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# Flight–crash events in turbulence

Edited by Harry L. Swinney, University of Texas at Austin, Austin, TX, and approved March 24, 2014 (received for review November 20, 2013)

## Significance

Irreversibility is a fundamental aspect of the evolution of natural systems, and quantifying its manifestations is a challenge in any attempt to describe nonequilibrium systems. In the case of fluid turbulence, an emblematic example of a system very far from equilibrium, we show that the motion of a single fluid particle provides a clear manifestation of time irreversibility. Namely, we observe that fluid particles tend to lose kinetic energy faster than they gain it. This is best seen by the presence of rare “flight–crash” events, where fast moving particles suddenly decelerate into a region where fluid motion is slow. Remarkably, the statistical signature of these events establishes a quantitative relation between the degree of irreversibility and turbulence intensity.

## Abstract

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.

- nonequilibrium systems
- turbulent mixing
- direct and inverse turbulent energy cascades
- nonequilibrium statistical mechanics
- Lagrangian description

In systems at thermal equilibrium, the probabilities of forward and backward transitions between any two states are equal, a property known as “detailed balance.” This fundamental property expresses time reversibility of equilibrium statistics (1). In the important class of nonequilibrium problems, where the dynamics of the system is coupled with a heat bath, the notion of detailed balance can be extended and fluctuation theorems successfully describe the behavior (2, 3). This class contains many experimental situations (4) where quantitative information on irreversibility was obtained (3). When a system driven by thermal noise is characterized by a probability current, the fluctuation–dissipation theorem and detailed balance was found to apply in a comoving reference frame (5).

In comparison, very little is known concerning the statistical properties of a small part embedded in a fluctuating, turbulent background. The fundamental notion of detailed balance is not expected to apply in such systems. Here we ask, what does the time irreversibility inherent to the large system imply for the statistical properties of small parts in the system and how do we measure the degree of irreversibility (6, 7) (or equivalently, how far is the system away from equilibrium) by monitoring a small part in the system? We focus here on fluid turbulence, a paradigm for ultimate far-from-equilibrium states, where irreversibility of fluctuations is a fundamental property (8, 9). We show that the simplest and most fundamental scalar quantity, namely, the kinetic energy of a fluid particle, enables a clear identification and quantification of the irreversibility of the turbulent flow.

The characteristic properties of turbulence rest on the vastly different scales: from the scale *ε*, a phenomenon called “energy cascade.” In 3D flows, where *ε*, however, cannot in itself be a measure of irreversibility in the system because *ε* is a dimensional quantity, so it can be made arbitrarily large by changing the units even if the system is very close to equilibrium. Moreover, it can be expressed as a moment of velocity differences at a single time (10, 13) without any reference to the evolution of the flow.

As we demonstrate below, the irreversibility induced by the energy flux through spatial scales can be revealed and quantified by following the change of the kinetic energy of small fluid elements (particles). The kinetic energy per unit mass of the fluid is simply

The recent advances in our ability to reliably measure the trajectories of small tracer particles in well-controlled laboratory flows (15⇓⇓–18), as well as to simulate accurately the motion of particle tracers using the Navier–Stokes equation (16) allow us to investigate these fundamental issues. In this work, we restrict ourselves to statistically stationary and homogeneous flows. The results shown here were obtained from a variety of flow configurations in 2D and 3D, including both laboratory experiments and direct numerical simulations of the Navier–Stokes equations. The datasets contain a large number of trajectories, with at least *Materials and Methods* and *SI Text* for details).

## Results

### “Flight–Crash” Events.

The phenomenon discussed here is illustrated in Fig. 1 *A* and *B*, which show the evolution of

### Statistics of Energy Difference.

The statistics of the energy increments, *SI Text*. The asymmetry revealed by Fig. 1 implies that the distribution of *A*. In all these flows, *B* shows that the third moment of *A*), i.e., it is independent of the difference in the direction of the energy flux in 2D and 3D. This demonstrates again that the energy flux *ε* by itself is not an appropriate measure of irreversibility and suggests the use of the dimensionless rate of change of the kinetic energy instead. A systematic statistical characterization of *C* shows the PDF of *τ* in the range *T* are the characteristic times at the dissipation scale *τ* is smaller. This feature is possibly related to intermittency, a characteristic phenomenon in turbulent fluids.

Could we understand the skewness of *C*. Our measurements of *τ*, shown in Fig. 2*D*, however, shows a more complicated dependence on *W* than the simple linear law **1**. This suggests that fluctuation theorems do not apply directly to tracer particles in turbulence. This we attribute to the properties of the forces acting on fluid particles, which are very different from the forces in usual thermodynamic systems (8).

### Statistics of Power Fluctuations: Quantifying Detailed Balance Violations.

As we demonstrate that the results obtained in the general context of stochastic thermodynamics do not apply to a small fluid element carried by the fluid, the asymmetry observed for the distribution of the energy differences along a trajectory (Fig. 2*C*) points to a more fundamental aspect, namely the breakdown of time reversibility in the system. In fact, as we show in the following, the third moment of

Let us consider the rate of change of the kinetic energy following a tracer particle, i.e., the power *p*, as shown in Fig. 3 *A* and *B*, are therefore a signature that detailed balance is violated. [We note that the statistics of the power *p* may be affected by specific, nonuniversal aspects of the forcing, especially in 2D, in which the external forcing acts at small scales and is fast-changing (*SI Text*).] This violation can then be quantified by odd moments of the fluctuations of *p*, which change sign when reversing *p* vanishes for stationary and homogeneous flows. The third moment, which can be measured reliably, is sufficient to quantify the violation of detailed balance.

As already explained, a proper measure must be dimensionless. A natural choice is the dimensionless power *Ir*, defined as*C* and *D* show that *Ir* increases with the Reynolds number in both 2D and 3D, hence with the separation of scales between forcing and dissipation. In 3D, it grows approximately as *SI Text*). The data from both experiments and numerics shown in Fig. 3*D* demonstrate that irreversibility grows also with this Reynolds number approximately as *E* and *F*. As a consequence, the skewness of the power fluctuations, defined as

Thus, remarkably, the measure of irreversibility, *Ir*, directly accessible in laboratory flows, depends on the Reynolds number to a simple power, independent of the specificity of the forcing, and even more surprisingly, of the directions of the energy flux.

We note that the dependence of the moments *n*, consistent with the observation that the PDFs of *SI Text*).

## Scaling of Power and Flight–Crash Events

The results documented in Figs. 2 and 3 establish a clear relation between the moments of the power, hence of the energy differences, and the Reynolds number. The aim of this section is to provide simple phenomenological arguments to interpret the dependence found in Fig. 3, which is observed both in 2D and 3D and seems universal.

First, let us note that the Lagrangian velocity difference does not have self-similar statistics, so it is reasonable to assume that there are events with different scaling exponents *γ* for the velocity change, *γ* contribute different velocity moments. Landau–Obukhov phenomenology (14) suggests *τ*. Note that this velocity change is much larger than the one suggested by Landau–Obukhov, *n*. As we now demonstrate, however, the scaling of the power suggests that such events provide the main contribution to the moments of the energy changes and power, which are determined by the correlation between **V** and *ε* is the third-order structure function of the energy difference, *W* by *A*, *Inset*. The very good collapse of the data in 3D gives support for the flight-crash picture discussed here. Using the same scaling in 2D leads to a very good collapse of the curves for very small *τ*. The difference between Fig. 2 *A* and *B* at longer times may be a manifestation of the very different physics occurring in 2D and 3D flows. The finite-time average power *τ* decreases and saturates at **[3]** using the saturation value **[3]** also explains the systematic dependence of *SI Text*).

## Discussion

Turbulence is characterized by large fluctuations. Notoriously, the local energy dissipation rate shows strong spatial fluctuations, which are known to play a key role in the origin of intermittency, or fluctuations of other quantities of the turbulent flow (10). Much work has been devoted to the modeling of fluctuations of the local dissipation rate and of its coarse-grained generalization, especially in relation to devising approximate numerical schemes (13). Here we are interested in the related but different question of the exchange between a very small subsystem, a fluid particle, and the surrounding turbulent flow, in the spirit of work done for small systems in contact with thermostats (3). The comparison of our results with those described by stochastic thermodynamics, therefore, reveals the general features of systems very far from thermal equilibrium.

The main achievement of this work has been to document and quantify the intrinsic irreversibility of turbulent flows. This led us to define a quantity *Ir*, which increases as a power law when the Reynolds number increases, i.e., when the flow becomes more turbulent. Remarkably, this way of characterizing turbulent flows is insensitive to the direction of energy flux through spatial scales.

The observation of asymmetry in the time dependence of the kinetic energy of particle in a turbulent flows should be contrasted with the spatial asymmetry, observed in particular in the case of a passive scalar with an imposed gradient (24). In this case, the asymmetric, ramp-and-cliff structure results from a large scale forcing (the gradient), and persists all the way to very small scales. A weaker analogy has been documented in shear flows (25, 26). The phenomenon documented here is very different, as the temporal asymmetry is found both for direct and inverse cascades. It would be interesting to understand whether the possible connection between *A* make any apparent relation with vortex tubes, observed many times before (15), unlikely.

Our results stress the main difference between the well-studied case of systems in contact with thermostats, and those involving a cascade through scales such as turbulence. In this context, our approach could shed new light on a variety of different problems, such as plasma turbulence (28), quantum turbulence (29), magnetohydrodynamics (30), and more generally, on any system that is irreversible and has a separation of scales. The investigation of such systems in the spirit of the present work is likely to lead to new concepts in the physics of the nonequilibrium.

## Materials and Methods

We describe briefly the different turbulent flows that we analyzed in both 3D and 2D and in both physical experiments and numerical simulations. More details can be found in *SI Text*.

### Experimental Setups.

The turbulent flows that we generated in laboratory experiments include the 3D von Kármán flows, 2D turbulence driven either electromagnetically or by surface ripples (Faraday waves).

#### von Kármán flows in 3D.

The 3D experiments were performed in a so-called von Kármán mixer, which generates high-Reynolds-number turbulent water flow between two counterrotating disks (15, 17). We measured 3D trajectories of tracer particles seeded in the flow using optical Lagrangian particle tracking (17, 18). The Reynolds number of the turbulence was in the *SI Text*).

#### Two-dimensional turbulence experiments.

Energy is injected into flows by driving horizontal vortices whose scale is much smaller than the size of the container. In electromagnetically driven turbulence such vortices are driven by the Lorenz force produced by the spatially varying vertical magnetic field and the electric current flowing across the fluid cell in electrolyte. In the Farady-wave-driven turbulence the vorticity is generated at the scale approximately at half of the wave period (31). The injected energy is then spread by the inverse energy cascade over a broad range of scales to form the Kolmogorov–Kraichnan spectrum. The 2D particle trajectories are tracked for a long time, up to 100 Lagrangian integral times (32).

### Direct Numerical Simulations.

The numerical work is based on simulating the Navier–Stokes equations,

where **x** and at time *t*. The velocity field is incompressible *P*. The flow is forced by using an external field **f**, which varies at a characteristic scale

The Lagrangian information is then obtained by integrating in time the equation of motion of fluid tracers

in which the tracer velocity **u** at position **X** and time *t*.

In all cases a statistically stationary flow was maintained by balancing the forcing and the dissipation terms. All the simulations discussed here were carried out in a periodic domain using standard pseudospectral methods.

In 3D, the simulations reported here used up to 384^{3} modes. The flow was forced at a scale *η*,

The simulations of 2D flows reported here were carried out with up to 8,192^{2} modes. Forcing acted at a small scale

## Acknowledgments

We thank the Kavli Institutes for Theoretical Physics (KITP), where the work started during the 2011 The Nature of Turbulence program (KITP) and continued during the 2012 New Directions in Turbulence program (KITP China). We also acknowledge partial support from the Max Planck Society, the Humboldt Foundation, and European Cooperation in Science and Technology Action MP0806 Particles in Turbulence. The work was supported by grants from the German Science Foundation (XU/91-3), the Minerva Foundation, the Bi-National Science Foundation, Agency of Natural Resources Grant Turbulent Evaporation and Condensation 2, the Australian Research Council Discovery Projects funding scheme (DP110101525), and a Discovery Early Career Research Award (DE120100364).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: eberhard.bodenschatz{at}ds.mpg.de.

Author contributions: H. Xu, A.P., G.F., and E.B. designed research; H. Xu, A.P., G.F., E.B., M.S., H. Xia, N.F., and G.B. performed research; H. Xu, A.P., G.F., E.B., M.S., H. Xia, N.F., and G.B. analyzed data; and H. Xu, A.P., G.F., and E.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1321682111/-/DCSupplemental.

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