Flight–crash events in turbulence

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. I n 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 fluctuationdissipation 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 qua...

Irreversibility of the two-dimensional enstrophy cascade

“…Similar results have been found for the inverse cascade of energy in two-dimensional turbulence, which is characterized by the same scaling law of 3D turbulence [5] and in compressible turbulence [6]. Despite the fact that in two dimensions the energy flows towards the large scales (instead of the small scales), time irreversibility manifests in two-dimensional turbulence as in three dimensions, with a negative skewness of the power computed along a Lagrangian trajectory [3].…”

“…The main motivation for this work is that the direct cascade is characterized by a single characteristic time [5] and therefore spatial separation (between injection and dissipation scales) does not correspond in a simple way to time-scale separation. This make the enstrophy cascade completely different from the energy cascades (direct and inverse) studied in [3]. On the basis of direct numerical simulations at different Reynolds numbers, we find that also in this case single-point statistics breaks time-reversal symmetry and that the degree of irreversibility grows with the Reynolds numbers of the flow.…”

“…The forcing term f injects energy and enstrophy at a characteristic scale from which they are transported towards large and small scales, respectively, generating the double cascade predicted by Kraichnan 50 years ago [5,8]. The inverse cascade of energy is characterized by a Kolmogorov-like spectrum with close-to-Gaussian statistics [9] and its time-reversal properties have been the subject of previous works [3,7]. In the direct cascade enstrophy is transferred at a rate η from the forcing scales f down to the small dissipative scales ν ∼ ν 1/2 /η 1/6 generating a power-law spectrum with an exponent close to the dimensional prediction −3 [10][11][12][13][14][15][16][17].…”

“…Recently, it has been shown on the basis of laboratory experiments and numerical simulations how irreversibility in turbulence manifests itself at the level of a single-point observable [3]. By looking at the evolution of the (kinetic) energy along a fluid trajectory, it has been shown that the particle acquires energy on a long time scale and loses it on a short time scale.…”

“…This reflects the fact that energy is injected in the flow by an external forcing at large slow scales and is dissipated by viscosity at small fast scales [4]. As a consequence, although in stationary conditions the mean temporal increment of energy vanishes, the full statistics is not time reversible and odd moments of energy increments are different from zero [3]. Time irreversibility can be quantified by looking at the statistics of the power along a trajectory p(t) = d/dt[v 2 (t)/2] and it has been shown that the third moment p 3 is negative and increases with the control parameter, the Reynolds number, of the flow.…”