Oscillatory motion and wake of a bubble rising in a thin-gap cell

Abstract

We investigate the characteristics of the oscillatory motion and wake of confined bubbles freely rising in a thin-gap cell ($h=3.1~\text{mm}$ width). Once the diameter $d$ of the bubble in the plane of the cell is known, the mean vertical velocity of the bubble $V_{b}$ is proportional to the gravitational velocity $(h/d)^{1/6}\sqrt{gd}$, where $g$ is the gravitational acceleration. This velocity is used to build the Reynolds number $Re=V_{b}d/{\it\nu}$ that characterizes the flow induced by the bubble in the surrounding liquid (of kinematic viscosity ${\it\nu}$), and which determines at leading order the mean deformation of the bubble given by the aspect ratio ${\it\chi}$ of the ellipse equivalent to the bubble contour. We then show that in the reference frame associated with the bubble (having a fixed origin and axes corresponding to the minor and major axes of the equivalent ellipse) the characteristics of its oscillatory motion in the plane of the cell display remarkable properties in the range $1200<Re<3000$ and $h/d<0.4$. In particular, the velocity of the bubble presents along its path an almost constant component along its minor axis (fluctuations in time of approximately 5 %), given by $V_{a}/V_{b}\simeq 0.92$ for all $Re$. The dimensionless amplitude of oscillation of the angular velocity is also constant for all $Re$, $\tilde{r}d/V_{b}\simeq 0.75$, while that of the transverse velocity of the bubble (along its major axis) is given by $\tilde{V}_{t}/V_{b}\simeq 0.32{\it\chi}$, reaching values comparable to those of the axial velocity $V_{a}$ for the most deformed bubbles (${\it\chi}\approx 3$). Furthermore, the frequency $f$ of oscillation scales with the inertial time scale based on the transverse velocity of the bubble $\tilde{V}_{t}$, corresponding to a constant Strouhal number $St^{\ast }=fd/\tilde{V}_{t}\simeq 0.27$. Using high-frequency particle image velocimetry, we investigate in detail the properties of the wake associated with the oscillatory motion of sufficiently confined bubbles. We observe that vortex shedding occurs for a maximal transverse velocity $V_{t}$ of the bubble, corresponding to a maximal drift angle of the bubble. Furthermore, the measured vorticity of the vortex at detachment corresponds to the estimation $V_{b}{\it\chi}^{3/2}/d$ of the vorticity produced at the bubble surface. Three stages then emerge concerning the evolution in time of the wake generated by the bubble. For one to two periods of oscillation $T_{x}$ following the release of a vortex, a rapid decay of the vorticity of the released vortex is observed. Meanwhile, the released vortex located initially at a distance of approximately one diameter from the bubble centre moves outwards from the bubble path and expands. At intermediate times, the vortex street undergoes vortex pairing. When viscous effects become predominant at a time of the order of the viscous time scale ${\it\tau}_{{\it\nu}}=h^{2}/(4{\it\nu})$, the vortex street becomes frozen and decays exponentially in place.