motivated by this, we use direct numerical simulations (DNS) to investigate the dynamics of settling, bidisperse particles in isotropic turbulence. In agreement with previous studies, we find that without gravity (i.e. $Fr=\infty$, where $Fr$ is the Froude number), bidispersity leads to an enhancement of the relative velocities, and a suppression of their spatial clustering. For $Fr&lt;1$, the relative velocities in the direction of gravity are enhanced by the differential settling velocities of the bidisperse particles, as expected. However, we also find that gravity can strongly enhance the relative velocities in the ‘horizontal’ directions (i.e. in the plane normal to gravity). This non-trivial behaviour occurs because fast settling particles experience rapid fluctuations in the fluid velocity field along their trajectory, leading to enhanced particle accelerations and relative velocities. Indeed, the results show that even when $Fr\ll 1$, turbulence can still play an important role, not only on the horizontal motion, but also on the vertical motion of the particles. This is related to the fact that $Fr$ only characterizes the importance of gravity compared with some typical acceleration of the fluid, yet accelerations in turbulence are highly intermittent. As a consequence, there is a significant probability for particles to be in regions of the flow where the Froude number based on the local, instantaneous fluid acceleration is ${&gt;}1$, even though the typically defined Froude number is $\ll 1$. This could imply, for example, that extreme events in the mixing of settling, bidisperse particles are only weakly affected by gravity even when $Fr\ll 1$. We also find that gravity drastically reduces the clustering of bidisperse particles. These results are strikingly different to the monodisperse case, for which recent results have shown that when $Fr&lt;1$, gravity strongly suppresses the relative velocities in all directions, and can enhance clustering. Finally, we consider the implications of these results for the collision rates of settling, bidisperse particles in turbulence. We find that for $Fr=0.052$, the collision kernel is almost perfectly predicted by the collision kernel for bidisperse particles settling in quiescent flow, such that the effect of turbulence may be ignored. However, for $Fr=0.3$, turbulence plays an important role, and the collisions are only dominated by gravitational settling when the difference in the particle Stokes numbers is ${\geqslant}O(1)$.

Rotations and accumulation of ellipsoidal microswimmers in isotropic turbulence

Aquatic micro-organisms and artificial microswimmers locomoting in turbulent flow encounter velocity gradients that rotate them, thereby changing their swimming direction and possibly providing cues about the local flow environment. Using numerical simulations of ellipsoidal particles in isotropic turbulence, we investigate the effects of body shape and swimming velocity on particle motion. Four particle shapes (sphere, rod, disc and triaxial ellipsoid) are investigated at five different swimming velocities in the range $0\leqslant V_{s}\leqslant 5u_{\unicode[STIX]{x1D702}}$, where $V_{s}$ is the swimming velocity and $u_{\unicode[STIX]{x1D702}}$ is the Kolmogorov velocity scale. We find that anisotropic, swimming particles preferentially sample regions of lower fluid vorticity than passive particles do, and hence they accumulate in these regions. While this effect is monotonic with swimming velocity, the particle enstrophy (variance of particle angular velocity) varies non-monotonically with swimming velocity. In contrast to passive particles, the particle enstrophy is a function of shape for swimming particles. The particle enstrophy is largest for triaxial ellipsoids swimming at a velocity smaller than $u_{\unicode[STIX]{x1D702}}$. We also observe that the average alignment of particles with the directions of the velocity gradient tensor are altered by swimming leading to a more equal distribution of rotation about different particle axes.

Analysis of the forward and backward in time pair-separation probability density functions for inertial particles in isotropic turbulence

In this paper we investigate, using theory and direct numerical simulations (DNS), the forward in time (FIT) and backward in time (BIT) probability density functions (PDFs) of the separation of inertial particle pairs in isotropic turbulence. In agreement with our earlier study (Bragg et al., Phys. Fluids, vol. 28, 2016, 013305), where we compared the FIT and BIT mean-square separations, we find that inertial particles separate much faster BIT than FIT, with the strength of the irreversibility depending upon the final/initial separation of the particle pair and their Stokes number $St$. However, we also find that the irreversibility shows up in subtle ways in the behaviour of the full PDF that it does not in the mean-square separation. In the theory, we derive new predictions, including a prediction for the BIT/FIT PDF for $St\geqslant O(1)$, and for final/initial separations in the dissipation regime. The prediction shows how caustics in the particle relative velocities in the dissipation range affect the scaling of the pair-separation PDF, leading to a PDF with an algebraically decaying tail. The predicted functional behaviour of the PDFs is universal, in that it does not depend upon the level of intermittency in the underlying turbulence. We also analyse the pair-separation PDFs for fluid particles at short times, and construct theoretical predictions using the multifractal formalism to describe the fluid relative velocity distributions. The theoretical and numerical results both suggest that the extreme events in the inertial particle-pair dispersion at the small scales are dominated by their non-local interaction with the turbulent velocity field, rather than due to the strong dissipation range intermittency of the turbulence itself. In fact, our theoretical results predict that for final/initial separations in the dissipation range, when $St\gtrsim 1$, the tails of the pair-separation PDFs decay faster as the Taylor Reynolds number $Re_{\unicode[STIX]{x1D706}}$ is increased, the opposite of what would be expected for fluid particles.

Developments and difficulties in predicting the relative velocities of inertial particles at the small-scales of turbulence

In this paper, we consider the development of theoretical models to predict the relative velocities of inertial particles in isotropic turbulence. In particular, we use our recently developed theory for the backward-in-time (BIT) relative dispersion of inertial particles in turbulence [Bragg et al., “Forward and backward in time dispersion of fluid and inertial particles in isotropic turbulence,” Phys. Fluids 28, 013305 (2016)] to develop the theoretical model by Pan and Padoan [“Relative velocity of inertial particles in turbulent flows,” J. Fluid Mech. 661, 73 (2010)]. We focus on the most difficult regime to model, the dissipation range, and find that the modified Pan and Padoan model (that uses the BIT dispersion theory) can lead to significantly improved predictions for the relative velocities, when compared with the Direct Numerical Simulation (DNS) data. However, when the particle separation distance, r, is less than the Kolmogorov lengthscale, η, the modified model overpredicts the DNS data. We explain how these overpredictions arise from two assumptions in the BIT dispersion theory that are in general not satisfied when the final separation of the BIT dispersing particles is &amp;lt;η. We then demonstrate the failure of both the original and modified versions of the Pan and Padoan model to predict the correct scale-invariant forms for the inertial particle relative velocity structure functions in the dissipation regime. It is shown how this failure, which is also present in other models, is associated with our current inability to correctly predict not only the quantitative but also the qualitative behavior of the radial distribution function in the dissipation range when St=O(1).

Large-eddy simulation of turbulent preferential concentration and collision of bidisperse heavy particles in isotropic turbulence

Particle-laden turbulent flows are ubiquitous in natural and engineering flows. The preferential concentration and pair-statistics of heavy particles at small and intermediate Stokes numbers are mainly related to the small-scale motions, which are missing in conventional large-eddy simulation (LES). Therefore, studies on the effects of subgrid scale (SGS) motions on particle-pair statistics are significant. This paper will study the effects of SGS motions on the collision-related statistics of a bidisperse system that involves two groups of particles at different Stokes numbers using a direct numerical simulation (DNS), filtered DNS (FDNS) and LES. Compared with a monodisperse system at a single Stokes number, the relative error of the collision rates in the bidisperse system is much smaller than that in the monodisperse one. For given filter widths and Stokes numbers, the relative errors between the FDNS and LES are much smaller than their relative errors to DNS, implying that the error caused by the filtering operation in LES plays a leading role in the overall error of the particle collision rate at Stokes numbers less than 3. A particle SGS model is, thus, necessary to consider the effect of SGS motions on bidisperse heavy particles.

On the accuracy of Lagrangian point-mass models for heavy non-spherical particles in isotropic turbulence

The interaction of isotropic turbulence with heavy particles, the size of which is on the order of the smallest turbulent eddies, is considered. Although this case is relevant to pulverized coal and biomass combustion, there is an apparent lack of reliable numerical models for the particle dynamics and the response of the turbulent flow field. By conducting particle resolving direct numerical simulations, we generate accurate results for the dynamics of spherical and non-spherical particles and the associated turbulence modulation effects. Subsequently, we verify the ability of two different two-way coupled Lagrangian models, one derived for spherical and one for ellipsoidal particles, to reproduce the reference results. It is found that the velocities of prolate ellipsoids are significantly reduced compared to spherical particles of the same volume. This behavior is captured by neither of the Lagrangian models since the spherical model does not account for the particle shape and the ellipsoidal model does not account for particle inertia. Additionally, the spectra of fluid kinetic energy indicate that the models fail to resolve the interaction in the high wave number regime, i.e., the interaction of the particle boundary layers and wakes with the smallest flow scales. This is reflected by the vast underprediction of the overall viscous dissipation by both models. As a consequence, the attenuation of kinetic energy contained in large-scale flow structures is underpredicted by the Lagrangian models. In summary, for the slightly non-spherical particles, accounting for inertia effects appears to be more important than inclusion of non-spherical effects for the kinetic energy balance of both phases. However, the ellipsoidal particle model reproduces the preferential alignment of the non-spherical particles within the turbulent flow which becomes more important in highly anisotropic turbulent flows such as in industrial furnaces.

The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 2. Simulations with gravitational effects

In Part 1 of this study (Ireland et al., J. Fluid Mech., vol. 796, 2016, pp. 617–658), we analysed the motion of inertial particles in isotropic turbulence in the absence of gravity using direct numerical simulation (DNS). Here, in Part 2, we introduce gravity and study its effect on single-particle and particle-pair dynamics over a wide range of flow Reynolds numbers, Froude numbers and particle Stokes numbers. The overall goal of this study is to explore the mechanisms affecting particle collisions, and to thereby improve our understanding of droplet interactions in atmospheric clouds. We find that the dynamics of heavy particles falling under gravity can be artificially influenced by the finite domain size and the periodic boundary conditions, and we therefore perform our simulations on larger domains to reduce these effects. We first study single-particle statistics that influence the relative positions and velocities of inertial particles. We see that gravity causes particles to sample the flow more uniformly and reduces the time particles can spend interacting with the underlying turbulence. We also find that gravity tends to increase inertial particle accelerations, and we introduce a model to explain that effect. We then analyse the particle relative velocities and radial distribution functions (RDFs), which are generally seen to be independent of Reynolds number for low and moderate Kolmogorov-scale Stokes numbers $St$. We see that gravity causes particle relative velocities to decrease by reducing the degree of preferential sampling and the importance of path-history interactions, and that the relative velocities have higher scaling exponents with gravity. We observe that gravity has a non-trivial effect on clustering, acting to decrease clustering at low $St$ and to increase clustering at high $St$. By considering the effect of gravity on the clustering mechanisms described in the theory of Zaichik &amp; Alipchenkov (New J. Phys., vol. 11, 2009, 103018), we provide an explanation for this non-trivial effect of gravity. We also show that when the effects of gravity are accounted for in the theory of Zaichik &amp; Alipchenkov (2009), the results compare favourably with DNS. The relative velocities and RDFs exhibit considerable anisotropy at small separations, and this anisotropy is quantified using spherical harmonic functions. We use the relative velocities and the RDFs to compute the particle collision kernels, and find that the collision kernel remains as it was for the case without gravity, namely nearly independent of Reynolds number for low and moderate $St$. We conclude by discussing practical implications of the results for the cloud physics and turbulence communities and by suggesting possible avenues for future research.