##manager.scheduler.building##: Edificio San Jose

##manager.scheduler.room##: Aula Magna

Date: 2019-07-11 05:30 PM – 05:45 PM

Last modified: 2019-06-10

#### Abstract

Recent public databases of velocity fields obtained in massive numerical simulations of the Navier-Stokes equations open the route to the analysis and the stochastic modeling of spatio-temporal maps of turbulence (see for instance http://turbulence.pha.jhu.edu/).

We begin with a statistical analysis of these empirical maps, and interpret the observed fluctuations of the velocity field in a phenomenological framework, mostly developed by Kolmogorov. The novelty of this study lies in the precise estimation of the singular characteristics of these fields along the temporal dimension, which include the non-differentiable nature of velocity, defined in a simple approach by an average Holder exponent close to 1/3, and additional intermittent (i.e. multifractal) corrections.

We then propose, construct and simulate, a random field able to reproduce these observed statistical trends. It consists in a two-dimensional fractional Gaussian field, known to be a comprehensive stochastic representation of this observed non-differentiable nature, and supplemented in a multiplicative way by a positive random measure (called a Multiplicative Chaos, see Rhodes and Vargas, Probability Surveys, 11, (2014), 315-392), with appropriate long-range dependence, that makes the overall field intermittent and highly non-Gaussian. We mention that the important asymmetrical property of the spatial statistics, that encodes the physics of energy transfers, can be considered as an additional difficulty. Nonetheless, we show that internal correlations imposed along the steps of construction of this synthetic spatio-temporal velocity map allow to reproduce in an accurate fashion this peculiar phenomenon. Various random fields entering in the construction are defined in a rigorous way, allowing the analytical computation of some of their marginals, in particular those which govern the physics of turbulence, as it has been done in other aspects of the stochastic modeling of this emblematic problem of nonlinear physics (Pereira et al. J. Fluid Mech. 794, 369 (2016)).