Numerical study of a Taylor bubble rising in stagnant liquids

The dynamics of a Taylor bubble rising in stagnant liquids is numerically investigated using a front tracking coupled with finite difference method. Parametric studies on the dynamics of the rising Taylor bubble including the final shape, the Reynolds number $({\text{Re}}_{T})$, the Weber number $({\text{We}}_{T})$, the Froude number (Fr), the thin liquid film thickness $(w/D)$, and the wake length $({l}_{w}/D)$ are carried out. The effects of density ratio $(\ensuremath{\eta})$, viscosity ratio $(\ensuremath{\lambda})$, E\"otv\"os number (Eo), and Archimedes number (Ar) are examined. The simulations demonstrate that the density ratio and the viscosity ratio under consideration have minimal effect on the dynamics of the Taylor bubble. E\"otv\"os number and Archimedes number influence the elongation of the tail and the wake structures, where higher Eo and Ar result in longer wake. To explain the sudden extension of the tail, a Weber number $({\text{We}}_{l})$ based on local curvature and velocity is evaluated and a critical ${\text{We}}_{l}$ is detected around unity. The onset of flow separation at the wake occurs in between $\text{Ar}=2\ifmmode\times\else\texttimes\fi{}{10}^{3}$ and $\text{Ar}=1\ifmmode\times\else\texttimes\fi{}{10}^{4}$, which corresponds to ${\text{Re}}_{T}$ between 13.39 and 32.55. Archimedes number also drastically affects the final shape of Taylor bubble, the terminal velocity, the thickness of thin liquid film, as well as the wall shear stress. It is found that $w/D=0.32\text{ }{\text{Ar}}^{\ensuremath{-}0.1}$.