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Turbulence of generalised flows in two dimensions

Published online by Cambridge University Press:  05 December 2019

Simon Thalabard Open the ORCID record for Simon Thalabard [Opens in a new window]  and
Jérémie Bec Open the ORCID record for Jérémie Bec [Opens in a new window]
Simon Thalabard*
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, IMPA, 22460-320Rio de Janeiro, Brazil
Jérémie Bec
Affiliation:
MINES ParisTech, PSL Research University, CNRS, CEMEF, 06904Sophia-Antipolis CEDEX, France
*
†Email address for correspondence: simon.thalabard@ens-lyon.org
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Abstract

This paper discusses the generalised least-action principle and the associated concept of generalised flow introduced by Brenier (J. Am. Math. Soc., vol. 2, 1989, pp. 225–255), from the perspective of turbulence modelling. In essence, Brenier’s generalised least-action principle is a probabilistic generalisation of Arnold’s geometric interpretation of ideal fluid mechanics, whereby strong solutions to the Euler equations are deduced from minimising an action over Lagrangian maps. While Arnold’s framework relies on the deterministic concept of Lagrangian flow, Brenier’s least-action principle describes solutions to the Euler equations in terms of non-deterministic generalised flows, namely probability measures over sets of Lagrangian trajectories. In concept, generalised flows seem naturally fit to describe turbulent Lagrangian trajectories in terms of stochastic processes, an approach that originates from Richardson’s seminal work on turbulent dispersion. Still, Brenier’s generalised least-action principle has so far hardly found any practical application in the realm of fluid mechanics, let alone for turbulence modelling. The purpose of the present paper is to provide a hydrodynamical perspective on Brenier’s principle, and to assess its skills at coarse graining the Lagrangian motion of fluid particles, in order to reconstruct the space–time dynamics of the underlying Eulerian velocity fields. In practice, we rely on a statistical-mechanics interpretation of the concept of generalised flows, whereby the latter become akin to statistical ensembles of suitably defined ‘permutation flows’. We then employ Monte Carlo techniques to numerically solve the generalised least-action principle, and analyse the Eulerian and Lagrangian statistical features of the associated generalised flows. For simplicity, we restrict ourselves to two dimensions of space and consider situations of increasing complexity, ranging from solid rotation and cellular flows to freely decaying two-dimensional turbulence. Our analysis highlights a major caveat of the generalised variational principle. When used over long time lags, e.g. longer than a well-defined hydrodynamic turnover time, it generates artificial generalised flows, with non-physical statistical features. We argue that this limitation is not specifically inherent to Brenier’s formulation, but rather to the variational framework being formulated as a two-end boundary-value problem. When appropriately used over sufficiently short times, the generalised least-action principle is however relevant, and generalised flows can be explicitly constructed, that represent a space–time coarse graining of the underlying dynamics. We show numerical evidence that Brenier’s principle may then even accommodate irreversible Eulerian behaviours. This suggests that, if carefully used, generalised variational formulations could provide new tools to coarse grain genuine multi-scale hydrodynamics.

JFM classification

Mathematical Foundations: Variational methods Turbulent Flows: Turbulence modelling Mathematical Foundations: Topological fluid dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A49
Copyright
© 2019 Cambridge University Press

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