Path instability of a no-slip spheroidal bubble in isotropic turbulence

Path instability of a millimetric spheroidal bubble in quiescent fluid and in isotropic turbulence is investigated by direct numerical simulation. An immersed boundary method along with a new formulation of the equation of bubble motion is utilized to impose the no-slip condition on the surface of an air bubble in fixed shape with the equivalent diameter of $1.0\sim 4.0\text{mm}$ in contaminated water. The range of Galilei number defined as the ratio of the gravitational force to the kinematic viscosity considered in this study is 100 $\sim$ 800. In still fluid, as the bubble size grows, the frequency of the zigzagging motion of the bubble increases while the range in the orientation angle variation of the bubble is hardly affected. The effect of background turbulence on path instability of a rising bubble, which typically shows zigzag pattern in still fluid, is investigated at three different Reynolds numbers, ${\text{Re}}_{\lambda }$, of 26, 45, and 73, or equivalently, for the ratio of fluid root-mean-square velocity to the bubble rise velocity $u^{\prime }/V_T$ ranging 0.030 $\sim$ 0.671. When a bubble rises in isotropic turbulence, the terminal rise velocity of the bubble does not show a noticeable difference. However, the pathways are significantly distorted by turbulence. Furthermore, the magnitude of zigzagging frequency and the degree of obliquity of the bubble become enhanced with ${\text{Re}}_{\lambda }$. We also observed wakes behind the bubble to find that the rear tails become weaker and tangled due to turbulence.

Figure 1

Distribution of Lagrangian points for (a) a sphere ($\chi =1$) with $N_l=1588$, (b) an oblate spheroid ($\chi =2$) with $N_l=2976$. The points are allocated on each cell along the surface in horizotnal $x$, $y$ axes and vertical $z$ axis.

Figure 2

Comparison against measurement data of (a) the averaged bubble Reynolds number ${\text{Re}}_b$ in still fluid and isotropic turbulence and (b) drag coefficient and Strouhal number. The Reynolds number is compared with experiments conducted in still fluid for the range of the bubble sizes considered in Tomiyama et al. [52]. The drag coefficient $C_d$ is compared with measurement data by Magnaudet and Eames [1] and the Strouhal number ${\text{St}}_r$ is compared with an estimated fit given by Tagawa et al. [8], Tsuge and Hibino [53].

Figure 3

Trajectories and horizontal projections of the spheroidal bubble in still fluid for (a) $S-1.0$, (b) $S-2.0$, (c) $S-3.0$, (d) $S-4.0$. The solid line and the dashed line in the lower panel represent $X/d_e$ and $Y/d_e$, respectively. A rectilinear motion is observed for $S-1.0$, and a zigzagging motion can be observed for bubbles larger than $1\text{mm}$.

Figure 4

Statistical results with respect to the bubble's locomotion in still fluid (a) bubble Reynolds number ${\mathrm{Re}}_b$, (b) two-point autocorrelations of the horizontal velocity ${\rho }_b$, (c) conversion frequency spectra $\phi$ from the autocorrelations, (d) variation of angle $\theta$ along its dominant horizontal direction.

Figure 5

Instantaneous vorticity contours ${\omega }_y$ around the rising bubble in still fluid for the four cases: (a) $S-1.0$, (b) $S-2.0$, (c) $S-3.0$, (d) $S-4.0$.

Figure 6

Three-dimensional enstrophy structures at ${\mathrm{\Omega }}_{En}$ and isovorticity ${\omega }_z$ surfaces around the rising bubble in still fluid for two cases: (a) $S-2.0$ (${\mathrm{\Omega }}_{En}=1600$), (b) $S-2.0$ (${\omega }_z=\pm 15$), (c) $S-3.0$ (${\mathrm{\Omega }}_{En}=1600$), (d) $S-3.0$ (${\omega }_z=\pm 15$). Values of red and blue colors for ${\omega }_z$ have the value of $+15$ and $-15$, respectively.

Figure 7

Trajectories and the horizontal projections of the spheroidal bubble in three kinds of isotropic turbulence at ${\text{Re}}_{\lambda }=$ 26 (red line), 45 (blue line), 73 (black line) compared with cases in still fluid (dashed line) for (a) $1.0\text{mm}$ bubble, (b) $2.0\text{mm}$, (c) $3.0\text{mm}$, (d) $4.0\text{mm}$, respectively. The lines in lower panel represent $Y/d_e$.

Figure 8

Statistics of motions for $2.0\text{mm}$, $4.0\text{mm}$ bubbles in turbulence ${\text{Re}}_{\lambda }=$ 26, 45, 73. (a) bubble Reynolds number ${\text{Re}}_b$, (b) two-point autocorrelations of the horizontal velocity $\rho$, (c) the frequency spectra $\phi$ from the autocorrelations, (d) temporal fluctuation of obliquity angle ${\theta }_x$ about the $X$ direction.

Figure 9

Various tatistics of bubble locomotion in dissipation $\epsilon$ (a) zigzagging frequency $f$ in real unit, (b) Strouhal number ${\text{St}}_r$ in ${\text{Re}}_{\lambda }$, (c) normalized horizontal velocity $v_h/V_T$, (d) averaged obliquity angle $\alpha$.

Figure 10

The horizontal mean-squared dispersion $H\left(t\right)$ in time for (a) $1.0\text{mm},2.0\text{mm}$ bubbles, (b) $3.0\text{mm},4.0\text{mm}$ bubbles. $H\left(t\right)$ shows a ballistic increase $\sim t^2$ at an early stage and then diffusive $\sim t$ later. The normalized Lagrangian integral time scale in each turbulence for each bubble size, $T_L{(g/d_e)}^{1/2}$ are 60.5 for $T1-1.0$, 7.2 for $T2-1.0$, 3.3 for $T3-1.0$, 88.0 for $T1-2.0$, 10.5 for $T2-2.0$, 4.8 for $T3-2.0$, 78.0 for $T1-3.0$, 9.3 for $T2-3.0$, 4.2 for $T3-3.0$, 90.5 for $T1-4.0$, 10.8 for $T2-4.0$, 4.9 for $T3-4.0$, respectively.

Figure 11

Instantaneous vorticity contours ${\omega }_y$ of flow around a bubble for 4 sizes in three sorts of isotropic turbulence.