Statistics of relative velocity for particles settling under gravity in a turbulent flow

We study the joint probability distributions of separation $R$ and radial component of the relative velocity $V_R$ of particles settling under gravity in a turbulent flow. We also obtain the moments of these distributions and analyze their anisotropy using spherical harmonics. We find that the qualitative nature of the joint distributions remains the same as no-gravity case. Distributions of $V_R$ for fixed values of $R$ show a power-law dependence on $V_R$ for a range of $V_R$; the exponent of the power law depends on the gravity. Effects of gravity are also manifested in the following ways: (a) Moments of the distributions are anisotropic; degree of anisotropy depends on particle's Stokes number, but does not depend on $R$ for small values of $R$. (b) Mean velocity of collision between two particles is decreased for particles having equal Stokes numbers but increased for particles having different Stokes numbers. For the later, collision velocity is set by the difference in their settling velocities.

Figure 1

(a) $m_0\left(R\right)/R^2$ as a function of $R$ for $\text{St}=3.02$ and three different values of $\text{Fr}$. (b) Real space correlation dimension $d_2$ as a function of $\text{St}$ for three different values of $\text{Fr}$.

Figure 2

(a) Contour plots of ${log}_{10}(\mathcal{P}(R,V_R)/R^2)$ for $\text{St}=2.11$ and $\text{Fr}=0.1$. (b) $\mathcal{P}(R,V_R)/R^2$ for $R=0.1, \text{St}=2.11$, plotted for three different values of $\text{Fr}$. Solid lines in (b) have slopes equal to $D_2-4$.

Figure 3

(a) Zeroth moment as a function of $cos\left(\theta \right)$ for $R=0.5$ normalized by its average over $cos\left(\theta \right)$ for $\text{Fr}=0.1$ and four different values of $\text{St}$. (b) $m_0^{\ell }$ for $\ell =0,2$, and 4 plotted as a function of $R$ for $\text{St}=3.02$. (c) Measure of anisotropy $m_0^2/m_0^0$ plotted as function of $R$ for $\text{Fr}=0.1$ and four different values of $\text{St}$.

Figure 4

(a) First moment as a function of $cos\left(\theta \right)$ for $R=0.5$ normalized by its average over $cos\left(\theta \right)$ for $\text{Fr}=0.1$ and four different values of $\text{St}$. (b) $m_1^{\ell }$ for $\ell =0$ and 2 plotted as a function of $R$ for $\text{St}=3.02$. (c) Measure of anisotropy $m_1^2/m_1^0$ plotted as function of $R$ for $\text{Fr}=0.1$ and four different values of $\text{St}$.

Figure 5

Mean $V_{\mathrm{n}}$ plotted as a function of ${\text{St}}_1$ and ${\text{St}}_2$, for $\text{Fr}=0.1$ (a), $\text{Fr}=0.3$ (b), and $\text{Fr}=\infty$ (c).

Figure 6

Mean of $V_{\mathrm{n}}$ at $R=a_1+a_2$ for ${\text{St}}_1={\text{St}}_2$ (top) and for ${\text{St}}_1=0.1$ as a function of ${\text{St}}_2$ (bottom), for different values of $\text{Fr}$.