Dynamics of two-dimensional bubbles

The dynamics of two-dimensional bubbles ascending under the influence of buoyant forces is numerically studied with a one-fluid model coupled with the front-tracking technique. The bubble dynamics are described by recording the position, shape, and orientation of the bubbles as functions of time. The qualitative properties of the bubbles and their terminal velocities are described in terms of the Eötvos (ratio of buoyancy to surface tension) and Archimedes numbers (ratio of buoyancy to viscous forces). The terminal Reynolds number result from the balance of buoyancy and drag forces and, consequently, is not an externally fixed parameter. In the cases that yield small Reynolds numbers, the bubbles follow straight paths and the wake is steady. A more interesting behavior is found at high Reynolds numbers where the bubbles follow an approximately periodic zigzag trajectory and an unstable wake with properties similar to the Von Karman vortex street is formed. The dynamical features of the motion of single bubbles are compared to experimental observations of air bubbles ascending in a water-filled Hele-Shaw cell. Although the comparison is not strictly valid in the sense that the effect of the lateral walls is not incorporated in the model, most of the dynamical properties observed are in good qualitative agreement with the numerical calculations. Hele-Shaw cells with different gaps have been used to determine the degree of approximation of the numerical calculation. It is found that for the relation between the terminal Reynolds number and the Archimedes number, the numerical calculations are closer to the observations of bubble dynamics in Hele-Shaw cells of larger gaps.

Figure 1

Sketch indicating the geometry and physical properties considered in the model. Initially ($t_o$), the bubble is circular with a diameter $D$; at subsequent times, the shape is obtained from the model (see text). The physical properties of the gas and liquid are used in the areas occupied by the bubbles and the surrounding fluid, respectively.

Figure 2

Map of qualitative behavior of the two-dimensional bubble motion. The black broken line indicates the critical Reynolds number. Below this line, the bubbles follow straight trajectories (circles) and above it the bubbles ascend in a zigzag pattern (filled circles). Cases with constant Archimedes numbers are joined by red (light gray) broken lines. The parameter ${\mu }_f/{\mu }_b$ = $2.2\times {10}^4$/Ar.

Figure 3

Terminal Reynolds number as a function of Archimedes number. Morton number is $2.5\times {10}^{-11}$. The line represents $\mathrm{Re}=\mathrm{Ar}$.

Figure 4

Strouhal number as a function of terminal Reynolds number. $\text{Mo}=2.5\times {10}^{-11}$. This case correspond to air bubbles in water. The line is $\mathrm{St}=A/\mathrm{Re}+B+C\mathrm{Re}$ with $A=-31.1$, $B=0.213$, and $C=1.5\times {10}^{-5}$.

Figure 5

Trajectory of the centroids of individual bubbles for $\mathrm{Eo}=0.84$ and $\mathrm{Ar}=14$ (red line), 28 (black broken line), 70 (green line), and 392 (blue dash-dotted line). Note that the horizontal dimension is amplified.

Figure 6

Horizontal and vertical Reynolds numbers as functions of time after the initial transient has died out. The blue trace (continuous line) is the Reynolds number defined with the vertical component of the velocity. The red trace (broken line) is the Reynolds number defined with the horizontal component of the velocity; $\mathrm{Eo}=0.84$, $\mathrm{Ar}=392$, and $\text{Mo}=2.5\times {10}^{-11}$.

Figure 7

Orientation and aspect ratio of a bubble rising during one oscillation cycle with $\mathrm{Eo}=0.84$, $\mathrm{Ar}=392$, and $\text{Mo}=2.5\times {10}^{-11}$. Upper row left: Inclination of the major axis of the elliptical bubble with respect to horizontal as a function of the horizontal coordinate. Time evolves in the direction of the arrow. The shape and orientation of the bubble at points labeled A and B are illustrated in the graphs in the lower rows. Upper row right: Aspect ratio as a function of the horizontal coordinate. Middle row: Shape and orientation of the bubble at points A and B. The major axis is denoted by a double arrow line, and the horizontal axis and the inclination angle $\theta$ are explicitly shown for reference. The lines denote isobars and their relative values near the bubble indicate the zones where pressure contributes to the deformation of the originally circular bubble. Lower row: vorticity field for positions A and B.

Figure 8

Orientation and aspect ratio as functions of the bubble trajectory for $\mathrm{Eo}=0.84$, $\mathrm{Ar}=392$, and $\text{Mo}=2.5\times {10}^{-11}$. The color (gray scale) in the trajectory indicates the magnitude of the vertical velocity according to the color (gray scale) bar.

Figure 9

(a) Horizontal position ($x^{*}=x/D)$ of the bubble as a function of time. [(b) and (c)] Projection of the resultant force on the axes of coordinates fixed in the laboratory as functions of time. The scale on the right-hand side of (c) indicates $F_y-g({\rho }_f-{\rho }_b)\pi D^2/4$, and (d) and (e) show, respectively, the projection of the resultant force on the axis of coordinates fixed on the bubble as functions of time. (f) Torque on the bubble as a function of time.

Figure 10

Vortex shedding around a single bubble, and velocity, pressure, and vorticity fields for $\mathrm{Eo}=0.84$, $\mathrm{Ar}=392$, and $\text{Mo}=2.5\times {10}^{-11}$; $a=10$, $b=4.4$.

Figure 11

Hele-Shaw cell used to obtain the experimental results of Sec. 6. (a) feeding needle, (b) feeding hose, (c) $\mathsf{Y}$ connector, (d) pressure head hose, (e) syringe pump, and (f) pressure head reservoir. The observation area is indicated by the broken line.

Figure 12

Relation between the Reynolds and Archimedes numbers for the Hele-Shaw cells with the three different gaps and for the numerical simulations. The black circles, blue squares, and green diamonds correspond to the experiments with gaps of 1 mm ($h^{*}=2.7$), 2 mm ($h^{*}=5.4$), and 2.5 mm ($h^{*}=6.75$), respectively. Red stars correspond to the results from the numerical simulations.

Figure 13

Comparison of the numerical trajectory of the bubble with experiment in a Hele-Shaw cell. The left panel is the numerical simulation ($\mathrm{Ar}=392$, $\mathrm{Eo}=0.84$, and $\text{Mo}=2.5\times {10}^{-11}$) and the right panel is the experimental observation in a cell with a gap of 2 mm ($h^{*}=5.4$).

Figure 14

Position of the centroid of a single bubble as recorded in a frame of reference ascending with the average vertical velocity; the left-hand side is the numerical simulation result ($\mathrm{Ar}=392$, $\mathrm{Eo}=0.84$, and $\text{Mo}=2.5\times {10}^{-11}$) and the right-hand side is the experimental observation.

Figure 15

Experimental Reynolds number as a function of time. The blue trace (black line) is the vertical Reynolds number. The red trace (gray line) is the horizontal Reynolds number.

Figure 16

Vortex shedding in the wake of the bubble. Left: The numerical velocity field. In the center, the experimental visualization in the cell with a gap of 2.0 mm ($h^{*}=5.4$) and, on the right, the experimental visualization in the cell with a gap of 2.5 mm ($h^{*}=6.8$) are presented.