The impact of hydrodynamic interactions on the preferential concentration of inertial particles in turbulence

We consider a dilute gas of inertial particles transported by the turbulent flow. Due to inertia the particles concentrate preferentially outside vortices. The pair-correlation function of the particles' concentration is known to obey at small separations a power-law with a negative exponent, if the hydrodynamic interactions between the particles are neglected. The divergence at zero separation is the signature of the random attractor asymptoted by the particles' trajectories at large times. However the hydrodynamic interactions produce a repulsion between the particles that is non-negligible at small separations. We introduce equations governing the repulsion and show it smoothens the singular attractor near the particles where the pair correlation function saturates. The effect is most essential at the Stokes number of order one, where the correlations decrease by a factor of a few.