[Une théorie phénoménologique des fluctuations de vitesse Eulérienne et Lagrangienne dans un écoulement turbulent]
Nous présentons une théorie phénoménologique des fluctuations de vitesse dans un écoulement turbulent pleinement développé isotrope et homogène. Nous mettons lʼaccent sur les fluctuations des incréments spatiaux (Eulérien) et temporels (Lagrangien) de vitesse. La nature universelle du phénomène dʼintermittence observé sur les mesures expérimentales et les simulations numériques est complètement pris en compte par les arguments développés par le formalisme multifractal, et ses extensions aux échelles dissipatives et au cadre Lagrangien. Cet article présente les prédictions de cette description multifractale et les compare aux données empiriques. En particulier, des prédictions explicites sont obtenues pour des grandeurs statistiques, comme les fonctions de densité de probabilité et les moments dʼordres supérieurs, des gradients de vitesse et de lʼaccélération. Dans le cadre Eulérien, à un nombre de Reynolds donné, nous montrons que ces prédictions ne dépendent que dʼune fonction à paramètres, appelée spectre de singularités, et dʼune constante régissant la transition entre les régimes inertiels et dissipatifs. Le spectre des singularités Lagrangien est relié à son homologue Eulérien par une transformation basée sur la nature incompressible, homogène et isotrope de lʼécoulement, alors que la constante restante est difficile à estimer à partir des données. Nous montrons finalement que lʼincrément est inadapté à quantifier précisément la nature singulière de la vitesse Lagrangienne. Cela est confirmé par lʼutilisation dʼincréments dʼordres supérieurs non biaisés par la présence de comportements linéaires, comme nous lʼobservons sur la vitesse le long dʼune trajectoire.
A phenomenological theory of the fluctuations of velocity occurring in a fully developed homogeneous and isotropic turbulent flow is presented. The focus is made on the fluctuations of the spatial (Eulerian) and temporal (Lagrangian) velocity increments. The universal nature of the intermittency phenomenon as observed in experimental measurements and numerical simulations is shown to be fully taken into account by the multiscale picture proposed by the multifractal formalism, and its extensions to the dissipative scales and to the Lagrangian framework. The article is devoted to the presentation of these arguments and to their comparisons against empirical data. In particular, explicit predictions of the statistics, such as probability density functions and high order moments, of the velocity gradients and acceleration are derived. In the Eulerian framework, at a given Reynolds number, they are shown to depend on a single parameter function called the singularity spectrum and to a universal constant governing the transition between the inertial and dissipative ranges. The Lagrangian singularity spectrum compares well with its Eulerian counterpart by a transformation based on incompressibility, homogeneity and isotropy and the remaining constant is shown to be difficult to estimate on empirical data. It is finally underlined the limitations of the increment to quantify accurately the singular nature of Lagrangian velocity. This is confirmed using higher order increments unbiased by the presence of linear trends, as they are observed on velocity along a trajectory.
Mot clés : Turbulence, Intermittence, Eulérien et Lagrangien
@article{CRPHYS_2012__13_9-10_899_0,
author = {Laurent Chevillard and Bernard Castaing and Alain Arneodo and Emmanuel L\'ev\^eque and Jean-Fran\c{c}ois Pinton and St\'ephane G. Roux},
title = {A phenomenological theory of {Eulerian} and {Lagrangian} velocity fluctuations in turbulent flows},
journal = {Comptes Rendus. Physique},
pages = {899--928},
publisher = {Elsevier},
volume = {13},
number = {9-10},
year = {2012},
doi = {10.1016/j.crhy.2012.09.002},
language = {en},
}
TY - JOUR AU - Laurent Chevillard AU - Bernard Castaing AU - Alain Arneodo AU - Emmanuel Lévêque AU - Jean-François Pinton AU - Stéphane G. Roux TI - A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows JO - Comptes Rendus. Physique PY - 2012 SP - 899 EP - 928 VL - 13 IS - 9-10 PB - Elsevier DO - 10.1016/j.crhy.2012.09.002 LA - en ID - CRPHYS_2012__13_9-10_899_0 ER -
%0 Journal Article %A Laurent Chevillard %A Bernard Castaing %A Alain Arneodo %A Emmanuel Lévêque %A Jean-François Pinton %A Stéphane G. Roux %T A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows %J Comptes Rendus. Physique %D 2012 %P 899-928 %V 13 %N 9-10 %I Elsevier %R 10.1016/j.crhy.2012.09.002 %G en %F CRPHYS_2012__13_9-10_899_0
Laurent Chevillard; Bernard Castaing; Alain Arneodo; Emmanuel Lévêque; Jean-François Pinton; Stéphane G. Roux. A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows. Comptes Rendus. Physique, Volume 13 (2012) no. 9-10, pp. 899-928. doi : 10.1016/j.crhy.2012.09.002. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2012.09.002/
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