Numerical study of wall effects on buoyant gas-bubble rise in a liquid-filled finite cylinder

Abstract

The wall effects on the axisymmetric rise and deformation of an initially spherical gas bubble released from rest in a liquid-filled, finite circular cylinder are numerically investigated. The bulk and gas phases are considered incompressible and immiscible. The bubble motion and deformation are characterized by the Morton number (Mo), Eötvös number (Eo), Reynolds number (Re), Weber number (We), density ratio, viscosity ratio, the ratios of the cylinder height and the cylinder radius to the diameter of the initially spherical bubble ( H*=H/d0, R*=R/d0). Bubble rise in liquids described by Eo and Mo combinations ranging from (1,0.01) to (277.5,0.092), as appropriate to various terminal state Reynolds numbers (ReT) and shapes have been studied. The range of terminal state Reynolds numbers includes 0.02<ReT<70 . Bubble shapes at terminal states vary from spherical to intermediate spherical-cap-skirted. The numerical procedure employs a front tracking finite difference method coupled with a level contour reconstruction of the front. This procedure ensures a smooth distribution of the front points and conserves the bubble volume. For the wide range of Eo and Mo examined, bubble motion in cylinders of height H*=8 and R*> or =3 , is noted to correspond to the rise in an infinite medium, both in terms of Reynolds number and shape at terminal state. In a thin cylindrical vessel (small R*), the motion of the bubble is retarded due to increased total drag and the bubble achieves terminal conditions within a short distance from release. The wake effects on bubble rise are reduced, and elongated bubbles may occur at appropriate conditions. For a fixed volume of the bubble, increasing the cylinder radius may result in the formation of well-defined rear recirculatory wakes that are associated with lateral bulging and skirt formation. The paper includes figures of bubble shape regimes for various values of R*, Eo, Mo, and ReT. Our predictions agree with existing results reported in the literature.

FIG. 1

Geometry of an axisymmetric bubble in a cylindrical tube.

FIG. 2

Characteristic spherical shape of the bubble at the attainment of terminal velocity for Eo = 1.0 and Mo = 0.01. Here, H* = 8 and R* = 3.

FIG. 3

Characteristic shape of an intermediate spherical-cap– skirted bubble at the attainment of terminal velocity for Eo = 97.1 and Mo = 0.971. Here, H* = 8 and R* = 3.

FIG. 4

Effects of changing R* on the rising bubble shape: Mo = 0.971, Eo = 97.1. The aspect ratios H*:R* for (a), (b), (c), (d), and (e) are 8:6, 8:4, 8:3, 8:2, and 8:1, respectively. Figure drawn to scale and the right-hand boundaries represent cylinder walls in cases (a)–(e).

FIG. 5

Instantaneous Reynolds number corresponding to various cases given in Fig. 4.

FIG. 6

Comparison of numerically predicted terminal bubble shape and wake structure with that of the experimental results given in [46][Fig. 1(a)]. Here, Eo = 158.4 (Bo = 39.6) and Mo = 0.065. Left-hand frame, numerical prediction; right-hand frame, experimental results (Reprinted with permission from [46]. Copyright 1976, American Institute of Physics).

FIG. 7

Comparison of the numerical results of rise of air bubble in the presence of lateral walls with that of experimental results given in [18]. Here, Mo = 2.6 × 10⁻¹¹, Eo = 11, and 1.0 ≤ R* ≤ 1.67.

FIG. 8

Bubble positions for $\frac{{\mu }_l}{{\mu }_g}=\frac{{\rho }_l}{{\rho }_g}=80$, Mo = 1.2 × 10⁻³, and Eo = 200. (a) Without conservation scheme, (b) with conservation scheme. The nondimensional times of the bubble positions from the bottom are τ = 0, 1, 2, 2.24, respectively.

FIG. 9

Bubble volume conservation with the level-contour reconstruction procedure. Eo = 97.1, Mo = 0.971.

FIG. 10

Schematic of the lengths used in defining the deformation factor for the bubble. A representative terminal bubble shape is shown.

FIG. 11

Deformation factor for bubbles in different regimes. Legend details for cases (a)–(e) are given in Table III. H* = 8, for all cases.

FIG. 12

Values of Re_T for various dimensionless wall distances. See Table III for legend details.

FIG. 13

Re_T-We space spanned by the bubbles in various regimes. The values of Re_T and We decrease as the wall proximity increases for each case. See Table III for legend details.

FIG. 14

Wall effects on the terminal velocity of the bubble in different regimes. See Table III for legend details.

FIG. 15

Velocity vectors and the streamlines for the ellipsoidal bubble at terminal state: Mo = 0.01, Eo = 10.0. Upper figures are velocity vectors in laboratory reference frame, and the lower ones are streamlines in the frame of reference of the bubble centroid. Here, (A) R* = 2, (B) R* = 1, (C) R* = 0.75, (D) R* = 0.6, (E) R* = 0.55.

FIG. 16

Values of the drag coefficient C_D for various dimensionless wall distances. See Table III for legend details.

FIG. 17

Velocity vectors and the streamlines for the intermediate spherical-cap–skirted bubble at the terminal state: Mo = 2.6 × 10⁻³, Eo = 54.4. Upper figures are velocity vectors in laboratory reference frame, and the lower ones are streamlines in the frame of reference of the bubble centroid. Here, (A) R* = 2, (B) R* = 1, (C) R* = 0.75, (D) R* = 0.6, (E) R* = 0.55.

FIG. 18

Nondimensional vorticity at the bubble surface at terminal state for two different R* values: (a) R* = 2 and (b) R* = 0.55. Here, Mo = 2.6 × 10⁻³, Eo = 54.4 and this corresponds to case (d) in Table III. Dashed–dotted line denotes zero vorticity.

FIG. 19

Dynamic pressure contours on the bubble surface at terminal state for two different R* values: (a) R* = 2 and (b) R* = 0.55. Here, Mo = 2.6 × 10⁻³, Eo = 54.4 and this corresponds to case (d) in Table III.

FIG. 20

Cross-sectional plots of the nondimensional vorticity at transverse cross-sectional planes located slightly below the lower surface of the bubble at terminal states for two different R* values: (a) R* = 2 and (b) R* = 0.55. Here, Mo = 2.6 × 10⁻³, Eo = 54.4, and this corresponds to the case (d) in Table III.

FIG. 21

Bubble shape regimes in the Eo - Re_T space for various wall distances: (A) R* = 2, (B) R* = 1, (C) R* = 0.75, (D) R* = 0.6, (E) R* = 0.55. The Mo values for cases (a)–(e) are given as follows: (a) Mo = 10, (b) Mo = 0.01, (c) Mo = 330, (d) Mo = 0.0026, (e) Mo = 0.092.