Influence of Reynolds number on the motion of settling, bidisperse inertial particles in turbulence

Using direct numerical simulations, we examine the effects of the Taylor Reynolds number, $R_{\lambda }\equiv u^{\prime }\lambda /\nu$ (where $u^{\prime }$, $\lambda$, and $\nu$ denote the fluid root-mean-squared velocity, the Taylor microscale and the fluid kinematic viscosity, respectively), and Froude number, $\text{Fr}\equiv a_{\eta }/g$ (where $a_{\eta }$ is the Kolmogorov acceleration, and $g$ is the acceleration due to gravity), on the motion of small, spherical, settling, bidisperse inertial particles (characterized by their Stokes number $\text{St}\equiv {\tau }_p/{\tau }_{\eta }$, which is the ratio of the particle response time to the Kolmogorov timescale) in isotropic turbulence. Particle accelerations play a key role in the relative motion of bidisperse particles, and we find that reducing Fr leads to an enhancement of the accelerations but a suppression of their intermittency. For Stokes numbers $\text{St}>1$, the effect of $R_{\lambda }$ on the accelerations is enhanced by gravity, since settling causes the particle accelerations to be affected by a larger range of flow scales. The results for the probability density function (PDF) of the particle relative velocities show that even when the particles are settling very fast, turbulence continues to play a key role in their vertical relative velocities, and increasingly so as $R_{\lambda }$ is increased. This occurs because although the settling velocity may be much larger than typical velocities of the turbulence, due to intermittency, there are significant regions of the flow where the turbulence contribution to the particle motion is of the same order as that from gravitational settling. Increasing $R_{\lambda }$ enhances the non-Gaussianity of the relative velocity PDFs, while reducing Fr has the opposite effect, and for fast settling particles, the PDFs become approximately Gaussian. Finally, we observe that low-order statistics such as the radial distribution function and the particle collision kernel are strongly affected by Fr and St, and especially by the degree of bidispersity of the particles. However, we also find that these low-order statistics are very weakly affected by $R_{\lambda }$ when $\text{St}\le O\left(1\right)$, irrespective of the degree of bidispersity. Therefore, although the mechanisms controlling the collision rates of monodisperse and bidisperse particles are different, they share the property of a weak sensitivity to $R_{\lambda }$ when $\text{St}\le O\left(1\right)$.

Figure 1

PDF of the vertical particle acceleration, normalized by $a_{\eta }$, for different St, Fr, and $R_{\lambda }$ combinations. Black, red, and blue lines correspond to $R_{\lambda }=90$, $R_{\lambda }=224$, and $R_{\lambda }=398$, respectively, and circle and triangle symbols denote $\text{Fr}=\infty$ and $\text{Fr}=0.052$, respectively.

Figure 2

PDF of the horizontal particle acceleration, normalized by $a_{\eta }$, for different St, Fr, and $R_{\lambda }$ combinations. Black, red and blue lines correspond to $R_{\lambda }=90$, $R_{\lambda }=224$, and $R_{\lambda }=398$, respectively, and circle and triangle symbols denote $\text{Fr}=\infty$ and $\text{Fr}=0.052$, respectively.

Figure 3

Variance of (a) vertical and (b) horizontal particle accelerations, as a function of St for different Fr and $R_{\lambda }$ combinations. Black, red, and blue lines correspond to $R_{\lambda }=90$, $R_{\lambda }=224$, and $R_{\lambda }=398$, respectively, and circle, square, and triangle symbols denote $\text{Fr}=\infty$, $\text{Fr}=0.3$, and $\text{Fr}=0.052$, respectively.

Figure 4

Kurtosis of (a) vertical and (b) horizontal particle accelerations, as a function of St for different Fr and $R_{\lambda }$ combinations. Black, red, and blue lines correspond to $R_{\lambda }=90$, $R_{\lambda }=224$ and $R_{\lambda }=398$, respectively, and circle, square, and triangle symbols denote $\text{Fr}=\infty$, $\text{Fr}=0.3$, and $\text{Fr}=0.052$, respectively.

Figure 5

PDF of (a, c, e) vertical and (b, d, f) horizontal relative velocity for ${\text{St}}_1=1$, and different ${\text{St}}_2$, Fr, and $R_{\lambda }$ combinations, and for particles with separation $r\in [0,2]\eta$. Black, red, and blue lines correspond to $R_{\lambda }=90$, $R_{\lambda }=224$, and $R_{\lambda }=398$, respectively, and circle and triangle symbols denote $\text{Fr}=\infty$ and $\text{Fr}=0.052$, respectively.

Figure 6

PDF of (a, c, e) vertical and (b, d, f) horizontal relative velocity for ${\text{St}}_1=1$, and different ${\text{St}}_2$, Fr, and $R_{\lambda }$ combinations, and for particles with separation $r\in [18,20]\eta$. Black, red, and blue lines correspond to $R_{\lambda }=90$, $R_{\lambda }=224$, and $R_{\lambda }=398$, respectively, and circle and triangle symbols denote $\text{Fr}=\infty$ and $\text{Fr}=0.052$, respectively.

Figure 7

PDF of (a, c, e) vertical, and (b, d, f) horizontal relative velocity for ${\text{St}}_1=0.1$, and different ${\text{St}}_2$, Fr, and $R_{\lambda }$ combinations, and for particles with separation $r\in [0,2]\eta$. Black, red, and blue lines correspond to $R_{\lambda }=90$, $R_{\lambda }=224$, and $R_{\lambda }=398$, respectively, and circle and triangle symbols denote $\text{Fr}=\infty$ and $\text{Fr}=0.052$, respectively.

Figure 8

Ratio between mean inward relative velocities and the standard deviation of the longitudinal relative velocities as a function of ${\text{St}}_2$, at $r/\eta =0.375$ for different ${\text{St}}_1$, Fr, and $R_{\lambda }$ combinations. Black, red, and blue lines correspond to $R_{\lambda }=90$, $R_{\lambda }=224$, and $R_{\lambda }=398$, respectively and circle, square, and triangle symbols denote $\text{Fr}=\infty$, $\text{Fr}=0.3$, and $\text{Fr}=0.052$, respectively.

Figure 9

RDF at $r/\eta =0.125$, as a function of ${\text{St}}_2$, and for different ${\text{St}}_1$, Fr, and $R_{\lambda }$ combinations. Black, red, and blue lines correspond to $R_{\lambda }=90$, $R_{\lambda }=224$, and $R_{\lambda }=398$, respectively, and circle, square, and triangle symbols denote $\text{Fr}=\infty$, $\text{Fr}=0.3$, and $\text{Fr}=0.052$, respectively.

Figure 10

Mean inward relative velocities at $r/\eta =0.125$, as a function of ${\text{St}}_2$, and for different ${\text{St}}_1$, Fr, and $R_{\lambda }$ combinations. Legend for this plot is the same as that in Fig. 9. Black, red, and blue lines correspond to $R_{\lambda }=90$, $R_{\lambda }=224$, and $R_{\lambda }=398$, respectively and circle, square, and triangle symbols denote $\text{Fr}=\infty$, $\text{Fr}=0.3$, and $\text{Fr}=0.052$, respectively.

Figure 11

Collision kernel at $d/\eta =0.125$ as a function of ${\text{St}}_2$, and for different ${\text{St}}_1$, Fr, and $R_{\lambda }$ combinations. Legend for this plot is the same as that in Fig. 9. Black, red, and blue lines correspond to $R_{\lambda }=90$, $R_{\lambda }=224$, and $R_{\lambda }=398$, respectively, and circle, square, and triangle symbols denote $\text{Fr}=\infty$, $\text{Fr}=0.3$, and $\text{Fr}=0.052$, respectively.