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.
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.
Mot clés : Turbulence, Hiérarchie de Lundgren–Monin–Novikov, Fonctions densité de probabilité
Rudolf Friedrich 1, 2; Anton Daitche 1; Oliver Kamps 2; Johannes Lülff 1; Michel Voßkuhle 3; Michael Wilczek 1
@article{CRPHYS_2012__13_9-10_929_0, author = {Rudolf Friedrich and Anton Daitche and Oliver Kamps and Johannes L\"ulff and Michel Vo{\ss}kuhle and Michael Wilczek}, title = {The {Lundgren{\textendash}Monin{\textendash}Novikov} hierarchy: {Kinetic} equations for turbulence}, journal = {Comptes Rendus. Physique}, pages = {929--953}, publisher = {Elsevier}, volume = {13}, number = {9-10}, year = {2012}, doi = {10.1016/j.crhy.2012.09.009}, language = {en}, }
TY - JOUR AU - Rudolf Friedrich AU - Anton Daitche AU - Oliver Kamps AU - Johannes Lülff AU - Michel Voßkuhle AU - Michael Wilczek TI - The Lundgren–Monin–Novikov hierarchy: Kinetic equations for turbulence JO - Comptes Rendus. Physique PY - 2012 SP - 929 EP - 953 VL - 13 IS - 9-10 PB - Elsevier DO - 10.1016/j.crhy.2012.09.009 LA - en ID - CRPHYS_2012__13_9-10_929_0 ER -
%0 Journal Article %A Rudolf Friedrich %A Anton Daitche %A Oliver Kamps %A Johannes Lülff %A Michel Voßkuhle %A Michael Wilczek %T The Lundgren–Monin–Novikov hierarchy: Kinetic equations for turbulence %J Comptes Rendus. Physique %D 2012 %P 929-953 %V 13 %N 9-10 %I Elsevier %R 10.1016/j.crhy.2012.09.009 %G en %F CRPHYS_2012__13_9-10_929_0
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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