Clusterlike instabilities in bubble-plume-driven flows

Continuous release of gas bubbles in large numbers from a localized source in a liquid column, popularly known as “bubble plumes”, is very relevant in nature and industries. The bubble plumes morphologically consist of a long continuous stem supporting a dispersed head. Through our direct numerical simulations using two-way coupled Euler-Lagrangian framework, we show that a bubble plume rising in a quiescent liquid column develops clusterlike instabilities for the Grashof numbers, $\text{Gr}>145$. For levels $\text{Gr}<100$, the stem is continuous with a small plume head, whereas at high buoyancy ($\text{Gr}>350$), the plume stem shows intermittently passing puffing instabilities in the form of bubble clusters. The clusters are a group of bubbles localized in space with high concentration that travel upward with speed $C_{ph}=0.45U_C$ and are separated by a distance of at least $5L_0$, where $U_C$ is the characteristic velocity and $L_0$ is the characteristic length based on the injection conditions. The bubble rise Reynolds numbers in the steady state for both the plume head and the stem shows $\text{Re}\propto {\text{Gr}}^{0.45\pm 0.03}$, and the proportionality constant is ten times higher in the plume stem than in the plume head. In the plume core, the spatial acceleration due to the bubble motion generates the turbulent production, whereas, at the plume edge, the small-scale fluctuations generate the mean vorticity. At high Gr, the clusters evolve due to the lift forces acting on the bubbles as a result of increase in the mean vorticity. While rising, bubbles entrain the liquid from the surroundings, and we found that the entrainment rate is not as strong as compared to the classical thermal plumes.

Figure 1

(a) Schematic of bubble plumes rising in a liquid column of size $B\times H\times W$, (b) a sectional view of computational mesh used, and (c) zoom-up view of bubbles rising from an entrance region of $L_0\times L_0$ at initial time.

Figure 2

Top view of a horizontal plane section taken at $y=40L_0$ and the simulated bubble locations are superimposed. The actual bubble size is enlarged by ten times to aid the view.

Figure 3

Bubble-plume morphology depends on Gr. Continuous stream at low Gr and sporadic clusters at high Gr. Snapshots for the plume when the cap (or head) reaches a vertical height of $y=78L_0$.

Figure 4

Validation study in comparison with [35]. The pressure gradient computed along the streamwise direction on a vortex core in a branching T-junction flow. Simulations are carried out for $\text{Re}=200$. A clear agreement between our DNS results and [35] can be noted.

Figure 5

Time evolution of bubble plumes for $\text{Gr}=49$ [top panels (a) to (e)] and $\text{Gr}=4900$ [bottom panels (f) to (j)]. Time increases from 14 s to 70 s by moving from panel (a) to (e) [and (f) to (j)]. Velocity disturbances created by rising bubbles on two perpendicular planes passing through mid-planes of the box are projected on the sidewalls. Note, contours of vertical velocity magnitude ($|u_y|$) range differently for the top and bottom panels. For visualization purpose, bubble diameter is increased by 100 times to show them as glyphs. Detailed three-dimensional flow features are shown as animations in the Supplemental Material [36].

Figure 6

Steady state rise Reynolds number of a bubble-plume head (${\text{Re}}_{\mathrm{head}}$) vs applied buoyancy (Gr). The 2D DNS data [6] in triangles and our 3D DNS data in squares for the plume head. Circles are for the bubble stem from our 3D DNS. Dash lines provide best fits: the red dash (bottom) line for the bubble-plume head (${\text{Re}}_{\mathrm{head}}\propto {\text{Gr}}^{0.45}$) and the magenta dash (upper) line for the plume stem. The classical thermal plume scaling relations are also shown for the plume head [the black (bottom) continuous line for [48] and the gray (middle) line 46], and the stem [the brown (upper) line for 44]. The cross symbol ($\times$) for bubble cap Reynolds number calculated by using a marker located on the topmost point of a rising bubble plume.

Figure 7

Source corrected scaling for bubble plumes: ${\text{Re}}_{\mathrm{head}}$ vs $(\text{Gr}-{\text{Gr}}_0)$. Here, ${\text{Gr}}_0\approx 2.42$ is the minimum buoyancy required to observe bubble rise.

Figure 8

Time averaged liquid vertical velocity, ${\overline{u}}_y/U_c$, along the midline ($y$ axis) of the box. Symbols for $\text{Gr}=1000$ (square) and $\text{Gr}=4900$ (circles). Note, ${\overline{u}}_y$ increases as $y^{1/5}$ in the region $y=5L_0$ to $y=30L_0$. In the inset, normalized vertical velocity gradient along $y$ axis as a function of Gr.

Figure 9

${\overline{u}}_y/U_c$ on a vertical midplane for different Gr. Note, dark red for high rise speed and blue for zero rise speed. Streamlines are superimposed.

Figure 10

Entrainment coefficient (${\alpha }_e$) as a function of $y$ for two different Gr: the red (thin) line for $\text{Gr}=200$ and the blue (thick) for $\text{Gr}=3000$. The entrainment effects are independent of Gr in the range considered. ${\alpha }_e$ saturates to a value of 0.05, far away from the injection source.

Figure 11

Normalized vertical velocity, $f^{\prime }/f^{\prime }\left(0\right)$, vs $\eta$ for various configurations. In panel (a), the dash lines for 2D thermal plumes, the continuous lines for 3D thermal plumes from [54], and the symbols for bubbly plumes (circles for 2D simulations [6] and diamonds for present 3D simulations). Panel (b) for the velocity profiles near the injection source at different heights. The red (upper) lines are for $\text{Pr}=0.7$ and the blue (bottom) lines are for $\text{Pr}=\infty$ (in the case of 2D plumes) and $\text{Pr}=10$ (in the case of 3D plumes). Simulations for $\text{Gr}=1000$.

Figure 12

Time series of vertical velocity on the axis at different heights: $y=L_0$ (the blue dark shaded bottom line), $y=10L_0$ (the black dark shaded upper line), $y=25L_0$ (the magenta light shaded bottom line), and $y=50L_0$ (the green light shaded top line). In inset, time series of velocity in $x$, $y$, and $z$ directions taken at a height $y=25L_0$ in the plume core. Simulations for $\text{Gr}=300$. Note, velocity is normalized with $U_c$.

Figure 13

Energy spectra, $E\left(f\right)$, computed from the vertical velocity. The blue bottom line for the signal taken at $y=L_0$ and the red top line for $y=10L_0$. The spectra show $E\left(f\right)\propto f^{-8/3}$ between dash-dot lines. Simulations for $\text{Gr}=3000$.

Figure 14

(a) Instantaneous bubble plume (white dots as bubbles) from our simulation with superimposed contours of vorticity (${\omega }_z$). Lateral variation of the profiles: time-averaged liquid velocities [panel (b) for ${\overline{u}}_x$ and (c) for ${\overline{u}}_y]$, vorticity $[{\omega }_z$ in panel (d)], and the Reynolds stresses $[\overline{u_x^{\prime }{u}_x^{\prime }}$ in panel (e), $\overline{u_y^{\prime }{u}_y^{\prime }}$ in panel (f), and $\overline{u_x^{\prime }{u}_y^{\prime }}$ in panel (g)]. Lines i to iv for the time-averaged profiles and line v for an instantaneous profile. Simulations for $\text{Gr}=3000$.

Figure 15

Normalized Reynolds stress component $\overline{u_y^{\prime }{u}_y^{\prime }}/{\overline{u}}_y^2$, as a function of $y$. Continuous line for the plume core and dash line for the plume edge. Simulation for $\text{Gr}=3000$.

Figure 16

Anisotropy in turbulent flutuations ($T_v$) as a function of lateral coordinate at different heights. Simulation for $\text{Gr}=3000$.

Figure 17

Production terms at different heights: $P_{xx}$ in panel (a), $P_{yy}$ in panel (b), and $P_{xy}$ in panel (c). Individual term contributions of $P_{yy}$ in panel (d). Simulation for $\text{Gr}=3000$.

Figure 18

Gr vs volume and time-averaged turbulent velocity fluctuations. Circles for ${\overline{u_y^{\prime }{u}_y^{\prime }}}^{1/2}$, squares for ${\overline{u_x^{\prime }{u}_x^{\prime }}}^{1/2}$, and diamonds for ${\overline{u_z^{\prime }{u}_z^{\prime }}}^{1/2}$. Note, below the critical Grashof number ${\text{Gr}}_c\approx 145$, the fluctuations scale as Gr, whereas the fluctuations scale as ${\text{Gr}}^{1/2}$ for $\text{Gr}>300$. The drop in intensity of fluctuations corresponds to the bubble cluster formation. The volume averages reported are considered in a small cylindrical portion around the axis from height $y=5L_0$ to $y=40L_0$.

Figure 19

Bubble clusters rise as a group with a phase speed $C_{ph}=0.45U_c$. Each of the snapshots is $4.8L_0$ in width and $19.2L_0$ in length. The snapshots are taken at time intervals of $t\approx 5.5\tau$. Simulations for $\text{Gr}=200$.

Figure 20

Normalized wavelength (circles) and frequency (squares) as functions of Gr. Bubble clusters travel with a constant phase velocity $C_{ph}\approx 0.45U_c$. The dash line indicates $\lambda =230{\text{Gr}}^{-0.61}{L}_0$.

Figure 21

PDFs of bubble rise velocities in a chosen cluster: bubbles in a cluster rise with a distribution of velocities. For low Gr a single peak and for a high Gr a double peak is noticed. The ensemble mean rise velocity of bubbles in a cluster increases with Gr. The vertical dash lines for $0.45U_c$ for the corresponding Gr.