The diffusive approach of the Richardson model (1926) for separating particle pairs fails to describe faithfully the real dynamics in 3D turbulence. Such a model requires a fast decorreation in time of the velocity field and assumes a single scaling exponent for the velocity fluctuations.
The diffusive approach is valid when diffusion is the dominant transport phenomena and thus the Peclet number has to be very small. In 3D turbulence, transport by diffusion is typically in the same order of magnitude with transport by advection, which indeed put the applicability of the diffusion approach into question.
I recently found in an online presentation (please see the attached picture) that this dimensionless number can be written as the ratio between the eddy-turn-over time at a given scale and the Lagrangian time along trajectories. This last time is the ratio between the considered scale r and the velocity fluctuation at that scale.
What is the origin of this definition? What is the physical explanation of this Lagrangian time? Thanks!
from wikipedia:
The Péclet number (Pe) is a dimensionless number relevant in the study of transport phenomena in fluid flows. It is named after the French physicist Jean Claude Eugène Péclet. It is defined to be the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient.
so in your presentation, in the statement " the Peclet number has to be very small" you need to add TURBULENT Peclet number, i.e. using turbulent diffusivity in the spirit of Richardson diffusivity. You also correctly posted that, re-phrasing G.I. Taylor - any "diffusivity-type approach" cannot work - because of a large number of scales involved, then also the rest of the question is not well defined.
as far as I can understand the snapshot (you need to post a link to the original presentation as well) the discussion is about turbulent Peclet number and it is a ratio of advection time scale, derived from the large, integral scales to the "diffusive" time scale taken from Richardson kind of law in the inertial range.
Thanks Alex for you answer. I found that formula in a presentation of Dr. Alessandra Lanotte few days ago. The time scale for advection is defined here as the eddy turnover time and the diffusive time as the ratio between the scale itself and the first order Eulerian longitudinal velocity structure function. I can't see the meaning of this definition??
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