Stochastic model for the alignment and tumbling of rigid fibers in two-dimensional turbulent shear flow

Nonspherical particles transported by an anisotropic turbulent flow preferentially align with the mean shear and intermittently tumble when the local strain fluctuates. Such an intricate behavior is here studied for small, inertialess, rod-shaped particles embedded in a two-dimensional turbulent flow with homogeneous shear. A Lagrangian stochastic model for the rods angular dynamics is introduced and compared to the results of direct numerical simulations. The model consists in superposing a short-correlated random component to the steady large-scale mean shear and can thereby be integrated analytically. Reproducing the single-time orientation statistics obtained numerically requires to take account of the mean shear, of anisotropic velocity gradient fluctuations, and of the presence of persistent rotating structures that combine together to bias cumulative Lagrangian statistics. The model is then used to address two-time statistics. The notion of tumbling rate is extended to diffusive dynamics by introducing the stationary probability flux of the rods unfolded angle, which provides information on the overall, cumulated rotation of the particle. The model is found to reproduce the long-term effects of an average shear on the mean and the variance of the fibers angular increment. Still, for intermediate times, the model fails catching violent fluctuations of the rods rotation that are due to trapping events in coherent, long-living eddies.