Noise in Fluids

Introduction

Fluids, in complex regimes, show random features. The aim of this project is to approach several questions around the randomness of fluids by means of a theory that could be called Stochastic Fluid Mechanics. The distinctive feature of this theory, opposite to others that investigated the stochastic features of fluids, is that it is based on the usual continuum mechanics equations, in particular the Navier-Stokes and Euler equations, but suitably modified by the presence of random elements, like an additive or a transport type noise.

Background

Stochastic equations of fluid dynamics have been studied already for three decades, and the number of foundational results is very large. However, two basic directions have been explored only partially:

1. The origin and the form of noise in fluids
2. The consequences of the presence of noise

Project Goals

This project will make progress in these two directions, describing the noise near boundaries due to vortex productions, including the question of intrinsic stochasticity at the boundary.

Noise Propagation

The project will also investigate the propagation of additive noise at small scales to a transport-stretching noise at large scales. The consequences of transport noise on:

• Eddy viscosity
• Enhanced dissipation
• Enhanced coalescence
• Other applications in turbulence and Geophysics

Core Ambition

The most ambitious core of the project is putting together these pieces in a picture that explains the mechanism of regularization by noise for the 3D Navier-Stokes equations. The additive noise at small scales is responsible for a transport-stretching noise at larger scales, which could prevent blow-up of high intensity vortex structures.

Recent Findings

We have already proved recently that a noise, of transport type only, has this regularization effect. However, stretching amplifies vorticity, and new progress is needed to cope with both processes. We aim to use the experimentally observed fact that small scale velocity should be approximately orthogonal to vorticity in high intensity regions.