PRL 114, 214502 (2015)
PHYSICAL
REVIEW
week ending 29 MAY 2015
LETTERS
Angular Statistics of Lagrangian Trajectories in Turbulence W outer J. T. B os,1 B enjam in Kadoch,2 and Kai Schneider3 1LMFA, CNRS UMR 5509, Ecole Centrale de Lyon, Universite de Lyon, Ecully, France ~IUSTI, CNRS UMR 7343, Aix-Marseille University, Marseille, France 'M2P2, CNRS UMR 7340 & CMl, Aix-Marseille University, Marseille, France (Received 20 November 2014; published 29 May 2015) The angle between subsequent particle displacement increments is evaluated as a function of the time lag in isotropic turbulence. It is shown that the evolution of this angle contains two well-defined power laws, reflecting the multiscale dynamics of high-Reynolds number turbulence. The probability density function of the directional change is shown to be self-similar and well approximated by an analytically derived model assuming Gaussianity and independence of the velocity and the Lagrangian acceleration. DOI: 10.1103/PhysRevLett. 114.214502
PACS numbers: 47.27.Jv, 47.27.Gs
Advances in experim ental devices and num erical sim u lations over the last tw o decades have opened the way to a Lagrangian characterization o f turbulent flows [1-3]. The structural description o f the statistical dynam ics o f turbu lence has thereby shifted from the investigation o f spatial correlations o f instantaneous velocity fields to the study of tem poral correlations along fluid particle trajectories. In the Lagrangian reference frame, the spatiotem poral complexity o f turbulence m anifests itself through the spiraling chaotic motion o f fluid particles, changing direction at every time scale. This directional change o f Lagrangian tracers, as a function o f the tim e lag betw een two observations, is the subject o f the present Letter. Instantaneous measures o f the curvature in turbulence have been investigated in the past ten years for academ ic turbulent flows, both in three [4-7] and in two space dim ensions [8,9], Curvature is dom inated by the small-scale structures and contains only little inform ation on the m ultiscale dynam ics o f turbulent flows. M ultiscale dynam ics can be m easured by Lagrangian structure functions [1,3], but those do not contain any direct inform ation on the curvature o f the trajectories. A time scale dependent m easure which is related to the curvature was only recently introduced by Burov eta l. [10]. M ore precisely, in this last work, the directional change o f a particle was introduced, and the characteristics o f this new m easure were investigated in various types o f random walks. In the present Letter, we will show how this m easure can characterize the time correlation o f the direction o f a fluid particle in a turbulent flow. In particular, we will show how the m ultiscale character o f a turbulent flow can be revealed by considering the tim e lag dependence o f the directional change. We define the Lagrangian spatial increm ent as S X {xQ, t , x ) = X ( x 0,t) - X ( x 0, t - T ) ,
(1)
where X (x 0, t) is the position o f a fluid particle at tim e t, passing through point x {) at the reference tim e t = t0 and advected by a velocity field u, i.e., d X / d t = u. The cosine 0031-900 7 /1 5 /1 1 4 (2 1 ) / 2 14502(5)
o f the angle 0 ( f , t ) between subsequent particle incre ments, introduced in [10], is
cos (0 (r, t ))
5 X ( x 0, t, t ) • dY(.r0, t + t , t )
|
PHYSICAL
REVIEW
week ending 29 MAY 2015
LETTERS
Angular Statistics of Lagrangian Trajectories in Turbulence W outer J. T. B os,1 B enjam in Kadoch,2 and Kai Schneider3 1LMFA, CNRS UMR 5509, Ecole Centrale de Lyon, Universite de Lyon, Ecully, France ~IUSTI, CNRS UMR 7343, Aix-Marseille University, Marseille, France 'M2P2, CNRS UMR 7340 & CMl, Aix-Marseille University, Marseille, France (Received 20 November 2014; published 29 May 2015) The angle between subsequent particle displacement increments is evaluated as a function of the time lag in isotropic turbulence. It is shown that the evolution of this angle contains two well-defined power laws, reflecting the multiscale dynamics of high-Reynolds number turbulence. The probability density function of the directional change is shown to be self-similar and well approximated by an analytically derived model assuming Gaussianity and independence of the velocity and the Lagrangian acceleration. DOI: 10.1103/PhysRevLett. 114.214502
PACS numbers: 47.27.Jv, 47.27.Gs
Advances in experim ental devices and num erical sim u lations over the last tw o decades have opened the way to a Lagrangian characterization o f turbulent flows [1-3]. The structural description o f the statistical dynam ics o f turbu lence has thereby shifted from the investigation o f spatial correlations o f instantaneous velocity fields to the study of tem poral correlations along fluid particle trajectories. In the Lagrangian reference frame, the spatiotem poral complexity o f turbulence m anifests itself through the spiraling chaotic motion o f fluid particles, changing direction at every time scale. This directional change o f Lagrangian tracers, as a function o f the tim e lag betw een two observations, is the subject o f the present Letter. Instantaneous measures o f the curvature in turbulence have been investigated in the past ten years for academ ic turbulent flows, both in three [4-7] and in two space dim ensions [8,9], Curvature is dom inated by the small-scale structures and contains only little inform ation on the m ultiscale dynam ics o f turbulent flows. M ultiscale dynam ics can be m easured by Lagrangian structure functions [1,3], but those do not contain any direct inform ation on the curvature o f the trajectories. A time scale dependent m easure which is related to the curvature was only recently introduced by Burov eta l. [10]. M ore precisely, in this last work, the directional change o f a particle was introduced, and the characteristics o f this new m easure were investigated in various types o f random walks. In the present Letter, we will show how this m easure can characterize the time correlation o f the direction o f a fluid particle in a turbulent flow. In particular, we will show how the m ultiscale character o f a turbulent flow can be revealed by considering the tim e lag dependence o f the directional change. We define the Lagrangian spatial increm ent as S X {xQ, t , x ) = X ( x 0,t) - X ( x 0, t - T ) ,
(1)
where X (x 0, t) is the position o f a fluid particle at tim e t, passing through point x {) at the reference tim e t = t0 and advected by a velocity field u, i.e., d X / d t = u. The cosine 0031-900 7 /1 5 /1 1 4 (2 1 ) / 2 14502(5)
o f the angle 0 ( f , t ) between subsequent particle incre ments, introduced in [10], is
cos (0 (r, t ))
5 X ( x 0, t, t ) • dY(.r0, t + t , t )
|
