Drag force clustering in bubble swarms

In this article, results of detailed numerical simulations are reported meant to provide a closure relation for the drag force acting on bubbles rising in a dense swarm. The formation of clusters of bubbles in a periodic domain and the effect thereof on the rise velocity and effective drag coefficient on the bubbles are studied. Using smaller bubble sizes than presented in our earlier work, we are also able to refine our correlation for the drag coefficient acting on bubbles rising in a swarm, such that it is applicable for a large range of bubble sizes. The simulations are performed with an advanced Front-Tracking model in which Lagrangian marker points are used to track the gas–liquid interface, while accounting for surface tension and substantial interface deformation. Simulations were performed using periodic domains to simulate rising air bubbles in water from 1.0 mm up to 6.0 mm in diameter. The effect of liquid phase viscosity was also studied to extend the range of validity of the drag correlation. For the 1.0 and 1.5 mm cases, strong horizontal clustering effects are observed. Especially, at high gas fractions, the bubbles tend to form rigid horizontal arrays, which have been shown to strongly increase the drag force acting on the bubbles in the cluster. For viscous liquids, the tendency to form horizontal clusters is lower, and even vertical clustering is observed. The bubble slip velocity was compared with the experimental results of Zenit et al., which agree very well taking into account the differences between simulations and experiments. Based on our simulations, a new drag correlation was proposed, taking into account Eötvös numbers ranging from 0.13 to 4.9, and Morton numbers in the range 3.8 = - log Mo <6.6, and gas hold-ups up to 40% (30% for Eo <0.3). At lower values for -log Mo, the Reynolds number drops to the order of unity, and the correlation overpredicts the drag coefficient, which defines the range of applicability of the currently proposed drag correlation. The correlation itself describes a linear increase of the normalized drag coefficient as a function of the gas hold-up. The strength of linear increase is stronger at lower Eötvös numbers