Scale-dependent statistics of inertial particle distribution in high Reynolds number turbulence

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 (${\text{Re}}_{\lambda }\gtrsim 200$) are performed. Lagrangian motion of inertial particles are tracked by the one-way coupled point-particle method assuming the Stokes drag. Up to ${10}^9$ inertial particles for each Stokes number ranging from 0.01 to 5.0 are computed. 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 ${\text{Re}}_{\lambda }$ and Stokes number St is assessed. For $\text{St}\sim 1$, both the scale-dependent skewness and flatness values become larger as the scale decreases, suggesting intermittent clustering at small scales. For $\text{St}\le 0.2$, the flatness at intermediate scales, i.e., 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 a signature of void regions. These results indicate that void regions at the intermediate scales are pronounced and intermittently distributed for such small Stokes numbers. As ${\text{Re}}_{\lambda }$ increases, the flatness increases weakly. For ${\text{Re}}_{\lambda }\ge 328$, the skewness shows negative values at large scales, suggesting that even for $\text{St}\sim 1$, void regions are pronounced at large scales, while clusters are pronounced at small scales.

Figure 1

Wavelet spectra $E\left[n_j\right]$ (black) and Fourier spectra $E_n\left(k\right)$ (red) of number density fluctuation at ${\text{Re}}_{\lambda }=204$ for the cases of (a) $\text{St}\le 0.2$ and (b) $\text{St}\ge 0.5$. The error bars for $E\left[n_j\right]$ indicate plus-minus one standard deviation obtained from particle position data at ten time instants.

Figure 2

Spatial distributions of total number density field $n\left(\mathit{x}\right)$ (a, b) and scale contributions $n_j\left(\mathit{x}\right)$ at $j=2$ ($k_2\eta =2.4\times {10}^{-2}$) (c, d), $j=4$ ($k_4\eta =9.7\times {10}^{-2}$) (e, f), and $j=8$ ($k_8\eta =1.6$) (g, h) in a $x_1\text{−}{x}_2$ cross section; (a, c, e, g) $\mathrm{St}=1.0$, (b, d, f, h) $\mathrm{St}=0.05$ for $R{e}_{\lambda }=204$.

Figure 3

Reynolds number dependence of scale-dependent skewness $S\left[n_j\right]$ and flatness $F\left[n_j\right]$ for $\mathrm{St}=1.0$. For $\mathrm{St}=1.0$, solid lines connect the symbols at the scales for which $\mathrm{SNR}\ge 10$, and dotted lines are used otherwise. The error bars indicate plus-minus one standard deviation obtained from particle position data at ten time instants.

Figure 4

Scale-dependent flatness $F\left[n_j\right]$ at $R{e}_{\lambda }=204$ for (a) $\mathrm{St}\le 0.2$ and (b) $\mathrm{St}\ge 0.5$. Solid lines connect the symbols at the scales for which $\mathrm{SNR}\ge 10$, and dotted lines are used otherwise. The error bars indicate plus-minus one standard deviation obtained from particle position data at ten time instants.

Figure 5

Scale-dependent skewness $S\left[n_j\right]$ at $R{e}_{\lambda }=204$ for (a) $\mathrm{St}\le 0.2$ and (b) $\mathrm{St}\ge 0.5$. Solid lines connect the symbols at the scales for which $\mathrm{SNR}\ge 10$, and dotted lines are used otherwise. The error bars indicate plus-minus one standard deviation obtained from particle position data at 10 time instants.

Figure 6

PDF of the normalized scale-dependent particle number density $n_j/\sigma \left[n_j\right]$ for (a) $\mathrm{St}=1.0$ and (b) $\mathrm{St}=0.05$ at $R{e}_{\lambda }=204$. Solid lines are used for the scales for which $\mathrm{SNR}\ge 10$, and dashed lines for the scales for which $\mathrm{SNR}<10$. The dotted lines correspond to the Gaussian distribution $\mathcal{N}(0,1)$.

Figure 7

Schematic figures of (a) cluster- and (b) void-pronounced structures for $n_j$.

Figure 8

Magnified spatial distributions of (a) total number density $n\left(\mathit{x}\right)$ and (b) scale contributions $n_j\left(\mathit{x}\right)$ at $j=4$ ($k_4\eta =9.7\times {10}^{-2}$) for $\mathrm{St}=0.05$ in the same $x_1\text{−}{x}_2$ cross section as Fig. 2.

Figure 9

Wavelet spectra $E\left[n_j\right]$ (a, b), scale-dependent skewness $S\left[n_j\right]$ (c, d), and scale-dependent flatness $F\left[n_j\right]$ (e, f) at $R{e}_{\lambda }=204$ and $\mathrm{St}=1.0$ for (a, c, e) $N_g=512$ and 1024 at fixed $N_p(=1.07\times {10}^9)$ and for (b, d, f) $N_p=1.68\times {10}^7, 1.34\times {10}^8$ and $1.07\times {10}^9$ at fixed $N_g(=1024)$. Dotted lines for $S\left[n_j\right]$ and $F\left[n_j\right]$ indicate $\mathrm{SNR}<10$.