Eulerian space-time correlations in turbulent flows: a renormalisation group approach

Calculating, from Navier-Stokes equation, the statistical properties of homogeneous and isotropic turbulence remains an unsolved issue. The theoretical challenge to overcome is that multi-point Eulerian correlations are determined by an infinite hierarchy of coupled dynamical equations which needs to be closed. The functional renormalisation group (FRG) offers an interesting theoretical tool to tackle this problem. In these lectures, I will present the bases of this formalism, which is a modern implementation of Wilson’s original ideas of the RG. I will then show how symmetries can be deeply exploited within this formalism. They indeed allow for a controlled closure of the flow equations for correlation functions in a specific limit which is the limit of large wave-numbers (small-scales). I will then explain how the solution of these equations can be derived, which provides the form of the space-time dependence of multi-point Eulerian correlations of the turbulent velocity in the limit of large wave-numbers. I will finally compare this result with data from experiments and direct numerical simulations.