Fluid particle accelerations in fully developed turbulence

Turbulent Flows Are Not Uniformly Multifractal

Understanding turbulence rests delicately on the conflict between Kolmogorov's 1941 theory of nonintermittent, space-filling energy dissipation characterized by a unique scaling exponent and the overwhelming evidence to the contrary of intermittency, multiscaling, and multifractality. Strangely, multifractality is not typically envisioned as a local flow property, variations in which might be clues exposing inroads into the fundamental unsolved issues of anomalous dissipation and finite time blowup. We present a simple construction of local multifractality and find that much of the dissipation field remains surprisingly monofractal à la Kolmogorov. Multifractality appears as small islands in this calm sea, its strength growing logarithmically with the local fluctuations in energy dissipation-a seemingly universal feature. These results suggest new ways to understand how singularities could arise and provide a fresh perspective on anomalous dissipation and intermittency. The simplicity and adaptability of our approach also holds great promise in applications ranging from climate sciences to medical data analysis.

Scaling of Acceleration Statistics in High Reynolds Number Turbulence

The scaling of acceleration statistics in turbulence is examined by combining data from the literature with new data from well-resolved direct numerical simulations of isotropic turbulence, significantly extending the Reynolds number range. The acceleration variance at higher Reynolds numbers departs from previous predictions based on multifractal models, which characterize Lagrangian intermittency as an extension of Eulerian intermittency. The disagreement is even more prominent for higher-order moments of the acceleration. Instead, starting from a known exact relation, we relate the scaling of acceleration variance to that of Eulerian fourth-order velocity gradient and velocity increment statistics. This prediction is in excellent agreement with the variance data. Our Letter highlights the need for models that consider Lagrangian intermittency independent of the Eulerian counterpart.

Clustering of fast gyrotactic particles in low-Reynolds-number flow

Systems of particles in turbulent flows exhibit clustering where particles form patches in certain regions of space. Previous studies have shown that motile particles accumulate inside the vortices and in downwelling regions, while light and heavy non-motile particles accumulate inside and outside the vortices, respectively. While strong clustering is generated in regions of high vorticity, clustering of motile particles is still observed in fluid flows where vortices are short-lived. In this study, we investigate the clustering of fast swimming particles in a low-Reynolds-number turbulent flow and characterize the probability distributions of particle speed and acceleration and their influence on particle clustering. We simulate gyrotactic swimming particles in a cubic system with homogeneous and isotropic turbulent flow. Here, the swimming velocity explored is relatively faster than what has been explored in other reports. The fluid flow is produced by conducting a direct numerical simulation of the Navier-Stokes equation. In contrast with the previous results, our results show that swimming particles can accumulate outside the vortices, and clustering is dictated by the swimming number and is invariant with the stability number. We have also found that highly clustered particles are sufficiently characterized by their acceleration, where the increase in the acceleration frequency distribution of the most clustered particles suggests a direct influence of acceleration on clustering. Furthermore, the acceleration of the most clustered particles resides in acceleration values where a cross-over in the acceleration PDFs are observed, an indicator that particle acceleration generates clustering. Our findings on motile particles clustering can be applied to understanding the behavior of faster natural or artificial swimmers.

The CoPhyLab comet-simulation chamber

The Comet Physics Laboratory (CoPhyLab) is an international research program to study the physical properties of cometary analog materials under simulated space conditions. The project is dedicated to studying, with the help of multiple instruments and the different expertise and background from the different partners, the physics of comets, including the processes inside cometary nuclei, the activity leading to the ejection of dust and gas, and the sub-surface and surface evolution of cometary nuclei when exposed to solar illumination. CoPhyLab will provide essential information on the formation and evolution of comets and insights into the origins of primitive Solar System bodies. To this end, we constructed a new laboratory that hosts several small-scale experiments and a large-scale comet-simulation chamber (L-Chamber). This chamber has been designed and constructed to host ice-dust samples with a diameter of up to 250 mm and a variable height between 100 and 300 mm. The cometary-analog samples will be kept at temperatures below 120 K and pressures around 10^-6 mbar to ensure cometary-like conditions. In total, 14 different scientific instruments are attached to the L-Chamber to study the temporal evolution of the physical properties of the sample under different insolation conditions. Due to the implementation of a scale inside the L-Chamber that can measure weight changes of the samples with high precision, the cooling system is mechanically decoupled from the sample holder and cooling of the samples occurs by radiation only. The constructed chamber allows us to conduct uninterrupted experiments at low temperatures and pressures up to several weeks.

Turbulence explains the accelerations of an eagle in natural flight

Turbulent winds and gusts fluctuate on a wide range of timescales from milliseconds to minutes and longer, a range that overlaps the timescales of avian flight behavior, yet the importance of turbulence to avian behavior is unclear. By combining wind speed data with the measured accelerations of a golden eagle (Aquila chrysaetos) flying in the wild, we find evidence in favor of a linear relationship between the eagle's accelerations and atmospheric turbulence for timescales between about 1/2 and 10 s. These timescales are comparable to those of typical eagle behaviors, corresponding to between about 1 and 25 wingbeats, and to those of turbulent gusts both larger than the eagle's wingspan and smaller than large-scale atmospheric phenomena such as convection cells. The eagle's accelerations exhibit power spectra and intermittent activity characteristic of turbulence and increase in proportion to the turbulence intensity. Intermittency results in accelerations that are occasionally several times stronger than gravity, which the eagle works against to stay aloft. These imprints of turbulence on the bird's movements need to be further explored to understand the energetics of birds and other volant life-forms, to improve our own methods of flying through ceaselessly turbulent environments, and to engage airborne wildlife as distributed probes of the changing conditions in the atmosphere.

Experimental observation of the elastic range scaling in turbulent flow with polymer additives

A minute amount of long-chain flexible polymer dissolved in a turbulent flow can drastically change flow properties, such as reducing the drag and enhancing mixing. One fundamental riddle is how these polymer additives interact with the eddies of different spatial scales existing in the turbulent flow and, in turn, alter the turbulence energy transfer. Here, we show how turbulent kinetic energy is transferred through different scales in the presence of the polymer additives. In particular, we observed experimentally the emerging of a previously unidentified scaling range, referred to as the elastic range, where increasing amount of energy is transferred by the elasticity of the polymers. In addition, the existence of the elastic range prescribes the scaling of high-order velocity statistics. Our findings have important implications to many turbulence systems, such as turbulence in plasmas or superfluids where interaction between turbulent eddies and other nonlinear physical mechanisms are often involved.

Flow of Spatiotemporal Turbulentlike Random Fields

We study the Lagrangian trajectories of statistically isotropic, homogeneous, and stationary divergence-free spatiotemporal random vector fields. We design this advecting Eulerian velocity field such that it gets asymptotically rough and multifractal, both in space and time, as it is demanded by the phenomenology of turbulence at infinite Reynolds numbers. We then solve numerically the flow equations for a differentiable version of this field. We observe that trajectories get also rough, characterized by nearly the same Hurst exponent as the one of our prescribed advecting field. Moreover, even when considering the simplest situation of the advection by a fractional Gaussian field, we evidence in the Lagrangian framework additional intermittent corrections. The present approach involves properly defined random fields, and asks for a rigorous treatment that would explain our numerical findings and deepen our understanding of this long lasting problem.

Higher-order statistics based multifractal predictability measures for anisotropic turbulence and the theoretical limits of aviation weather forecasting

Theoretical predictability measures of turbulent atmospheric flows are essential in estimating how realistic the current storm-scale strategic forecast skill expectations are. Atmospheric predictability studies in the past have usually neglected intermittency and anisotropy, which are typical features of atmospheric flows, rendering their application to the storm-scale weather regime ineffective. Furthermore, these studies are frequently limited to second-order statistical measures, which do not contain information about the rarer, more severe, and, therefore, more important (from a forecasting and mitigation perspective) weather events. Here we overcome these rather severe limitations by proposing an analytical expression for the theoretical predictability limits of anisotropic multifractal fields based on higher-order autocorrelation functions. The predictability limits are dependent on the order of statistical moment (q) and are smaller for larger q. Since higher-order statistical measures take into account rarer events, such more extreme phenomena are less predictable. While spatial anisotropy of the fields seems to increase their predictability limits (making them larger than the commonly expected eddy turnover times), the ratio of anisotropic to isotropic predictability limits is independent of q. Our results indicate that reliable storm-scale weather forecasting with around 3 to 5 hours lead time is theoretically possible.

Implicit atomistic viscosities in smoothed dissipative particle dynamics

We apply a standard nonequilibrium dynamics microscopic analysis of transport coefficients to the smoothed dissipative particle dynamics (SDPD) method of steady-shear flow conditions. Extending the research of Ellero et al. [Phys. Rev. E 82, 046702 (2010)PRESCM1539-375510.1103/PhysRevE.82.046702] for smoothed particle hydrodynamics (SPH), we focus, in particular, on velocity and acceleration statistics and on mean-density phenomena. Implicit and explicit fluctuations affect non-Gaussian statistics and effective viscosities whereas only explicit fluctuations affect large-scale dissipation through the fluctuation-dissipation relation. SDPD facilitates the simulation of mesoscopic systems as the resolution scale is defined by the scaling of the random fluctuations. In the kinetic regime, SDPD recovers the behavior of SPH. In the diffusive regime, non-Gaussian behavior occurs, in contrast to SPH. We observe the formation of isotropic randomly oriented structures with high density which are related to the magnitude of thermal fluctuations. It is demonstrated that SDPD produces non-Gaussian acceleration PDF corresponding to that of a turbulent flow field.

Turbulent mixing: A perspective

Mixing of initially distinct substances plays an important role in our daily lives as well as in ecological and technological worlds. From the continuum point of view, which we adopt here, mixing is complete when the substances come together across smallest flow scales determined in part by molecular mechanisms, but important stages of the process occur via the advection of substances by an underlying flow. We know how smooth flows enable mixing but less well the manner in which a turbulent flow influences it; but the latter is the more common occurrence on Earth and in the universe. We focus here on turbulent mixing, with more attention paid to the postmixing state than to the transient process of initiation. In particular, we examine turbulent mixing when the substance is a scalar (i.e., characterized only by the scalar property of its concentration), and the mixing process does not influence the flow itself (i.e., the scalar is "passive"). This is the simplest paradigm of turbulent mixing. Within this paradigm, we discuss how a turbulently mixed state depends on the flow Reynolds number and the Schmidt number of the scalar (the ratio of fluid viscosity to the scalar diffusivity), point out some fundamental aspects of turbulent mixing that render it difficult to be addressed quantitatively, and summarize a set of ideas that help us appreciate its physics in diverse circumstances. We consider the so-called universal and anomalous features and summarize a few model studies that help us understand them both.

Persistent accelerations disentangle Lagrangian turbulence

Particles in turbulence frequently encounter extreme accelerations between extended periods of quiescence. The occurrence of extreme events is closely related to the intermittent spatial distribution of intense flow structures such as vorticity filaments. This mixed history of flow conditions leads to very complex particle statistics with a pronounced scale dependence, which presents one of the major challenges on the way to a non-equilibrium statistical mechanics of turbulence. Here, we introduce the notion of persistent Lagrangian acceleration, quantified by the squared particle acceleration coarse-grained over a viscous time scale. Conditioning Lagrangian particle data from simulations on this coarse-grained acceleration, we find remarkably simple, close-to-Gaussian statistics for a range of Reynolds numbers. This opens the possibility to decompose the complex particle statistics into much simpler sub-ensembles. Based on this observation, we develop a comprehensive theoretical framework for Lagrangian single-particle statistics that captures the acceleration, velocity increments as well as single-particle dispersion.

Particles in turbulence, as encountered in the atmosphere or the oceans, experience strongly varying local flow conditions over time. Bentkamp et al. show that this statistical complexity can be broken down into simpler parts, allowing for insights into the space-time structure of turbulent flows.

V-ONSET (Vertical Octagonal Noncorrosive Stirred Energetic Turbulence): A vertical water tunnel with a large energy dissipation rate to study bubble/droplet deformation and breakup in strong turbulence

A vertical water tunnel facility has been constructed to study the dynamics of turbulent multiphase flow. The new system features several unique designs that allow us to study bubble deformation and breakup in strong turbulence: (i) The mean flow can be adjusted to balance the rising velocity of buoyant bubbles/droplets so that they can stay in the view area for an extended period of time. (ii) Turbulence is generated and controlled using a 3D-printed jet array that can fire 88 random high-speed momentum jets with the individual jet velocity of up to 12 m/s. This component allows us to attain turbulence with a high energy dissipation rate (≥0.1 m²/s³), which is orders-of-magnitude higher than most of the existing turbulent multiphase flow facilities. (iii) Turbulence generated in the test section is nearly homogeneous and isotropic, and the turbulent fluctuations are also decoupled from the mean flow. The resulting turbulence intensity can be varied between 20% and 80% with the speed of the mean flow at around 0.2 m/s. (iv) This system has an octagonal test section that allows six cameras to image and reconstruct the 3D shape of deforming bubbles/droplets in turbulence. The same set of cameras was also used for tracking tracers in the surrounding turbulent flow. Both the reconstruction and particle tracking were completed using our in-house codes that were parallelized to run on high-performance computing clusters efficiently.

Langevin dynamics encapsulate the microscopic and emergent macroscopic properties of midge swarms

In contrast to bird flocks, fish schools and animal herds, midge swarms maintain cohesion but do not possess global order. High-speed imaging techniques are now revealing that these swarms have surprising properties. Here, I show that simple models found on the Langevin equation are consistent with this wealth of recent observations. The models predict correctly that large accelerations, exceeding 10 g, will be common and they predict correctly the coexistence of core condensed phases surrounded by dilute vapour phases. The models also provide new insights into the influence of environmental conditions on swarm dynamics. They predict that correlations between midges increase the strength of the effective force binding the swarm together. This may explain why such correlations are absent in laboratory swarms but present in natural swarms which contend with the wind and other disturbances. Finally, the models predict that swarms have fluid-like macroscopic mechanical properties and will slosh rather than slide back and forth after being abruptly displaced. This prediction offers a promising avenue for future experimentation that goes beyond current quasi-static testing which has revealed solid-like responses.

Extended 3D-PTV for direct measurements of Lagrangian statistics of canopy turbulence in a wind tunnel

Direct estimation of Lagrangian turbulence statistics is essential for the proper modeling of dispersion and transport in highly obstructed canopy flows. However, Lagrangian flow measurements demand very high rates of data acquisition, resulting in bottlenecks that prevented the estimation of Lagrangian statistics in canopy flows hitherto. We report on a new extension to the 3D Particle Tracking Velocimetry (3D-PTV) method, featuring real-time particle segmentation that outputs centroids and sizes of tracer particles and performed on dedicated hardware during high-speed digital video acquisition from multiple cameras. The proposed extension results in four orders of magnitude reduction in data transfer rate that enables to perform substantially longer experimental runs, facilitating measurements of convergent statistics. The extended method is demonstrated through an experimental wind tunnel investigation of the Lagrangian statistics in a heterogeneous canopy flow. We observe that acceleration statistics are affected by the mean shear at the top of the canopy layer and that Lagrangian particle dispersion at small scales is dominated by turbulence in the wake of the roughness elements. This approach enables to overcome major shortcomings from Eulerian-based measurements which rely on assumptions such as the Taylor's frozen turbulence hypothesis, which is known to fail in highly turbulent flows.

Gyrotactic phytoplankton in laminar and turbulent flows: A dynamical systems approach

Gyrotactic algae are bottom heavy, motile cells whose swimming direction is determined by a balance between a buoyancy torque directing them upwards and fluid velocity gradients. Gyrotaxis has, in recent years, become a paradigmatic model for phytoplankton motility in flows. The essential attractiveness of this peculiar form of motility is the availability of a mechanistic description which, despite its simplicity, revealed predictive, rich in phenomenology, easily complemented to include the effects of shape, feedback on the fluid and stochasticity (e.g., in cell orientation). In this review we consider recent theoretical, numerical and experimental results to discuss how, depending on flow properties, gyrotaxis can produce inhomogeneous phytoplankton distributions on a wide range of scales, from millimeters to kilometers, in both laminar and turbulent flows. In particular, we focus on the phenomenon of gyrotactic trapping in nonlinear shear flows and in fractal clustering in turbulent flows. We shall demonstrate the usefulness of ideas and tools borrowed from dynamical systems theory in explaining and interpreting these phenomena.

Turbulent particle pair diffusion: A theory based on local and non-local diffusional processes

A re-appraisal of the Richardson's 1926 dataset [Richardson, L. F. Proc. Roy. Soc. Lond. A 100, 709-737, (1926)] displays an unequivocal non-local scaling for the pair diffusion coefficient, [Formula: see text], quite different to the previously assumed locality scaling law [Formula: see text], where σl is the pair separation. Consequently, the foundations of turbulent pair diffusion theory are re-examined here and it is shown that pair diffusion is governed by both local and non-local diffusional processess inside the inertial subrange. In the context of generalised energy spectra, E(k) ∼ k-p for 1 < p ≤ 3, the new theory predicts two non-Richardson regimes depending on the size of the inertial subrange: (1) in the limit of asymptotically infinite subrange, non-local scaling laws is obtained, [Formula: see text], with γ intermediate between the purely local and the purely non-local scalings, i.e. (1 + p)/2 < γ ≤ 2; and (2) for finite (short) inertial subrange, local scaling laws are obtained, [Formula: see text]. The theory features a novel mathematical approach expressing the pair diffusion coefficient through a Fourier integral decomposition.

Non-Linear Diffusion and Power Law Properties of Heterogeneous Systems: Application to Financial Time Series

In this work, we show that it is possible to obtain important ubiquitous physical characteristics when an aggregation of many systems is taken into account. We discuss the possibility of obtaining not only an anomalous diffusion process, but also a Non-Linear diffusion equation, that leads to a probability distribution, when using a set of non-Markovian processes. This probability distribution shows a power law behavior in the structure of its tails. It also reflects the anomalous transport characteristics of the ensemble of particles. This ubiquitous behavior, with a power law in the diffusive transport and the structure of the probability distribution, is related to a fast fluctuating phenomenon presented in the noise parameter. We discuss all the previous results using a financial time series example.

Dispersion of Air Bubbles in Isotropic Turbulence

Bubbles play an important role in the transport of chemicals and nutrients in many natural and industrial flows. Their dispersion is crucial to understanding the mixing processes in these flows. Here we report on the dispersion of millimetric air bubbles in a homogeneous and isotropic turbulent flow with a Taylor Reynolds number from 110 to 310. We find that the mean squared displacement (MSD) of the bubbles far exceeds that of fluid tracers in turbulence. The MSD shows two regimes. At short times, it grows ballistically (∝τ^{2}), while at larger times, it approaches the diffusive regime where the MSD∝τ. Strikingly, for the bubbles, the ballistic-to-diffusive transition occurs one decade earlier than for the fluid. We reveal that both the enhanced dispersion and the early transition to the diffusive regime can be traced back to the unsteady wake-induced motion of the bubbles. Further, the diffusion transition for bubbles is not set by the integral timescale of the turbulence (as it is for fluid tracers and microbubbles), but instead, by a timescale of eddy crossing of the rising bubbles. The present findings provide a Lagrangian perspective towards understanding mixing in turbulent bubbly flows.

Microbubbles and Microparticles are Not Faithful Tracers of Turbulent Acceleration

We report on the Lagrangian statistics of acceleration of small (sub-Kolmogorov) bubbles and tracer particles with Stokes number St≪1 in turbulent flow. At a decreasing Reynolds number, the bubble accelerations show deviations from that of tracer particles; i.e., they deviate from the Heisenberg-Yaglom prediction and show a quicker decorrelation despite their small size and minute St. Using direct numerical simulations, we show that these effects arise due the drift of these particles through the turbulent flow. We theoretically predict this gravity-driven effect for developed isotropic turbulence, with the ratio of Stokes to Froude number or equivalently the particle drift velocity governing the enhancement of acceleration variance and the reductions in correlation time and intermittency. Our predictions are in good agreement with experimental and numerical results. The present findings are relevant to a range of scenarios encompassing tiny bubbles and droplets that drift through the turbulent oceans and the atmosphere. They also question the common usage of microbubbles and microdroplets as tracers in turbulence research.

Influence of compressibility on the Lagrangian statistics of vorticity-strain-rate interactions

The objective of this study is to investigate the influence of compressibility on Lagrangian statistics of vorticity and strain-rate interactions. The Lagrangian statistics are extracted from "almost" time-continuous data sets of direct numerical simulations of compressible decaying isotropic turbulence by employing a cubic spline-based Lagrangian particle tracker. We study the influence of compressibility on Lagrangian statistics of alignment in terms of compressibility parameters-turbulent Mach number, normalized dilatation-rate, and flow topology. In comparison to incompressible turbulence, we observe that the presence of compressibility in a flow field weakens the alignment tendency of vorticity toward the largest strain-rate eigenvector. Based on the Lagrangian statistics of alignment conditioned on dilatation and topology, we find that the weakened tendency of alignment observed in compressible turbulence is because of a special group of fluid particles that have an initially negligible dilatation-rate and are associated with stable-focus-stretching topology.

Centripetal focusing of gyrotactic phytoplankton

A suspension of gyrotactic microalgae Chlamydomonas augustae swimming in a cylindrical water vessel in solid-body rotation is studied. Our experiments show that swimming algae form an aggregate around the axis of rotation, whose intensity increases with the rotation speed. We explain this phenomenon by the centripetal orientation of the swimming direction towards the axis of rotation. This centripetal focusing is contrasted by diffusive fluxes due to stochastic reorientation of the cells. The competition of the two effects lead to a stationary distribution, which we analytically derive from a refined mathematical model of gyrotactic swimmers. The temporal evolution of the cell distribution, obtained via numerical simulations of the stochastic model, is in quantitative agreement with the experimental measurements in the range of parameters explored.

Turbulent transport with intermittency: Expectation of a scalar concentration

Scalar transport by turbulent flows is best described in terms of Lagrangian parcel motions. Here we measure the Eulerian distance travel along Lagrangian trajectories in a simple point vortex flow to determine the probabilistic impulse response function for scalar transport in the absence of molecular diffusion. As expected, the mean squared Eulerian displacement scales ballistically at very short times and diffusively for very long times, with the displacement distribution at any given time approximating that of a random walk. However, significant deviations in the displacement distributions from Rayleigh are found. The probability of long distance transport is reduced over inertial range time scales due to spatial and temporal intermittency. This can be modeled as a series of trapping events with durations uniformly distributed below the Eulerian integral time scale. The probability of long distance transport is, on the other hand, enhanced beyond that of the random walk for both times shorter than the Lagrangian integral time and times longer than the Eulerian integral time. The very short-time enhancement reflects the underlying Lagrangian velocity distribution, while that at very long times results from the spatial and temporal variation of the flow at the largest scales. The probabilistic impulse response function, and with it the expectation value of the scalar concentration at any point in space and time, can be modeled using only the evolution of the lowest spatial wave number modes (the mean and the lowest harmonic) and an eddy based constrained random walk that captures the essential velocity phase relations associated with advection by vortex motions. Preliminary examination of Lagrangian tracers in three-dimensional homogeneous isotropic turbulence suggests that transport in that setting can be similarly modeled.

Structure of sheared and rotating turbulence: Multiscale statistics of Lagrangian and Eulerian accelerations and passive scalar dynamics

The acceleration statistics of sheared and rotating homogeneous turbulence are studied using direct numerical simulation results. The statistical properties of Lagrangian and Eulerian accelerations are considered together with the influence of the rotation to shear ratio, as well as the scale dependence of their statistics. The probability density functions (pdfs) of both Lagrangian and Eulerian accelerations show a strong and similar dependence on the rotation to shear ratio. The variance and flatness of both accelerations are analyzed and the extreme values of the Eulerian acceleration are observed to be above those of the Lagrangian acceleration. For strong rotation it is observed that flatness yields values close to three, corresponding to Gaussian-like behavior, and for moderate and vanishing rotation the flatness increases. Furthermore, the Lagrangian and Eulerian accelerations are shown to be strongly correlated for strong rotation due to a reduced nonlinear term in this case. A wavelet-based scale-dependent analysis shows that the flatness of both Eulerian and Lagrangian accelerations increases as scale decreases, which provides evidence for intermittent behavior. For strong rotation the Eulerian acceleration is even more intermittent than the Lagrangian acceleration, while the opposite result is obtained for moderate rotation. Moreover, the dynamics of a passive scalar with gradient production in the direction of the mean velocity gradient is analyzed and the influence of the rotation to shear ratio is studied. Concerning the concentration of a passive scalar spread by the flow, the pdf of its Eulerian time rate of change presents higher extreme values than those of its Lagrangian time rate of change. This suggests that the Eulerian time rate of change of scalar concentration is mainly due to advection, while its Lagrangian counterpart is only due to gradient production and viscous dissipation.

Lagrangian statistics in weakly forced two-dimensional turbulence

Measurements of Lagrangian single-point and multiple-point statistics in a quasi-two-dimensional stratified layer system are reported. The system consists of a layer of salt water over an immiscible layer of Fluorinert and is forced electromagnetically so that mean-squared vorticity is injected at a well-defined spatial scale ri. Simultaneous cascades develop in which enstrophy flows predominately to small scales whereas energy cascades, on average, to larger scales. Lagrangian correlations and one- and two-point displacements are measured for random initial conditions and for initial positions within topological centers and saddles. Some of the behavior of these quantities can be understood in terms of the trapping characteristics of long-lived centers, the slow motion near strong saddles, and the rapid fluctuations outside of either centers or saddles. We also present statistics of Lagrangian velocity fluctuations using energy spectra in frequency space and structure functions in real space. We compare with complementary Eulerian velocity statistics. We find that simultaneous inverse energy and enstrophy ranges present in spectra are not directly echoed in real-space moments of velocity difference. Nevertheless, the spectral ranges line up well with features of moment ratios, indicating that although the moments are not exhibiting unambiguous scaling, the behavior of the probability distribution functions is changing over short ranges of length scales. Implications for understanding weakly forced 2D turbulence with simultaneous inverse and direct cascades are discussed.

Characterization of intermittency in zooplankton behaviour in turbulence

We consider Lagrangian velocity differences of zooplankters swimming in still water and in turbulence. Using cumulants, we quantify the intermittency properties of their motion recorded using three-dimensional particle tracking velocimetry. Copepods swimming in still water display an intermittent behaviour characterized by a high probability of small velocity increments, and by stretched exponential tails. Low values arise from their steady cruising behaviour while heavy tails result from frequent relocation jumps. In turbulence, we show that at short time scales, the intermittency signature of active copepods clearly differs from that of the underlying flow, and reflects the frequent relocation jumps displayed by these small animals. Despite these differences, we show that copepods swimming in still and turbulent flow belong to the same intermittency class that can be modelled by a log-stable model with non-analytical cumulant generating function. Intermittency in swimming behaviour and relocation jumps may enable copepods to display oriented, collective motion under strong hydrodynamic conditions and thus, may contribute to the formation of zooplankton patches in energetic environments.

Wake-Driven Dynamics of Finite-Sized Buoyant Spheres in Turbulence

Particles suspended in turbulent flows are affected by the turbulence and at the same time act back on the flow. The resulting coupling can give rise to rich variability in their dynamics. Here we report experimental results from an investigation of finite-sized buoyant spheres in turbulence. We find that even a marginal reduction in the particle's density from that of the fluid can result in strong modification of its dynamics. In contrast to classical spatial filtering arguments and predictions of particle models, we find that the particle acceleration variance increases with size. We trace this reversed trend back to the growing contribution from wake-induced forces, unaccounted for in current particle models in turbulence. Our findings highlight the need for improved multiphysics based models that account for particle wake effects for a faithful representation of buoyant-sphere dynamics in turbulence.

Intermittent Lagrangian velocities and accelerations in three-dimensional porous medium flow

Intermittency of Lagrangian velocity and acceleration is a key to understanding transport in complex systems ranging from fluid turbulence to flow in porous media. High-resolution optical particle tracking in a three-dimensional (3D) porous medium provides detailed 3D information on Lagrangian velocities and accelerations. We find sharp transitions close to pore throats, and low flow variability in the pore bodies, which gives rise to stretched exponential Lagrangian velocity and acceleration distributions characterized by a sharp peak at low velocity, superlinear evolution of particle dispersion, and double-peak behavior in the propagators. The velocity distribution is quantified in terms of pore geometry and flow connectivity, which forms the basis for a continuous-time random-walk model that sheds light on the observed Lagrangian flow and transport behaviors.

Bursts in discontinuous Aeolian saltation

Close to the onset of Aeolian particle transport through saltation we find in wind tunnel experiments a regime of discontinuous flux characterized by bursts of activity. Scaling laws are observed in the time delay between each burst and in the measurements of the wind fluctuations at the fluid threshold Shields number θ_c. The time delay between each burst decreases on average with the increase of the Shields number until sand flux becomes continuous. A numerical model for saltation including the wind-entrainment from the turbulent fluctuations can reproduce these observations and gives insight about their origin. We present here also for the first time measurements showing that with feeding it becomes possible to sustain discontinuous flux even below the fluid threshold.

Long-range ordering of turbulent stresses in two-dimensional flow

Using filter-space techniques, we study the spatial structure of the turbulent stress that couples motion on different length scales in a quasi-two-dimensional laboratory flow. As the length scale increases, we observe the appearance of long-range, system-spanning spatial order of this stress, even though the flow field remains disordered. Suggestively, this ordering occurs only in the range of scales over which we find net inverse energy transfer to larger scales. However, we find that a field built from wave vectors with random phases also displays ordering, suggesting that at least some of the ordering we observe is purely kinematic. Our results help to clarify the role played by geometric alignment in the turbulent energy cascade and highlight the importance of the scale-dependent rate of strain in the energy-transfer process.

Multicolor particle shadow accelerometry

This paper describes the extension of multicolor particle shadow velocimetry (CPSV) to the measurement of local acceleration in an Eulerian frame of reference. A validation experiment was conducted on a pendulous disk undergoing unsteady rigid body rotation. Angular velocity and acceleration profiles by CPSA are presented along with a comparison to recordings by an accelerometer mounted on the pendulum. CPSA is also demonstrated in a fully-developed turbulent pipe flow. Profiles of standard deviation of the local acceleration in the near wall region (0< y + <75) are compared to similar measurements by Christensen and Adrian. A favorable comparison is found between CPSA and particle image accelerometry (PIA). The effect of acceleration time delay, or the time between two velocity estimates, on local acceleration estimates is discussed.

Flight-crash events in turbulence

The statistical properties of turbulence differ in an essential way from those of systems in or near thermal equilibrium because of the flux of energy between vastly different scales at which energy is supplied and at which it is dissipated. We elucidate this difference by studying experimentally and numerically the fluctuations of the energy of a small fluid particle moving in a turbulent fluid. We demonstrate how the fundamental property of detailed balance is broken, so that the probabilities of forward and backward transitions are not equal for turbulence. In physical terms, we found that in a large set of flow configurations, fluid elements decelerate faster than accelerate, a feature known all too well from driving in dense traffic. The statistical signature of rare "flight-crash" events, associated with fast particle deceleration, provides a way to quantify irreversibility in a turbulent flow. Namely, we find that the third moment of the power fluctuations along a trajectory, nondimensionalized by the energy flux, displays a remarkable power law as a function of the Reynolds number, both in two and in three spatial dimensions. This establishes a relation between the irreversibility of the system and the range of active scales. We speculate that the breakdown of the detailed balance characterized here is a general feature of other systems very far from equilibrium, displaying a wide range of spatial scales.

Experimental investigation of a local recirculation photobioreactor for mass cultures of photosynthetic microorganisms

The present work deals with the experimental fluid mechanics analysis of a wavy-bottomed cascade photobioreactor, to characterize the extent and period of recirculatory and straight-flowing streams establishing therein as a function of reactor inclination and liquid flow rate. The substream characterization via Feature Tracking (FT) showed that a local recirculation zone establishes in each vane only at inclinations ≤6° and that its location changes from the lower (≤3°) to the upper part of each vane (6°). A straight-flowing stream flows opposite (above or below) the local recirculation stream. The recirculation time ranges from 0.86 s to 0.23 s, corresponding, respectively, to the minimum flow rate at the minimum inclination and to the maximum flow rate at the maximum inclination where recirculation was observed. The increase of photosynthetic activity, resulting from the entailed "flash effect", was estimated to range between 102 and 113% with respect to equivalent tubular and bubble column photobioreactors.

Turbulent fluid acceleration generates clusters of gyrotactic microorganisms

The motility of microorganisms is often biased by gradients in physical and chemical properties of their environment, with myriad implications on their ecology. Here we show that fluid acceleration reorients gyrotactic plankton, triggering small-scale clustering. We experimentally demonstrate this phenomenon by studying the distribution of the phytoplankton Chlamydomonas augustae within a rotating tank and find it to be in good agreement with a new, generalized model of gyrotaxis. When this model is implemented in a direct numerical simulation of turbulent flow, we find that fluid acceleration generates multifractal plankton clustering, with faster and more stable cells producing stronger clustering. By producing accumulations in high-vorticity regions, this process is fundamentally different from clustering by gravitational acceleration, expanding the range of mechanisms by which turbulent flows can impact the spatial distribution of active suspensions.

Macroscopic quantum-type potentials in theoretical systems biology

We review in this paper the use of the theory of scale relativity and fractal space-time as a tool particularly well adapted to the possible development of a future genuine systems theoretical biology. We emphasize in particular the concept of quantum-type potentials, since, in many situations, the effect of the fractality of space—or of the underlying medium—can be reduced to the addition of such a potential energy to the classical equations of motion. Various equivalent representations—geodesic, quantum-like, fluid mechanical, stochastic—of these equations are given, as well as several forms of generalized quantum potentials. Examples of their possible intervention in high critical temperature superconductivity and in turbulence are also described, since some biological processes may be similar in some aspects to these physical phenomena. These potential extra energy contributions could have emerged in biology from the very fractal nature of the medium, or from an evolutive advantage, since they involve spontaneous properties of self-organization, morphogenesis, structuration and multi-scale integration. Finally, some examples of applications of the theory to actual biological-like processes and functions are also provided.

Design, implementation and application of distributed order PI control

In this paper, a series of distributed order PI controller design methods are derived and applied to the robust control of wheeled service robots, which can tolerate more structural and parametric uncertainties than the corresponding fractional order PI control. A practical discrete incremental distributed order PI control strategy is proposed basing on the discretization method and the frequency criterions, which can be commonly used in many fields of fractional order system, control and signal processing. Besides, an auto-tuning strategy and the genetic algorithm are applied to the distributed order PI control as well. A number of experimental results are provided to show the advantages and distinguished features of the discussed methods in fairways.

Acceleration of passive tracers in compressible turbulent flow

In compressible turbulence at high Reynolds and Mach numbers, shocklets emerge as a new type of flow structure in addition to intense vortices as in incompressible turbulence. Using numerical simulation of compressible homogeneous isotropic turbulence, we conduct a Lagrangian study to explore the effects of shocklets on the dynamics of passive tracers. We show that shocklets cause very strong intermittency and short correlation time of tracer acceleration. The probability density function of acceleration magnitude exhibits a -2.5 power-law scaling in the high compression region. Through a heuristic model, we demonstrate that this scaling is directly related to the statistical behavior of strong negative velocity divergence, i.e., the local compression. Tracers experience intense acceleration near shocklets, and most of them are decelerated, usually with large curvatures in their trajectories.

Emergent dynamics of laboratory insect swarms

Collective animal behaviour occurs at nearly every biological size scale, from single-celled organisms to the largest animals on earth. It has long been known that models with simple interaction rules can reproduce qualitative features of this complex behaviour. But determining whether these models accurately capture the biology requires data from real animals, which has historically been difficult to obtain. Here, we report three-dimensional, time-resolved measurements of the positions, velocities, and accelerations of individual insects in laboratory swarms of the midge Chironomus riparius. Even though the swarms do not show an overall polarisation, we find statistical evidence for local clusters of correlated motion. We also show that the swarms display an effective large-scale potential that keeps individuals bound together, and we characterize the shape of this potential. Our results provide quantitative data against which the emergent characteristics of animal aggregation models can be benchmarked.

Particle and particle pair dispersion in turbulence modeled with spatially and temporally correlated stochastic processes

In this paper we present a model for modeling the diffusion and relative dispersion of particles in homogeneous isotropic turbulence. We use a Heisenberg-like Hamiltonian to incorporate spatial correlations between fluid particles, which are modeled by stochastic processes correlated in time. We are able to reproduce the ballistic regime in the mean square displacement of single particles and the transition to a normal diffusion regime for long times. For the dispersion of particle pairs we find a t2 dependence of the mean square separation at short times and a t dependence for long ones. For intermediate times indications for a Richardson t3 law are observed in certain situations. Finally, the influence of inertia of real particles on the dispersion is investigated.

Statistical description of turbulent dispersion

We derive a comprehensive statistical model for dispersion of passive or almost passive admixture particles such as fine particulate matter, aerosols, smoke, and fumes in turbulent flow. The model rests on the Markov limit for particle velocity. It is in accordance with the asymptotic structure of turbulence at large Reynolds number as described by Kolmogorov. The model consists of Langevin and diffusion equations in which the damping and diffusivity are expressed by expansions in powers of the reciprocal Kolmogorov constant C_{0}. We derive solutions of O(C_{0}^{0}) and O(C_{0}^{-1}). We truncate at O(C_{0}^{-2}) which is shown to result in an error of a few percentages in predicted dispersion statistics for representative cases of turbulent flow. We reveal analogies and remarkable differences between the solutions of classical statistical mechanics and those of statistical turbulence.

Multicanonical distribution: statistical equilibrium of multiscale systems

A multicanonical formalism is introduced to describe the statistical equilibrium of complex systems exhibiting a hierarchy of time and length scales, where the hierarchical structure is described as a set of nested "internal heat reservoirs" with fluctuating "temperatures." The probability distribution of states at small scales is written as an appropriate averaging of the large-scale distribution (the Boltzmann-Gibbs distribution) over these effective internal degrees of freedom. For a large class of systems the multicanonical distribution is given explicitly in terms of generalized hypergeometric functions. As a concrete example, it is shown that generalized hypergeometric distributions describe remarkably well the statistics of acceleration measurements in Lagrangian turbulence.

Analysis of intermittency in under-resolved smoothed-particle-hydrodynamics direct numerical simulations of forced compressible turbulence

We perform three-dimensional under-resolved direct numerical simulations of forced compressible turbulence using the smoothed particle hydrodynamics (SPH) method and investigate the Lagrangian intermittency of the resulting hydrodynamic fields. The analysis presented here is motivated by the presence of typical stretched tails in the probability density function (PDF) of the particle accelerations previously observed in two-dimensional SPH simulations of uniform shear flow [Ellero et al., Phys. Rev. E 82, 046702 (2010)]. In order to produce a stationary isotropic compressible turbulent state, the real-space stochastic forcing method proposed by Kida and Orszag is applied, and the statistics of particle quantities are evaluated. We validate our scheme by checking the behavior of the energy spectrum in the supersonic case where the expected Burgers-like scaling is obtained. By discretizing the continuum equations along fluid particle trajectories, the SPH method allows us to extract Lagrangian statistics in a straightforward fashion without the need for extra tracer particles. In particular, Lagrangian PDF of the density, particle accelerations as well as their Lagrangian structure functions and local scaling exponents are analyzed. The results for low-order statistics of Lagrangian intermittency in compressible turbulence demonstrate the implicit subparticle-scale modeling of the SPH discretization scheme.

Lagrangian acceleration of passive tracers in statistically steady rotating turbulence

The statistical properties of the Lagrangian acceleration vector of passive tracers in statistically steady rotating turbulence is studied by particle tracking velocimetry. Direct effects of the background rotation are the suppression of high-acceleration events parallel to the (vertical) rotation axis, the enhancement of high-acceleration events for the horizontal acceleration, and the strong amplification of the autocorrelation of the acceleration component perpendicular to both the rotation vector Ω and local velocity vector u. The autocorrelation of the acceleration component in the plane set up by Ω and u is only mildly enhanced.

Stochastic rolling of a rigid sphere in weak adhesive contact with a soft substrate

We study the rolling motion of a small solid sphere on a fibrillated rubber substrate in an external field in the presence of a Gaussian noise. From the nature of the drift and the evolution of the displacement fluctuation of the ball, it is evident that the rolling is controlled by a complex non-linear friction at a low velocity and a low noise strength (K), but by a linear kinematic friction at a high velocity and a high noise strength. This transition from a non-linear to a linear friction control of motion can be discerned from another experiment in which the ball is subjected to a periodic asymmetric vibration in conjunction with a random noise. Here, as opposed to that of a fixed external force, the rolling velocity decreases with the strength of the noise suggesting a progressive fluidization of the interface. A state (K) and rate (V) dependent friction model is able to explain both the evolution of the displacement fluctuation as well as the sigmoidal variation of the drift velocity with K. This research sets the stage for studying friction in a new way, in which it is submitted to a noise and then its dynamic response is studied using the tools of statistical mechanics. Although more works would be needed for a fuller realization of the above-stated goal, this approach has the potential to complement direct measurements of friction over several decades of velocities and other state variables. It is striking that the non-Gaussian displacement statistics as observed with the stochastic rolling is similar to that of a colloidal particle undergoing Brownian motion in contact with a soft microtubule.

Dynamic multiscaling in two-dimensional fluid turbulence

We obtain, by extensive direct numerical simulations, time-dependent and equal-time structure functions for the vorticity, in both quasi-Lagrangian and Eulerian frames, for the direct-cascade regime in two-dimensional fluid turbulence with air-drag-induced friction. We show that different ways of extracting time scales from these time-dependent structure functions lead to different dynamic-multiscaling exponents, which are related to equal-time multiscaling exponents by different classes of bridge relations; for a representative value of the friction we verify that, given our error bars, these bridge relations hold.