Flight-crash events in turbulence

Nonequilibrium statistical mechanics of the turbulent energy cascade: Irreversibility and response functions

The statistical properties of turbulent flows are fundamentally different from those of systems at equilibrium due to the presence of an energy flux from the scales of injection to those where energy is dissipated by the viscous forces: a scenario dubbed "direct energy cascade." From a statistical mechanics point of view, the cascade picture prevents the existence of detailed balance, which holds at equilibrium, e.g., in the inviscid and unforced case. Here, we aim at characterizing the nonequilibrium properties of turbulent cascades in a shell model of turbulence by studying an asymmetric time-correlation function and the relaxation behavior of an energy perturbation, measured at scales smaller or larger than the perturbed one. We contrast the behavior of these two observables in both nonequilibrium (forced and dissipated) and equilibrium (inviscid and unforced) cases. Finally, we show that equilibrium and nonequilibrium physics coexist in the same system, namely, at scales larger and smaller, respectively, of the forcing scale.

Multimode correlations and the entropy of turbulence in shell models

We suggest a new focus for turbulence studies-multimode correlations-which reveal the hitherto hidden nature of turbulent state. We apply this approach to shell models describing basic properties of turbulence. The family of such models allows one to study turbulence close to thermal equilibrium, which happens when the interaction time weakly depends on the mode number. As the number of modes increases, the one-mode statistics approaches Gaussian (like in weak turbulence), the occupation numbers grow, while the three-mode cumulant describing the energy flux stays constant. Yet we find that higher multimode cumulants grow with the order. We derive analytically and confirm numerically the scaling law of such growth. The sum of all squared dimensionless cumulants is equal to the relative entropy between the full multimode distribution and the Gaussian approximation of independent modes; we argue that the relative entropy could grow as the logarithm of the number of modes, similar to the entanglement entropy in critical phenomena. Therefore, the multimode correlations give the new way to characterize turbulence states and possibly divide them into universality classes.

Heavy inertial particles in rotating turbulence: Distribution of particles in flow and evolution of Lagrangian trajectories

We revisit the problem of heavy particles suspended in homogeneous box turbulence flow subjected to rotation along the vertical axis, which introduces anisotropy along the vertical and horizontal planes. We investigate the effects of the emergent structures due to rotation, on the spatial distribution and temporal statistics of the particles. The distribution of particles in the flow are studied using the joint probability distribution function (JPDFs) of the second and third principle invariants of the velocity gradient tensor, Q and R. At high rotation rates, the JPDFs of Lagrangian Q-R plots show remarkable deviations from the well-known teardrop shape. The cumulative probability distribution functions for times during which a particle remains in vortical or straining regions show exponentially decaying tails except for the deviations at the highest rotation rate. The average residence times of the particles in vortical and straining regions are also affected considerably due to the addition of rotation. Furthermore, we compute the temporal velocity autocorrelation and connect it to the Lagrangian anisotropy in presence of rotation. The spatial and temporal statistics of the particles are determined by a complex competition between the rotation rate and inertia of the particle.

Eulerian vs Lagrangian Irreversibility in an Experimental Turbulent Swirling Flow

In a turbulent fluid, the time-reversal symmetry is explicitly broken by viscosity, and spontaneously broken in the inviscid limit. Recently, Drivas [J. Nonlinear Sci. 29, 65 (2019).JNSCEK0938-897410.1007/s00332-018-9476-8] proved the equivalence of two different local indicators of time irreversibility: (i) an Eulerian one, based on regularity properties of the velocity field [Duchon and Robert, Nonlinearity 13, 249 (2000).NONLE50951-771510.1088/0951-7715/13/1/312]; (ii) a Lagrangian one, based on symmetry properties of the trajectories under time reversal [Jucha et al., Phys. Rev. Lett. 113, 054501 (2014).PRLTAO0031-900710.1103/PhysRevLett.113.054501]. We test this equivalence in a turbulent Von Kármán experiment at a resolution of the order of the Kolmogorov scale using a high resolution 4D-PTV technique. We use the equivalence to perform the first joined Eulerian-Lagrangian exploration of the dynamics leading to time irreversibility, and find that it is linked with vortex interaction, suggesting a link between irreversibility and singularity.

Emerging scale invariance in a model of turbulence of vortices and waves

This note is devoted to broken and emerging scale invariance of turbulence. Pumping breaks the symmetry: the statistics of every mode explicitly depend on the distance from the pumping. And yet the ratios of mode amplitudes, called Kolmogorov multipliers, are known to approach scale-invariant statistics away from the pumping. This emergent scale invariance deserves an explanation and a detailed study. We put forward the hypothesis that the invariance of multipliers is due to an extreme non-locality of their interactions (similar to the appearance of mean-field properties in the thermodynamic limit for systems with long-range interaction). We analyse this phenomenon in a family of models that connects two very different classes of systems: resonantly interacting waves and wave-free incompressible flows. The connection is algebraic and turns into an identity for properly discretized models. We show that this family provides a unique opportunity for an analytic (perturbative) study of emerging scale invariance in a system with strong interactions. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Broken Mirror Symmetry of Tracer's Trajectories in Turbulence

Topological properties of physical systems play a crucial role in our understanding of nature, yet their experimental determination remains elusive. We show that the mean helicity, a dynamical invariant in ideal flows, quantitatively affects trajectories of fluid elements: the linking number of Lagrangian trajectories depends on the mean helicity. Thus, a global topological invariant and a topological number of fluid trajectories become related, and we provide an empirical expression linking them. The relation shows the existence of long-term memory in the trajectories: the links can be made of the trajectory up to a given time, with particles positions in the past. This property also allows experimental measurements of mean helicity.

Constrained Reversible System for Navier-Stokes Turbulence

Following a Gallavotti's conjecture, stationary states of Navier-Stokes fluids are proposed to be described equivalently by alternative equations besides the Navier-Stokes equation itself. We discuss a model system symmetric under time reversal based on the Navier-Stokes equations constrained to keep the enstrophy constant. It is demonstrated through highly resolved numerical experiments that the reversible model evolves to a stationary state which reproduces quite accurately all statistical observables relevant for the physics of turbulence extracted by direct numerical simulations (DNS) at different Reynolds numbers. The possibility of using reversible models to mimic turbulence dynamics is of practical importance for the coarse-grained version of Navier-Stokes equations, as used in large-eddy simulations. Furthermore, the reversible model appears mathematically simpler, since enstrophy is bounded to be constant for every Reynolds number. Finally, the theoretical interest in the context of statistical mechanics is briefly discussed.

Fluctuation theorem and extended thermodynamics of turbulence

Turbulent flows are out-of-equilibrium because the energy supply at large scales and its dissipation by viscosity at small scales create a net transfer of energy among all scales. This energy cascade is modelled by approximating the spectral energy balance with a nonlinear Fokker-Planck equation consistent with accepted phenomenological theories of turbulence. The steady-state contributions of the drift and diffusion in the corresponding Langevin equation, combined with the killing term associated with the dissipation, induce a stochastic energy transfer across wavenumbers. The fluctuation theorem is shown to describe the scale-wise statistics of forward and backward energy transfer and their connection to irreversibility and entropy production. The ensuing turbulence entropy is used to formulate an extended turbulence thermodynamics.

Statistics of Lagrangian trajectories in a rotating turbulent flow

We investigate the Lagrangian statistics of three-dimensional rotating turbulent flows through direct numerical simulations. We find that the emergence of coherent vortical structures because of the Coriolis force leads to a suppression of the "flight-crash" events reported by Xu et al. [Proc. Natl. Acad. Sci. (USA) 111, 7558 (2014)PNASA60027-842410.1073/pnas.1321682111]. We perform systematic studies to trace the origins of this suppression in the emergent geometry of the flow and show why such a Lagrangian measure of irreversibility may fail in the presence of rotation.

Phase transition in time-reversible Navier-Stokes equations

We present a comprehensive study of the statistical features of a three-dimensional (3D) time-reversible truncated Navier-Stokes (RNS) system, wherein the standard viscosity ν is replaced by a fluctuating thermostat that dynamically compensates for fluctuations in the total energy. We analyze the statistical features of the RNS steady states in terms of a non-negative dimensionless control parameter R_{r}, which quantifies the balance between the fluctuations of kinetic energy at the forcing length scale ℓ_{f} and the total energy E_{0}. For small R_{r}, the RNS equations are found to produce "warm" stationary statistics, e.g., characterized by the partial thermalization of the small scales. For large R_{r}, the stationary solutions have features akin to standard hydrodynamic ones: they have compact energy support in k space and are essentially insensitive to the truncation scale k_{max}. The transition between the two statistical regimes is observed to be smooth but rather sharp. Using insights from a diffusion model of turbulence (Leith model), we argue that the transition is in fact akin to a continuous second-order phase transition, where R_{r} indeed behaves as a thermodynamic control parameter, e.g., a temperature. A relevant order parameter can be suitably defined in terms of a (normalized) enstrophy, while the symmetry-breaking parameter h is identified as (one over) the truncation scale k_{max}. We find that the signatures of the phase transition close to the critical point R_{r}^{★} can essentially be deduced from a heuristic mean-field Landau free energy. This point of view allows us to reinterpret the relevant asymptotics in which the dynamical ensemble equivalence conjectured by Gallavotti [Phys. Lett. A 223, 91 (1996)PYLAAG0375-960110.1016/S0375-9601(96)00729-3] could hold true. We argue that Gallavotti's limit is precisely the joint limit R_{r}→[over >]R_{r}^{★} and h→[over >]0, with the overset symbol ">" indicating that those limits are approached from above. The limit therefore relates to the statistical features at the critical point. In this regime, our numerics indicate that the low-order statistics of the 3D RNS are indeed qualitatively similar to those observed in direct numerical simulations of the standard Navier-Stokes equations with viscosity chosen so as to match the average value of the reversible thermostat. This result suggests that Gallavotti's equivalence conjecture could indeed be of relevance to model 3D turbulent statistics, and provides a clear guideline for further numerical investigations involving higher resolutions.

Power fluctuations in turbulence

To generate or maintain a turbulent flow, one needs to introduce kinetic energy. This energy injection necessarily fluctuates and these power fluctuations act on all turbulent excited length scales. If the power is injected using forces proportional to the velocity, such as those common in shear flows, or with a force acting at the largest scales only, the spectrum of these fluctuations is shown to have a universal inertial range, proportional to the energy spectrum.

Heavy inertial particles in turbulent flows gain energy slowly but lose it rapidly

We present an extensive numerical study of the time irreversibility of the dynamics of heavy inertial particles in three-dimensional, statistically homogeneous, and isotropic turbulent flows. We show that the probability density function (PDF) of the increment, W(τ), of a particle's energy over a time scale τ is non-Gaussian, and skewed toward negative values. This implies that, on average, particles gain energy over a period of time that is longer than the duration over which they lose energy. We call this slow gain and fast loss. We find that the third moment of W(τ) scales as τ^{3} for small values of τ. We show that the PDF of power-input p is negatively skewed too; we use this skewness Ir as a measure of the time irreversibility and we demonstrate that it increases sharply with the Stokes number St for small St; this increase slows down at St≃1. Furthermore, we obtain the PDFs of t^{+} and t^{-}, the times over which p has, respectively, positive or negative signs, i.e., the particle gains or loses energy. We obtain from these PDFs a direct and natural quantification of the slow gain and fast loss of the energy of the particles, because these PDFs possess exponential tails from which we infer the characteristic loss and gain times t_{loss} and t_{gain}, respectively, and we obtain t_{loss}<t_{gain} for all the cases we have considered. Finally, we show that the fast loss of energy occurs with greater probability in the strain-dominated region than in the vortical one; in contrast, the slow gain in the energy of the particles is equally likely in vortical or strain-dominated regions of the flow.

Time irreversibility in reversible shell models of turbulence

Turbulent flows governed by the Navier-Stokes equations (NSE) generate an out-of-equilibrium time irreversible energy cascade from large to small scales. In the NSE, the energy transfer is due to the nonlinear terms that are formally symmetric under time reversal. As for the dissipative term: first, it explicitly breaks time reversibility; second, it produces a small-scale sink for the energy transfer that remains effective even in the limit of vanishing viscosity. As a result, it is not clear how to disentangle the time irreversibility originating from the non-equilibrium energy cascade from the explicit time-reversal symmetry breaking due to the viscous term. To this aim, in this paper we investigate the properties of the energy transfer in turbulent shell models by using a reversible viscous mechanism, avoiding any explicit breaking of the [Formula: see text] symmetry. We probe time irreversibility by studying the statistics of Lagrangian power, which is found to be asymmetric under time reversal also in the time-reversible model. This suggests that the turbulent dynamics converges to a strange attractor where time reversibility is spontaneously broken and whose properties are robust for what concerns purely inertial degrees of freedoms, as verified by the anomalous scaling behavior of the velocity structure functions.

Active turbulence in a gas of self-assembled spinners

Colloidal particles subject to an external periodic forcing exhibit complex collective behavior and self-assembled patterns. A dispersion of magnetic microparticles confined at the air-liquid interface and energized by a uniform uniaxial alternating magnetic field exhibits dynamic arrays of self-assembled spinners rotating in either direction. Here, we report on experimental and simulation studies of active turbulence and transport in a gas of self-assembled spinners. We show that the spinners, emerging as a result of spontaneous symmetry breaking of clock/counterclockwise rotation of self-assembled particle chains, generate vigorous vortical flows at the interface. An ensemble of spinners exhibits chaotic dynamics due to self-generated advection flows. The same-chirality spinners (clockwise or counterclockwise) show a tendency to aggregate and form dynamic clusters. Emergent self-induced interface currents promote active diffusion that could be tuned by the parameters of the external excitation field. Furthermore, the erratic motion of spinners at the interface generates chaotic fluid flow reminiscent of 2D turbulence. Our work provides insight into fundamental aspects of collective transport in active spinner materials and yields rules for particle manipulation at the microscale.

Irreversibility of the two-dimensional enstrophy cascade

We study the time irreversibility of the direct cascade in two-dimensional turbulence by looking at the time derivative of the square vorticity along Lagrangian trajectories, a quantity called metenstrophy. By means of extensive direct numerical simulations we measure the time irreversibility from the asymmetry of the probability density function of the metenstrophy and we find that it increases with the Reynolds number of the cascade, similarly to what is found in three-dimensional turbulence. A detailed analysis of the different contributions to the enstrophy budget reveals a remarkable difference with respect to what is observed for the energy cascade, in particular the role of the statistics of the forcing to determine the degree of irreversibility.

Angular Dynamics of a Small Particle in Turbulence

We compute the angular dynamics of a neutrally buoyant nearly spherical particle immersed in an unsteady fluid. We assume that the particle is small, that its translational slip velocity is negligible, and that unsteady and convective inertia are small perturbations. We derive an approximation for the torque on the particle that determines the first inertial corrections to Jeffery's equation. These corrections arise as a consequence of local vortex stretching and can be substantial in turbulence, where local vortex stretching is strong and closely linked to the irreversibility of turbulence.

Single-Particle Motion and Vortex Stretching in Three-Dimensional Turbulent Flows

Three-dimensional turbulent flows are characterized by a flux of energy from large to small scales, which breaks the time reversal symmetry. The motion of tracer particles, which tend to lose energy faster than they gain it, is also irreversible. Here, we connect the time irreversibility in the motion of single tracers with vortex stretching and thus with the generation of the smallest scales.

Braid Entropy of Two-Dimensional Turbulence

The evolving shape of material fluid lines in a flow underlies the quantitative prediction of the dissipation and material transport in many industrial and natural processes. However, collecting quantitative data on this dynamics remains an experimental challenge in particular in turbulent flows. Indeed the deformation of a fluid line, induced by its successive stretching and folding, can be difficult to determine because such description ultimately relies on often inaccessible multi-particle information. Here we report laboratory measurements in two-dimensional turbulence that offer an alternative topological viewpoint on this issue. This approach characterizes the dynamics of a braid of Lagrangian trajectories through a global measure of their entanglement. The topological length of material fluid lines can be derived from these braids. This length is found to grow exponentially with time, giving access to the braid topological entropy . The entropy increases as the square root of the turbulent kinetic energy and is directly related to the single-particle dispersion coefficient. At long times, the probability distribution of is positively skewed and shows strong exponential tails. Our results suggest that may serve as a measure of the irreversibility of turbulence based on minimal principles and sparse Lagrangian data.

New class of turbulence in active fluids

Turbulence is a fundamental and ubiquitous phenomenon in nature, occurring from astrophysical to biophysical scales. At the same time, it is widely recognized as one of the key unsolved problems in modern physics, representing a paradigmatic example of nonlinear dynamics far from thermodynamic equilibrium. Whereas in the past, most theoretical work in this area has been devoted to Navier-Stokes flows, there is now a growing awareness of the need to extend the research focus to systems with more general patterns of energy injection and dissipation. These include various types of complex fluids and plasmas, as well as active systems consisting of self-propelled particles, like dense bacterial suspensions. Recently, a continuum model has been proposed for such "living fluids" that is based on the Navier-Stokes equations, but extends them to include some of the most general terms admitted by the symmetry of the problem [Wensink HH, et al. (2012) Proc Natl Acad Sci USA 109:14308-14313]. This introduces a cubic nonlinearity, related to the Toner-Tu theory of flocking, which can interact with the quadratic Navier-Stokes nonlinearity. We show that as a result of the subtle interaction between these two terms, the energy spectra at large spatial scales exhibit power laws that are not universal, but depend on both finite-size effects and physical parameters. Our combined numerical and analytical analysis reveals the origin of this effect and even provides a way to understand it quantitatively. Turbulence in active fluids, characterized by this kind of nonlinear self-organization, defines a new class of turbulent flows.

Time irreversibility of the statistics of a single particle in compressible turbulence

We investigate time irreversibility from the point of view of a single particle in Burgers turbulence. Inspired by the recent work for incompressible flows [Xu et al., Proc. Natl. Acad. Sci. USA 111, 7558 (2014)], we analyze the evolution of the kinetic energy for fluid markers and use the fluctuations of the instantaneous power as a measure of time irreversibility. For short times, starting from a uniform distribution of markers, we find the scaling 〈[E(t)-E(0)](n)〉∝t and 〈p(n)〉∝Re(n-1) for the power as a function of the Reynolds number. Both observations can be explained using the "flight-crash" model, suggested by Xu et al. Furthermore, we use a simple model for shocks that reproduces the moments of the energy difference, including the pre-factor for 〈E(t)-E(0)〉. To complete the single-particle picture for Burgers we compute the moments of the Lagrangian velocity difference and show that they are bifractal. This arises in a similar manner to the bifractality of Eulerian velocity differences. In the above setting, time irreversibility is directly manifest as particles eventually end up in shocks. We additionally investigate time irreversibility in the long-time limit when all particles are located inside shocks and the Lagrangian velocity statistics are stationary. We find the same scalings for the power and energy differences as at short times and argue that this is also a consequence of rare "flight-crash" events related to shock collisions.

Time-reversal-symmetry breaking in turbulence

In three-dimensional turbulent flows, the flux of energy from large to small scales breaks time symmetry. We show here that this irreversibility can be quantified by following the relative motion of several Lagrangian tracers. We find by analytical calculation, numerical analysis, and experimental observation that the existence of the energy flux implies that, at short times, two particles separate temporally slower forwards than backwards, and the difference between forward and backward dispersion grows as t^{3}. We also find the geometric deformation of material volumes, defined by four points spanning an initially regular tetrahedron, to show sensitivity to the time reversal with an effect growing linearly in t. We associate this with the structure of the strain rate in the flow.