  Non-spherical particles transported by an anisotropic turbulent flow
preferentially align with the mean shear and intermittently tumble when the
local strain fluctuates. Such an intricate behaviour is here studied for
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. To reproduce the single-time orientation statistics obtained
numerically, it is found that one has to properly account for the combined
effect of the mean shear, for anisotropic velocity gradient fluctuations, and
for the presence of persistent rotating structures in the flow that bias
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. The model is found
to reproduce the long-term effects of an average shear on the mean and the
variance of the fibres angular increment. Still, it does not reproduce an
intricate behaviour observed in numerics for intermediate times: the unfolded
angle is there very similar to a L\'evy walk with distributions of increments
displaying intermediate power-law tails.
