Contour Shape Dependency of Circulation Statistics in Homogeneous and Isotropic Turbulence

Statistical moments of the turbulent circulation are complex geometry-dependent functionals of closed oriented contours and present a hard challenge for theoretical understanding. Conveniently defined circulation moment ratios, however, are empirically known to have appreciable geometric dependency only at lower moment orders and for contours which are sized near the bottom of the inertial range, in the situation where they span minimal surfaces of equivalent areas. Resorting to ideas addressed in the framework of the vortex gas model of circulation statistics, which integrates structural and multifractal aspects of the turbulent velocity field, we are able to reproduce, with reasonable accuracy, the observed contour shape dependency of circulation moment ratios, up to high order statistics. A key phenomenological point in our discussion is the assumption that the energy dissipation field, closely related to the local density of thin vortex tubes, is sharply bounded from above at finite Reynolds numbers.