We investigate the settling speeds and root mean square (r.m.s.) velocities of inertial particles in isotropic turbulence with gravity using experiments with water droplets in air turbulence from 32 loudspeaker jets and direct numerical simulations (DNS). The dependence on particle inertia, gravity and the scales of both the smallest and largest turbulent eddies is investigated. We isolate the mechanisms of turbulence settling modification and find that the reduced settling speeds of large particles in experiments are due to nonlinear drag effects. We demonstrate using DNS that reduced settling speeds with linear drag (e.g. see Nielsen, J. Sedim. Petrol., vol. 63, 1993, pp. 835–838) only arise in artificial flows that, by design, eliminate preferential sweeping by the eddies. Gravity and inertia both reduce the particle r.m.s. velocities and falling particles are more responsive to vertical than to horizontal fluctuations. The model by Wang & Stock (J. Atmos. Sci., vol. 50, 1993, pp. 1897–1913) captures these trends

In this study, we analyse the statistics of both individual inertial particles and inertial particle pairs in direct numerical simulations of homogeneous isotropic turbulence in the absence of gravity. The effect of the Taylor microscale Reynolds number, R λ , on the particle statistics is examined over the largest range to date (from R λ = 88 to 597), at small, intermediate and large Kolmogorov-scale Stokes numbers St. We first explore the effect of preferential sampling on the single-particle statistics and find that low-St inertial particles are ejected from both vortex tubes and vortex sheets (the latter becoming increasingly prevalent at higher Reynolds numbers) and preferentially accumulate in regions of irrotational dissipation. We use this understanding of preferential sampling to provide a physical explanation for many of the trends in the particle velocity gradients, kinetic energies and accelerations at low St, which are well represented by the model of Chun et al. (J. Fluid Mech., vol. 536, 2005, pp. 219-251). As St increases, inertial filtering effects become more important, causing the particle kinetic energies and accelerations to decrease. The effect of inertial filtering on the particle kinetic energies and accelerations diminishes with increasing Reynolds number and is well captured by the models of Abrahamson (Chem. We then consider particle-pair statistics, and focus our attention on the relative velocities and radial distribution functions (RDFs) of the particles, with the aim of understanding the underlying physical mechanisms contributing to particle collisions. The relative velocity statistics indicate that preferential sampling effects are important for St 0.1 and that path-history/non-local effects become increasingly important for St 0.2. While higher-order relative velocity statistics are influenced by the increased intermittency of the turbulence at high Reynolds numbers, the lower-order relative velocity statistics are only weakly sensitive to changes in Reynolds number at low St. The Reynolds-number trends in these quantities at intermediate and large St are explained based on the influence of the available flow scales on the path-history and inertial filtering effects. We find that the RDFs peak near St of order unity, that they exhibit power-law scaling for low and intermediate St and that they are largely independent of Reynolds number for low and intermediate St. We use the model of † ) to explain the physical mechanisms responsible for these trends, and find that this model is able to capture the quantitative behaviour of the RDFs extremely well when direct numerical simulation data for the structure functions are specified, in agreement with Bragg & Collins (New J. Phys., vol. 16, 2014a, 055013). We also observe that at large St, changes in the RDF are related to changes in the scaling exponents of the relative velocity variances. The particle collision kernel closely matches that computed by Rosa et al. (New J. Phys., vol. 15, 2013, 045032) and is found to be largely insen...

In Part I of this study (Ireland et al. 2015), we analyzed the motion of inertial particles in isotropic turbulence in the absence of gravity using direct numerical simulation (DNS). Here, in Part II, we introduce gravity and study its effect on single-particle and particlepair 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 which 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 analyze 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 & Alipchenkov (2009), 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 & Alipchenkov (2009), the results compare favorably 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 cloud physics and turbulence communities and by suggesting possible avenues for future research.

In this paper we investigate both theoretically and numerically the forward in time (FIT) and backward in time (BIT) dispersion of fluid and inertial particle pairs in isotropic turbulence. Fluid particles are known to separate faster BIT than FIT in three-dimensional turbulence, and we find that inertial particles do the same.However, we find that the irreversibility in the inertial particle dispersion is in general much stronger than that for fluid particles. For example, the ratio of the BIT to FIT mean-square separation can be up to an order of magnitude larger for inertial particles than for the fluid particles. We also find that for both the inertial and fluid particles the irreversibility becomes stronger as the scale of their separation decreases. Regarding the physical mechanism for the irreversibility, we argue that whereas the irreversibility of fluid particle-pair dispersion can be understood in terms of a directional bias arising from the energy transfer process in turbulence, inertial particles experience an additional source of irreversibility arising from the non-local contribution to their velocity dynamics, a contribution which vanishes in the limit St → 0, where St is the particle Stokes number. For each given initial (final, in the backward in time case) separation r 0 there is an optimum value of St for which the dispersion irreversibility is strongest, as such particles are optimally affected by both sources of irreversibility. We derive analytical expressions for the BIT, meansquare separation of inertial particles and compare the predictions with numerical data obtained from a Re λ ≈ 580 DNS of particle-laden isotropic turbulent flow. The small-time theory, which in the dissipation range is valid for times ≤ max[Stτ η , τ η ] (where τ η is the Kolmogorov timescale), we find excellent agreement between the theoretical predictions and the DNS. The theory for long-times is in good agreement with the DNS provided that St is small enough so that the inertial particle motion at long-times may be considered as a perturbation about the fluid particle motion, a condition which would in fact be satisfied for arbitrary St at sufficiently long-times in the limit Re λ → ∞.

In this paper, we consider the physical mechanism for the clustering of inertial particles in the inertial range of isotropic turbulence. We analyze the exact, but unclosed, equation governing the radial distribution function (RDF) and compare the mechanisms it describes for clustering in the dissipation and inertial ranges. We demonstrate that in the limit Str 1, where Str is the Stokes number based on the eddy turnover timescale at separation r, the clustering in the inertial range can be understood to be due to the preferential sampling of the coarse-grained fluid velocity gradient tensor at that scale. When Str O(1) this mechanism gives way to a non-local clustering mechanism. These findings reveal that the clustering mechanisms in the inertial range are analogous to the mechanisms that we identified for the dissipation regime (see New J. Phys. 16:055013, 2014). Further, we discuss the similarities and differences between the clustering mechanisms we identify in the inertial range and the "sweep-stick" mechanism developed by Coleman & Vassilicos (Phys. Fluids 21:113301, 2009). We argue that when Str 1 the sweep-stick mechanism is equivalent to our mechanism in the inertial range if the particles are suspended in Navier-Stokes turbulence, but that the sweep-stick mechanism breaks down for Str O(1). The argument also explains why the sweep-stick mechanism is unable to predict particle clustering in kinematic simulations. We then consider the closed, model equation for the RDF given in Zaichik & Alipchenkov (Phys. Fluids. 19:113308, 2007) and use this, together with the results from our analysis, to predict the analytic form of the RDF in the inertial range for Str 1, which, unlike that in the dissipation range, is not scale-invariant. The results are in good agreement with direct numerical simulations, provided the separations are well within the inertial range.