Rethinking the Reynolds Transport Theorem, Liouville Equation, and Perron-Frobenius and Koopman Operators - Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2020

Rethinking the Reynolds Transport Theorem, Liouville Equation, and Perron-Frobenius and Koopman Operators

Résumé

The Reynolds transport theorem provides a generalised conservation law for any conserved quantity carried by fluid flow through a continuous domain, and underpins all integral and differential analyses of flow systems. It is also intimately linked to the Liouville equation for the conservation of a local probability density function (pdf), and to the Perron-Frobenius and Koopman evolution operators. All of these tools can be interpreted as continuous temporal maps between fluid elements or domains, connected by the integral curves (pathlines) described by a velocity vector field. We present new formulations of these theorems and operators in different spaces. These include (a) spatial maps between different positions in a time-independent flow field, connected by a velocity gradient tensor field, and (b) parametric maps -- expressed using an extended exterior calculus -- between different positions in a manifold, connected by a vector or tensor field. The analyses reveal the existence of multivariate continuous (Lie) symmetries induced by a vector or tensor field associated with a conserved quantity, which will be manifested in all subsidiary conservation laws such as the Navier-Stokes and energy equations. The analyses significantly expand the scope of methods for the reduction of fluid flow and dynamical systems.
Fichier principal
Vignette du fichier
1810.06022.pdf (1.46 Mo) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-02411477 , version 1 (26-12-2020)

Identifiants

Citer

Robert K. Niven, Laurent Cordier, Eurika Kaiser, Michael Schlegel, Bernd R. Noack. Rethinking the Reynolds Transport Theorem, Liouville Equation, and Perron-Frobenius and Koopman Operators. 2020. ⟨hal-02411477⟩
278 Consultations
715 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More