Quantifying the entanglement complexity in turbulent flows

Turbulence be regarded as a dynamically disordered whirl of vortical structures on multiple scales. Complementing statistical measures of the complex flows in terms of correlation functions, here we consider the use of a locally-averaged average crossing number (ACN) of filamentary vortical structures to quantify the dynamical entanglement of vorticity. Using data from direct numerical simulations of homogenous isotropic turbulence which show that vortex filaments can entangle with each other into higher-order structures of ever-increasing complexity, we calculate the ACN for different Reynolds numbers. We describe the evolution of the ACN and compare the topological complexity of such structures in Navier-Stokes turbulence with that of a synthetic Gaussian Random Field with the same two-point correlation in order to distinguish between dynamical and kinematic effects on the topology of the vortex filaments.