Path instabilities and drag in the settling of single spheres

Abstract

The settling behavior of individual spheres in a quiescent fluid was studied experimentally. The dynamics of the spheres was analyzed in the parameter space of particle-to-fluid density ratio ($\Gamma$) and Galileo number ($\mathrm{Ga}$), with $\Gamma \in (1.1, 7.9)$ and $\mathrm{Ga} \in (100, 340)$. The experimental results showed for the first time that the mean trajectory angle with the vertical exhibits a complex behavior as $\mathrm{Ga}$ and $\Gamma$ are varied. Numerically predicted regimes such as Vertical Periodic and Planar Rotating were validated at high $\Gamma$ values. In particular, for the denser spheres, a clear transition from planar to non-planar trajectories was observed, accompanied by the emergence of semi-helical trajectories corresponding to the Planar Rotating Regime. The spectra of trajectory oscillations were also quantified as a function of $\mathrm{Ga}$, confirming the existence of oblique oscillating regimes at both low and high frequencies. The amplitudes of the perpendicular velocities in these regimes were also quantified and compared with numerical simulations in the literature. The terminal velocity and drag of the spheres were found to depend on the particle-to-fluid density ratio, and correlations between the drag coefficient and particle Reynolds number ($Re_p$) as a function of Ga were established, allowing for the estimation of drag and settling velocity using $\mathrm{Ga}$, a control parameter, rather than the response parameter $Re_p$.