Forces on aligned rising spherical bubbles at low-to-moderate Reynolds number

摘要

In this paper, the dynamic of a pair of equal-sized spherical gas bubbles rising in vertical line within a Newtonian liquid at low-to-moderate Reynolds numbers (Re ≤ 50) is studied. The dynamic momentum balance includes buoyancy, quasi-steady, and unsteady (history) drag, as well as inertial and added-mass body acceleration forces acting on the trailing bubble. This equation has been obtained under the following assumptions: (i) the bubble interaction occurs through the steady non-uniform wake induced by the leading bubble and (ii) the flow structure behind the leading bubble is known, so that proper expressions for the trailing bubble hydrodynamic force and its rising velocity can be derived. We propose an approximate analytical model for predicting the hydrodynamic force and the rise velocity of the trailing bubble. For this aim, we first use the well-known asymptotic far wake velocity solution (AWVS) for an axisymmetric body complementing it with an adequate drag expression. Then, the AWVS is modified via a Galilean transformation by introducing an artificial origin whose position is determined by fitting numerical data of known velocity profiles. Comparisons between the proposed models predictions with those reported experimental and numerical data for dimensionless distance between bubbles s/d in the interval 2 ≤ s/d ≤ 12.5 are presented. The results show that the added-mass body acceleration and the history forces are negligible compared to the other considered forces.