Preferential concentration vs. clustering in inertial particle transport by random velocity fields

The concept of preferential concentration in the transport of inertial particles by random velocity fields is extended to account for the possibility of zero correlation time and compressibility of the velocity field. It is shown that, in the case of an uncorrelated in time random velocity field, preferential concentration takes the form of a condition on the field history leading to the current particle positions. This generalized form of preferential concentration appears to be a necessary condition for clustering in the uncorrelated in time case. The standard interpretation of preferential concentration is recovered considering local time averages of the velocity field. In the compressible case, preferential concentration occurs in regions of negative divergence of the field. In the incompressible case, it occurs in regions of simultaneously high strain and negative field skewness.