We provide evidence that the small-scale statistics of the acceleration of a test particle in high-Reynolds number Lagrangian turbulence is correctly described by Tsallis statistics with entropic index q=3/2. We present theoretical arguments why Tsallis statistics can naturally arise in Lagrangian turbulence and why at the smallest scales q=3/2 is relevant. A generalized Heisenberg-Yaglom formula is derived from the nonextensive model.
The advection and mixing of a scalar quantity by fluid flow is an important problem in engineering and natural sciences. If the fluid is turbulent, the statistics of the passive scalar exhibit complex behavior. This paper is concerned with two Lagrangian scalar turbulence models based on the recent fluid deformation model that can be shown to reproduce the statistics of passive scalar turbulence for a range of Reynolds numbers. For these models, we demonstrate how events of extreme passive scalar gradients can be recovered by computing the instanton, i.e., the saddle-point configuration of the…
Inpired by recent measurements of the velocity and acceleration statistics of Lagrangian tracer particles embedded in a turbulent quantum liquid we propose a new superstatistical model for the dynamics of tracer particles in quantum turbulence. Our model consists of random sequences S/N/S/... where the particle spends some time in the superfluid (S) and some time in the normal fluid (N). This model leads to a superposition of power law distributions generated in the superfluid and Gaussian distributions in the normal liquid, in excellent agreement with experimental measurements. We include mem…
A study of the relationship between Lagrangian statistics and flow topology in fluid turbulence is presented. The topology is characterized using the Weiss criterion that provides a simplified tool to partition the flow into topologically different regions: elliptic (vortex dominated), hyperbolic (deformation dominated), and intermediate (turbulent background). The flow corresponds to forced two-dimensional Navier-Stokes turbulence in doubly periodic and circular bounded domains with non-slip boundary conditions. In the double periodic domain, the probability density function (pdf) of the Weis…
We present a numerical investigation of two-dimensional decaying turbulence in the Lagrangian framework. Focusing on single particle statistics, we investigate Lagrangian trajectories in a freely evolving turbulent velocity field. The dynamical evolution of the tracer particles is strongly dominated by the emergence and evolution of coherent structures. For a statistical analysis we focus on the Lagrangian acceleration as a central quantity. For more geometrical aspects we investigate the curvature along the trajectories. We find strong signatures for self-similar universal behavior.
Mean-field-based Lagrangian framework is developed for the fluid turbulence theory. The space- time vector flow is naturally introduced from the mean velocity, which provides the Lagrangian picture based on the mean field in totally equivalent manner to that of the instantaneous field. The proposed framework is applied, on one hand, to one-point closure model, yielding an objective expression of the turbulence visco-elastic effect. Application to two-point closure, on the other hand, is also discussed, where natural extension of known Lagrangian correlation is discovered on the basis of extend…
We present a fluid dynamics video which illustrates the Lagrangian aspects of local heat transfer in turbulent Rayleigh-Benard convection. The data are obtained from a direct numerical simulation.
The angle between subsequent particle displacement increments is evaluated as a function of the timelag in isotropic turbulence. It is shown that the evolution of this angle contains two well-defined power-laws, reflecting the multi-scale dynamics of high-Reynolds number turbulence. The proba-bility density function of the directional change is shown to be self-similar and well approximated by an analytically derived model assuming Gaussianity and independence of the velocity and the Lagrangian acceleration.
We consider the dynamics of small tracer particles in turbulent quantum liquids. The complicated interaction processes of vortex filaments, the quantum constraints on vorticity and the varying influence of both the superfluid and the normal fluid on the tracer particle effectively lead to a superstatistical Langevin-like model that in a certain approximation can be solved analytically. An analytic expression for the PDF of velocity v of the tracer particle is derived that exhibits not only the experimentally observed $v^-3$ tails but also the correct behavior near the center of the distributio…
The Lagrangian and Eulerian transversal velocity structure functions of fully developed fluid turbulence are found basing on the Navier-Stokes equation. The structure functions are shown to obey the scaling relations inside the inertial range. The scaling exponents are calculated analytically without using dimensional considerations. The obtained values are in a very good agreement with recent numerical and experimental data.
We consider a superstatistical dynamical model for the 3-d movement of a Lagrangian tracer particle embedded in a high-Reynolds number turbulent flow. The analytical model predictions are in excellent agreement with recent experimental data for flow between counter-rotating disks. In particular, we calculate the Lagrangian scaling exponents zeta_j for our system, and show that they agree well with the measured exponents reported in [X. Hu et al., PRL 96, 114503 (2006)]. Moreover, the model correctly predicts the shape of velocity difference and acceleration probability densities, the fast deca…
We study the correlation function and mean linear response function of the velocity Fourier mode of statistically steady-state, homogeneous and isotropic turbulence in the Eulerian and Lagrangian coordinates through direct numerical simulation (DNS). As the Lagrangian velocity, we here adopt Kraichnan's Lagrangian history framework where Lagrangian particles are labelled with current positions and their velocity are measured at some time before. This Lagrangian velocity is numerically calculated with a method known as passive vector method. Our first goal is to study relation between the corre…
In Lagrangian turbulence one is faced with the puzzle that 2D Navier-Stokes flows are nearly as intermittent as in three dimensions although no intermittency is present in the inverse cascade in 2D Eulerian turbulence. In addition, an inertial range is very difficult to detect and it is questionable whether it exists at all. Here, we investigate the transition of Eulerian to Lagrangian probability density functions (PDFs) which leads to a new type of Lagrangian structure function. This possesses an extended inertial range similar to the case of tracer particles in a frozen turbulent velocity f…
A detailed comparison between data from experimental measurements and numerical simulations of Lagrangian velocity structure functions in turbulence is presented. By integrating information from experiments and numerics, a quantitative understanding of the velocity scaling properties over a wide range of time scales and Reynolds numbers is achieved. The local scaling properties of the Lagrangian velocity increments for the experimental and numerical data are in good quantitative agreement for all time lags. The degree of intermittency changes when measured close to the Kolmogorov time scales o…
We report on probability-density-functions (PDF) of the mass density in numerical simulations of highly compressible hydrodynamic flows and the corresponding structure formation of Lagrangian particles advected by the flows. Numerical simulations were performed with $512^3$ collocation points and 2 million tracer particles integrated over several dynamical times. We propose a connection between the PDF of the Lagrangian tracer particles and the predicted log-normal distribution of the density fluctuations in isothermal systems.
We report on simulations of two-dimensional turbulence in the inverse energy cascade regime. Focusing on the statistics of Lagrangian tracer particles, scaling behavior of the probability density functions of velocity fluctuations is investigated. The results are compared to the three-dimensional case. In particular an analysis in terms of compensated cumulants reveals the transition from a strong non-Gaussian behavior with large tails to Gaussianity. The reported computation of correlation functions for the acceleration components sheds light on the underlying dynamics of the tracer particles…
The Lagrangian velocity structure functions in the inertial range of fully developed fluid turbulence are derived basing on the Navier-Stokes equations. For time $τ$ much smaller than the correlation time, the structure functions are shown to obey the scaling relations $K_n(τ)\propto τ^{ζ_n}$. The scaling exponents $ζ_n$ are calculated analytically. The obtained values are in amazing agreement with the unique experimental results of the Bodenschatz group \cite{Bod2}. New notion -- the Lagrangian position structure functions $R_n(τ)$ is introduced. All the $R_n$ of the order $n>3$ are shown to …
New aspects of turbulence are uncovered if one considers flow motion from the perspective of a fluid particle (known as the Lagrangian approach) rather than in terms of a velocity field (the Eulerian viewpoint). Using a new experimental technique, based on the scattering of ultrasounds, we have obtained a direct measurement of particle velocities, resolved at all scales, in a fully turbulent flow. It enables us to approach intermittency in turbulence from a dynamical point of view and to analyze the Lagrangian velocity fluctuations in the framework of random walks. We find experimentally that …
The Recent Fluid Deformation Closure (RFDC) model of lagrangian turbulence is recast in path-integral language within the framework of the Martin-Siggia-Rose functional formalism. In order to derive analytical expressions for the velocity-gradient probability distribution functions (vgPDFs), we carry out noise renormalization in the low-frequency regime and find approximate extrema for the Martin-Siggia-Rose effective action. We verify, with the help of Monte Carlo simulations, that the vgPDFs so obtained yield a close description of the single-point statistical features implied by the origina…
For the phenomenological description of magnetohydrodynamic turbulence competing models exist, e.g. Boldyrev [Phys.Rev.Lett. \textbf{96}, 115002, 2006] and Gogoberidze [Phys.Plas. \textbf{14}, 022304, 2007], which predict the same Eulerian inertial-range scaling of the turbulent energy spectrum although they employ fundamentally different basic interaction mechanisms. {A relation is found that links} the Lagrangian frequency spectrum {with} the autocorrelation timescale of the turbulent fluctuations, $τ_\mathrm{ac}$, and the associated cascade timescale, $τ_{\mathrm{cas}}$. Thus, the Lagrangia…
We show by direct numerical simulation (DNS) that the Lagrangian cross correlation of velocity gradients in homogeneous isotropic turbulence increases at short times, whereas its auto-correlation decreases. Kinematic considerations allow to show that two invariants of the turbulent velocity field determine the short-time velocity gradient correlations. In order to get a more intuitive understanding of the dynamics for longer times, heuristic models are proposed involving the combined action of local shear and rotation. These models quantitatively reproduce the effects and disentangle the diffe…
Using the Lagrangian transport of momentum, the Reynolds shear stress can be expressed in terms of basic turbulence parameters. In this view, the Reynolds stress gradient represents the lateral transport of streamwise momentum, balanced by the u2 transport, pressure and shear force terms in the momentum equation. We extend this formalism to other turbulence parameters such as diagonal components of the Reynolds stress, and also to more complex flows (boundary layer flows with adverse pressure gradients and pipe flows with swirl). Data from direct numerical simulations (DNS) are used to validat…
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…
Based on direct numerical simulations with point-like inertial particles, with Stokes numbers, $\textrm{St}=0, 0.5$, $3$, and $6$, transported by homogeneous and isotropic turbulent flows, we present in this letter for the first time evidence for the existence of Markov property in Lagrangian turbulence. We show that the Markov property is valid for a finite step size larger than a Stokes number-dependent Einstein-Markov coherence time scale. This enables the description of multi-scale statistics of Lagrangian particles by Fokker-Planck equations, which can be embedded in an interdisciplinary …
Using time-resolved measurements of the orientation of rigid inertial fibers in a turbulence-tank, we investigate the autocorrelation of their tumbling rate. The correlation time ($τ_d$) is well predicted by Kolmogorov inertial-range scaling based on the fiber length (L) when the fiber inertia can be neglected. For inertial fibers, we propose a simple model considering fiber inertia (measured by a tumbling Stokes number) and a viscous torque which accurately predicts both the correlation time and the variance of the tumbling rate.
We study the effect of different forcing functions and of the local gradient Richardson number Rig on the vertical mixing of Lagrangian tracers in stably stratified turbulence under the Boussinesq approximation, and present a wave and continuous-time random walk model for single- and two- particle vertical dispersion. The model consists of a random superposition of linear waves with their amplitude based on the observed Lagrangian spectrum of vertical velocity, and a random walk process to capture overturning that depends on the statistics of Rig among other Eulerian quantities. The model is i…
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 auto-correlation of the acceleration component perpendicular to both the rotation vector Ω and local velocity vector u. The auto-correlation of the acceleration component in the plan…
Lagrangian studies of the local temperature mixing and heat transport in turbulent Rayleigh-Benard convection are presented, based on three-dimensional direct numerical simulations. Contrary to vertical pair distances, the temporal growth of lateral pair distances agrees with the Richardson law, but yields a smaller Richardson constant due to correlated pair motion in plumes. Our results thus imply that Richardson dispersion is also found in anisotropic turbulence. We find that extremely large vertical accelerations appear less frequently than lateral ones and are not connected with rising or …
The Lagrangian velocity statistics of dissipative drift-wave turbulence are investigated. For large values of the adiabaticity (or small collisionality), the probability density function of the Lagrangian acceleration shows exponential tails, as opposed to the stretched exponential or algebraic tails, generally observed for the highly intermittent acceleration of Navier-Stokes turbulence. This exponential distribution is shown to be a robust feature independent of the Reynolds number. For small adiabaticity, algebraic tails are observed, suggesting the strong influence of point-vortex-like dyn…
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 acquisiti…
Lagrangian properties obtained from a Particle Tracking Velocimetry experiment in a turbulent flow at intermediate Reynolds number are presented. Accurate sampling of particle trajectories is essential in order to obtain the Lagrangian structure functions and to measure intermittency at small temporal scales. The finiteness of the measurement volume can bias the results significantly. We present a robust way to overcome this obstacle. Despite no fully developed inertial range we observe strong intermittency at the scale of dissipation. The multifractal model is only partially able to reproduce…
In this paper, the approach for investigation of asymptotic ($Re\to \infty$) scaling exponents of Eulerian structure functions (J. Schumacher et al, New. J. of Physics {\bf 9}, 89 (2007)) is generalized to studies of Lagrangian structure functions in turbulence. The novel "bridging relation" based on the derived expression for the fluctuating, moment-order - dependent dissipation time $τ_{η,n}$ enabled us to calculate scaling exponents ($κ_{n}$) of the moments of Lagrangian velocity differences $S_{n,L}(τ)=\bar{(u(t+τ)-u(t))^{n}}\propto τ^{κ_{n}}$ in a good agreement with experimental and nume…
Cellular suspensions such as dense bacterial flows exhibit a turbulence-like phase under certain conditions. We study this phenomenon of "active turbulence" statistically by using numerical tools. Following Wensink et al. [Proc. Natl. Acad. Sci. U.S.A. 109, 14308 (2012)], we model active turbulence by means of a generalized Navier-Stokes equation. Two-point velocity statistics of active turbulence, both in the Eulerian and the Lagrangian frame, is explored. We characterize the scale-dependent features of two-point statistics in this system. Furthermore, we extend this statistical study with me…
We present a formal connection between Lagrangian and Eulerian velocity increment distributions which is applicable to a wide range of turbulent systems ranging from turbulence in incompressible fluids to magnetohydrodynamic turbulence. For the case of the inverse cascade regime of two-dimensional turbulence we numerically estimate the transition probabilities involved in this connection. In this context we are able to directly identify the processes leading to strongly non-Gaussian statistics for the Lagrangian velocity increments.
The current claim by Grebenev et al. [J. Phys. A: Math. Theor. 52, 335501 (2019)], namely that the inviscid and unclosed 2D Lundgren-Monin-Novikov (LMN) equations on a zero-vorticity Lagrangian path admit conformal invariance, is based on a flawed and misleading analysis published earlier by Grebenev et al. (2017). All false results and conclusions made before in the Eulerian picture were now extended by Grebenev et al. (2019) to the Lagrangian picture. Although we have already commented on these errors and consistently refuted their previous study (Frewer & Khujadze, 2018), we deem it necessa…
The distribution of particle accelerations in turbulence is intermittent, with non-Gaussian tails that are quite different for light and heavy particles. In this article we analyse a closure scheme for the acceleration fluctuations of light and heavy inertial particles in turbulence, formulated in terms of Lagrangian correlation functions of fluid tracers. We compute the variance and the flatness of inertial particle accelerations and we discuss their dependency on the Stokes number. The closure incorporates effects induced by the Lagrangian correlations along the trajectories of fluid tracers…
The Lagrangian view of passive scalar turbulence has recently produced interesting results and interpretations. Innovations in theory, experiments, simulations and data analysis of Lagrangian turbulence are reviewed here in brief. Part of the review is closely related to the so-called Kraichnan model for the advection of the passive scalar in synthetic turbulence. Possible implications for a better understanding of the passive scalar mixing in Navier-Stokes turbulence are also discussed.
We define and characterize multi-time Lagrangian structure functions using data stemming from two swirling flows with mean flow and turbulent fluctuations: A Taylor-Green numerical flow, and a von Kármán laboratory experiment. Data is obtained from numerical integration of tracers in the former case, and from three-dimensional particle tracking velocimetry measurements in the latter. Multi-time statistics are shown to decrease the contamination of large scales in the inertial range scaling. A time scale at which contamination from the mean flow becomes dominant is identified, with this scale s…
Self-consistent closure theory for passive-scalar turbulence has been developed on the basis of the Hessian of the scalar field. As a primitive indicator of spatial structure of the scalar, we employ the Hessian into the core of the theory to properly characterize the time scale intrinsic to the scalar field itself. The resultant closure model is now endowed with several realistic features, i.e. the scale-locality of the interscale interaction, the detailed conservation, and the memory-fading effect. Applying the current theory to the inertial-convective range eventually leads to self-consiste…
The behavior of the second-order Lagrangian structure functions on state-of-the-art numerical data both in two and three dimensions is studied. On the basis of a phenomenological connection between Eulerian space-fluctuations and the Lagrangian time-fluctuations, it is possible to rephrase the Kolmogorov $4/5$-law into a relation predicting the linear (in time) scaling for the second order Lagrangian structure function. When such a function is directly observed on current experimental or numerical data, it does not clearly display a scaling regime. A parameterization of the Lagrangian structur…
We report a detailed study of Eulerian and Lagrangian statistics from high resolution Direct Numerical Simulations of isotropic weakly compressible turbulence. Reynolds number at the Taylor microscale is estimated to be around 600. Eulerian and Lagrangian statistics is evaluated over a huge data set, made by $1856^3$ spatial collocation points and by 16 million particles, followed for about one large-scale eddy turn over time. We present data for Eulerian and Lagrangian Structure functions up to ten order. We analyse the local scaling properties in the inertial range and in the viscous range. …
Turbulent compressible flows are traditionally simulated using explicit time integrators applied to discretized versions of the Navier-Stokes equations. However, the associated Courant-Friedrichs-Lewy condition severely restricts the maximum time step size. Exploiting the Lagrangian nature of the Boltzmann equation's material derivative, we now introduce a feasible three-dimensional semi-Lagrangian lattice Boltzmann method (SLLBM), which circumvents this restriction. While many lattice Boltzmann methods for compressible flows were restricted to two dimensions due to the enormous number of disc…
We introduce a simple variational principle for coherent material vortices in two-dimensional turbulence. Vortex boundaries are sought as closed stationary curves of the averaged Lagrangian strain. Solutions to this problem turn out to be mathematically equivalent to photon spheres around black holes in cosmology. The fluidic photon spheres satisfy explicit differential equations whose outermost limit cycles are optimal Lagrangian vortex boundaries. As an application, we uncover super-coherent material eddies in the South Atlantic, which yield specific Lagrangian transport estimates for Agulha…
Motivated by results from recent particle tracking experiments in turbulence (Xu et al., Nat. Phys. 7, 709 (2011)), we study the Lagrangian time correlations of vorticity alignments with the three eigenvectors of the deformation-rate tensor. We use data from direct numerical simulations (DNS), and explore the predictions of a Lagrangian model for the velocity gradient tensor. We find that the initial increase of correlation of vorticity direction with the most extensive eigen-direction observed by Xu et al. is reproduced accurately using the Lagrangian model, as well as the evolution of correl…
We show that Lagrangian measurements in active turbulence bear imprints of turbulent and anomalous streaky hydrodynamics leading to a self-selection of persistent trajectories - Levy walks - over diffusive ones. This emergent dynamical heterogeneity results in a super-diffusive first passage distribution which could lead to biologically advantageous motility. We then go beyond single-particle statistics to show that for the pair-dispersion problem as well, active flows are at odds with inertial turbulence. Our study, we believe, will readily inform experiments in establishing the extent of uni…
The statistical properties of fluid particles transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. Single trajectory statistics is investigated in a time range spanning more than three decades, from less than a tenth of the Kolmogorov timescale up to one large-eddy turnover time. Acceleration and velocity statistics show a neat quantitative agreement with recent experimental results. Trapping effects in vortex filaments give rise to enhanced small-scale intermittency on Lagrangian observables.
Lagrangian statistics of passive tracers in rotating turbulence is investigated by Particle Tracking Velocimetry. A confined and steadily-forced turbulent flow is subjected to five different rotation rates. The PDFs of the velocity components clearly reveal the anisotropy induced by background rotation. Although the statistical properties of the horizontal turbulent flow field are approximately isotropic, in agreement with previously reported results by van Bokhoven and coworkers [Phys. Fluids 21, 096601 (2009)], the velocity component parallel to the (vertical) rotation axis gets strongly red…
Stretching in continuum mechanics is naturally described using the Cauchy-Green strain tensors. These tensors quantify the Lagrangian stretching experienced by a material element, and provide a powerful way to study processes in turbulent fluid flows that involve stretching such as vortex stretching and alignment of anisotropic particles. Analyzing data from a simulation of isotropic turbulence, we observe preferential alignment between anisotropic particles and vorticity. We show that this alignment arises because both of these quantities independently tend to align with the strongest Lagrang…
The universality of intermittency in hydrodynamic turbulence is considered based on a recent model for the velocity gradient tensor evolution. Three possible versions of the model are investigated differing in the assumed correlation time-scale and forcing strength. Numerical tests show that the same (universal) anomalous relative scaling exponents are obtained for the three model variants. It is also found that transverse velocity gradients are more intermittent than longitudinal ones, whereas dissipation and enstrophy scale with the same exponents. The results are consistent with the univers…
Based on geometric considerations, longitudinal and transverse Lagrangian velocity increments are introduced as components along, and perpendicular to, the displacement of fluid particles during a time scale τ. It is argued that these two increments probe preferentially the stretching and spinning of material fluid elements, respectively. This property is confirmed (in the limit of vanishing τ) by examining the variances of these increments conditioned on the local topology of the flow. Interestingly, these longitudinal and transverse Lagrangian increments are found to share some qualitative f…
