Spatiotemporal velocity-velocity correlation function in fully developed turbulence

TL;DR

This paper presents a theoretical prediction for the spatiotemporal velocity-velocity correlation function in homogeneous isotropic turbulence, derived from Navier-Stokes equations using non-perturbative renormalization group methods, and confirms it with numerical simulations.

Contribution

It provides the first analytical fixed-point solution for the space-time correlation function in turbulence from a field-theoretic approach, validated by numerical data.

Findings

Excellent agreement with simulations in inertial and dissipative ranges

Analytical solution derived from non-perturbative renormalization group

Advances understanding of turbulence statistics from fundamental equations

Abstract

Turbulence is an ubiquitous phenomenon in natural and industrial flows. Since the celebrated work of Kolmogorov in 1941, understanding the statistical properties of fully developed turbulence has remained a major quest. In particular, deriving the properties of turbulent flows from a mesoscopic description, that is from Navier-Stokes equation, has eluded most theoretical attempts. Here, we provide a theoretical prediction for the {\it space and time} dependent velocity-velocity correlation function of homogeneous and isotropic turbulence from the field theory associated to Navier-Stokes equation with stochastic forcing. This prediction is the analytical fixed-point solution of Non-Perturbative Renormalisation Group flow equations, which are exact in a certain large wave-number limit. This solution is compared to two-point two-times correlation functions computed in direct numerical simulations. We obtain a remarkable agreement both in the inertial and in the dissipative ranges.