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An Isaac Newton Institute Workshop

Pattern Formation in Fluid Mechanics

Lagrangian and Eulerian time-scales in passive-scalar turbulence

Authors: Dhrubaditya MITRA (Observatoire de la Cote d'Azur), Rahul Pandit (Indian Institute of Science, Bangalore)

Abstract

A crucial ingradient in understanding turbulence is the characteristic life-time of turbulent eddies. This life-time may be described by (suitable defined) time-dependent correlation function of velocities. It is well known experimentally that the Eulerian different time correlation functions are plagued by the so called "sweeping effect" in which the small scale eddies are transported without distortion by the large scale eddies. Whereas the Lagrangian velocities are conjecture to be free of this sweeping.

We show these explicitly in the Kraichnan model of passive scalar advection. In the Kraichnan model a passive-scalar is advected by a white-in-time velocity which has power-law correlation in space and Gaussian statistics. This is one of the few models in turbulence where explicit analytical calculations can be performed. We show analytically that in an Eulerian framework sweeping appears in differnt time correlation functions of the passive-scalar. We next show how to remove this sweeping using the quasi-Lagrangian framework. We show further that the Lagrangian time-correlation function in the Kraichnan decays exponentially in time with a characteristic time scale which depends via a power-law on the characteristic length scale.

Next using the multifractal model we show how this simpler picture is changed in the case of Navier--Stokes turbulence, and as a consequence we may have to attribute an infinity of time scales to a single characteristic length scale.