On the use of the Weierstrass-Mandelbrot function to describe the fractal component of turbulent velocity

It is shown that the Weierstrass-Mandelbrot function simulates the irregularity in a turbulent velocity record and yields correct forms for the energy and dissipation spectra. In particular, the universal properties of a corresponding multi-fractal function are demonstrated by showing its ability to reproduce and explain turbulent flow spectra measured near the walls of straight and curved channels and in the obstructed space between a pair of disks corotating in an axisymmetric enclosure. The simulation capabilities of the multi-fractal function strongly suggest that turbulence is fractal in the frequency range of the turbulent energy spectrum where the slope of the logarithm of the spectrum, G , is -3 < G < -1. The scale-independent frequency range of the energy spectrum correctly represented by the multi-fractal function includes the isotropic dissipation subrange (-3 < G < -5/3), the inertial subrange ( G = -5/3), and the "inner" portion of the anisotropic large-scale subrange (-5/3 < G < -1).