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Mechanisms for the clustering of inertial particles in the inertial range of isotropic turbulence

In this paper, we consider the physical mechanism for the clustering of inertial particles in the inertial range of isotropic turbulence. We analyze the exact, but unclosed, equation governing the radial distribution function (RDF) and compare the mechanisms it describes for clustering in the dissipation and inertial ranges. We demonstrate that in the limit ${St_r\ll1}$, where $St_r$ is the Stokes number based on the eddy turnover timescale at separation $r$, the clustering in the inertial range can be understood to be due to the preferential sampling of the coarse-grained fluid velocity gradient tensor at that scale. When ${St_r\gtrsim\mathcal{O}(1)}$ this mechanism gives way to a non-local clustering mechanism. These findings reveal that the clustering mechanisms in the inertial range are analogous to the mechanisms that we identified for the dissipation regime (see \emph{New J. Phys.} \textbf{16}:055013, 2014). Further, we discuss the similarities and differences between the clustering mechanisms we identify in the inertial range and the "sweep-stick" mechanism developed by Coleman \& Vassilicos (Phys. Fluids 21:113301, 2009). We argue that when ${St_r\ll1}$ the sweep-stick mechanism is equivalent to our mechanism in the inertial range if the particles are suspended in Navier-Stokes turbulence, but that the sweep-stick mechanism breaks down for ${St_r\gtrsim\mathcal{O}(1)}$. The argument also explains why the sweep-stick mechanism is unable to predict particle clustering in kinematic simulations. We then consider the closed, model equation for the RDF given in Zaichik \& Alipchenkov (Phys. Fluids. 19:113308, 2007) and use this, together with the results from our analysis, to predict the analytic form of the RDF in the inertial range for ${St_r\ll1}$, which, unlike that in the dissipation range, is not scale-invariant. The results are in good agreement with direct...