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 $\left({\text{Re}}_T\right)$, the Weber number $\left({\text{We}}_T\right)$, the Froude number (Fr), the thin liquid film thickness $\left(w/D\right)$, and the wake length $\left(l_w/D\right)$ are carried out. The effects of density ratio $\left(\eta \right)$, viscosity ratio $\left(\lambda \right)$, Eötvös 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. Eötvös 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 $\left({\text{We}}_l\right)$ 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 $\text{Ar}=2\times {10}^3$ and $\text{Ar}=1\times {10}^4$, 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 $w/D=0.32{\text{Ar}}^{-0.1}$.

Figure 1

Computational domain for simulating a Taylor bubble rising in stagnant liquids.

Figure 2

(Color online) The regime map of experimentally observed terminal shapes of single spherical bubble rising in different flow regimes: (a) spherical, (b) oblate ellipsoidal, (c) disk-shaped ellipsoidal, (d) oblate ellipsoidal cap, (e) spherical cap with closed and steady wake, (f) spherical cap with open and unsteady wake, (g) smooth and steady skirted shape, and (h) wavy and unsteady skirted shape [34, 35]. In the figure, three filled squares together with simulated shapes represent three simulations conducted: (i) $\text{Eo}=3.1$, $\text{Mo}=1.2\times {10}^{-1}$; (ii) $\text{Eo}=327.0$, $\text{Mo}=36.3$; and (iii) $\text{Eo}=32.7$, $\text{Mo}=1.4\times {10}^{-4}$.

Figure 3

(Color online) The simulated bubble shapes for (i) $\text{Eo}=3.1$, $\text{Mo}=1.2\times {10}^{-1}$; (ii) $\text{Eo}=327.0$, $\text{Mo}=36.3$; and (iii) $\text{Eo}=32.7$, $\text{Mo}=1.4\times {10}^{-4}$. In these cases, the density ratio and viscosity ratio are set to $\eta =100$ and $\lambda =100$, respectively.

Figure 4

(Color online) Velocity vectors (left half) and streamlines (right half) of a rising Taylor bubble at a frame of reference moving with the bubble nose. (a) $L_0=4r_0$, (b) $L_0=5r_0$, and (c) enlarged view of (a) near the tail region.

Figure 5

(Color online) Effect of density ratio $\left(\eta \right)$ on the bubble terminal shapes, velocity vectors (left half), streamlines (right half), and wake lengths. (a) $\eta =25$ (case 1), (b) $\eta =50$ (case 2), and (c) $\eta =100$ (case 5).

Figure 6

(Color online) Effect of viscosity ratio $\left(\lambda \right)$ on the bubble terminal shapes, velocity vectors (left half), streamlines (right half), and wake lengths. (a) $\lambda =10$ (case 3), (b) $\lambda =50$ (case 4), and (c) $\lambda =100$ (case 5).

Figure 7

(Color online) Effect of Eötvös number (Eo) on the bubble shapes and wake patterns. (a) $\text{Eo}=304$ (case 6), (b) $\text{Eo}=203$ (case 5), (c) $\text{Eo}=152$ (case 7), and (d) $\text{Eo}=122$ (case 8). At a frame of reference moving with the bubble, vectors (left half) and streamlines (right half) of each figure show the velocity field within and around each Taylor bubble. The enlarged view at the tail regions for different Eo’s is given in (e). The tail tips from the top to the bottom represent a progressive reduction in surface tension forces.

Figure 8

(Color online) Final shapes and wake structures under the effect of Archimedes number (Ar): (a) $\text{Ar}=1\times {10}^2$ (case 9), (b) $\text{Ar}=6\times {10}^2$ (case 10), (c) $\text{Ar}=2\times {10}^3$ (case 11), (d) $\text{Ar}=1\times {10}^4$ (case 12), (e) $\text{Ar}=4\times {10}^4$ (case 5), (f) $\text{Ar}=9\times {10}^4$ (case 13), and (g) $\text{Ar}=2\times {10}^5$ (case 14). In each figure, the left half provides vectors at a frame of reference moving with the bubble and on the right half, it shows the corresponding streamlines.

Figure 9

(a) Comparison of simulated results with the correlations from Viana et al. [37] for cases where $\text{Eo}>40$. (b) Correlation between $w/D$ and Ar.

Figure 10

(Color online) Enlarged view of the tails for different Ar’s. The lines from the top to the bottom represent cases in Fig. 8 with Ar’s of $1\times {10}^2$, $2\times {10}^3$, $1\times {10}^4$, $4\times {10}^4$, and $9\times {10}^4$, and the corresponding ${\text{We}}_l$’s are 0.11, 0.29, 0.77, 1.29, and 1.50, respectively.

Figure 11

Phase diagram of the tail shape in Eo and Ar space of a Taylor bubble rising in a quiescent fluid.

Figure 12

(a) Wall shear stress under the effect of Ar. $z/D=0$ represents the bubble nose. (b) Maximum value of wall shear stress versus ${\text{log}}_{10}\text{Ar}$.