Statistical mechanics of Beltrami flows in axisymmetric geometry: Theory reexamined

TL;DR

This paper reexamines the statistical mechanics of axisymmetric Beltrami flows, deriving equilibrium states, justifying the selective decay principle, and connecting theoretical predictions with experimental turbulence data.

Contribution

It introduces a simplified thermodynamic framework for axisymmetric Euler flows, linking entropy maximization to energy minimization and providing new insights into turbulence behavior.

Findings

Equilibrium states are Gaussian with Beltrami mean flow.

Entropy maximization is equivalent to energy minimization under constraints.

Theoretical predictions align with von Karman turbulence experiments.

Abstract

A simplified thermodynamic approach of the incompressible axisymmetric Euler equations is considered based on the conservation of helicity, angular momentum and microscopic energy. Statistical equilibrium states are obtained by maximizing the Boltzmann entropy under these sole constraints. We assume that these constraints are selected by the properties of forcing and dissipation. The fluctuations are found to be Gaussian while the mean flow is in a Beltrami state. Furthermore, we show that the maximization of entropy at fixed helicity, angular momentum and microscopic energy is equivalent to the minimization of macroscopic energy at fixed helicity and angular momentum. This provides a justification of this selective decay principle from statistical mechanics. These theoretical predictions are in good agreement with experiments of a von Karman turbulent flow and provide a way to measure the temperature of turbulence and check Fluctuation-Dissipation Relations (FDR). Relaxation equations are derived that could provide an effective description of the dynamics towards the Beltrami state and the progressive emergence of a Gaussian distribution. They can also provide a numerical algorithm to determine maximum entropy states or minimum energy states.