Fractal dimension of velocity signals in high Reynolds number hydrodynamic turbulence

A. Scotti
Masters Essay, The John Hopkins University
May 1995, Baltimore MD

ABSTRACT: In this essay we address questions concerning the fractal nature of velocity signals as measured in turbulent flows. In particular, we investigate the geometrical nature of the graph (x, f(x)) of the function that gives one component of the velocity at position x (or at time t= x/V). A better understanding of the geometry of such graphs is crucial in developing new subgrid-scale models, and this is the main motivation for this study. Special emphasis is given to the effects that a limited resolution of the signal, or natural small-scale cut-offs, have on the estimate of the fractal dimension, and a new procedure to account for such finite-size effects is proposed and tested on several artificial fractal graphs. We consider data from three flows, namely atmospheric turbulence, boundary layer turbulence and turbulence in the wake behind a cylinder. The results clearly indicate that a high Reynolds number, turbulent velocity signals have a fractal dimension of D~1.7±0.05, very near the value of D = 5/3 expected for Gaussian processes with a -5/3 power-law in their power spectrum.