Path Instabilities of a Rising or Falling Sphere in a Fluid at Rest

Abstract

Multi-phase flows are ubiquitous in nature and in everyday life surrounding us, impacting us in almost all possible ways. The presence of particles in a flow can change the flow behaviour in an unpredictable manner. The simplest example of a particle-laden flow, that one can think of, is the settling of a single sphere under gravity in a quiescent fluid. This seemingly simple problem has very high relevance in various practical applications ranging from sedimentation of particles for water treatment, process industries, transport of a dense suspension(slurry) through a pipe and even in land reclamation. The settling/ascension of a single sphere, even after having been subject to extensive study for more than a century, remains far from being understood completely. The path and the wake of a falling/rising sphere in a quiescent fluid may be subject to various instabilities depending upon two dimensionless quantities which are sufficient to characterize the motion. One being the Galileo number (Ga), which is the ratio of the net gravity force to the viscous force and the second one being the mass density ratio, which is the ratio of the density of the solid to the density of the fluid. Depending upon Ga and mass density ratio, the sphere can take up various regimes of motion such as vertical, oblique, zigzagging, helical to name a few. This is mainly due to wake instabilities that trigger such path instabilities. Based on Ga and mass density ratio, various regime maps have been proposed in literature. There have been several disagreements regarding the characterization of such paths taken by the sphere. This is due to the strong solid-fluid coupling and the inherent complexity due to triggering of the instabilities in such cases, which is far from being trivial to model numerically and also to test experimentally. The disagreements between different numerical works and different experimental works make the problem hard pressing and tempting to study. Moreover, the settling behaviour of a single sphere can also aid in understanding the collective effects displayed in the settling of dilute suspensions. The goal of the present study is to shed light on the confusion/disagreements in literature until now and characterize various path instabilities. A detailed experimental investigation is conducted to cover the parameter space (regime map) by employment of over 250 different combinations of Ga and mass density ratio to cover as many regimes of motions as possible within the given time framework. The motion of a sphere is tracked in time using high-speed cameras and corresponding path/regime of motion, higher-order statistics like velocity and physical characteristics such as the Strouhal number/ drag coefficient has been computed. The results validate well for some simple regimes of motion for which results from the previous studies perfectly agree with each other. With the confidence obtained after the validation, the current work attempts to draw points of consensus and disagreements with these earlier works for other more controversial regimes. Some regimes, which had only been observed using numerical simulations, have been observed experimentally for the first time. Also, intriguing bi-stable regimes (coexistence of two regimes) have been observed. Moreover, attempt is also made to characterize the suppression of the high-frequency oscillations with increase in the sphere inertia. An update of the regime maps is proposed with the results obtained from the experiments conducted. The results obtained will also serve as an excellent tool for validation of new numerical models, using which the Ga-mass density ratio parameter space can be covered in great detail. Recommendations for future work are given.