The Lundgren–Monin–Novikov hierarchy: Kinetic equations for turbulence

Rudolf Friedrich; Anton Daitche; Oliver Kamps; Johannes Lülff; Michel Voßkuhle; Michael Wilczek

doi:10.1016/j.crhy.2012.09.009

Structures and statistics of fluid turbulence / Structures et statistiques de la turbulence des fluides

The Lundgren–Monin–Novikov hierarchy: Kinetic equations for turbulence
[La hiérarchie de Lundgren–Monin–Novikov : Des équations cinétiques de la turbulence]

Rudolf Friedrich ^1,² ; Anton Daitche ¹ ; Oliver Kamps ² ; Johannes Lülff ¹ ; Michel Voßkuhle ³ ; Michael Wilczek ¹

¹ Institute for Theoretical Physics, Westfälische Wilhelms-Universität, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany
² Center for Nonlinear Science, Westfälische Wilhelms-Universität, Corrensstr. 2, 48149 Münster, Germany
³ Laboratoire de physique, ENS de Lyon, 46, allée dʼItalie, 69007 Lyon, France

Comptes Rendus. Physique, Volume 13 (2012) no. 9-10, pp. 929-953.

Résumé
Abstract

Nous présentons un aperçu des travaux récents sur la description statistique des écoulements turbulents en terme de fonctions densité de probabilité (PDFs) dans le cadre de la hiérarchie de Lundgren–Monin–Novikov (LMN). Dans ce cadre, des équations dʼévolution pour les PDFs sont dérivées à partir des équations fondamentales décrivant la dynamique des fluides. Le problème de fermeture se pose soit sous la forme dʼune couplage aux PDFs multipoints, soit sous la forme de moyennes conditionnelles qui entrent dans les équations dʼévolution en tant que fonctions inconnues. Nous nous concentrons principalement sur le dernier cas et utilisons les données des simulations numériques directes (DNS) pour calculer les termes non fermés. Nous donnerons donc une introduction de base aux techniques analytiques. Ensuite, nous présenterons quelques applications, en particulier à la statistique de la vorticité en deux dimensions, aux statistiques de vitesse et de vorticité en un point en trois dimensions, à la statistique de la température dans la convection de Rayleigh–Bénard et à la turbulence de Burgers.

We present an overview of recent works on the statistical description of turbulent flows in terms of probability density functions (PDFs) in the framework of the Lundgren–Monin–Novikov (LMN) hierarchy. Within this framework, evolution equations for the PDFs are derived from the basic equations of fluid motion. The closure problem arises either in terms of a coupling to multi-point PDFs or in terms of conditional averages entering the evolution equations as unknown functions. We mainly focus on the latter case and use data from direct numerical simulations (DNS) to specify the unclosed terms. Apart from giving an introduction into the basic analytical techniques, applications to two-dimensional vorticity statistics, to the single-point velocity and vorticity statistics of three-dimensional turbulence, to the temperature statistics of Rayleigh–Bénard convection and to Burgers turbulence are discussed.

Publié le : 2012-11-01

DOI : 10.1016/j.crhy.2012.09.009

Keywords: Turbulence, Lundgren–Monin–Novikov hierarchy, Probability density functions
Mot clés : Turbulence, Hiérarchie de Lundgren–Monin–Novikov, Fonctions densité de probabilité

Affiliations des auteurs :

Rudolf Friedrich ^{1, 2} ; Anton Daitche ¹ ; Oliver Kamps ² ; Johannes Lülff ¹ ; Michel Voßkuhle ³ ; Michael Wilczek ¹

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Rudolf Friedrich; Anton Daitche; Oliver Kamps; Johannes Lülff; Michel Voßkuhle; Michael Wilczek. The Lundgren–Monin–Novikov hierarchy: Kinetic equations for turbulence. Comptes Rendus. Physique, Volume 13 (2012) no. 9-10, pp. 929-953. doi : 10.1016/j.crhy.2012.09.009. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2012.09.009/

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