Turbulent pair dispersion as a ballistic cascade phenomenology

Abstract

Since the pioneering work of Richardson in 1926, later refined by Batchelor and Obukhov in 1950, it is predicted that the rate of separation of pairs of fluid elements in turbulent flows with initial separation at inertial scales, grows ballistically first (Batchelor regime), before undergoing a transition towards a super-diffusive regime where the meansquare separation grows as t 3 (Richardson regime). Richardson empirically interpreted this super-diffusive regime in terms of a non-Fickian process with a scale dependent diffusion coefficient (the celebrated Richardson's "4/3rd" law). However, the actual physical mechanism at the origin of such a scale dependent diffusion coefficient remains unclear. The present article proposes a simple physical phenomenology for the time evolution of the mean square relative separation in turbulent flows, based on a scale dependent ballistic scenario rather than a scale dependent diffusive. It is shown that this phenomenology accurately retrieves most of the known features of relative dispersion for particles mean square separation ; among others : (i) it is quantitatively consistent with most recent numerical simulations and experiments for mean square separation between particles (both for the short term Batchelor regime and the long term Richardson regime, and for all initial separations at inertial scales), (ii) it gives a simple physical explanation of the origin of the super diffusive t 3 Richardson regime which naturally builts itself as an iterative process of elementary short-term-scale-dependent ballistic steps, (iii) it shows that the Richardson constant is directly related to the Kolmogorov constant (and eventually to a ballistic persistence parameter) and (iv) in a further extension of the phenomenology, taking into account third order corrections, it robustly describes the temporal asymmetry between forward and backward dispersion, with an explicit connection to the cascade of energy flux across scales. An important aspect of this phenomenology is that it simply and robustly connects long term super-diffusive features to elementary short term mechanisms, and at the same time it connects basic Lagrangian features of turbulent relative dispersion (both at short and long times) to basic Eulerian features of the turbulent field : second order Eulerian statistics control the growth of separation (both at short and long times) while third order Eulerian statistics control the temporal asymmetry of the dispersion process, which can then be directly identified as the signature of the energy cascade and associated to well known exact results as the Karman-Howarth-Monin relation.

Extreme-value statistics from Lagrangian convex hull analysis for homogeneous turbulent Boussinesq convection and MHD convection

“…For an introduction to the rich field of Lagrangian particle-pair dispersion, we refer the reader to the review of Salazar and Collins [56]. In addition to this review, several more recent works [57][58][59] propose new dispersion phenomenologies based on locally ballistic dynamics, an alternative to the classical idea of turbulent diffusion exhibiting scale-dependent diffusivity [19]. Here we briefly recall the basic argument for scaling regimes of pair dispersion.…”

“…Because the observable dispersion regimes and their duration can change as a consequence of different 0 D or hull ℓ , a direct comparison of both figures is difficult. In Navier-Stokes turbulence, the initial turnover time, 0 t , has been shown [57,59] to signal the transition from the ballistic to the inertial range of dispersion, and thus to provide a reference scale of dispersion. We therefore use the initial turnover time 0 t to normalize the dispersion of particles contained in a convex hull, with initial length scale hull ℓ .…”

“…and the rms velocity fluctuations on this scale v 0 D . Recent theoretical[57][58][59] and experimental[60,61] works make use of a key time scale linked to 0 D , the initial nonlinear turnover time v…”

Pair dispersion in inhomogeneous turbulent thermal convection

“…the review articles by Sawford [13] and Salazar et al [14]). It has important applications including oceanic plankton or atmospheric pollutant dispersion [15,16] for which thermal convection is a key ingredient. Three regimes can be distinguished.…”

Potential landscape and flux field theory for turbulence and nonequilibrium fluid systems

“…It is therefore natural to approach the turbulence problem from the perspective of nonequilibrium statistical mechanics [12,13,14,15,16]. The statistical properties of turbulence in connection with the deviation from equilibrium has been investigated from the angles of fluctuation-dissipation theorem (FDT) [17,18,19], fluctuation theorem [20,21,22,23], large deviation theory [24,25,26], and time asymmetry in Lagrangian statistics [27,28,29,30,31] among others.…”