Unsteady development of a deformable bubble rising in a quiescent liquid

An accurate finite-volume based numerical method for the simulation of an isothermal two-phase flow, consisting of a rising deformable bubble translating in a quiescent, unbounded liquid, is presented. This direct simulation method is built on a sharp interface concept and developed on an Eulerian, Cartesian fixed-grid with a cut-cell scheme and marker points to track the moving interface. The unsteady Navier–Stokes equations in both liquid and gas phases are solved separately. The mass continuity and momentum flux conditions are explicitly matched at the true surface phase boundary to determine the evolving interface shape and movement of the bubble. The highlights of this method are that it utilizes a combined Eulerian–Lagrangian approach, and is capable of treating the interface as a sharp discontinuity. A fixed underlying grid is used to represent the control volume. The interface, however, is denoted by a separate set of marker particles which move along with the interface. A quadratic curve fitting algorithm with marker points is used to yield smooth and accurate information of the interface curvatures. This numerical scheme can handle a wide range of density and viscosity ratios. The bubble is assumed to be spherical and at rest initially, but deforms as it rises through the liquid pool due to buoyancy. Additionally, the flow is assumed to be axisymmetric and incompressible. The bubble deformation and dynamic motion are characterized by the Reynolds number, the Weber number, the density ratio and the viscosity ratio. The effects of these parameters on the translational bubble dynamics and shape are given and the physical mechanisms are explained and discussed. Results for the shape, velocity profile and various forces acting on the bubble are presented here as a function of time until the bubble reaches terminal velocity. The range of Reynolds numbers investigated is 1 < Re < 100, and that of Weber number is 1 < We < 10.