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The small-scale structure of acceleration correlations and its role in the statistical theory of turbulent dispersion

Published online by Cambridge University Press:  26 April 2006

M. S. Borgas  and
B. L. Sawford
M. S. Borgas
Affiliation:
CSIRO Division of Atmospheric Research, Private Bag No. 1, PO Mordialloc, Vic. 3195, Australia
B. L. Sawford
Affiliation:
CSIRO Division of Atmospheric Research, Private Bag No. 1, PO Mordialloc, Vic. 3195, Australia
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Abstract

Some previously accepted results for the form of one- and two-particle Langrangian turbulence statistics within the inertial subrange are corrected and reinterpreted using dimensional methods and kinematic constraints. These results have a fundamental bearing on the statistical theory of turbulent dispersion.

One-particle statistics are analysed in an inertial frame [Sscr ] moving with constant velocity (which is different for different realizations) equal to the velocity of the particle at the time of labelling. It is shown that the inertial-subrange form of the Lagrangian acceleration correlation traditionally derived from dimensional arguments constrained by the property of stationarity, ${\cal C}_0^{(a)}\overline{\epsilon}/\tau $, where ${\cal C}_0^{(a)}$ is a universal constant, $\overline{\epsilon}$ is the mean rate of dissipation of turbulence kinetic energy and τ is the time lag, is kinematically inconsistent with the corresponding velocity statistics unless ${\cal C}_0^{(a)} = 0$. On the other hand, velocity and displacement correlations in the inertial subrange are non-trivial and the traditional results are confirmed by the present analysis. Remarkably, the universal constant ${\cal C}_0$ which characterizes these latter statistics in the inertial subrange is shown to be entirely prescribed by the inner (dissipation scale) acceleration covariance; i.e. there is no contribution to velocity and displacement statistics from inertial-subrange acceleration structure, but rather there is an accumulation of small-scale effects.

In the two-particle case the (cross) acceleration covariance is deduced from dimensional arguments to be of the form $\overline{\epsilon}t_1^{-1}{\cal R}_2(t_1/t_2)$ in the inertial subrange. In contrast to the one-particle case this is non-trivial since the two-particle acceleration covariance is non-stationary and there is therefore no condition which constraints [Rscr ]2 to a form which is kinematically inconsistent with the corresponding velocity and displacement statistics. Consequently it is possible for two-particle inertial-subrange acceleration structure to make a non-negligible contribution to relative velocity and dispersion statistics. This is manifested through corrections to the universal constant appearing in these statistics, but does not otherwise affect inertial-subrange structure. Nevertheless, these corrections destroy the simple correspondence between relative- and one-particle statistics traditionally derived by assuming that two-particle acceleration correlations are negligible within the inertial subrange.

A simple analytic expression which is proposed as an example of the form of [Rscr ]2 provides an excellent representation in the inertial subrange of Lagrangian stochastic simulations of relative velocity and displacement statistics.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 228 , July 1991 , pp. 295 - 320
Copyright
© 1991 Cambridge University Press

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