Circulation Statistics in Homogeneous and Isotropic Turbulence

TL;DR

This thesis introduces the Vortex Gas Model, a new statistical framework for turbulence circulation, validated through simulations and field theory, explaining multifractal behavior and extreme events in isotropic turbulence.

Contribution

The paper develops and validates a novel Vortex Gas Model incorporating volume exclusion and multifractal characteristics, advancing the statistical understanding of turbulence circulation.

Findings

Vortex Gas Model captures intermittent and polarized vortex behavior.

Model reproduces linearization of circulation moments observed in simulations.

Field theory approach analyzes extreme circulation events via Instantons.

Abstract

This is the final version of a Thesis presented to the PostGrad Program in Physics of the Physics Institute of the Federal University of Rio de Janeiro (UFRJ), as a necessary requirement for the title of Ph.D. in Science (Physics). The development of the Vortex Gas Model (VGM) introduces a novel statistical framework for describing the characteristics of velocity circulation. In this model, the underlying foundations rely on the statistical attributes of two fundamental constituents. The first is a GMC field that governs intermittent behavior and the second constituent is a Gaussian Free field responsible for the partial polarization of the vortices in the gas. The model is revisited in a more sophisticated language, where volume exclusion among vortices is addressed. These additions were subsequently validated through numerical simulations of turbulent Navier-Stokes equations. This revised approach harmonizes with the multifractal characteristics exhibited by circulation statistics, offering a compelling elucidation for the phenomenon of linearization of the statistical circulation moments, observed in recent numerical simulation. In the end, a field theoretical approach, known as Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD) functional method is carried out in the context of circulation probability density function. This approach delves into the realm of extreme circulation events, often referred to as Instantons, through two distinct methodologies: The First investigates the linear solutions and, by a renormalization group argument a time-rescaling symmetry is discussed. Secondly, a numerical strategy is implemented to tackle the nonlinear instanton equations in the axisymmetric approximation. This approach addresses the typical topology exhibited by the velocity field associated with extreme circulation events.