On the relationship between the non-local clustering mechanism and preferential concentration

Preferential concentration' (\emph{Phys. Fluids} \textbf{A3}:1169--78, 1991) refers to the clustering of inertial particles in the high-strain, low-rotation regions of turbulence. The centrifuge mechanism’ of Maxey (\emph{J. Fluid Mech.} $\mathbf{\text{174}}$:441–65, 1987) appears to explain this phenomenon. In a recent paper, Bragg & Collins (\emph{New J. Phys.} $\mathbf{\text{16}}$:055013, 2014) showed that the centrifuge mechanism is dominant only in the regime $St\ll 1$, where $St$ is the Stokes number based on the Kolmogorov time scale. Outside this regime, the centrifuge mechanism gives way to a non-local, path-history symmetry breaking mechanism. However, despite the change in the clustering mechanism, the instantaneous particle positions continue to correlate with high-strain, low-rotation regions of the turbulence. In this paper, we analyze the exact equation governing the radial distribution function and show how the non-local clustering mechanism is influenced by, but not dependent upon, the preferential sampling of the fluid velocity gradient tensor along the particle path-histories in such a way as to generate a bias for clustering in high-strain regions of the turbulence. We also show how the non-local mechanism still generates clustering, but without preferential concentration, in the limit where the timescales of the fluid velocity gradient tensor measured along the inertial particle trajectories approaches zero (such as white-noise flows or for particles in turbulence settling under strong gravity). Finally, we use data from a direct numerical simulation of inertial particles suspended in Navier-Stokes turbulence to validate the arguments presented in this study.