Mixing and Dispersion in Flows Dominated by Rotation and Buoyancy by Herman J. H. Clercx & GertJan F. Van Heijst

Author:Herman J. H. Clercx & GertJan F. Van Heijst
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(5)

where is the universal constant in the inertial range scaling law for the Eulerian second-order velocity structure function, which has a well-known value of approximately 2.1, see Sreenivasan (1995) (interestingly, this relation directly bridges Eulerian and Lagrangian approaches). Note that the first equation in (5) arises from a purely kinematic argument, trivially stating that at short times, a Taylor expansion of particles separation time dependency gives

(6)

with the second order Eulerian structure function estimated at a scale given by the initial separation . In the classical cascade model of turbulence, corresponds to the eddy turnover time at the scale and may be interpreted as the time for which the two fluid elements “remember” their initial relative velocity as they move in the same eddy of size . For times on the order of , this eddy breaks up, and the growth of the pair separation is then expected to undergo a transition to Richardson-Obukhov scaling. According to this prediction, at short times (namely ), relative dispersion is ballistic and initial separation dependent, while it accelerates for at the same time as it loses the memory of its initial separation. For times much larger (namely ) a Brownian-like dispersion is recovered as the two particles evolve then without any correlation. The aim of this study was to investigate this scenario by a systematic analysis of dispersion of pairs of particles in a highly turbulent flow, emphasizing the role of initial separation of the particles.

In his seminal 1926 article, Richardson gave an interpretation of turbulent super-diffusion in terms of a non-Fickian process which could be locally modeled as a normal diffusion process, but with a scale dependent diffusion coefficient which depends on particles separation D, according to the celebrated Richardson’s 4/3rd law: . Richardson (1926) conjectured such a scale dependent diffusive scenario from an empirical short time, scale by scale, analysis of local diffusion properties over a wide range of phenomena, from diffusion of oxygen into nitrogen, to the diffusion of cyclones in the atmosphere (Fig. 5b), such that at each scale the mean square separation could be locally written as . It is now accepted that his derivation of the 4 / 3rd law was at the same time fortuitous and the result of his unique intuition (Sawford 2001). Richardson also showed that such a non-Fickian diffusion resulted in a cubic super-diffusive growth of the mean square separation of pairs of particles according to the law , where is the turbulent energy dissipation rate and g a universal constant since known as the Richardson constant.

Fig. 5 a Qualitative illustration of the non-normal dispersion of a dense cluster of particles as proposed in Richardson’s original 1926 article. b Original empirical derivation of the “4/3rd” law by Richardson. (Both figures are taken from Richardson’s seminal article on relative dispersion (Richardson 1926)

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