Journal of Fluid Mechanics

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

M. S.  Borgas a1 and B. L.  Sawford a1
a1 CSIRO Division of Atmospheric Research, Private Bag No. 1, PO Mordialloc, Vic. 3195, Australia

Article author query
borgas ms   [Google Scholar] 
sawford bl   [Google Scholar] 


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 [script S] 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 [script R]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 [script R]2 provides an excellent representation in the inertial subrange of Lagrangian stochastic simulations of relative velocity and displacement statistics.

(Received September 4 1989)
(Revised December 20 1990)