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4 - A Lagrangian View of Turbulent Dispersion and Mixing

Published online by Cambridge University Press:  05 February 2013

By
Jean-François Pinton  and
Brian L. Sawford
Edited by
Peter A. Davidson ,
Yukio Kaneda  and
Katepalli R. Sreenivasan
Jean-François Pinton
Affiliation:
École Normale Supérieure de Lyon
Peter A. Davidson
Affiliation:
University of Cambridge
Yukio Kaneda
Affiliation:
Aichi Institute of Technology, Japan
Katepalli R. Sreenivasan
Affiliation:
New York University
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Summary

Introduction

For good practical reasons, most experimental observations of turbulent flow are made at fixed points x in space at time t and most numerical calculations are performed on a fixed spatial grid and at fixed times. On the other hand, it is possible to describe the flow in terms of the velocity and concentration (and other quantities of interest) at a point moving with the flow. This is known as a Lagrangian description of the flow ((Monin and Yaglom, 1971)). The position of this point x+(t; x0, t0) is a function of time and of some initial point x0 and time t0 at which it was identified or “labelled”. Its velocity is the velocity of the fluid where it happens to be at time t, u+ (t; x0, t0) = u(x+(t), t). We will use the superscript (+) to denote Lagrangian quantities, and quantities after the semi-colon are independent parameters. We refer to a point moving in this way as a fluid particle.

Flow statistics obtained at fixed points and times are known as Eulerian statistics. On the other hand, statistics obtained at specific times by sampling over trajectories, which at some reference times passed through fixed points, are known as Lagrangian statistics. For example, the mean displacement at time t of those particles that passed through the point x0 at time t0 is just 〈x+ (t; x0, t0) − x0〉. In both cases, the measurement time t can be earlier or later than the reference time.

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Ten Chapters in Turbulence , pp. 132 - 175
Publisher: Cambridge University Press
Print publication year: 2012

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