Size distribution spectrum of noninertial particles in turbulence

Collision-coalescence growth of noninertial particles in three-dimensional homogeneous isotropic turbulence is studied. Smoluchowski's coagulation equation describes the evolution of the size distribution of particles in this system. By applying a methodology based on turbulence theory, the equation is shown to have a steady-state solution, which corresponds to the Kolmogorov-type power-law spectrum. Direct numerical simulations of turbulence and Lagrangian particles are conducted. The result shows that the size distribution in a statistically steady state agrees accurately with the theoretical prediction.

Figure 1

(a) Nondimensional constant $K_0$ in (19) as a function of the cutoff parameter $q$ for forced locality defined in (16). The data points are from $q=1.004$ to $q=16$. (b) Same as (a) except that the horizontal axis is $\beta ={\log }_2\left(q\right)$. The blue line $\left(\propto {\beta }^{-1/2}\right)$ is the asymptotic form of $K_0$ for $\beta \approx 0$ given in (26). The short black line indicates a slope of $-1/2$.

Figure 2

(a) Time-averaged kinetic energy spectra normalized by Kolmogorov units: ${\epsilon }^{-1/4}{\nu }_{\mathrm{a}}^{-5/4}E\left(k\right)$. (b) Time-averaged dissipation spectra normalized by Kolmogorov units: ${\epsilon }^{-1/4}{\nu }_{\mathrm{a}}^{-5/4}{\left(k\eta \right)}^{2}E\left(k\right)$. The results for runs 1–3 are indicated by the red, green, and blue curves, respectively. Note that the red and green curves are almost identical. The line in the left-hand panel indicates a slope of $-5/3$.

Figure 3

(a) Time-averaged size distribution $n\left(\sigma \right)$ in a statistically steady state. The results for runs 1–3 are indicated by the red, green, and blue curves, respectively. The line indicates a slope of $-2$ from the theoretical prediction. (b) Same as (a), but the distributions are normalized by (34). The horizontal line indicates the nondimensional constant $K_0=0.38$ from the theoretical prediction (23).

Figure 4

(a) Snapshots of the size distribution $n(\sigma ,t)$ for run 1 at $t=500$ s (red), 2000 s (green), 4000 s (blue), 6000 s (purple), and 8000 s (cyan) from left to right. The short line indicates a slope of $-2$ from the theoretical prediction. (b) Same as (a), except that the distributions are normalized by (34). The horizontal line indicates the nondimensional constant $K_0=0.38$ from the theoretical prediction (23).