We study the joint probability distributions of separation R and radial component of the relative velocity V-R of particles settling under gravity in a turbulent flow. We also obtain the moments of these distributions and analyze their anisotropy using spherical harmonics. We find that the qualitative nature of the joint distributions remains the same as no-gravity case. Distributions of V-R for fixed values of R show a power-law dependence on V-R for a range of V-R; the exponent of the power law depends on the gravity. Effects of gravity are also manifested in the following ways: (a) Moments of the distributions are anisotropic; degree of anisotropy depends on particle's Stokes number, but does not depend on R for small values of R. (b) Mean velocity of collision between two particles is decreased for particles having equal Stokes numbers but increased for particles having different Stokes numbers. For the later, collision velocity is set by the difference in their settling velocities.
Small particles in suspension in a turbulent fluid have trajectories that do not follow the pathlines of the flow exactly. We investigate the statistics of the angle of deviation φ between the particle and fluid velocities. We show that, when the effects of particle inertia are small, the probability distribution function (PDF) Pφ of this deviation angle shows a power-law region in which Pφ∼φ-4. We also find that the PDFs of the trajectory curvature κ and modulus θ of the torsion have power-law tails that scale, respectively, as Pκ∼κ-5/2, as κ→∞, and Pθ∼θ-3, as θ→∞: These exponents are in agreement with those previously observed for fluid pathlines. We propose a way to measure the complexity of heavy-particle trajectories by the number NI(t,St) of points (up until time t) at which the torsion changes sign. We present numerical evidence that nI(St)≡limt→∞NI(t,St)t∼St-Δ for large St, with Δ≃0.5.
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(tau), of a particle's energy over a time scale tau 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(tau) scales as tau(3) for small values of tau. 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 similar or equal to 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.
We obtain the probability distribution functions (PDFs) of the time that a Lagrangian tracer or a heavy inertial particle spends in vortical or strain-dominated regions of a turbulent flow, by carrying out direct numerical simulations of such particles advected by statistically steady, homogeneous, and isotropic turbulence in the forced, three-dimensional, incompressible Navier-Stokes equation. We use the two invariants, Q and R, of the velocity-gradient tensor to distinguish between vortical and strain-dominated regions of the flow and partition the Q-R plane into four different regions depending on the topology of the flow; out of these four regions two correspond to vorticity-dominated regions of the flow and two correspond to strain-dominated ones. We obtain Q and R along the trajectories of tracers and heavy inertial particles and find out the time t(pers) for which they remain in one of the four regions of the Q-R plane. We find that the PDFs of tpers display exponentially decaying tails for all four regions for tracers and heavy inertial particles. From these PDFs we extract characteristic time scales, which help us to quantify the time that such particles spend in vortical or strain-dominated regions of the flow.
We perform direct numerical simulations of a bidisperse suspension of heavy spherical particles in forced, homogeneous, and isotropic three-dimensional turbulence. We compute the joint distribution of relative particle distances and longitudinal relative velocities between particles of different inertia. For a pair of particles with small difference in their inertias we compare our results with recent theoretical predictions [Meibohm et al., Phys. Rev. E 96, 061102 (2017)] for the shape of this distribution. We also compute the moments of relative velocities as a function of particle separation and compare with the theoretical predictions. We observe good agreement. For a pair of particles that are very different from each other-one is heavy and the other one has negligible inertia-we give a theory to calculate their root-mean-square relative velocity. This theory also agrees well with the results of our simulations.
We use direct numerical simulations to calculate the joint probability density function of the relative distance R and relative radial velocity component V-R for a pair of heavy inertial particles suspended in homogeneous and isotropic turbulent flows. At small scales the distribution is scale invariant, with a scaling exponent that is related to the particle-particle correlation dimension in phase space, D-2. It was argued [K. Gustavsson and B. Mehlig, Phys. Rev. E 84, 045304 (2011); J. Turbul. 15, 34 (2014)] that the scale invariant part of the distribution has two asymptotic regimes: (1) vertical bar V-R vertical bar << R, where the distribution depends solely on R, and (2) vertical bar V-R vertical bar >> R, where the distribution is a function of vertical bar V-R vertical bar alone. The probability distributions in these two regimes are matched along a straight line: vertical bar V-R vertical bar = z*R. Our simulations confirm that this is indeed correct. We further obtain D-2 and z* as a function of the Stokes number, St. The former depends nonmonotonically on St with aminimum at about St approximate to 0.7 and the latter has only a weak dependence on St.
The rate of collision and the relative velocities of the colliding particles in turbulent flows are a crucial part of several natural phenomena, e.g. rain formation in warm clouds and planetesimal formation in protoplanetary discs. The particles are often modelled as passive, but heavy and inertial. Within this model, large relative velocities emerge due to formation of singularities (caustics) of the gradient matrix of the velocities of the particles. Using extensive direct numerical simulations of heavy particles in both two (direct and inverse cascade) and three-dimensional turbulent flows, we calculate the rate of formation of caustics, J as a function of the Stokes number (St). The best approximation to our data is J similar to exp(-C/St), in the limit St -> 0 where C is a non-universal constant. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.
We study small-scale and high-frequency turbulent fluctuations in three-dimensional flows under Fourier-mode reduction. The Navier-Stokes equations are evolved on a restricted set of modes, obtained as a projection on a fractal or homogeneous Fourier set. We find a strong sensitivity (reduction) of the high-frequency variability of the Lagrangian velocity fluctuations on the degree of mode decimation, similarly to what is already reported for Eulerian statistics. This is quantified by a tendency towards a quasi-Gaussian statistics, i.e., to a reduction of intermittency, at all scales and frequencies. This can be attributed to a strong depletion of vortex filaments and of the vortex stretching mechanism. Nevertheless, we found that Eulerian and Lagrangian ensembles are still connected by a dimensional bridge-relation which is independent of the degree of Fourier-mode decimation.
We study the collision rates of settling spheres and elongated spheroids in homogeneous, isotropic turbulence by means of direct numerical simulations aiming to understand microscale-particle encounters in oceans and lakes. We explore a range of aspect ratios and sizes relevant to the dynamics of plankton and microplastics in water environments. The results presented here confirm that collision rates between elongated particles in a quiescent fluid are more frequent than those among spherical particles in turbulence due to oblique settling. We also demonstrate that turbulence generally enhances collisions among elongated particles as compared to those expected for a random distribution of the same particles settling in a quiescent fluid, although we also find a decrease in collision rates in turbulence for particles of the highest density and moderate aspect ratios ( A = 5 ) . The increase in the collision rate due to turbulence is found to quickly decrease with aspect ratio, reach a minimum for aspect ratios approximately equal to 5, and then slowly increase again, with an increase up to 50% for the largest aspect ratios investigated. This non-monotonic trend is explained as the result of two competing effects: the increase in the surface area with aspect ratio (beneficial to increase encounter rates) and the alignment of nearby prolate particles in turbulence (reducing the probability of collision). Turbulence mixing is, therefore, partially balanced by rod alignment at high particle aspect ratios.
We present high-resolution (1024 3) simulations of super-/hypersonic isothermal hydrodynamic turbulence inside an interstellar molecular cloud (resolving scales of typically 20-100 au), including a multidisperse population of dust grains, i.e. a range of grain sizes is considered. Due to inertia, large grains (typical radius a ≳ 1.0μm) will decouple from the gas flow, while small grains (al∼ 0.1μm) will tend to better trace the motions of the gas. We note that simulations with purely solenoidal forcing show somewhat more pronounced decoupling and less clustering compared to simulations with purely compressive forcing. Overall, small and large grains tend to cluster, while intermediate-size grains show essentially a random isotropic distribution. As a consequence of increased clustering, the grain-grain interaction rate is locally elevated; but since small and large grains are often not spatially correlated, it is unclear what effect this clustering would have on the coagulation rate. Due to spatial separation of dust and gas, a diffuse upper limit to the grain sizes obtained by condensational growth is also expected, since large (decoupled) grains are not necessarily located where the growth species in the molecular gas is.
We present an overview of the statistical properties of turbulence in two-dimensional (2D) fluids. After a brief recapitulation of well-known results for statistically homogeneous and isotropic 2D fluid turbulence, we give an overview of recent progress in this field for such 2D turbulence in conducting fluids, fluids with polymer additives, binary-fluid mixtures, and superfluids; we also discuss the statistical properties of particles advected by 2D turbulent fluids.
We revisit the issue of Lagrangian irreversibility in the context of recent results [H. Xu et al., Proc. Natl. Acad. Sci. USA, 111, 7558 (2014)] on flight-crash events in turbulent flows and show how extreme events in the Eulerian dissipation statistics are related to the statistics of power fluctuations for tracer trajectories. Surprisingly, we find that particle trajectories in intense dissipation zones are dominated by energy gains sharper than energy losses, contrary to flight crashes, through a pressure-gradient driven take-off phenomenon. Our conclusions are rationalized by analyzing data from simulations of three-dimensional intermittent turbulence, as well as from nonintermittent decimated flows. Lagrangian irreversibility is found to persist even in the latter case, wherein fluctuations of the dissipation rate are shown to be relatively mild and to follow probability distribution functions with exponential tails.
We use pseudospectral direct numerical simulations to solve the three-dimensional (3D) Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model of superfluid helium. We then explore the statistical properties of inertial particles, in both coflow and counterflow superfluid turbulence (ST) in the 3D HVBK system; particle motion is governed by a generalization of the Maxey-Riley-Gatignol equations. We first characterize the anisotropy of counterflow ST by showing that there exist large vortical columns. The light particles show confined motion as they are attracted toward these columns, and they form large clusters; by contrast, heavy particles are expelled from these vortical regions. We characterize the statistics of such inertial particles in 3D HVBK ST: (1) The mean angle (SIC)(tau) between particle positions, separated by the time lag r, exhibits two different scaling regions in (a) dissipation and (b) inertial ranges, for different values of the parameters in our model; in particular, the value of (SIC)(tau), at large r, depends on the magnitude of U-ns. (2) The irreversibility of 3D HVBK turbulence is quantified by computing the statistics of energy increments for inertial particles. (3) The probability distribution function (PDF) of energy increments is of direct relevance to recent experimental studies of irreversibility in superfluid turbulence; we find, in agreement with these experiments, that, for counterflow ST, the skewness of this PDF is less pronounced than its counterparts for coflow ST or for classical fluid turbulence.
We study the spreading of viruses, such as SARS-CoV-2, by airborne aerosols, via a first-passage-time problem for Lagrangian tracers that are advected by a turbulent flow: By direct numerical simulations of the three-dimensional (3D) incompressible Navier-Stokes equation, we obtain the time t(R) at which a tracer, initially at the origin of a sphere of radius R, crosses the surface of the sphere for the first time. We obtain the probability distribution function P(R, t(R)) and show that it displays two qualitatively different behaviors: (a) for R << L-I, P(R, t(R)) has a power-law tail similar to t(R)(-alpha), with the exponent alpha = 4 and L-I the integral scale of the turbulent flow; (b) for L-I less than or similar to R, the tail of P(R, t(R)) decays exponentially. We develop models that allow us to obtain these asymptotic behaviors analytically. We show how to use P(R, t(R)) to develop social-distancing guidelines for the mitigation of the spreading of airborne aerosols with viruses such as SARS-CoV-2.