Kinematics of a bubble freely rising in a thin-gap cell with additional in-plane confinement

We analyze the behavior of air bubbles freely rising at high Reynolds numbers in a planar thin-gap cell filled with distilled water. The gap thickness of the cell is fixed to $h\simeq 2.8$ mm (or $h\simeq 1$ mm in additional experiments) and its in-plane width $W$ is varied from 2.4 to 21 cm. This allows us to investigate the evolution from unconfined thin-gap situations (i.e., large $W$ and $h\ll W$) controlled by the bubble characteristic lengths (diameter in the cell plane $d>h$ and thickness close to the gap size $h$) to doubly confined situations controlled by the channel dimensions. As the bubble size $d$ increases, and beyond a critical value that depends on $W$, we observe a mean rise velocity of the bubble, $V_b$, lower than that for larger $W$, along with a modification of the bubble shape. The departure occurs for oscillating bubbles of approximate elliptical shape, which becomes closer to circular as the lateral confinement increases. We further investigate how the bubble oscillatory motion is impacted by the transverse confinement. Assuming that the wall effect is related to the strength of the downward flow generated by the bubble, we introduce the relative velocity $U_{\mathrm{rel}}=V_b/\xi$, where $\xi =1-d/W$ is the confinement ratio and found $U_{\mathrm{rel}}\simeq V_{b,\infty }$ for all the cell widths considered, where $V_{b,\infty }$ is the mean rise velocity in the absence of the transverse confinement (i.e., for $W$ sufficiently large). This provides an estimation, at leading order, of the bubble velocity, $V_b\simeq 0.8\xi {(h/d)}^{1/6}\sqrt{gd}$, that generalizes the expression proposed by Filella et al. J. Fluid Mech. 778, 60 (2015) and accounts for the additional drag experienced by the bubble due to the lateral walls. We then show that, for given $d$ and $\xi$, the frequency and amplitudes of the oscillatory motion can be predicted using the characteristic length and velocity scales, $d$ and $U_{\mathrm{rel}}$. As the bubble size is increased further, the bubble behavior becomes fully dominated by the channel dimensions. Cylindrical-capped shapes emerge, corresponding to a radius of curvature $R_c$ at the front of the bubble, $R_c\simeq 0.31W$, independent of the bubble size and of the gap thickness. At the same time, the mean rise velocity of the bubble saturates at a constant value, corresponding to a constant Froude number, $\mathrm{Fr}=V_b/\sqrt{gW}$, that depends on the gap thickness $h$ of the cell.

Figure 1

Scheme of the experimental setup, consisting in a thin-gap cell (thickness $h$, height $H$, and width $W_c$), in which internal walls can be included in order to reduce the effective width to $W$.

Figure 2

(a) The different bubble behaviors observed in the $d\text{−}W$ plane for a cell with $h\simeq 2.8$ mm. The color of the data points is associated with the value of the cell width $W$ indicated by the vertical axis, and each symbol (being it empty or solid) represents a regime: $W=21$ cm, empty black; $W=15$ cm, solid black; $W=9$ cm, orange; $W=7$ cm, blue; $W=4$ cm, green; and $W=2.4$ cm, red. (b) Image superpositions illustrating the different regimes described in panel (a), which can be identified with the background color and the solid black symbol on the top. $\diamond$, path oscillations and weak shape oscillations about a vertical path; $\circ$, path oscillations and moderate shape oscillations (usually with a drift in the path); $▿$, regular large oscillations of both path and shape (about the vertical); $▵$, large and complex shape oscillations that induce irregular path deviations from the vertical; $▹$, quasirectilinear path with periodic shape oscillations; $\square$, cylindrical-capped bubbles; and $◃$, slugs. The various regimes are denoted $I$ to $VII$ as shown. The black vertical lines in regimes $III$, $V$, and $VII$ represent the lateral walls.

Figure 3

Mean rise velocity of the bubble as a function of its planar diameter $d$ and of Ar, for different cell widths $W$ ($h\simeq 2.8$ mm). Same convention for colors and symbols as in Fig. 2. Magenta crosses correspond to the results from Filella et al. [12]. The dotted horizontal lines correspond to the constant velocity values used in Sec. 6 for regimes $VI$ and $VII$. For a given $W$, the critical and maximum diameters, $d_{cV}$ and $d_{mV}$, and the corresponding velocity values are indicated with solid and dashed lines, respectively.

Figure 4

(a) Diameters $d_{cV}$ (solid black), $d_{mV}$ (empty black), $d_{c\chi }$ (solid red), and $d_{m\chi }$ (empty red) along with the corresponding velocity and aspect ratio values, for different cell widths $W$ ($h\simeq 2.8$ mm). (b) Values of $L^{*}$ corresponding to the critical diameters $d_{cV}$ for two gap thicknesses ($h\simeq 2.8$ and 1 mm) and various $W$.

Figure 5

Comparison of the bubble contour with the ellipse used to characterize its shape for (a) an oscillating bubble and (b) a cylindrical-capped bubble. In the latter case, we also show additional characteristic dimensions of the bubble that are used in Sec. 6: the radius of curvature at the bubble front, $R_c$, and the height $H_b$ and the width $W_b$ of the bubble.

Figure 6

(a) Mean aspect ratio of the bubble as a function of the planar diameter of the bubble $d$ and of the Archimedes number Ar, for different cell widths $W$ ($h\simeq 2.8$ mm). For a given $W$, the critical and maximum diameters, $d_{c\chi }$ and $d_{m\chi }$, and the corresponding $\chi$ values are indicated with solid and dashed lines, respectively. (b) Mean aspect ratio $\chi$ compared to the normalized distance left between the bubble and the wall $\xi$ for different $\mathrm{Ar}$ and $W$. Same convention as in Fig. 2.

Figure 7

(a) $\mathrm{Re}=V_{b}d/\nu$ and (b) ${\mathrm{Re}}_f=\mathrm{Re}/\xi$ characterizing the intensity of the counterflow generated by a rising bubble for different cell widths $W$ ($h\simeq 2.8$ mm), as functions of the Archimedes number of the bubble. Same convention as in Fig. 2. The solid black line in panel (a) corresponds to the law (1) from Filella et al. [12], and the solid red line in panel (b) corresponds to expression (10).

Figure 8

Aspect ratio $\chi$ of oscillating bubbles as a function of $\sqrt{\mathrm{We}}$ for different cell widths $W$ ($h\simeq 2.8$ mm). Same convention as in Fig. 2. The solid black line corresponds to the expression $\chi \simeq 1.12\sqrt{\mathrm{We}}$ from Filella et al. [12].

Figure 9

Translational components of the bubble oscillatory motion: (a) Vertical velocity amplitude normalized with the relative velocity $U_{\mathrm{rel}}$ and (b) amplitude of horizontal displacement normalized with $d$, as functions of the Archimedes number, for different cell widths $W$ ($h\simeq 2.8$ mm). Same convention as in Fig. 2. Magenta crosses correspond to the results from Filella et al. [12] for an unconfined cell ($W=40$ cm).

Figure 10

(a) Amplitude of the inclination angle ${\tilde{\beta }}_n$ of the bubble as a function of its aspect ratio $\chi$, for different cell widths $W$ ($h\simeq 2.8$ mm). (b) Strouhal number, ${\mathrm{St}}_{\mathrm{rel}}$, based on the frequency of path oscillation and on the relative velocity $U_{\mathrm{rel}}$ as a function of the Archimedes number of the bubble, for different cell widths $W$ ($h\simeq 2.8$ mm). Same convention as in Fig. 2. In both figures, magenta crosses correspond to the values ${\tilde{\beta }}_{n,\infty }$ and ${\mathrm{St}}_{\infty }$ reported in Filella et al. [12] for an unconfined cell ($W=40$ cm).

Figure 11

Evolutions of (a) $W_b/W$ (upper curve), $R_c/W$ (lower curve), and (b) $H_b/W$, describing the shape of confined cylindrical-capped bubbles, when their volume increases ($h\simeq 2.8$ mm). The horizontal dashed line in panel (a) corresponds to $R_c\simeq 0.31W$. Same convention as in Fig. 2 (except for $R_c$, where stars are used as symbols).

Figure 12

(a) Illustration of the shape evolution of confined cylindrical-capped bubbles, when their volume is increased, showing the invariance of the radius of curvature at the bubble front. The position of the channel walls are shown with thick black lines, and three channel widths $W$ are presented. The colored numbers indicate the Archimedes number of the corresponding bubble contour ($h\simeq 2.8$ mm). (b) and (c) Superposition of the bubble contours for different $W$ and gap thicknesses $h$: blue, $W=7$ cm; green, $W=4$ cm; red, $W=2.4$ cm; solid bubble contours, $h\simeq 2.8$ mm; dashed contours, $h\simeq 1$ mm; black dotted contours, $h=6.35$ mm and $W=7.9$ cm from Collins [18].

Figure 13

(a) Froude number as a function of the cell width obtained in different experiments for different gap thicknesses. The data points from this work and those from Ref. [16] correspond to averages over different sizes of cylindrical-capped bubbles. The horizontal lines correspond to the average values for Fr over several cell widths. (b) Froude number observed for cylindrical-capped bubbles as a function of the gap thickness $h$ normalized with the capillary length $l_c=\sqrt{\sigma /\left({\rho }_{l}g\right)}\simeq 2.7$ mm. The results from this work correspond to the average over several cell widths.

Figure 14

(a) $C_{D,W}$ and (b) $C_{D,W}$ normalized with expression (A5), characterizing the drag experienced by a rising oscillating bubble for different cell widths $W$ ($h\simeq 2.8$ mm), as functions of the Archimedes number of the bubble. Same convention as in Fig. 2.

Figure 15

Visualization of the vortex streets generated by a bubble with $\mathrm{Ar}\simeq 2650$ ($h\simeq 2.8$ mm) in two cells having different widths [panel (a), $W=15$ cm; panel (b), $W=4.5$ cm], illustrating the difference in number of vortices released over the same distance. Both figures are plotted using the same scale, as indicated.