Turbulence is a major problem of Physics and Mathematics. Among various fields it is linked with, there is the area of Stochastic Partial Differential Equations, central to this project. The project title, Noise in Fluids, summarizes that we consider various fluid dynamic models (e.g. Navier-Stokes and Euler equations, heat conduction in fluids, particles and polymers in fluids, kinetic equations) and incorporate in them stochastic elements representing a modelisation of turbulent features.
The main purpose of this approach is understanding the reaction of different systems and components to the presence of turbulence. Let us list some of them.
A classical problem is the dissipative effect (the increase of viscosity) that small-scale turbulence has on large scales, identified by Bousinnesq and implemented by Large Eddy Simulation models. We have developed a mathematical scheme where this problem can be rigorously investigated. The extension to 3D models is particularly challenging.
A turbulent fluid then acts on other fields and objects. The modified diffusion properties of temperature is the most natural effect, analogous to the increase of viscosity mentioned above. But the effect on vector fields, like a magnetic field in a conducting fluid, is more complex, involving a deep understanding of the stochastic stretching mechanism.
And a fluid may act on particles embedded into it, like air on water droplets, raising the question of enhanced collision rate due to turbulence. We consider this problem, as well as the problem of stretching of polymers in fluids, problems which increase our understanding of transport and stretching mechanisms.
Ultimately, turbulence in confined plasma, dispersion barriers and zonal flows are even more difficult problems that we hope to approach with suitable stochastic modeling. Here, inverse cascade mechanisms are one of the main concern.
On the theoretical side, referring to the millenium prize problem of singularities for the Navier-Stokes equations, a relevant question is whether turbulence may prevent the development of singularities, thanks to its chaotic rearrangements. A simplified portrait of this question is whether noisy modeled turbulent small scales may prevent the emergence of singularities, which is then one of our main objectives.
