ICTS

We resurrect an old Heisenberg – Chandrasekhar technique and insight from randomly stirred models to study the crossover between Kolmogorov and Bolgiano-Obukhov scaling for fully developed turbulence in a stably stratified fluid. The crossover scales and their dependence on the Richardson number are clearly established.

We use a new measure of many-body chaos for classical systems---cross-correlators---to show that in a thermalised fluid (obtained from the Galerkin-truncated three-dimensional Euler and one-dimensional Burgers equations) characterised by a temperature $T$ and $N_G$ degrees of freedom, the Lyapunov exponent $\lambda$ scales as $N_G\sqrt{T}$. This scaling, obtained from detailed numerical simulations and theoretical estimates, provides compelling evidence not only for recent conjectures $\lambda \sim \sqrt{T}$ for chaotic, equilibrium, classical many-body systems, as well as, numerical results from frustrated spin systems, but also, remarkably, show that $\lambda$ scales linearly with the degrees of freedom in a finite-dimensional, classical, chaotic system.

The finite amount of dissipation of kinetic energy in turbulent fluid, where viscosity seemingly plays a vanishingly small role, is one of the main properties of turbulence, known as the dissipative anomaly. This property is grounded in the singular nature and deep irreversibility of turbulent flows, and plays a central role in our understanding of turbulence. We present new numerical measurements aiming at understanding how time irreversibility influences the relative motion of tracers transported by the flow. In particular, a strong interlink is found between the long-range, intermittent Lagrangian correlations of energy dissipation and violent separations between pairs of fluid particle trajectories.

The large-scale dynamics of 3D turbulence has been investigated since Leonardo da Vinci (1505) who wondered why vortices, generated at the pillars of a bridge in the Arno river (Florence), tended to be long-lived.

More than four centuries later, Kármán in 1938 and Kolmogorov in 1941 speculated that, in the limit of vanishing viscosity and in the absence of forces, the (mean) energy of 3D incompressible turbulence would asymptotically decline at long time as a power-law $E(t) \propto t^{-n} $. More recently, in 2019, the investigations of Lászlό Székelyhidi and colleagues, using weak (distributional) solutions of the 3D incompressible Euler equations, showed that the total energy decay could be compatible with an almost arbitrary law. Three-dimensional simulations of strong solutions to the Navier-Stokes equations (Matsumoto, 2020) are beginning to give some evidence for non-power law decline of the energy.

What about the 1D Burgers equation? Phenomenology and simulations seemed, until recently, to indicate again a power-law decay. Very recently a rigorous proof was given in our group for this power-law decay under the condition that the initial potential is either the Brownian motion in space (or its fractional generalization introduced by Kolmogorov). 

Very recently Frisch, Pandit, and Roy managed to extend this proof and the simulations and prove non-powerlaw decay for a composite Brownian motion, called here the large-scale multifractal Brownian motion. This seems to be the first case where multifractals appear at large scales.

The Navier-Stokes-Voigt (NSV) model of viscoelastic incompressible fluid has been proposed as a regularization of the three-dimensional Navier-Stokes equations for the purpose of direct numerical simulations. In this talk I will derive several statistical properties of the invariant measures associated with the solutions of the three-dimensional NSV equations. Moreover, I will show that for fixed viscosity, ν > 0, as the regularizing parameter, α, tends to zero, there exists a subsequence of probability invariant measures converging, in a suitable sense, to a strong stationary statistical solution of the three-dimensional Navier-Stokes equations, which is a regularized version of the notion of stationary statistical solutions - a generalization of the concept of invariant measure introduced and investigated by Foias. This fact is also supported by direct numerical simulations, employing the NSV analogue of the Sabra shell model, for small values of the regularizing parameter α compared to the Kolmogorov length scale. However, for larger values of α simulations exhibit multiscaling inertial range, and the dissipation range is displaying low intermittency. These facts provide evidence that the NSV regularization may reduce the stiffness of direct numerical simulations of turbulent flows, with a small impact on the energy containing scales.

Preliminary to the main topic, there has been an intriguing result of L. Arnold, H. Crauel and V. Wisthutz 1983 about stabilization by noise. M. Capinsly, during a talk by Arnold group in Bremen in 1987, conjectured that such result could have a counterpart for SPDEs of fluid mechanics and could lead to a result of improved mixing rate due to transport noise. Recently, Lucio Galeati, Dejun Luo and myself have proved results in this direction. The particular result of this talk is concerned with a difficult case: the heat loss through a boundary when the fluid is turbulent but zero at the boundary, being viscous. We idealize the fluid transport by a white noise in time with suitable space covariance and show that the decay of temperature is improved by the noise. Other related results, for mixing of Euler flows and delayed blow-up of certain PDE models, will be mentioned. 

We study the applicability of artificial intelligence tools to different problems in fluid dynamics, from the search of an optimal navigation strategy in complex environments to data reconstruction from partial measurements of turbulent flows. To solve navigation problems we follow the Reinforcement Learning approach. Here, we focus on finding the path that minimizes the navigation time between two given points in a fluid flow, known as the Zermelo’s problem [1]. Concerning data-assimilation, we explore the capability of Generative Adversarial Network (GAN) to generate missing data in turbulent configurations. In particular, we investigate on a quantitative basis, their use in reconstructing 2d damaged snapshots extracted from a large database of numerical configurations of 3d turbulence in the presence of rotation, a case with multi-scale random features where both large-scale organized structures and small-scale highly intermittent and non- Gaussian fluctuations are present [2,3].

References
[1] Biferale, L., Bonaccorso, F., Buzzicotti, M., Di Leoni, P. C., & Gustavsson, K. (2019). Zermelo’s problem: Optimal point-to-point navigation in 2D turbulent flows using Reinforcement Learning. Chaos: An Interdisciplinary Journal of Nonlinear Science 29.10 (2019): 103138.
[2] Buzzicotti, M., Bonaccorso, F., Di Leoni, P. C., & Biferale, L. (2020). Reconstruction of turbulent data with deep generative models for semantic inpainting from TURB-Rot database. arXiv preprint arXiv:2006.09179 (in press Physycal Review FLuids) 
[3] Biferale, L., Bonaccorso, F., Buzzicotti, M. and di Leoni, P.C., 2020. TURB-Rot. A large database of 3d and 2d snapshots from turbulent rotating flows. arXiv:2006.07469. 

Visual manifestations of intermittency in computations of three-dimensional Navier-Stokes fluid turbulence appear as the familiar low-dimensional or `thin' filamentary sets on which vorticity and strain accumulate as energy cascades down to small scales. The first task of this talk is to investigate how weak solutions of the Navier-Stokes equations can be associated with a cascade and, as a consequence, with an infinite sequence of inverse length scales. It turns out that this sequence converges to a finite limit. The second task is to show how these results scale with integer dimension D=1, 2 & 3. In the light of the occurrence of thin sets, I discuss the mechanism of how the fluid might find the smoothest, most dissipative class of solutions rather than the most singular.

Weak solutions of the incompressible Euler equations which are weak limits of vanishing viscosity Navier-Stokes solutions inherit, in two dimensions,  conservation properties which are not available for general weak solutions. Research has focused on the behavior of energy and the $p$-moments of vorticity, always in fluid domains with no boundary. In this talk I will report on recent work in this direction

Incompressible fluids are described by the Navier-Stokes equations. Yet, it is not yet known whether the Cauchy problem is well posed for these equations, and whether solutions of finite energy are regular or unique. 

From a practical point of view, two interesting questions arise: if there is a blow-up solution, what is his shape?
Is there a loss of unicity after the blow-up?
This talk describes experimental and numerical investigations that are devised to provide helpful intuitions to mathematicians regarding these two questions. In particular, I will show how to detect and characterize areas of “lesser regularity” and how to connect them with possible loss of unicity.