Paths and wakes of deformable nearly spheroidal rising bubbles close to the transition to path instability

We report on a series of results provided by three-dimensional numerical simulations of nearly spheroidal bubbles freely rising and deforming in a still liquid in the regime close to the transition to path instability. These results improve upon those of recent computational studies [Cano-Lozano et al., Int. J. Multiphase Flow 51, 11 (2013); Phys. Fluids 28, 014102 (2016)] in which the neutral curve associated with this transition was obtained by considering realistic but frozen bubble shapes. Depending on the dimensionless parameters that characterize the system, various paths geometries are observed by letting an initially spherical bubble starting from rest rise under the effect of buoyancy and adjust its shape to the surrounding flow. These include the well-documented rectilinear axisymmetric, planar zigzagging, and spiraling (or helical) regimes. A flattened spiraling regime that most often eventually turns into either a planar zigzagging or a helical regime is also frequently observed. Finally, a chaotic regime in which the bubble experiences small horizontal displacements (typically one order of magnitude smaller than in the other regimes) is found to take place in a region of the parameter space where no standing eddy exists at the back of the bubble. The discovery of this regime provides evidence that path instability does not always result from a wake instability as previously believed. In each regime, we examine the characteristics of the path, bubble shape, and vortical structure in the wake, as well as their couplings. In particular, we observe that, depending on the fluctuations of the rise velocity, two different vortex shedding modes exist in the zigzagging regime, confirming earlier findings with falling spheres. The simulations also reveal that significant bubble deformations may take place along zigzagging or spiraling paths and that, under certain circumstances, they dramatically alter the wake structure. The instability thresholds that can be inferred from the computations compare favorably with experimental data provided by various sets of recent experiments guaranteeing that the bubble surface is free of surfactants.

Figure 1

Phase diagram in the $(\text{Bo},\text{Ga})$ plane: solid line, approximate critical curve determined from three-dimensional simulations performed using frozen realistic bubble shapes in Ref. [23]; dash-dotted and dashed lines, critical curves obtained in Ref. [24] using the LSA approach with a freely moving and a fixed bubble, respectively; and symbols, experimental data corresponding to incipient path instability observed in various liquids $(+$ [12], $\times$ [13], $•$ [4], $⧫$ [14], $▴$ [9], $▾$ [10], and $▸$ [11]).

Figure 2

Detail of a typical grid in the vicinity of the bubble: (a) refined region in the vicinity of the bubble, especially in the wake region (the last refinement level of the actual grid has been omitted in the figure to improve readability), and (b) refinement on both sides of the bubble surface.

Figure 3

Rise of an initially slightly prolate bubble corresponding to $\text{Bo}=2.36$ and $\text{Ga}=108.6$: (a) evolution of the aspect ratio and (b) variation of the frequency of mode $(2,0)$ with the time-averaged aspect ratio. In (b) the solid line corresponds to the inviscid prediction of [35].

Figure 4

Phase diagram showing the entire set of simulations and positioning them in the $(\text{Bo},\text{Ga})$ plane with respect to the two neutral curves displayed in Fig. 1. Open (closed) symbols refer to runs in which the rectilinear bubble path was found to be stable (unstable). The dotted line represents the critical curve beyond which a standing eddy exists [23] and the thin solid lines correspond to iso-$\text{Mo}$ lines (shown from left to right are water at $28{}^{\circ }\mathrm{C}$ and silicon oils T00, T02, T05, and T11, respectively, all at a temperature of $22{}^{\circ }\mathrm{C})$. The various bubbles are numbered as in Table 1.

Figure 5

(a) Three-dimensional reconstruction of the path followed by bubble number 22 in Table 1. (b) Streamlines of the flow around the bubble showing its axisymmetry and the presence of a standing eddy.

Figure 6

(a) Three-dimensional reconstruction and (b) top view of a bubble trajectory typical from the zigzagging regime (bubble number 19 in Table 1). (c) Two perpendicular views of the temporal evolution of the bubble shape at positions indicated by bullets in (a) and (b); the solid line corresponds to the average vertical $\left(x\right)$ axis of the path.

Figure 7

Evolution of the streamwise vorticity isocontours ${\omega }_x=\pm 0.24$ at five successive positions corresponding to the bullets shown in Figs. 6 and 6 for bubble number 19. The left and right panels correspond to two perpendicular views. In the right panel, the dotted lines mark the hidden part of the primary isocontour at positions 1, 3, and 5.

Figure 8

Two perpendicular views of the wake structure in the zigzagging regime revealed by the ${\lambda }_2$ criterion (bubble number 19). The bubble is visible on the right.

Figure 9

Two perpendicular views of the wake structure revealed by the ${\lambda }_2$ criterion at the end of the rise of bubble number 16: (a) complete wake and (b) zoom-in of the near wake.

Figure 10

(a) Three-dimensional reconstruction and (b) top view of the trajectory corresponding to bubble number 26. (c) Two perpendicular views of the temporal evolution of the bubble shape corresponding to the bullets in (a) and (b).

Figure 11

Evolution of the streamwise vorticity isocontours ${\omega }_x=\pm 0.24$ at four successive positions corresponding to the bullets shown in Fig. 10 for bubble number 26. The left and right panels correspond to two perpendicular views.

Figure 12

(a) Three-dimensional reconstruction and (b) top view of a bubble trajectory typical from the flattened spiraling regime (bubble number 5). (c) Two perpendicular views (on the left, the $x\text{−}z$ plane and on the right, the $x\text{−}y$ plane) of the evolution of the bubble shape corresponding to the bullets in (a) and (b).

Figure 13

Same as Fig. 12 for bubble number 16.

Figure 14

Evolution of the streamwise vorticity isocontours ${\omega }_x=\pm 0.24$ at positions corresponding to the bullets shown in Figs. 12 and 12 for bubble number 5. The left and right panels display two perpendicular views and the solid line indicates the $x$ axis.

Figure 15

Two perpendicular views of the wake structure past bubble number 5 revealed by the ${\lambda }_2$ criterion.

Figure 16

Same as Fig. 14 for bubble number 16 (at a much earlier time than in Fig. 9).

Figure 17

Same as Fig. 15 for bubble number 16 (at a much earlier time than in Fig. 9).

Figure 18

Horizontal cuts showing the shape of the streamwise vorticity isocontours, from ${\omega }_x=-0.24$ (black) up to ${\omega }_x=0.24$ (white) for bubble number 16 in cross sections located (from right to left) $0.5D,2D$, and $4D$ apart from the bubble centroid.

Figure 19

(a) Three-dimensional reconstruction and (b) top view of a bubble trajectory typical from the chaotic regime (bubble number 3). (c) Evolution of the streamwise vorticity isocontours ${\omega }_x=\pm 0.24$ corresponding to the bullets shown in (a) and (b); time and $x$ coordinate increase from top to bottom; on the left is the $x\text{−}z$ plane and on the right, the $x\text{−}y$ plane. The solid lines in (c) indicate the $x$ axis.

Figure 20

Evolution of the Reynolds number for different bubbles in Table 1: (a) rectilinear regime, (b) chaotic regime, (c) planar zigzagging regime, (d) and (e) flattened spiraling and then planar zigzagging regime, and (f) spiraling (helical) regime. In (c) the inset shows a detailed view of the oscillations and the numbers identify the instants of time 1–5 of Figs. 6 and 7.

Figure 21

Comparison between present predictions (open and closed squares, same definition as in Fig. 4) and critical conditions determined in various works. Data are shown for the DNS of Ref. [29] $(☆)$; the experiments with ultrapure water of Refs. [14] $\left(⧫\right)$ and [4] $(•)$; and the experiments in various silicon oils of Refs. [9] $\left(▴\right)$, [10] $\left(▾\right)$, and [11] $(▸)$. In these data sets, the open (closed) symbols correspond to the largest stable (smallest unstable) bubble. The three curves are identical to those in Fig. 4.

Figure 22

Phase diagram summarizing the different styles of path observed in present simulations (rectilinear, $▵$; planar zigzag, $•$; flattened spiral, $▾$; spiral (helical), $⧫$; and chaotic, $\blacksquare )$. The three curves are identical to those in Fig. 4.