Functional renormalisation group for turbulence

Turbulence is a complex nonlinear and multi-scale phenomenon. Although the fundamental underlying equations, the Navier–Stokes equations, have been known for two centuries, it remains extremely challenging to extract from them the statistical properties of turbulence. Therefore, for practical purposes, a sustained effort has been devoted to obtaining an effective description of turbulence, that we may call turbulence modelling, or statistical theory of turbulence. In this respect, the renormalisation group (RG) appears as a tool of choice, since it is precisely designed to provide effective theories from fundamental equations by performing in a systematic way the average over fluctuations. However, for Navier–Stokes turbulence, a suitable framework for the RG, allowing in particular for non-perturbative approximations, has been missing, which has thwarted RG applications for a long time. This framework is provided by the modern formulation of the RG called the functional renormalisation group (FRG). The use of the FRG has enabled important progress in the theoretical understanding of homogeneous and isotropic turbulence. The major one is the rigorous derivation, from the Navier–Stokes equations, of an analytical expression for any Eulerian multi-point multi-time correlation function, which is exact in the limit of large wavenumbers. We propose in this JFM Perspectives article a survey of the FRG method for turbulence. We provide a basic introduction to the FRG and emphasise how the field-theoretical framework allows one to systematically and profoundly exploit the symmetries. We stress that the FRG enables one to describe fully developed turbulence forced at large scales, which was not accessible by perturbative means. We show that it yields the energy spectrum and second-order structure function with accurate estimates of the related constants, and also the behaviour of the spectrum in the near-dissipative range. Finally, we expound the derivation of the spatio-temporal behaviour of $n$ -point correlation functions, and largely illustrate these results through the analysis of data from experiments and direct numerical simulations.