Orientation and slip statistics of spheroids settling in turbulence

Abstract

Small spheroids settling through turbulence are turned broadside-on by fluid inertia and tilted away by velocity gradients. We describe their orientation and slip statistics through the crossover from random orientation to alignment, in the overdamped limit. With quasi-static Stokes drag the leading inertial torque is an exact gradient flow on the orientation sphere, its potential quadratic in the cosine of the tilt and only rescaled by mobility anisotropy. The orientation performs an overdamped random walk in a quadratic well, and isotropic Gaussian white noise gives exactly a Boltzmann distribution. For forcing of finite correlation time, a closure keeps this form with its width set by linear response; it recovers the uniform density as settling vanishes and, for fixed-orientation forcing statistics, the aligned coefficient of rapid-settling theory, and errs most in the crossover. The root-mean-square lateral slip peaks where the alignment rate approaches the Kolmogorov rate, as faster settling strengthens the sideways drift at a given tilt but suppresses the tilt; the nearly shape-independent ratio, which tensor-forced simulations reproduce, lets a measured peak constrain the torque coefficient or dissipation rate for given forcing statistics. Dissipation intermittency enters as a lognormal superposition of Boltzmann distributions, which fattens the tails but leaves the leading aligned mean-square tilt unchanged. The superposition reproduces the shape of plate orientation distributions from direct numerical simulations, where a single Boltzmann distribution fails; published moments confirm the predicted spread beyond the weakest torques, and at strong alignment the measured fourth-order tilt moment lies between the two predictions.