# Universality and Thermodynamics of Turbulence

## TL;DR

This paper demonstrates a universal log-scale behavior in turbulence velocity structure functions across experiments and simulations, linking it to multifractal thermodynamics and phase transitions in velocity fields.

## Contribution

It introduces the concept of log-universality in Eulerian velocity structure functions and connects it with multifractal thermodynamics and phase transitions.

## Key findings

- Velocity structure functions follow a log-universal curve.
- Log-universality relates to multifractal free energy properties.
- A phase transition in velocity fields is associated with fluctuating dissipation scales.

## Abstract

We investigate universality of the Eulerian velocity structure functions using velocity fields obtained from the stereoscopic particle image velocimetry (SPIV) technique in experiments and the direct numerical simulations (DNS) of the Navier-Stokes equations. We show that the numerical and experimental velocity structure functions up to order 9 follow a log-universality; we find that they collapse on a universal curve, if we use units that include logarithmic dependence on the Reynolds number. We then investigate the meaning and consequences of such log-universality, and show that it is connected with the properties of a "multifractal free energy", based on an analogy between multifractal and themodynamics. We show that in such a framework, the existence of a fluctuating dissipation scale is associated with a phase transition describing the relaminarisation of rough velocity fields with different Holder exponents. Such a phase transition has been already observed using the Lagrangian velocity structure functions, but was so far believed to be out of reach for the Eulerian data.