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

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.

Boltzmann equation and hydrodynamic equations: their equilibrium and non-equilibrium behaviour

This short article summarizes the key features of equilibrium and non-equilibrium aspects of Boltzmann and hydrodynamic equations. Under equilibrium, the Boltzmann equation generates uncorrelated random velocity that corresponds to k2 energy spectrum for the Euler equation. The latter spectrum is produced using initial configuration with many Fourier modes of equal amplitudes but with random phases. However, for a large-scale vortex as an initial condition, earlier simulations exhibit a combination of k-5/3 (in the inertial range) and k2 (for large wavenumbers) spectra, with the range of k2 spectrum increasing with time. These simulations demonstrate an approach to equilibrium or thermalization of Euler turbulence. In addition, they also show how initial velocity field plays an important role in determining the behaviour of the Euler equation. In non-equilibrium scenario, both Boltzmann and Navier-Stokes equations produce similar flow behaviour, for example, Kolmogorov's k-5/3 spectrum in the inertial range. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

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.