Journal of Fluid Mechanics

  • Journal of Fluid Mechanics / Volume 691 / January 2012 , pp 69-94
  • Copyright © Cambridge University Press 2011 The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution-NonCommercial-ShareAlike licence <http://creativecommons.org/licenses/by-nc-sa/2.5/>. The written permission of Cambridge University Press must be obtained for commercial re-use.
  • DOI: http://dx.doi.org/10.1017/jfm.2011.443 (About DOI), Published online: 13 December 2011
  • OPEN ACCESS

Table of Contents - Volume 691 - 2012

Papers

The effect of a non-zero Lagrangian time scale on bounded shear dispersion

Matthew S. Spydella1 c1 and Falk Feddersena1

a1 Integrative Oceanography Division, Scripps Institution of Oceanography, La Jolla, CA 92093-0209, USA

Abstract

Previous studies of shear dispersion in bounded velocity fields have assumed random velocities with zero Lagrangian time scale (i.e. velocities are $\delta $-function correlated in time). However, many turbulent (geophysical and engineering) flows with mean velocity shear exist where the Lagrangian time scale is non-zero. Here, the longitudinal (along-flow) shear-induced diffusivity in a two-dimensional bounded velocity field is derived for random velocities with non-zero Lagrangian time scale ${\tau }_{L} $. A non-zero ${\tau }_{L} $ results in two-time transverse (across-flow) displacements that are correlated even for large (relative to the diffusive time scale ${\tau }_{D} $) times. The longitudinal (along-flow) shear-induced diffusivity ${D}_{S} $ is derived, accurate for all ${\tau }_{L} $, using a Lagrangian method where the velocity field is periodically extended to infinity so that unbounded transverse particle spreading statistics can be used to determine ${D}_{S} $. The non-dimensionalized ${D}_{S} $ depends on time and two parameters: the ratio of Lagrangian to diffusive time scales ${\tau }_{L} / {\tau }_{D} $ and the release location. Using a parabolic velocity profile, these dependencies are explored numerically and through asymptotic analysis. The large-time ${D}_{S} $ is enhanced relative to the classic Taylor diffusivity, and this enhancement increases with $ \sqrt{{\tau }_{L} } $. At moderate ${\tau }_{L} / {\tau }_{D} = 0. 1$ this enhancement is approximately a factor of 3. For classic shear dispersion with ${\tau }_{L} = 0$, the diffusive time scale ${\tau }_{D} $ determines the time dependence and large-time limit of the shear-induced diffusivity. In contrast, for sufficiently large ${\tau }_{L} $, a shear time scale ${\tau }_{S} = \mathop{ ({\tau }_{L} {\tau }_{D} )}\nolimits ^{1/ 2} $, anticipated by a simple analysis of the particle’s domain-crossing time, determines both the ${D}_{S} $ time dependence and the large-time limit. In addition, the scalings for turbulent shear dispersion are recovered from the large-time ${D}_{S} $ using properties of wall-bounded turbulence.

(Received January 20 2011)

(Reviewed September 28 2011)

(Accepted October 06 2011)

(Online publication December 13 2011)

Key Words:

  • mixing;
  • mixing and dispersion

Correspondence:

c1 Email address for correspondence: mspydell@ucsd.edu

References