The Fractal Dimension of Iso - Vorticity Structures in 3 - Dimensional Turbulence

The fractal dimension of iso-vorticity surfaces is estimated from a 3-dimensional simulation of homogeneous turbulence at moderate Reynolds numbers, performed by Vincent and Meneguzzi. The results are found to be compatible with a recently proposed theory which predicts a crossover from a 2-dimensional geometry at small scales to a fractal geometry at larger scales, with a dimension D = 2.5 + 112, with 1 being the exponent characterizing the scaling of velocity differences. One promising venue for the search of universal aspects in fluid turbulence is the study of universal geometrical structures. Mandelbrot [l] has suggested that such universal structures may be fractal, exhibiting the self-similarity on a range of scales which is at the heart of the scaling properties of turbulence. Some encouraging progress in identifying candidates for universal fractal structures has been achieved experimentally [2], but only little help was provided from theoretical studies; most of these were completely phenomenological [3,4], and theories based directly on the equations of fluid mechanics failed to provide insight towards the existence and the characterization of fractal structures in turbulence. Recently, an approach based on fluid mechanics and on concepts of geometric measure theory was proposed [5], and some progress in calculating the dimensions of level sets of both passive scalars [6,7] and vorticity magnitude [8] were reported. The results concerning the properties of passive scalars were compared directly with experiments [7]. In this letter we offer a comparison of the predictions on the geometries of iso-vorticity surfaces to the results of numerical simulations of homogeneous turbulence at moderate Reynolds numbers [91. We begin with a short summary of the theoretical predictions. Consider the Navier-Stokes equation (1) where v, p and F are the viscosity, pressure and external forcing, respectively. Equation (1) is considered in three spatial dimensions, in a volume V (of linear scale L) and is