Vortex clustering, polarisation and circulation intermittency in classical and quantum turbulence

Twisting vortex lines regularize Navier-Stokes turbulence

Fluid flows are intrinsically characterized via the topology and dynamics of underlying vortex lines. Turbulence in common fluids like water and air, mathematically described by the incompressible Navier-Stokes equations (INSE), engenders spontaneous self-stretching and twisting of vortex lines, generating a complex hierarchy of structures. While the INSE are routinely used to describe turbulence, their regularity remains unproven; the implicit assumption being that the self-stretching is ultimately regularized by viscosity, preventing any singularities. Here, we uncover an inviscid regularizing mechanism stemming from self-stretching itself, by analyzing the flow topology as perceived by an observer aligned with the vorticity vector undergoing amplification. While, initially, vorticity amplification occurs via increasing twisting of vortex lines, a regularizing anti-twist spontaneously emerges to prevent unbounded growth. By isolating a vortex, we additionally demonstrate the genericity of this self-regularizing anti-twist. Our work, directly linking dynamics of vortices to turbulence statistics, reveals how the Navier-Stokes dynamics avoids the development of singularities even without the aid of viscosity.

Designing turbulence with entangled vortices

Matter entanglement is a common chaotic structure found in both quantum and classical systems. For classical turbulence, viscous vortices are like sinews in fluid flows, storing and dissipating energy and accommodating strain and stress throughout a complex vortex network. However, to explain how the statistical properties of turbulence arise from elemental vortical structures remains challenging. Here, we use the quantum vortex tangle as a skeleton to generate an instantaneous classical turbulent field with intertwined vortex tubes. Combining the quantum skeleton and tunable vortex thickness makes the synthetic turbulence satisfy key statistical laws, offering valuable insights for elucidating energy cascade and extreme events. By manipulating the elemental structures, we customize turbulence with desired statistical features. This bottom-up approach of designing turbulence provides a testbed for analyzing and modeling turbulence.

Vortex polarization and circulation statistics in isotropic turbulence

We carry out an in-depth analysis of a recently introduced vortex gas model of homogeneous and isotropic turbulence. Direct numerical simulations are used to provide a concrete physical interpretation of one of the model's constituent fields: the degree of vortex polarization. Our investigations shed light on the complexity underlying vortex interactions and reveal, furthermore, that despite some striking similarities, classical and quantum turbulence exhibit distinct structural characteristics, even at inertial range scales. Crucially, these differences arise due to correlations between the polarization and circulation intensity within vortex clusters.

Exploring the Equivalence between Two-Dimensional Classical and Quantum Turbulence through Velocity Circulation Statistics

We study the statistics of velocity circulation in two-dimensional classical and quantum turbulence. We perform numerical simulations of the incompressible Navier-Stokes and the Gross-Pitaevskii (GP) equations for the direct and inverse cascades. Our GP simulations display clear energy spectra compatible with the double cascade theory of two-dimensional classical turbulence. In the inverse cascade, we found that circulation intermittency in quantum turbulence is the same as in classical turbulence. We compare GP data to Navier-Stokes simulations and experimental data from Zhu et al. [Phys. Rev. Lett. 130, 214001 (2023)PRLTAO0031-900710.1103/PhysRevLett.130.214001]. In the direct cascade, for nearly incompressible GP flows, classical and quantum turbulence circulation displays the same self-similar scaling. When compressibility becomes important, quasishocks generate quantum vortices and the equivalence of quantum and classical turbulence only holds for low-order moments. Our results establish the boundaries of the equivalence between two-dimensional classical and quantum turbulence.

Experimental evidence of random shock-wave intermittency

We report the experimental observation of intermittency in a regime dominated by random shock waves on the surface of a fluid. We achieved such a nondispersive surface-wave field using a magnetic fluid subjected to a high external magnetic field. We found that the small-scale intermittency of the wave-amplitude fluctuations is due to shock waves, leading to much more intense intermittency than previously reported in three-dimensional hydrodynamics turbulence or in wave turbulence. The statistical properties of intermittency are found to be in good agreement with the predictions of a Burgers-like intermittency model. Such experimental evidence of random shock-wave intermittency could lead to applications in various fields.

Statistics of extreme turbulent circulation events from multifractality breaking

Recent numerical explorations of extremely intense circulation fluctuations at high Reynolds number flows have brought to light novel aspects of turbulent intermittency. Vortex gas modeling ideas, which are related to a picture of turbulence as a dilute system of vortex tube structures, have been introduced alongside such developments, leading to accurate descriptions of the core and the intermediate tails of circulation probability distribution functions (cPDFs), as well as the scaling exponents associated to statistical moments of circulation. We extend the predictive reach of the vortex gas picture of turbulence by emphasizing that multifractality breaking, one of its salient phenomenological ingredients, is the key concept to disclose the asymptotic form of cPDF tails. A remarkable analytical agreement is found with previous results derived within the framework of the instanton approach to circulation intermittency, a functional formalism devised to single out the statistically dominant velocity configurations associated to extreme circulation events.

Circulation statistics and the mutually excluding behavior of turbulent vortex structures

The small-scale statistical properties of velocity circulation in classical homogeneous and isotropic turbulent flows are assessed through a modeling framework that brings together the multiplicative cascade and the structural descriptions of turbulence. We find that vortex structures exhibit short-distance repulsive correlations, which is evidenced when they are "tomographically" investigated, by means of planar cuts of the flow, as two-dimensional vortex gases. This phenomenon is suggested from model improvements which allow us to obtain an accurate multiscale description of the intermittent fluctuations of circulation. Its crucial new ingredient, the conjectured hard disk behavior of the effective planar vortices, is then found to be strongly supported from a study of their spatial distributions in direct numerical simulations of the Navier-Stokes equations.

The area rule for circulation in three-dimensional turbulence

An important idea underlying a plausible dynamical theory of circulation in three-dimensional turbulence is the so-called area rule, according to which the probability density function (PDF) of the circulation around closed loops depends only on the minimal area of the loop, not its shape. We assess the robustness of the area rule, for both planar and nonplanar loops, using high-resolution data from direct numerical simulations. For planar loops, the circulation moments for rectangular shapes match those for the square with only small differences, these differences being larger when the aspect ratio is farther from unity and when the moment order increases. The differences do not exceed about 5% for any condition examined here. The aspect ratio dependence observed for the second-order moment is indistinguishable from results for the Gaussian random field (GRF) with the same two-point correlation function (for which the results are order-independent by construction). When normalized by the SD of the PDF, the aspect ratio dependence is even smaller ( < 2%) but does not vanish unlike for the GRF. We obtain circulation statistics around minimal area loops in three dimensions and compare them to those of a planar loop circumscribing equivalent areas, and we find that circulation statistics match in the two cases only when normalized by an internal variable such as the SD. This work highlights the hitherto unknown connection between minimal surfaces and turbulence.