Power law wave-number spectra of scum on the surface of a flowing fluid.

Abstract

An initial cloud of particles floating on the surface of a flowing fluid will often tend to a fractal pattern. If the wave-number spectrum of the pattern has an observable power law dependence ${k}^{\ensuremath{-}\ensuremath{\rho}}$, then the exponent $\ensuremath{\rho}$ is predicted to be $\ensuremath{\rho}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}D}_{2}\ensuremath{-}1$, where ${D}_{2}$ is the correlation dimension of the fractal attractor. Numerical, experimental, and theoretical results are shown to support this prediction, but it is also found that, when the observable range in $k$ is limited, the predicted power law scaling can be obscured by fluctuations in the $k$ spectrum. The expected behavior can, however, be restored by use of averaging.