Circulation in high Reynolds number isotropic turbulence is a bifractal

Abstract

The turbulence problem at the level of scaling exponents is hard in part because of the multifractal scaling of small scales, which demands that each moment order be treated and understood independently. This conclusion derives from studies of velocity structure functions, energy dissipation, enstrophy density (that is, square of vorticity), etc. However, it is likely that there exist other physically pertinent quantities with uncomplicated structure in the inertial range, potentially resulting in huge simplifications in the turbulence theory. We show that velocity circulation around closed loops is such a quantity. By using a large databases of isotropic turbulence, generated from numerical simulations of the Navier-Stokes equations over a wide range of Reynolds numbers, we show that circulation exhibits a bifractal behavior at the highest Reynolds number considered: space filling for moments up to order $3$ and a mono-fractal with an unchanging dimension of about $2.5$ for higher orders; this change in character roughly at the third-order moment is reminiscent of a "phase transition". We explore the possibility that circulation becomes effectively space filling at much higher Reynolds numbers even though it may technically be regarded as a bifractal. We confirm that the circulation properties depend on only the area of the loop, not its shape; and, for a figure-$8$ loop, the relevant area is the scalar sum of the two segments of the loop.