Interaction of a rigid buoyant sphere and a deforming bubble with a vortex ring: The role of deformability

In bubbly turbulent flows, the deformation of the bubble is known to play an important role in the interaction between the carrier phase and the dispersed phase. In the present paper, we experimentally study an idealized version of the problem where we look at the interaction of a deforming air bubble (nonzero Weber number) and a rigid buoyant particle (zero Weber number) with a single vortex ring (in water) for a large range of ring-circulation-based Reynolds numbers ($\text{Re}=\mathrm{\Gamma }/\nu$=6000–67 300; $\mathrm{\Gamma }$=ring circulation, $\nu$=kinematic viscosity of water). We are able to obtain the particle to fluid density ratio (${\rho }_{\text{particle}}/{\rho }_{\text{water}}\approx 0.008$) to be very low and of the same order as that of the bubble (${\rho }_{\text{bubble}}/{\rho }_{\text{water}}\approx 0.001$), with the main difference being the distinct difference in their deformability. The buoyant low density (rigid) particle and the deforming bubble are directly engulfed into the core of the vortex ring and enable measurements of the effects of particles or bubbles on the vortex ring. On the particle (and bubble) dynamics side, distinct differences can be seen right from the capture phase, as unlike the rigid particle, the bubble undergoes a large axial elongation and, consequently, a quicker capture than the particle. After capture by the ring, the bubble experiences an azimuthal pressure gradient, which leads to its elongation into a thin long bubble whose diameter is relatively small compared to the vortex core size. In sharp contrast, the particle does not deform and remains effectively localized along the rings azimuthal direction. These differences in the shape of the bubble and particle within the ring lead to distinct differences in the ring's convection speed, azimuthal vorticity, and enstrophy at later times. In particular, the measurements indicate that the localized large perturbation to the vorticity in the rigid particle case (an equivalent of a rigid bubble, zero Weber number) leads to a larger disruption of the vortex ring as seen in terms of higher reduction in the rings convection speed and azimuthal enstrophy than for the deforming bubble (nonzero Weber number) case. These results could have implications in bubbly turbulent flows such as in bubble drag reduction (BDR), where many studies [like Van Gils et al., J. Fluid Mech. 722, 317 (2013)] indicate that deforming bubbles are better for drag reduction than nondeforming bubbles at higher flow Re, where the dominant mechanism is the lifting of structures away from the wall. The present results suggest that in cases where vortex disruption is the primary mechanism for BDR [for example, at lower flow Re, Sugiyama et al., J. Fluid Mech. 608, 21 (2008)], better drag reduction would occur with nondeforming bubbles. The current study thus brings us insight into the role of deformability in the interaction of bubbles with a vortical structure, which could be important in helping us model and understand bubble-turbulence interactions.

Figure 1

A schematic of a bubble (and buoyant particle) of diameter $D_b$ and a vortex ring of diameter $2R_o$ and core diameter of $2a_o$, traveling vertically upward in $z$ direction with velocity $u_{co}$.

Figure 2

Drag coefficient ($\widetilde{C_D}$) of freely rising bubbles (for bubble diameter $<$ 1.5 mm): A comparison is shown between the present results in tap water (contaminated) with an empirical relation for a rigid sphere from Clift et al. [48], experimental results for contaminated bubbles (Takagi et al. [49]), numerical results for a clean bubble (Mei et al. [52]), and experimental results for clean bubbles (Takagi et al. [49], Duineveld [50], Tagawa et al. [51]).

Figure 3

Schematic of the side view of the present experimental setup for a single bubble (particle) interacting with a vortex ring. A piston-cylinder arrangement was used to generate the vortex ring and a bubble (and buoyant particle) was released near the vortex ring. The vortex having a diameter $2R_o$ and core diameter $2a_o$, while convecting upward with a convection speed of $u_{\text{co}}$, captures the bubble (particle) with diameter $D_b$. The dynamics of the interactions were recorded from both the side- and top-view visualizations.

Figure 4

Time sequence of images of the side view and top view visualizations of a bubble (particle) interacting with a single (a) laminar (Re=6000), and (b) turbulent (Re=25 800) vortex ring. As seen in the figure, the bubble undergoes large elongation before and after capture by the ring and breaks into several smaller bubbles within the ring. On the other hand, as expected, the rigid particle does not deform. Here, the nondimensional times ($t{u}_{co}/R_o$) corresponding to the instances shown are (i) 2.4, (ii) 2.59, (iii) 3.89, and (iv) 5.9, respectively. In all cases shown, $D_b/2a_o\approx 1.4$.

Figure 5

(a) The nondimensional radial distance ($r/a_o$) of the bubble (particle) from the vortex core center with time ($t^{\prime }{u}_{co}/R_o$) as the bubble (particle) travels from $r/a_o$=3 to $r/a_o\approx \mathrm{1/2}$, shown for a laminar (Re=6000) and turbulent (Re=25 800) ring; here, $t^{\prime }$ is the dimensional time and is taken to be zero when the bubble (particle) is at $r/a_o$=3. (b) The variation of the nondimensional capture time ($t_c{u}_{co}/R_o$) with ring Re for the particle and bubble cases. The capture time computed from a simplified model initially proposed by Oweis et al. [53] is also shown. In this figure and subsequent figures in the text, rigid particle ($P$) and deforming bubble ($B$) are subscripted by $L$ and $T$, and these represent a laminar (Re=6000) and a turbulent (Re=25 800) ring case, respectively.

Figure 6

The variation of the (a) sphericity $\mathrm{\Phi }$ and (b) crosswise sphericity ${\mathrm{\Phi }}_{\perp }$, with ring Re for deforming bubble and rigid particle cases. (c) The drag coefficient ($C_D$) variation with ring Re for the spherical particle and elongated bubble cases during their capture process, obtained from the expressions of Haberman and Morton [54] and Hölzer and Sommerfeld [55], the latter based on the measured elongated shape.

Figure 7

(a) Bubble's nondimensional azimuthal elongation ($L_b^{*}$) inside the vortex with nondimensional time ($t{u}_{co}/R_o$), shown for a laminar (Re=6000) and turbulent (Re=25 800) ring; here, $L_b^{*}=L_b/2\pi R^{\prime }$, where $L_b$ is the instantaneous bubble's elongated length and $2\pi R^{\prime }$ is ring's instantaneous perimeter. Here, rigid particle ($P$) and deforming bubble ($B$) cases are subscripted by $\mathit{L}$ and $\mathit{T}$, and these represent the laminar and turbulent cases, respectively. (b) The number of broken bubbles ($N_b$) inside the vortex ring shown with nondimensional time ($t{u}_{co}/R_o$). (c) The translation of the rigid particle and a broken bubble along the azimuthal direction ($\theta$) with time ($t{u}_{co}/R_o$). (d) The variation in the rate of rotation of the particle (${\mathrm{\Omega }}_{\text{particle}}$) normalized by the average angular rotation rate of the core [${\mathrm{\Omega }}_{\text{core}}$] is shown with time ($t{u}_{co}/R_o$). In (c) and (d), top-view images of a particle within the vortex ring are attached for a better demonstration of the translational and rotational motions.

Figure 8

(a) The variation of the ring-equivalent nondimensional diameter ($D_{\text{ring}}^{*}$) with nondimensional time ($t{u}_{co}/R_o$). Here, $D_{\text{ring}}^{*}$=$D_{\text{ring}}/D_{{}_{\text{ring},O}}$; $D_{\text{ring}}$ is ring's instantaneous diameter and $D_{{}_{\text{ring},O}}$ is the diameter before bubble (particle) capture. (b) The variation in aspect ratio ($\beta$) of the ring with nondimensional time ($t{u}_{co}/R_o$). In these plots, both the base and interaction cases are shown for a laminar (Re=6000) and turbulent (Re=25 800) ring. Here, rigid particle ($P$) and deforming bubble ($B$) cases are subscripted by $L$ and $T$, representing a laminar and a turbulent case, respectively.

Figure 9

(a) The nondimensional vertical position ($z^{*}$=$z/R_o$) of the vortex ring with nondimensional time ($t^{*}$=$t{u}_{co}/R_o$) shown for laminar (Re=6000) and turbulent (Re=25,800) ring base cases and interaction cases with both bubble and particle. The blue dashed line represents the base ring's position obtained from the viscous vortex ring model [60] and shows a good match with the experimental base ring position data. (b) The percentage reduction in ring's convection speed ($\mathrm{\Delta }{u}_c$) with nondimensional time ($t{u}_{co}/R_o$) shown for Re=6000 and 25 800, cases. Here, the reduction in convection speed is calculated as $\mathrm{\Delta }{u}_c(\%)=100\times (u_{c_{\text{base}}}-u_{c_{\text{interact}}})/u_{c_{\text{base}}}$; here, $u_{c,\text{base}}$ and $u_{c,\text{interact}}$ are the base and interaction case ring's instantaneous convection speed. In the subplot, $\mathrm{\Delta }{u}_c$ (%) obtained at $t^{*}$ of 25 is shown for all Re cases studied. In these plots, rigid particle ($P$) and deforming bubble ($B$) are subscripted by $\mathit{L}$ and $\mathit{T}$, and these represent the laminar and turbulent ring cases, respectively.

Figure 10

Comparison of the time sequence of nondimensional azimuthal vorticity ($\omega R_o/u_{co}$) for the interaction of (a) particle and (b) bubble with a laminar (Re=6000) vortex ring. In each image, the particle (bubble) is marked as a hatched area (magenta online). Here, solid and dashed line contours represent positive and negative vorticity, respectively. The nondimensional time ($t{u}_{co}/R_o$) corresponding to each of the images are 2.7, 6.1, and 13, as shown on the left side. The nondimensional azimuthal vorticity contour levels are shown on the right.

Figure 11

Comparison of the time sequence of nondimensional azimuthal vorticity ($\omega R_o/u_{co}$) for the interaction of (a) particle, and (b) bubble with a turbulent (Re=25 800) vortex ring. The nondimensional time ($t{u}_{co}/R_o$) corresponding to each of the images are 2.7, 6.1, and 14, as shown on the left side. All other markings are as in Fig. 10.

Figure 12

(a) and (b) show the nondimensional azimuthal enstrophy ($E/E_o$) with nondimensional time ($t{u}_{co}/R_o$) for the bubble and particle, respectively, at laminar Re=6000. Here, instantaneous enstrophy ($E$) is normalized by the enstrophy ($E_o$) before the bubble (particle) capture. (c) and (d) show the nondimensional circulation ($\mathrm{\Gamma }/{\mathrm{\Gamma }}_o$) with nondimensional time ($t{u}_{co}/R_o$) for the bubble and particle, respectively, at laminar Re=6000. Here, the instantaneous circulation ($\mathrm{\Gamma }$) is normalized by the circulation (${\mathrm{\Gamma }}_o$) before bubble (particle) capture. In these plots, the enstrophy and circulation are shown for the left core (LC), right core (RC), and average of both cores (BC).

Figure 13

The variation in the percentage reduction in (a) enstrophy ($\mathrm{\Delta }E$) and (b) circulation ($\mathrm{\Delta }\mathrm{\Gamma }$) at $t{u}_{co}/R_o$=15 for bubble and particle cases at all Re studied. Here, the reductions are shown for the left core (LC), right core (RC), and average of both cores (BC).

Figure 14

The variation in the percentage reduction in (a) convection speed ($\mathrm{\Delta }{u}_c$), and (b) both core averaged azimuthal enstrophy ($\mathrm{\Delta }E$), with $D_b/2a_o$, for bubble and particle cases at Re of 6000. Here, the $D_b/2a_o$ of 1.1 and 0.8 correspond to ${\text{We}}_{\text{bubble}}$ of 9 and 6, respectively.