Tumbling of Small Axisymmetric Particles in Random and Turbulent Flows

We analyze the tumbling of small nonspherical, axisymmetric particles in random and turbulent flows. We compute the orientational dynamics in terms of a perturbation expansion in the Kubo number, and obtain the tumbling rate in terms of Lagrangian correlation functions. These capture preferential sampling of the fluid gradients, which in turn can give rise to differences in the tumbling rates of disks and rods. We show that this is a weak effect in Gaussian random flows. But in turbulent flows persistent regions of high vorticity cause disks to tumble much faster than rods, as observed in direct numerical simulations [, Phys. Rev. Lett. 109, 134501 (2012)]. For larger particles (at finite Stokes numbers), rotational and translational inertia affects the tumbling rate and the angle at which particles collide, due to the formation of rotational caustics.

Figure 1

Numerical results for Lagrangian correlations $\mathrm{Tr}⟨{\mathbb{S}}_0^2{\mathbb{S}}_t⟩{\tau }_K^3$ (cyan, ▿), $⟨\mathrm{Tr}{\mathbb{S}}_0{\mathbb{O}}_0{\mathbb{S}}_t⟩{\tau }_K^3$ (green, □) and $\mathrm{Tr}⟨{\mathbb{O}}_0^2{\mathbb{S}}_t⟩{\tau }_K^3$ (magenta, ▵) obtained using the JHTDB [17, 18] (see text) in units of ${\tau }_K\equiv 1/\sqrt{\mathrm{Tr}⟨{\mathbb{A}}^{\mathit{T}}\mathbb{A}⟩}$.

Figure 2

Left: tumbling rate ${\stackrel{.}{n}}^2$ for a disk (blue) and a rod (red) in turbulent flow as a function of time, using the JHTDB [17, 18]. Also shown is $\mathrm{Tr}{\mathbb{O}}_t^2{\mathbb{S}}_t$ (green). Right: average squared tumbling rate $⟨{\stackrel{.}{n}}^2⟩$ in turbulence as a function of the aspect ratio $\lambda$ (red, ∘). Equation (4) with data from Fig. 1 is shown as solid red. Also shown is ${\stackrel{.}{n}}^2$ averaged conditional on large values of $\mathrm{Tr}{\mathbb{O}}_t^2{\mathbb{S}}_t$ (green, □) (23% of the sampled data) and conditional on small values of $\mathrm{Tr}{\mathbb{O}}_t^2{\mathbb{S}}_t$ (magenta, △) (77% of the sampled data). Inset: alignment distributions for disks (blue, ⋄) and rods (red, ∘).

Figure 3

Left: $⟨{\stackrel{.}{n}}^2⟩$ as a function of St in a Gaussian random flow (see text). Symbols show results of numerical simulations; solid lines show theory (8). Parameters: $\mathrm{Ku}=0.1$, $\lambda =\sqrt{0.1}$ (blue, ⋄), $\lambda =1$ (green, □), $\lambda =\sqrt{10}$ (red, ∘). Right: distribution of the relative angle $\mathrm{\Delta }\phi$ between the orientation vectors $\mathit{n}$ of two particles close together (at separation $R=0.1\eta$). Black dashed curve shows $\sin \mathrm{\Delta }\phi$. Parameters: $\mathrm{Ku}=1$, $\lambda =\sqrt{10}$. $\mathrm{St}=0$ (cyan, ▿), $\mathrm{St}=1$ (green, □), and $\mathrm{St}=10$ (magenta, ▵).