Critical comments to results of investigations of drop collisions in turbulent clouds

One issue associated with the use of Large-Eddy Simulation (LES) to investigate the dispersion of small inertial particles in turbulent flows is the accuracy with which particle statistics and concentration can be reproduced. The motion of particles in LES fields may differ significantly from that observed in experiments or direct numerical simulation (DNS) because the force acting on the particles is not accurately estimated, due to the availability of the only filtered fluid velocity, and because errors accumulate in time leading to a progressive divergence of the trajectories. This may lead to different degrees of inaccuracy in the prediction of statistics and concentration. We identify herein an ideal subgrid correction of the a-priori LES fluid velocity seen by the particles in turbulent channel flow. This correction is computed by imposing that the trajectories of individual particles moving in filtered DNS fields exactly coincide with the particle trajectories in a DNS. In this way the errors introduced by filtering into the particle motion equations can be singled out and analyzed separately from those due to the progressive divergence of the trajectories. The subgrid correction term, and therefore the filtering error, is characterized in the present paper in terms of statistical moments. The effects of the particle inertia and of the filter type and width on the properties of the correction term are investigated.

Understanding the dispersion and the deposition of inertial particles in turbulent flows is a domain of research of utmost practical interest. With advances in computing resources, Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) have become powerful tools for the investigation of particle-laden turbulent flows with the hybrid Eulerian-Lagrangian approach playing a key role in predicting inertial particle dispersion and deposition. Computational intractability that arises due to the need of solving all the scales has restricted DNS to the very low Reynolds number turbulent flows that are not often of practical interest. LES, by solving only the large energy-containing eddies and modeling the small quasi-universal scales, is relaxed from this restriction. Thus, tackling high Reynolds number turbulent flow becomes possible. The use of large-eddy simulation has increased over the years as a promising tool to address these types of problems with the required accuracy at an affordable computing cost. In LES of dispersed turbulent multiphase flows, it has been common that tracking inertial particles in turbulent flows is carried out using only the filtered velocity field. This turned out to be inaccurate for cases dealing with very small, turbulence-responsive particles. For these cases, the timedependent velocity field seen by the inertial particles can be stochastically constructed in a Lagrangian framework. This can be achieved through the use of a stochastic diffusion process such as Langevin models.

We present an extensive numerical study of the time irreversibility of the dynamics of heavy inertial particles in three-dimensional, statistically homogeneous, and isotropic turbulent flows. We show that the probability density function (PDF) of the increment, W(τ), of a particle's energy over a time scale τ is non-Gaussian, and skewed toward negative values. This implies that, on average, particles gain energy over a period of time that is longer than the duration over which they lose energy. We call this slow gain and fast loss. We find that the third moment of W(τ) scales as τ^{3} for small values of τ. We show that the PDF of power-input p is negatively skewed too; we use this skewness Ir as a measure of the time irreversibility and we demonstrate that it increases sharply with the Stokes number St for small St; this increase slows down at St≃1. Furthermore, we obtain the PDFs of t^{+} and t^{-}, the times over which p has, respectively, positive or negative signs, i.e., the particle gains or loses energy. We obtain from these PDFs a direct and natural quantification of the slow gain and fast loss of the energy of the particles, because these PDFs possess exponential tails from which we infer the characteristic loss and gain times t_{loss} and t_{gain}, respectively, and we obtain t_{loss}<t_{gain} for all the cases we have considered. Finally, we show that the fast loss of energy occurs with greater probability in the strain-dominated region than in the vortical one; in contrast, the slow gain in the energy of the particles is equally likely in vortical or strain-dominated regions of the flow.

Direct numerical simulation is used to investigate effects of turbulent flow in the confined geometry of a face-centred cubic porous unit cell on the transport, clustering and deposition of fine particles at different Stokes numbers ( $St = 0.01, 0.1, 0.5, 1, 2$ ) and at a pore Reynolds number of 500. Particles are advanced using one-way coupling and the collision of particles with pore walls is modelled as perfectly elastic with specular reflection. Tools for studying inertial particle dynamics and clustering developed for homogeneous flows are adapted to take into account the embedded, curved geometry of the pore walls. The pattern and dynamics of clustering are investigated using the volume change of Voronoi tesselation in time to analyse the divergence and convergence of the particles. Similar to the case of homogeneous, isotropic turbulence, the cluster formation is present at large volumes, while cluster destruction is prominent at small volumes and these effects are amplified with the Stokes number. However, unlike homogeneous, isotropic turbulence, the formation of a large number of very small volumes was observed at all Stokes numbers and attributed to the collision of particles with the pore wall. Multiscale wavelet analysis of the particle number density indicates that the peak of the energy density spectrum, representative of enhanced particle clustering, shifts towards larger scales with an increase in the Stokes number. Scale-dependent skewness and flatness quantify the intermittent void and cluster distribution, with cluster formation observed at small scales for all Stokes numbers, and void regions at large scales for large Stokes numbers.

Non-spherical particles suspended in fluid flows are subject to hydrodynamic torques generated by fluid velocity gradients. For small axisymmetric particles, the most popular formulation of hydrodynamic torques is that given by Jeffery (Proc R Soc Lond A 102:161–179, 1922), which is valid for uniform shear flow in the viscous Stokes regime. In the lack of simple alternative formulations outside the Stokes regime, the Jeffery formulation has been widely applied to inertial particles in turbulent flows, where it is bound to produce inaccurate results. In this paper we quantify the statistical error incurred when the Jeffery formulation is used to study the motion of elongated axisymmetric particles under nonlinear shear flow conditions. Considering the archetypical case of prolate ellipsoidal particles in turbulent channel flow, we show that error for ellipsoids of the same length, l, as the Kolmogorov scale of the flow, $$\eta _K$$ , is indeed small (order $$1\%$$ ) but increases exponentially up to $$l \simeq 10 \eta _K$$ before becoming almost independent of elongation.

Preferential concentration of inertial particles in turbulent flow is studied by high resolution direct numerical simulations of two-dimensional turbulence. The formation of network-like regions of high particle density, characterized by a length scale which depends on the Stokes number of inertial particles, is observed. At smaller scales, the size of empty regions appears to be distributed according to a universal scaling law.

We present a model for the relative velocity of inertial particles in turbulent flows that provides new physical insight into this problem. Our general formulation shows that the relative velocity has contributions from two terms, referred to as the ‘generalized acceleration’ and ‘generalized shear’, because they reduce to the well-known acceleration and shear terms in the Saffman–Turner limit. The generalized shear term represents particles' memory of the flow velocity difference along their trajectories and depends on the inertial particle pair dispersion backward in time. The importance of this backward dispersion in determining the particle relative velocity is emphasized. We find that our model with a two-phase separation behaviour, an early ballistic phase and a later tracer-like phase, as found by recent simulations for the forward (in time) dispersion of inertial particle pairs, gives good fits to the measured relative speeds from simulations at low Reynolds numbers. In the monodisperse case with identical particles, the generalized acceleration term vanishes and the relative velocity is determined by the generalized shear term. At large Reynolds numbers, our model gives a St1/2-dependence of the relative velocity on the Stokes number St in the inertial range for both the ballistic behaviour and the Richardson separation law. This leads to the same inertial-range scaling for the two-phase separation that well fits the simulation results. Our calculations for the bidisperse case show that, with the friction timescale of one particle fixed, the relative speed as a function of the other particle's friction time has a dip when the two timescales are similar. This indicates that similar-size particles tend to have stronger velocity correlation than different ones. We find that the primary contribution at the dip, i.e. for similar particles, is from the generalized shear term, while the generalized acceleration term is dominant for particles of very different sizes. Future numerical studies are motivated to check the accuracy of the assumptions made in our model and to investigate the backward-in-time dispersion of inertial particle pairs in turbulent flows.

The distribution of inertial particles in turbulent flows is highly nonuniform and is governed by the dynamics of turbulent structures of the underlying carrier flow field which, in turn, is affected by the presence of a loading of dispersed particles. The issue is discussed here focusing on the coupling between near-bed coherent structures and suspended solid particles dynamics, in wall-bounded turbulent multiphase flows, bounded by rough boundaries. The friction Reynolds number of the unladen flow is Reτ=180 and the dispersed phase spans one order of magnitude of particle diameter. The analysis takes into account fluid-particle interaction (two-way coupling) in the frame of the Particle-Source-In-Cell (PSIC) method, using Direct Numerical Simulations (DNS) for the carrier phase coupled with Lagrangian Particle Tracking (LPT) for the dispersed phase. The effect of the wall’s roughness is taken into account modelling the elastic rebound of particles onto it, instead of using a virtual rebound model.

Interaction between turbulent structures and particles in roughened channel

Understanding and predicting the collision rate of small inertial particles in turbulent flows is of importance in many meteorological and industrial processes. For instance, predicting the time of rain initiation of warm clouds, which are known to be turbulent in their core, is still an open problem. Droplets in such clouds, which can be treated as inertial particles, are believed to grow to rain drop by coalescence due to collisions.

This paper investigates the effects of particle shape and Stokes number on the behaviour of non-spherical particles in turbulent channel flow. Although there are a number of studies concerning spherical particles in turbulent flows, most important applications occurring in process, energy, and pharmaceutical industries deal with non-spherical particles. The computation employs a unique and novel four-way coupling with the Lagrangian point-particle approach. The fluid phase at low Reynolds number (Reτ = 150) is modelled by direct numerical simulation, while particles are tracked individually. Inter-particle and particle-wall collisions are also taken into account. To explore the effects of particles on the flow turbulence, the statistics of the fluid flow such as the fluid velocity, the terms in the turbulence kinetic energy equation, the slip velocity between the two phases and velocity correlations are analysed considering ellipsoidal particles with different inertia and aspect ratio. The results of the simulations show that the turbulence is considerably attenuated, even in the very dilute regime. The reduction of the turbulence intensity is predominant near the turbulence kinetic energy peak in the near wall region, where particles preferentially accumulate. Moreover, the elongated shape of ellipsoids strengthens the turbulence attenuation. In simulations with ellipsoidal particles, the fluid-particle interactions strongly depend on the orientation of the ellipsoids. In the near wall region, ellipsoids tend to align predominantly within the streamwise (x) and wall-normal (y) planes and perpendicular to the span-wise direction, whereas no preferential orientation in the central region of the channel is observed. Important conclusions from this work include the effective viscosity of the flow is not affected, the direct dissipation by the particles is negligible, and the primary mechanism by which the particles affect the flow is by altering the turbulence structure around the turbulence kinetic energy peak.

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).

An asymptotic solution is derived for the motion of inertial particles exposed to Stokes drag in an unsteady random flow. This solution provides the finite-time Lyapunov exponents as a function of Stokes number and Lagrangian strain- and rotation-rates autocovariances. The sum of these exponents, which corresponds to a concentration-weighted divergence of particle velocity field, is considered as a measure of clustering. For inertial particles dispersed in an isotropic turbulent flow our analysis predicts maximum clustering at an intermediate Stokes number and minimal clustering at small and large Stokes numbers. Direct numerical simulations are performed for quantitative validation of our analysis, showing a reasonable agreement between the two.

Heavy particles suspended in turbulent flow possess inertia and are ejected from violent vortical structures by centrifugal forces. Once piled up along particle paths, this small-scale mechanism leads to an effective large-scale drift. This phenomenon, known as ‘turbophoresis’, causes particles to leave highly turbulent regions and migrate towards calmer regions, explaining why particles transported by non-homogeneous flows tend to concentrate near the minima of turbulent kinetic energy. It is demonstrated here that turbophoretic effects are just as crucial in statistically homogeneous flows. Although the average turbulent activity is uniform, instantaneous spatial fluctuations are responsible for inertial-range inhomogeneities in the particle distribution. Direct numerical simulations are used to probe particle accelerations, specifically how they correlate to local turbulent activity, yielding an effective coarse-grained dynamics that accounts for particle detachment from the fluid and ejection from excited regions through a space- and time-dependent non-Fickian diffusion. This leads to cast fluctuations in particle distributions in terms of a scale-dependent Péclet number ${\textit {Pe}}_\ell$ , which measures the importance of turbulent advection compared with inertial turbophoresis at a given scale $\ell$ . Multifractal statistics of energy dissipation indicate that $ {\textit {Pe}}_\ell \sim \ell ^\delta /\tau _{p}$ with $\delta \approx 0.84$ . Numerical simulations support this behaviour and emphasise the relevance of the turbophoretic Péclet number in characterising how particle distributions, including their radial distribution function, depends on $\ell$ . This approach also explains the presence of voids with inertial-range sizes, and the fact that their volumes have a non-trivial distribution with a power-law tail $p(\mathcal {V}) \propto \mathcal {V}^{-\alpha }$ , with an exponent $\alpha$ that tends to 2 as ${\textit {Pe}}_\ell \to 0$ .

The distribution of inertial particles in turbulent flows is highly nonuniform and is governed by the local dynamics of the turbulent structures of the underlying carrier flow field. In wall-bounded flows, wall roughness strongly affects the turbulent flow field, nevertheless its effects on the particle transport in two-phase turbulent flows has been still poorly investigated. The issue is discussed here by addressing direct numerical simulations of a dilute dispersion of heavy particles in a turbulent channel flow, bounded by irregular two-dimensional rough surfaces, in the one-way coupling regime.