  Multiscale statistical analyses of inertial particle distributions are
presented to investigate the statistical signature of clustering and void
regions in particle-laden incompressible isotropic turbulence.
Three-dimensional direct numerical simulations of homogeneous isotropic
turbulence at high Reynolds number ($Re_\lambda \gtrsim 200$) with up to $10^9$
inertial particles are performed for Stokes numbers ranging from $0.05$ to
$5.0$. Orthogonal wavelet analysis is then applied to the computed particle
number density fields. Scale-dependent skewness and flatness values of the
particle number density distributions are calculated and the influence of
Reynolds number $Re_\lambda$ and Stokes number $St$ is assessed. For $St \sim
1.0$, both the scale-dependent skewness and flatness values become larger as
the scale decreases, suggesting intermittent clustering at small scales. For
$St \le 0.2$, the flatness at intermediate scales, i.e. for scales larger than
the Kolmogorov scale and smaller than the integral scale of the flow, increases
as $St$ increases, and the skewness exhibits negative values at the
intermediate scales. The negative values of the skewness are attributed to void
regions. These results indicate that void regions at the intermediate sales are
pronounced and intermittently distributed for such small Stokes numbers. As
$Re_\lambda$ increases, the flatness increases slightly. For $Re_\lambda \ge
328$, the skewness shows negative values at large scales, suggesting that void
regions are pronounced at large scales, while clusters are pronounced at small
scales.
