Lagrangian turbulence and the Brownian motion paradox

Abstract

The unique properties of three-dimensional hydrodynamic turbulence depend on the nature of the long-range time correlations as well as the spatial correlations. Although Kolmogorov’s second similarity hypothesis predicts a power-law spatial scaling exponent for the Eulerian velocity fluctuations in agreement with experiments, it also leads, via the Lagrangian velocity time structure function relationship, to particle dispersion predictions that are inconsistent with enhanced diffusion. Recently, a new computational technique has been developed which can generate random power-law correlated fields in any number of dimensions with unlimited scale range. This new method is used to explore the consequences of a proposed set of assumptions about the nature of the time correlations and their relationship to the spatial correlations. In particular, the Brownian motion paradox is examined and it is shown that it can be resolved if the time domain constraint part of Kolmogorov’s second hypothesis is relaxed and replaced with an assumption of space-time isotropy. The proposed modification preserves the observed one-dimensional k−5/3 spatial energy spectrum, allows for enhanced diffusion consistent with Richardson’s law, is consistent with Taylor’s frozen turbulence assumption under the appropriate conditions, and yields an ω−5/3 frequency spectrum for the velocity fluctuations in a frame at rest with respect to the turbulence.