Particle Pair Separation Research Articles

The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate

The finite size Lyapunov exponent (FSLE) has been used extensively since the late 1990s to diagnose turbulent regimes from Lagrangian experiments and to detect Lagrangian coherent structures in geophysical flows and two-dimensional turbulence. Historically, the FSLE was defined in terms of its computational method rather than via a mathematical formulation, and the behavior of the FSLE in the turbulent inertial ranges is based primarily on scaling arguments. Here, we propose an exact definition of the FSLE based on conditional averaging of the finite amplitude growth rate (FAGR) of the particle pair separation. With this new definition, we show that the FSLE is a close proxy for the inverse structural time, a concept introduced a decade before the FSLE. The (in)dependence of the FSLE on initial conditions is also discussed, as well as the links between the FAGR and other relevant Lagrangian metrics, such as the finite time Lyapunov exponent and the second-order velocity structure function.

Particle radial distribution function and relative velocity measurement in turbulence at small particle-pair separations

Abstract

New scaling laws predicting turbulent particle pair diffusion, overcoming the limitations of the prevalent Richardson–Obukhov theory

Both the evolution of particle pair separation distance l in a turbulent flow and how different length scales affect l are major unresolved challenges. The reigning theory in this topic is that of Richardson and Obukhov (R-O theory). We propose a new theory of pair diffusion in homogeneous, isotropic turbulence hypothesizing that not only structures of size l, but much larger ones also induce significant pair separation—ignored in the R-O theory. We arrive at new scaling laws for the pair diffusivity K, leading to K∼lγ where γ depends on the size of the inertial subrange: for a short inertial subrange, we find from our simulations that K∼l1.44, and for an infinite inertial subrange, we find that K∼l1.556—these relations agree closely with data. We assert that the celebrated “R-O constant” gl is neither physically meaningful nor a constant as universally assumed; our theory leads to two new physically relevant constants: GK for pair diffusivity and Gl for pair separation—which asymptote to GK≈0.73 and Gl≈0.01 at high Reynolds numbers. We find that the particle dispersion is smaller by an order of magnitude compared to R-O prediction; this is significant in many applications such as sprays, and, in particular, the spread of biological contagions (e.g., COVID19) which persist longer and drift farther compared to R-O prediction. We find that the turbulent dispersion does not depend on the fine structure timescale—a striking result which would greatly facilitate turbulent diffusion modeling.

Local anisotropy of laboratory two-dimensional turbulence affects pair dispersion

Experimental investigation of particle pair separation is conducted in two types of laboratory two-dimensional turbulence under a broad range of experimental conditions. In the range of scales corresponding to the inverse energy cascade inertial interval, the particle pair separation exhibits diffusive behaviour. The analysis of the pair velocity correlations suggests the existence of coherent bundles or clusters of non-diverging fluid particles. Such bundles are also detected using a recently developed topological tool based on the concept of braids. The bundles are observed as meandering streams whose width is determined by the turbulence forcing scale. In such locally anisotropic turbulence, the particle pair dispersion depends on the initial particle separation and on the width of the bundles.

Tracer particle dispersion around elementary flow patterns

The motion of tracer particles is kinematically simulated around three elementary flow patterns; a Burgers vortex, a shear-layer structure with coincident vortices and a node-saddle topology. These patterns are representative for their broader class of coherent structures in turbulence. Therefore, examining the dispersion in these elementary structures can improve the general understanding of turbulent dispersion at short time scales. The shear-layer structure and the node-saddle topology exhibit similar pair dispersion statistics compared to the actual turbulent flow for times up to $3{-}10\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$, where, $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ is the Kolmogorov time scale. However, oscillations are observed for the pair dispersion in the Burgers vortex. Furthermore, all three structures exhibit Batchelor’s scaling. Richardson’s scaling was observed for initial particle pair separations $r_{0}\leqslant 4\unicode[STIX]{x1D702}$ for the shear-layer topology and the node-saddle topology and was related to the formation of the particle sheets. Moreover, the material line orientation statistics for the shear-layer and node-saddle topology are similar to the actual turbulent flow statistics, up to at least $4\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$. However, only the shear-layer structure can explain the non-perpendicular preferential alignment between the material lines and the direction of the most compressive strain, as observed in actual turbulence. This behaviour is due to shear-layer vorticity, which rotates the particle sheet generated by straining motions and causes the particles to spread in the direction of compressive strain also. The material line statistics in the Burgers vortex clearly differ, due to the presence of two compressive principal straining directions as opposed to two stretching directions in the shear-layer structure and the node-saddle topology. The tetrad dispersion statistics for the shear-layer structure qualitatively capture the behaviour of the shape parameters as observed in actual turbulence. In particular, it shows the initial development towards planar shapes followed by a return to more volumetric tetrads at approximately $10\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$, which is associated with the particles approaching the vortices inside the shear layer. However, a large deviation is observed in such behaviour in the node-saddle topology and the Burgers vortex. It is concluded that the results for the Burgers vortex deviated the most from actual turbulence and the node-saddle topology dispersion exhibits some similarities, but does not capture the geometrical features associated with material lines and tetrad dispersion. Finally, the dispersion around the shear-layer structure shows many quantitative (until 2–$4\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$) and qualitative (until $20\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$) similarities to the actual turbulence.

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.

Evidence of a forward energy cascade and Kolmogorov self-similarity in submesoscale ocean surface drifter observations

Using two-point velocity and position data from the near-simultaneous release of O(100) GPS-tracked surface drifters in the northern Gulf of Mexico, we examine the applicability of classical turbulent scaling laws to upper ocean velocity fields. The dataset allows direct estimates of both velocity structure functions and the temporal evolution of the distribution of particle pair separations. On 100 m-10 km spatial scales, and time scales of order 1-10 days, all metrics of the observed surface fluctuations are consistent with standard Kolmogorov turbulence theory in an energy cascade inertial-range regime. The sign of the third-order structure function is negative and proportional to the separation distance for scales ≲10 km where local, fluctuating Rossby numbers are found to be larger than 0.1. The scale-independent energy dissipation rate, or downscale spectral flux, estimated from Kolmogorov’s 4/5th law in this regime closely matches nearby microscale dissipation measurements in the near-surface. In contrast, similar statistics derived from a like-sized set of synthetic drifters advected by purely geostrophic altimetric AVISO data agree well with Kolmogorov-Kraichnan scaling for 2D turbulence in the forward enstrophy cascade range.

Effect of anchoring energy and elastic anisotropy on spherical inclusions in a nematic liquid crystal

This paper explores how pairs of spherical particles with homeotropic (normal) surface anchoring cluster when immersed in nematic liquid crystal. By means of the Landau-de Gennes continuum theory we calculate how the equilibrium separation of a particle pair depends on the anchoring energy at the particle surface and the elastic anisotropy of the liquid crystal. We find that, for modest to strong anchoring strengths, the particle separation depends linearly on the elastic anisotropy and the inverse of the anchoring strength. Thus, the anchoring strength can be estimated by measuring the particle-pair separation.

Suppression of particle dispersion by sweeping effects in synthetic turbulence

Synthetic models of Eulerian turbulence like so-called kinematic simulations (KS) are often used as computational shortcuts for studying Lagrangian properties of turbulence. These models have been criticized by Thomson and Devenish (2005), who argued on physical grounds that sweeping decorrelation effects suppress pair dispersion in such models. We derive analytical results for Eulerian turbulence modeled by Gaussian random fields, in particular for the case with zero mean velocity. Our starting point is an exact integrodifferential equation for the particle pair separation distribution obtained from the Gaussian integration-by-parts identity. When memory times of particle locations are short, a Markovian approximation leads to a Richardson-type diffusion model. We obtain a time-dependent pair diffusivity tensor of the form K(ij)(r,t)=S(ij)(r)τ(r,t), where S(ij)(r) is the structure-function tensor and τ(r,t) is an effective correlation time of velocity increments. Crucially, this is found to be the minimum value of three times: the intrinsic turnover time τ(eddy)(r) at separation r, the overall evolution time t, and the sweeping time r/v(0) with v(0) the rms velocity. We study the diffusion model numerically by a Monte Carlo method. With inertial ranges like the largest achieved in most current KS (about 6 decades long), our model is found to reproduce the t(9/2) power law for pair dispersion predicted by Thomson and Devenish and observed in the KS. However, for much longer ranges, our model exhibits three distinct pair-dispersion laws in the inertial range: a Batchelor t(2) regime, followed by a Kraichnan-model-like t(1) diffusive regime, and then a t(6) regime. Finally, outside the inertial range, there is another t(1) regime with particles undergoing independent Taylor diffusion. These scalings are exactly the same as those predicted by Thomson and Devenish for KS with large mean velocities, which we argue hold also for KS with zero mean velocity. Our results support the basic conclusion of Thomson and Devenish (2005) that sweeping effects make Lagrangian properties of KS fundamentally differ from those of hydrodynamic turbulence for very extended inertial ranges.

ADVECTION OF PASSIVE TRACERS IN THE ATMOSPHERE: BATCHELOR SCALING

Extensive numerical experiments are performed on tracer dispersion in global reanalysis wind fields. Particle trajectories are computed both along an isobaric (500 hPa) and an isentropic (315 K) surface in a time interval of one year. Besides mean quantities such as advection of the center of mass and growth of tracer clouds, special attention is paid to the evaluation of particle pair separation dynamics. The characteristic behavior for intermediate time scales is Batchelor's dispersion along both surfaces, where the zonal extent of the tracer cloud increases linearly in time. The long-time evolution after 70–80 days exhibits a slower, diffusive dispersion (Taylor regime), in agreement with expectations. Richardson–Obukhov scaling (superdiffusion with an exponent of 3/2) could not be identified in the numerical tests. The results confirm the classical prediction by Batchelor that the initial pair-separation determines subsequent time evolution of tracers. The quantitative dependence on the initial distance differs however from the prediction of the theory.

Pair reversal in homogeneous isotropic turbulence

We show that the separation of particle pairs in the inertial subrange of homogeneous isotropic turbulence is strongly influenced by the pairs that separate quasi-diffusively. We quantify the influence of the diffusive separators by considering the probability that a pair will 'reverse' direction across a given separation i.e. its separation will decrease (before eventually increasing) and derive an analytical expression for the expected number of reversals across this separation for a quasi-one-dimensional model of relative dispersion in the inertial subrange with Gaussian turbulence. We compare this theoretical result with three different Lagrangian stochastic models in which the influence of the diffusive and ballistic separators (the latter dominated by velocity memory) can be varied by means of the value of C0, the constant of proportionality in the Lagrangian velocity structure function, which appears explicitly in Lagrangian stochastic models. We also compare these results with data from a direct numerical simulation of turbulence. The results indicate the importance of the transverse relative velocity component (i.e. the ability of pairs to rotate), which is absent in Q1D models, in determining the correct quantitative relative dispersion statistics.

Pair Dispersion in Turbulence: The Subdominant Role of Scaling

The mixing properties of turbulent flows are, at first order, related to the dynamics of separation of particle pairs. Scaling laws for the evolution in time of the mean distance between particle pairs <r(2)>(t) have been proposed since the pioneering work of Richardson. We analyze a model which shares some features with 3D experimental and numerical turbulence, and suggest that pure scaling laws are only subdominant. The dynamics is dominated by a very wide distribution of "delay times" t(d), the duration for which particle pairs remain together before their separation increases significantly. The delay time distribution is exponential for small separations and evolves towards a flat distribution at large separations. The observed <r(2)>(t) behavior is best understood as an average over separations that individually follow the Richardson-Obukhov scaling, r(2) ∝ t(3), but each only after a fluctuating time delay t(d), where t(d) is distributed uniformly.

Relative dispersion in the Liguro-Provençal basin: From sub-mesoscale to mesoscale

Relative dispersion in the Liguro-Provençal basin (a subregion of the Mediterranean Sea) is investigated using clusters of surface drifters deployed during two Marine Rapid Environment Assessment (MREA) experiments covering different months in 2007 and 2008, respectively. The clusters have initial radii of less than 1 km, or an order of magnitude below a typical deformation radius (approximately 10–20 km). The data set consists of 45 original pairs and more than 50 total pairs (including chance ones) in the spatial range between 1 and 200 km. Relative dispersion is estimated using the mean square separation of particle pairs and the Finite Scale Lyapunov Exponents (FSLEs). The two metrics show broadly consistent results, indicating in particular a clear exponential behaviour with an e-folding time scale between 0.5 and 1 days, or Lyapunov exponent λ in the range of 0.7–1 days −1. The exponential phase extends for 4–7 days in time and between 1 and 10–20 km in separation space. To our knowledge, this is only the third time that an exponential regime is observed in the world ocean from drifter data. This result suggests that relative dispersion in the Liguro-Provençal basin is nonlocal, namely controlled mainly by mesoscale dynamics, and that the effects of the sub-mesoscale motions are negligible in comparison. NCOM model results are used to complement the data and to quantify errors arising from the sparse sampling in the observations.

Sedimentation instabilities: Impact of the fluid compressibility and viscosity

The effect of an interstitial fluid on the mixing of sedimenting grains is studied numerically in a closed rectangular Hele-Shaw cell. We investigate the impact of the fluid compressibility and fluid viscosity on the dynamics and structures of the granular Rayleigh-Taylor instability. First we discuss the effect of the fluid compressibility on the initial fluid pressure evolution and on the dynamics of the particles. Here, the emerging patterns do not seem highly affected by the compressibility change studied. To characterize the patterns and motion the combined length of the particle trajectories in relation to the movement of the center of mass is analyzed, and the separation of particle pairs is measured as a function of the fluid viscosity.

Optoelectrofluidic field separation based on light-intensity gradients.

Optoelectrofluidic field separation (OEFS) of particles under light -intensity gradient (LIG) is reported, where the LIG illumination on the photoconductive layer converts the short-ranged dielectrophoresis (DEP) force to the long-ranged one. The long-ranged DEP force can compete with the hydrodynamic force by alternating current electro-osmosis (ACEO) over the entire illumination area for realizing effective field separation of particles. In the OEFS system, the codirectional illumination and observation induce the levitation effect, compensating the attenuation of the DEP force under LIG illumination by slightly floating particles from the surface. Results of the field separation and concentration of diverse particle pairs (0.82-16 mum) are well demonstrated, and conditions determining the critical radius and effective particle manipulation are discussed. The OEFS with codirectional LIG strategy could be a promising particle manipulation method in many applications where a rapid manipulation of biological cells and particles over the entire working area are of interest.

Single-particle, particle-pair, and multiparticle dispersion of fluid particles in forced stably stratified turbulence

The dispersion of fluid particles in statistically stationary stably stratified turbulence is studied by means of direct numerical simulations. Due to anisotropy of the flow, horizontal and vertical dispersion show different behavior. Single-particle dispersion in horizontal direction is similar to that in isotropic turbulence for short times, but shows a long-time growth rate proportional to t2.1±0.1, larger than the classical linear diffusion limit. In vertical direction, three successive regimes can be identified: a classical t2-regime, a plateau that scales as N−2, and a diffusion limit where dispersion is proportional to t. By forcing the flow and performing long-time simulations, we are able to observe this last regime, which was predicted but not observed before in stratified turbulence. This diffusive regime is caused by molecular diffusion of the active scalar (density). The mean squared separation of particle pairs (relative dispersion) in vertical direction shows two plateaus that are not present in isotropic turbulence. They can be associated with the characteristic layered structure of the flow. In the long-time limit again a linear regime is found as for single-particle dispersion. Pair dispersion in horizontal direction behaves similar to that in isotropic turbulence except for long times. Finally, the study of multiparticle statistics in stably stratified turbulent flows is reported. The evolution of tetrads gives an impression of the shape of particle clouds. It is found that with increasing stratification, the volume of the tetrads decreases, and they become flatter and more elongated.

Lagrangian statistics of particle pairs in homogeneous isotropic turbulence

We present a detailed investigation of the particle pair separation process in homogeneous isotropic turbulence. We use data from direct numerical simulations up to Rλ∼280 following the evolution of about two million passive tracers advected by the flow over a time span of about three decades. We present data for both the separation distance and the relative velocity statistics. Statistics are measured along the particle pair trajectories both as a function of time and as a function of their separation, i.e., at fixed scales. We compare and contrast both sets of statistics in order to gain insight into the mechanisms governing the separation process. We find very high levels of intermittency in the early stages, that is, for travel times up to order ten Kolmogorov time scales. The fixed scale statistics allow us to quantify anomalous corrections to Richardson diffusion in the inertial range of scales for those pairs that separate rapidly. It also allows a quantitative analysis of intermittency corrections for the relative velocity statistics.

Particle pair separation in kinematic simulations

The separation of pairs of particles within turbulent flow fields constructed using the 'kinematic simulation' method is explored. A consequence of the way the flow is constructed is that, in contrast to real turbulence, there is no 'sweeping' of the smaller eddies by the larger eddies. The implications of this are investigated. A simple phenoenological argument is presented which predicts that the mean-square separation of the particle pairs should grow like t 6 in kinematic simulation. Simulations support this result for the case where a large mean velocity is added to the flow to exaggerate the sweeping problem and the inertial subrange is sufficiently long. In the absence of a large mean velocity, the situation is more complex with the simple phenomenological argument failing in the parts of the flow where the velocity is much smaller than the r.m.s. velocity and where there is no sweeping problem. The separation process then follows t 6 in the bulk of the flow but follows Richardson's classical t 3 law in regions where the velocity is much smaller than the r.m.s. velocity. Because of the way the size of these regions varies in time, the resulting mean-square separation grows like t 9/2

Relative dispersion in isotropic turbulence. Part 2. A new stochastic model with Reynolds-number dependence

A new model for Lagrangian particle-pair separation in turbulent flows is developed and compared with data from direct numerical simulations (DNS) of isotropic turbulence. The model formulation emphasizes (i) non-Gaussian behaviour in Eulerian and Lagrangian statistics, (ii) the occurrence of large separation velocities, (iii) the role of straining and streaming flow structure as recognized in kinematic simulations of turbulence, and (iv) the role of conditionally averaged accelerations in stochastic modelling of turbulent relative dispersion. Previous stochastic models of relative dispersion have produced unrealistic behaviour, particularly in the dissipation subrange where viscous effects are important, which have led to questions on the adequacy of stochastic modelling. However, this failure can now be recognized as inadequate detail in formulation, which is explained and rectified in this paper. The model is quasi-one-dimensional in nature, and is focused on the statistics of particle-pair separation distance and its rate of change, referred to as the separation speed. Detailed comparisons are presented at several Reynolds numbers using the DNS database reported in a companion paper (Part 1). Up to fourth-order moments for these quantities are examined, as well as the separation-distance probability density function, which is discussed in the context of recent claims of Richardson scaling in the literature. The model is able to account for the spatial representation of straining regions as well as incompressibility of the flow, and is shown to reproduce strong non-Gaussianity and intermittency in the Lagrangian separation statistics observed in DNS. Comparisons with recent physical experiments are also remarkably consistent. This work demonstrates that stochastic models when properly formulated are effective and efficient representations of the dispersion process and this general class of models therefore possess great utility for calculations of both one-particle and two-particle dispersion. The techniques developed in this paper will facilitate such general model development.

Dispersion of mass by two-dimensional homogeneous and incompressible Çinlar flows

We consider the dispersion of passive tracers in stationary, homogeneous and incompressible Çinlar flows on the plane. The associated velocity field is generated by the superposition of eddies of various size, arrival time and location which form a Poisson point process. Our focus is on the dispersion of a tracer cloud, which is measured through the variance of its centroid and the mean of the dispersion tensor. We also study single particle dispersion and particle pair separations in conjunction with dispersion. Monte Carlo simulations of all these measures first establish the relation of the dispersion to the parameters of the flow model. Second, the physical predictions on the behavior of these measures with respect to time as well as their relationship to each other are confirmed.