L. Chen, S. V. Garimella, J. A. Reizes, and E. Leonardi, “The Development of a Bubble Rising in a Viscous Liquid,” Journal of Fluid Mechanics, Vol. 387, pp. 61‑96, 1999.
The rise and deformation of a gas bubble in an otherwise stationary liquid contained in a closed, right vertical cylinder is investigated using a modiﬁed volume-of-ﬂuid (VOF) method incorporating surface tension stresses. Starting from a perfectly spherical bubble which is initially at rest, the upward motion of the bubble in a gravitational ﬁeld is studied by tracking the liquid–gas interface. The gas in the bubble can be treated as incompressible. The problem is simulated using primitive variables in a control-volume formulation in conjunction with a pressure–velocity coupling based on the simple algorithm. The modiﬁed VOF method used in this study is able to identify and physically treat features such as bubble deformation, cusp formation, breakup and joining. Results in a two-dimensional as well as a three-dimensional coordinate framework are presented. The bubble deformation and its motion are characterized by the Reynolds number, the Bond number, the density ratio, and the viscosity ratio. The effects of these parameters on the bubble rise are demonstrated. Physical mechanisms are discussed for the computational results obtained, especially the formation of a toroidal bubble. The results agree with experiments reported in the literature.
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