Scale-similar clustering of heavy particles in the inertial range of turbulence

Heavy particle clustering in turbulence is discussed from both phenomenological and analytical points of view, where the $-4/3$ power law of the pair-correlation function is obtained in the inertial range. A closure theory explains the power law in terms of the balance between turbulence mixing and preferential-concentration mechanism. The obtained $-4/3$ power law is supported by a direct numerical simulation of particle-laden turbulence.

Figure 1

Balance between turbulence mixing and preferential concentration is schematically shown in $k$ space. The budget of the number-density variance in the wave-number band $[k_2,k_1]$ (gray area) reads $-\left(A1\right)+\left(A2\right)+\left(B\right)=0$, expressing the balance between total flux $-\left(A1\right)+\left(A2\right)$ and source $\left(B\right)$ due to preferential concentration.

Figure 2

Pair correlation function (PCF) from the DNS. The inset shows the compensated PCF $\left[{\tau }_p^{-2}{\epsilon }^{-2/3}{r}^{4/3}\mathrm{\Theta }\left(r\right)\right]$ which gently varies from 13 to 16 for $75\lesssim r/\eta \lesssim 200$, being comparable with theoretical estimation $16\pm 2$ of Eq. (16) (green band).