Spatial Velocity Correlations Research Articles

Harmonic chain driven by active Rubin bath: transport properties and steady-state correlations

Characterizing the properties of an extended system driven by active reservoirs is a question of increasing importance. Here, we address this question in two steps. We start by investigating the dynamics of a probe particle connected to an ‘active Rubin bath’: a linear chain of overdamped run-and-tumble particles. We derive exact analytical expressions for the effective noise and dissipation kernels, acting on the probe and show that the active nature of the bath leads to a modified fluctuation–dissipation relation. In the next step, we study the properties of an activity-driven system, modelled by a chain of harmonic oscillators connected to two such active reservoirs at the two ends. We show that the system reaches a non-equilibrium stationary state (NESS), remarkably different from that generated due to a thermal gradient. We characterize this NESS by computing the kinetic temperature profile, spatial and temporal velocity correlations of the oscillators, and the average energy current flowing through the system. It turns out that the activity drive leads to the emergence of two characteristic length-scales, proportional to the activities of the reservoirs. Strong signatures of activity are also manifest in the anomalous short-time decay of the velocity autocorrelations. Finally, we find that the energy current shows a non-monotonic dependence on the activity drive and reversal in direction, corroborating previous findings.

Active fluctuations in the harmonic chain: phonons, entropons and velocity correlations

Non-equilibrium random fluctuations of a non-thermal nature are a defining feature of active matter. In this work, we study the collective excitations of active systems at high density, focusing on a one-dimensional chain of elastically coupled inertial particles, where activity is modeled by an Ornstein–Uhlenbeck process. The excitation spectrum reveals two types of fluctuations: thermally excited phonons, analogous to those in passive crystals, and entropons, which are associated with entropy production due to active forces. These fluctuations exhibit distinct properties: only entropons generate spatial velocity correlations and violate the standard fluctuation-response relation. We derive exact expressions for equal-time velocity and displacement correlations, as well as for the structure factor, identifying the contributions from both phonons and entropons. Finally, we explore the dynamical properties of these excitations through steady-state two-time correlations, such as the intermediate scattering function and mean-square displacement. Both phonon and entropon fluctuations are characterized by a long-wavelength overdamped regime and a short-wavelength underdamped regime. In the large persistence case, entropons decay more slowly than phonons, and activity generally suppresses the oscillations typical of the underdamped regime.

Inhomogeneous entropy production in active crystals with point imperfections

The presence of defects in solids formed by active particles breaks their discrete translational symmetry. As a consequence, many of their properties become space-dependent and different from those characterizing perfectly ordered structures. Motivated by recent numerical investigations concerning the nonuniform distribution of entropy production and its relation to the configurational properties of active systems, we study theoretically and numerically the spatial profile of the entropy production rate (EPR) when an active solid contains an isotopic mass defect. The theoretical study of such an imperfect active crystal is conducted by employing a perturbative analysis that considers the perfectly ordered harmonic solid as a reference system. The perturbation theory predicts a nonuniform profile of the entropy production extending over large distances from the position of the impurity. The EPR decays exponentially to its bulk value with a typical healing length that coincides with the correlation length of the spatial velocity correlations characterizing the perfect active solids in the absence of impurities. The theory is validated against numerical simulations of an active Brownian particle crystal in two dimensions with Weeks–Chandler–Andersen repulsive interparticle potential.

Stationary state of harmonic chains driven by boundary resetting

We study the nonequilibrium steady state of an ordered harmonic chain of N oscillators connected to two walls which undergo diffusive motion with stochastic resetting. The intermittent resetting of the walls effectively emulates two nonequilibrium reservoirs that exert temporally correlated forces on the boundary oscillators. These reservoirs are characterized by the diffusion constants and resetting rates of the walls. We find that, for any finite N, the velocity distribution of the bulk oscillators remains non-Gaussian, as evidenced by a non-zero bulk kurtosis that decays . We calculate the spatio-temporal correlation of the velocity of the oscillators both analytically as well as using numerical simulation. The signature of the boundary resetting is present at the bulk in terms of the two-time velocity correlation of a single oscillator and the equal-time spatial velocity correlation. For the resetting driven chain, the two-time velocity correlation decay as at the large time, and there exists a non-zero equal-time spatial velocity correlation when . A non-zero average energy current will flow through the system when the boundary walls reset to their initial positions at different rates. This average energy current can be computed exactly in the thermodynamic limit. Numerically we show that the distribution of the instantaneous energy current at the boundary is independent of the system size. However, the distribution of the instantaneous energy current in the bulk approaches a stationary distribution in the thermodynamic limit.

Modeling spatial patterns in a moving crowd of people using data-driven approach—A concept of Interplay Floor Field

On the basis of data-driven methodology, especially velocity correlations of pedestrians moving in a crowd, we have proposed a new model of pedestrian dynamics with an easy-adjustable space discretization. The model is based on Cellular Automata (CA) with an adaptive lattice and takes into account proxemics patterns among pedestrians. The proposed model uses agents located on the CA lattice, constructed using the concept of Floor Field (FF), namely, a set of gradient potential fields influencing the movement of agents. In the model, we have proposed three kinds of such fields: Static FF—responsible for navigation to agents’ Points of Interest (POIs), Wall FF—a repulsive influence with obstacles, and Interplay FF—to model agents’ volume and proxemics effects. The third field, which models mutual relations between pedestrians taking into account the rules of proxemics is an added value in the area of crowd modeling and simulation.During the creation of the proposed model, we have based on a set of experimental data provided by Jülich Supercomputing Centre and our previous analysis of spatial patterns and velocity correlations. Thus, the proposed model includes, among other things the most probable positions of the nearest neighbors in a crowd—including the shape of “the forbidden zone” around a pedestrian and the finding of rules explaining observed mutual spatial relations between neighbors in a crowd. In the described model the cell size is an adjustable parameter, with available values from 1 cm to 50 cm. It is possible with the usage of embodied agents which always occupy only one cell of Cellular Automata, actual pedestrian volume is modeled with the Interplay FF. An interesting property of the model is that changing the cell size does not involve changing the rules of the model, e.g. its transition function.

Spatial correlations and relative velocities of polydisperse droplets in homogeneous isotropic turbulence

We investigate the motions of polydisperse droplets in homogeneous and isotropic turbulence at Reynolds numbers Reλ=200–300. The emphasize is put on the parameter dependences of spatial velocity correlations (SVCs) and relative velocities (RVs) of droplets, which are relevant to particle transport and dispersion in turbulence and have been less studied in experiments. The Kolmogorov-scale Stokes number is Stp=10−1–101, and the settling parameter, i.e., the ratio of particle settling velocity to fluid velocity fluctuations, is SvL=0.5–2.0. Using high-resolution measurements, we can resolve the motions of turbulence and droplet over a wide range of scales (10−1η to 102η, η is Kolmogorov length). The parabolic behavior in droplet SVCs near the origin is observed, similar to turbulence. The droplet SVCs are smaller than turbulence for all scales and decrease with both Stp and SvL. At large scales, the droplet RVs, smaller than those of turbulence due to the inertial filtering effect, also decrease with Stp and SvL. At small scales, the path-history effect leads to larger droplet RVs than fluid RVs. Interestingly, we find RVs present a non-monotonic trend with Stp and reach a valley at Stp≈1.0. It may originate from particle clustering and preferential sweeping effects, which both prevail at Stp≈1.0. It is also found that droplet motions are less intermittent than turbulence. This is in contrast to the previous observations by simulations with the gravity effect being ignored. The intermittency of droplet RVs decreases with SvL due to the diminished droplet–turbulence interactions, and it presents opposite trends with Stp for small and large scales. Finally, the balance between the effects of path histories and turbulent structures makes the velocity statistics of droplets quasi-independent from the scale in the range of the dissipative scale (r≲5η).

Role of rotational inertia for collective phenomena in active matter.

We investigate the effect of rotational inertia on the collective phenomena of underdamped active systems and show that the increase of the moment of inertia of each particle favors non-equilibrium phase coexistence, known as motility induced phase separation, and counteracts its suppression due to translational inertia. Our conclusion is supported by a non-equilibrium phase diagram (in the plane spanned by rotational inertial time and translational inertial time) whose transition line is understood theoretically through scaling arguments. In addition, rotational inertia increases the correlation length of the spatial velocity correlations in the dense cluster. The fact that rotational inertia enhances collective phenomena, such as motility induced phase separation and spatial velocity correlations, is strongly linked to the increase of rotational persistence. Moreover, large moments of inertia induce non-monotonic temporal (cross) correlations between translational and rotational degrees of freedom truly absent in non-equilibrium systems.

Collective effects in confined active Brownian particles.

We investigate a two-dimensional system of active particles confined to a narrow annular domain. Despite the absence of explicit interactions among the velocities or the active forces of different particles, the system displays a transition from a disordered and stuck state to an ordered state of global collective motion where the particles rotate persistently clockwise or anticlockwise. We describe this behavior by introducing a suitable order parameter, the velocity polarization, measuring the global alignment of the particles' velocities along the tangential direction of the ring. We also measure the spatial velocity correlation function and its correlation length to characterize the two states. In the rotating phase, the velocity correlation displays an algebraic decay that is analytically predicted together with its correlation length, while in the stuck regime, the velocity correlation decays exponentially with a correlation length that increases with the persistence time. In the first case, the correlation (and, in particular, its correlation length) does not depend on the active force but the system size only. The global collective motion, an effect caused by the interplay between finite-size, periodicity, and persistent active forces, disappears as the size of the ring becomes infinite, suggesting that this phenomenon does not correspond to a phase transition in the usual thermodynamic sense.

Spatial velocity correlations in inertial systems of active Brownian particles.

Recently, it has been discovered that systems of active Brownian particles (APB) at high density organise their velocities into coherent domains showing large spatial structures in the velocity field. This collective behavior occurs spontaneously, i.e. is not caused by any specific interparticle force favoring the alignment of the velocities. This phenomenon was investigated in the absence of thermal noise and in the overdamped regime where inertial forces could be neglected. In this work, we demonstrate through numerical simulations and theoretical analysis that velocity alignment is a robust property of ABP and persists even in the presence of inertial forces and thermal fluctuations. We also show that a single dimensionless parameter, such as the Péclet number customarily employed in the description of self-propelled particles, is not sufficient to fully characterize this phenomenon either in the regimes of large viscosity or small mass. Indeed, the size of the velocity domains, measured through the correlation length of the spatial velocity correlation, remains constant when the swim velocity increases and decreases as the rotational diffusion becomes larger. We find that, contrary to the common belief, the spatial velocity correlation not only depends on inertia but is also non-symmetrically affected by mass and inverse viscosity variations. We conclude that in self-propelled systems, at variance with passive systems, variations in the inertial time (mass over solvent viscosity) and mass act as independent control parameters. Finally, we highlight the non-thermal nature of the spatial velocity correlations that are fairly insensitive both to solvent and active temperatures.

Three dimensional flows beneath a thin layer of 2D turbulence induced by Faraday waves

Faraday waves occur on a fluid being subject to vertical shaking. Although it is well known that form and shape of the wave pattern depend on driving amplitude and frequency, only recent studies discovered the existence of a horizontal velocity field at the surface, called Faraday flow. This flow exhibits attributes of two-dimensional turbulence and is replicated in this study. Despite the increasing attention towards the inverse energy flux in the Faraday flow and other not strictly two-dimensional (2D) systems, little is known about the velocity fields developing beneath the fluid surface. In this study, planar velocity fields are measured by means of particle image velocimetry with high spatio-temporal resolution on the water surface and at different depths below it. A sudden drop in velocity and turbulent kinetic energy is observed at half a Faraday wavelength below the surface revealing that the surface flow is the main source of turbulent fluid motion. The flow structures below the surface comprise much larger spatial scales than those on the surface leading to very long-tailed temporal and spatial velocity (auto-) correlation functions. The three-dimensionality of the flow is estimated by the compressibility, which increases strongly with depth while the divergence changes its appearance from intermittent and single events to a large scale pattern resembling 2D cut-planes of convection rolls. Our findings demonstrate that the overall fluid flow beneath the surface is highly three-dimensional and that an inverse cascade and aspects of a confined 2D turbulence can coexist with a three-dimensional flow.Graphic abstract

Compressibility Effects on Large Structures and Entrainment Length Scales in Mixing Layers

In this work, three-component planar velocity measurements are analyzed to identify the effects of compressibility on mixing layer turbulence, with a focus on the evolution of large-scale structures, dominant spatial eigenmodes, and entrainment length scales. Velocity measurements obtained via stereoscopic particle image velocimetry for five different dual-stream air planar mixing layers with convective Mach numbers of , 0.38, 0.55, 0.69, and 0.88 are analyzed. Results of two-point correlations reveal that length scales of the streamwise velocity fluctuations increase in both the streamwise and transverse directions, whereas the length scales of the transverse velocity fluctuations decrease in the transverse direction. To further investigate these trends, the spatial organization, shape, and dynamics of large-scale turbulent structures are examined via two-dimensional spatial velocity correlations, proper orthogonal decomposition, and linear stochastic estimation. These quantitative techniques support qualitative findings in the literature that report coherent, round roller structures in incompressible mixing layers becoming flattened and elongated longitudinally with increased compressibility. This evolution is likely due to a dominant pulsing motion present for the higher cases and can be linked to compressibility effects on the entrainment mechanisms. Length scales of entrainment are determined using autocorrelations of normal flow velocity components along instantaneously identified turbulent/non-turbulent interfaces and are found to decrease with increasing . This result can heuristically be interpreted as small-scale nibbling being dominant for higher , whereas larger-scale engulfment contributes more in nearly incompressible mixing layers.

Hidden velocity ordering in dense suspensions of self-propelled disks

The present paper studies the spontaneous velocity alignment and the time-intermittency of the kinetic-energy in dense phases of active matter. The dynamical properties are described by considering the spatial velocity correlations and constructing a non-equilibrium phase diagram of active force and density which explores homogeneous and inhomogeneous phases

Recirculating flow-induced anomalous transport in meandering open-channel flows

Channel meanders in rivers induce complex three-dimensional (3D) flow characteristics such as secondary flows and flow recirculation. Helical secondary flows promote transverse dispersion, and flow recirculation zones trap tracers. As a result, these meander-driven flows may cause anomalous transport manifested by unusually elevated levels of tracer concentration at early and late times of breakthrough curves (BTCs). In this study, we perform 3D numerical simulations in meandering channels across a wide range of channel sinuosity to investigate the impact of meander geometry on flow and transport. We solve 3D Reynolds-averaged Navier-Stoke equations blended with a SST k − ω turbulence model to obtain velocity and turbulence fields. We then incorporate the obtained flow fields into a Lagrangian particle tracking model to simulate solute transport. Sinuosity higher than 1.5 leads to the onset of horizontal recirculating flows along the apex outer banks, and these recirculation zones expand as sinuosity increases. The analysis of the transport simulations elucidates that the interplay between the secondary flows and recirculation zones induces anomalous transport. The helical secondary flows bring particles into the recirculation zones by promoting transverse dispersion, and the recirculating flows delay particle transport by the trapping effect. We show that the tail power-law slope and truncated time of BTCs as well as Lagrangian tortuosity distributions change dramatically with the emergence of recirculation zones. These analyses demonstrate that the recirculation zones are acting as the primary driver of anomalous transport. Finally, we successfully predict the observed anomalous transport with a Spatial Markov Model (SMM), which is an upscaled transport model that incorporates Lagrangian velocity distribution and spatial velocity correlation. The successful predictions show that the Lagrangian velocity statistics effectively capture the underlying mechanisms of anomalous transport in meandering open-channel flows.

Swarming in the Dirt: Ordered Flocks with Quenched Disorder.

The effect of quenched (frozen) disorder on the collective motion of active particles is analyzed. We find that active polar systems are far more robust against quenched disorder than equilibrium ferromagnets. Long-ranged order (a nonzero average velocity ⟨v⟩) persists in the presence of quenched disorder even in spatial dimensions d=3; in d=2, quasi-long-ranged order (i.e., spatial velocity correlations that decay as a power law with distance) occurs. In equilibrium systems, only quasi-long-ranged order in d=3 and short-ranged order in d=2 are possible. Our theoretical predictions for two dimensions are borne out by simulations.

Hydrodynamic theory of flocking in the presence of quenched disorder

The effect of quenched (frozen) orientational disorder on the collective motion of active particles is analyzed. We find that, as with annealed disorder (Langevin noise), active polar systems are far more robust against quenched disorder than their equilibrium counterparts. In particular, long ranged order (i.e., the existence of a non-zero average velocity $\langle {\bf v} \rangle$) persists in the presence of quenched disorder even in spatial dimensions $d=3$, while it is destroyed even by arbitrarily weak disorder in $d \le 4$ in equilibrium systems. Furthermore, in $d=2$, quasi-long-ranged order (i.e., spatial velocity correlations that decay as a power law with distance) occurs when quenched disorder is present, in contrast to the short-ranged order that is all that can survive in equilibrium. These predictions are borne out by simulations in both two and three dimensions.

An experimental investigation of coupled underexpanded supersonic twin-jets

High-resolution particle image velocimetry measurements of coupled underexpanded twin-jets are presented. Two nozzle pressure ratios are examined, which are selected due to a change in coupled plume mode indicated by a discontinuous jump in screech frequency. Estimates of the turbulent flow statistics, shear-layer thickness, merge point, inter-nozzle mixing, and integral length scales are provided. The higher nozzle pressure ratio case shows a strong standing-wave present in the velocity fluctuation amplitude and integral length scale. The ratios of standing, acoustic, and hydrodynamic wavelength are compared and find a close fit to Panda’s relation for screech. This indicates that screech in the twin-jet system operates with similar length-scale and frequency characteristics to single jets and provides evidence to suggest screech is an integral part of the twin-jet coupling process. Second-order spatial velocity correlation maps reveal the larger modal structure. A symmetric mode is found for the higher pressure ratio and a weakly symmetric mode for the lower. Comparison is made between where the standing-wave is present and where it is not. It is found that the standing-wave, not the shock structure, is the driver of turbulence coherence modulation near the jet. In regions that are affected only by the standing-wave, it is found that it contributes to both the turbulence intensity and coherence modulation.

Epithelial vertex models with active biochemical regulation of contractility can explain organized collective cell motility.

Collective motions of groups of cells are observed in many biological settings such as embryo development, tissue formation, and cancer metastasis. To effectively model collective cell movement, it is important to incorporate cell specific features such as cell size, cell shape, and cell mechanics, as well as active behavior of cells such as protrusion and force generation, contractile forces, and active biochemical signaling mechanisms that regulate cell behavior. In this paper, we develop a comprehensive model of collective cell migration in confluent epithelia based on the vertex modeling approach. We develop a method to compute cell-cell viscous friction based on the vertex model and incorporate RhoGTPase regulation of cortical myosin contraction. Global features of collective cell migration are examined by computing the spatial velocity correlation function. As active cell force parameters are varied, we found rich dynamical behavior. Furthermore, we find that cells exhibit nonlinear phenomena such as contractile waves and vortex formation. Together our work highlights the importance of active behavior of cells in generating collective cell movement. The vertex modeling approach is an efficient and versatile approach to rigorously examine cell motion in the epithelium.

A Kriging-based elliptic extended anisotropic model for the turbulent boundary layer wall pressure spectrum

The present study addresses the computation of the wall pressure spectrum for a turbulent boundary layer flow without pressure gradient, at high Reynolds numbers, using a new model, the Kriging-based elliptic extended anisotropic model (KEEAM). A space–time solution to the Poisson equation for the wall pressure fluctuations is used. Both the turbulence–turbulence and turbulence–mean shear interactions are taken into account. It involves the mean velocity field and space–time velocity correlations which are modelled using Reynolds stresses and velocity correlation coefficients. We propose a new model, referred to as the extended anisotropic model, to evaluate the latter in all regions of the boundary layer. This model is an extension of the simplified anisotropic model of Gavin (PhD thesis, 2002, The Pennsylvania State University, University Park, PA) which was developed for the outer part of the boundary layer. It relies on a new expression for the spatial velocity correlation function and new parameters calibrated using the direct numerical simulation results of Sillero et al. (Phys. Fluids, vol. 26, 2014, 105109). Spatial correlation coefficients are related to space–time coefficients with the elliptic model of He & Zhang (Phys. Rev. E, vol. 73, 2006, 055303). The turbulent quantities necessary for the pressure computation are obtained by Reynolds-averaged Navier–Stokes solutions with a Reynolds stress turbulence model. Then, the pressure correlations are evaluated with a self-adaptive sampling strategy based on Kriging in order to reduce the computation time. The frequency and wavenumber–frequency wall pressure spectra obtained with the KEEAM agree well with empirical models developed for turbulent boundary layer flows without pressure gradient.

Anomalous diffusion and q-Weibull velocity distributions in epithelial cell migration.

In multicellular organisms, cell motility is central in all morphogenetic processes, tissue maintenance, wound healing and immune surveillance. Hence, the control of cell motion is a major demand in the creation of artificial tissues and organs. Here, cell migration assays on plastic 2D surfaces involving normal (MDCK) and tumoral (B16F10) epithelial cell lines were performed varying the initial density of plated cells. Through time-lapse microscopy quantities such as speed distributions, velocity autocorrelations and spatial correlations, as well as the scaling of mean-squared displacements were determined. We find that these cells exhibit anomalous diffusion with q-Weibull speed distributions that evolves non-monotonically to a Maxwellian distribution as the initial density of plated cells increases. Although short-ranged spatial velocity correlations mark the formation of small cell clusters, the emergence of collective motion was not observed. Finally, simulational results from a correlated random walk and the Vicsek model of collective dynamics evidence that fluctuations in cell velocity orientations are sufficient to produce q-Weibull speed distributions seen in our migration assays.

Anomalous transport in disordered fracture networks: Spatial Markov model for dispersion with variable injection modes

We investigate tracer transport on random discrete fracture networks that are characterized by the statistics of the fracture geometry and hydraulic conductivity. While it is well known that tracer transport through fractured media can be anomalous and particle injection modes can have major impact on dispersion, the incorporation of injection modes into effective transport modeling has remained an open issue. The fundamental reason behind this challenge is that—even if the Eulerian fluid velocity is steady—the Lagrangian velocity distribution experienced by tracer particles evolves with time from its initial distribution, which is dictated by the injection mode, to a stationary velocity distribution. We quantify this evolution by a Markov model for particle velocities that are equidistantly sampled along trajectories. This stochastic approach allows for the systematic incorporation of the initial velocity distribution and quantifies the interplay between velocity distribution and spatial and temporal correlation. The proposed spatial Markov model is characterized by the initial velocity distribution, which is determined by the particle injection mode, the stationary Lagrangian velocity distribution, which is derived from the Eulerian velocity distribution, and the spatial velocity correlation length, which is related to the characteristic fracture length. This effective model leads to a time-domain random walk for the evolution of particle positions and velocities, whose joint distribution follows a Boltzmann equation. Finally, we demonstrate that the proposed model can successfully predict anomalous transport through discrete fracture networks with different levels of heterogeneity and arbitrary tracer injection modes.