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 less complex statistical 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 database of isotropic turbulence, generated from numerical simulations of the Navier-Stokes equations over a wide range of Reynolds numbers, we show that circulation exhibits, to excellent accuracy, a bifractal behavior at the highest Reynolds number considered: space filling for low-order moments, close but not identical to the 1941 paradigm of Kolmogorov, and a monofractal with a dimension of about 2.2 for higher orders. This change in character, occurring around the third moment for the highest Reynolds number considered here, is reminiscent of a “phase transition.” We explore the possibility that the transition point moves to higher-order moments as the Reynolds numbers increases—even though one may continue to regard the structure as bifractal for moments of sufficiently high order. We confirm that the circulation properties depend essentially on the area of the loop, not its shape, and that the relevant contour area in figure-eight loops is the scalar area and not the vector area. These results demonstrate an intrinsic simplicity in the statistical structure of turbulence when considering circulation around closed loops, thus motivating a paradigm shift in turbulence research.