Fractal dimension of velocity signals in high-Reynolds-number hydrodynamic turbulence.

In this paper the fractal nature of velocity signals as measured in turbulent flows is investigated. In particular, we study the geometrical nature of the graph (x,f(x)) of the function f that gives one component of the velocity at position x. Special emphasis is given to the effects that a limited resolution of the signal, or natural small-scale cutoffs, have on the estimate of the fractal dimension, and a procedure to account for such finite-size effects is proposed and tested on artificial fractal graphs. We then consider experimental data from three turbulent flows: the wake behind a circular cylinder, the atmospheric surface layer, and the rough-wall zero-pressure-gradient boundary layer developing on the test-section ceiling of the 80\ifmmode\times\else\texttimes\fi{}120 ${\mathrm{ft}}^{2}$ full-scale NASA Ames wind tunnel (the world's largest wind tunnel). The results clearly indicate that at high Reynolds numbers, turbulent velocity signals have a fractal dimension of D\ensuremath{\simeq}1.7\ifmmode\pm\else\textpm\fi{}0.05, very near the value of D=5/3 expected for Gaussian processes with a -5/3 power law in their power spectrum.