Characteristics of mean flow and turbulence in bubble-in-chain induced flows

This paper presents the experimental results from an air bubble-in-chain induced flow at different bubble release frequencies, resulting in a range of 1510–2318 for the Reynolds number based on bubble parameters. Mean flow and turbulent statistics were measured using planar particle image velocimetry. The bubble rise velocities were validated to be the superposition of the mean vertical velocity of water and the terminal velocity of the isolated bubbles. We tested the scaling parameters for the bubble-in-chain induced flow using those typically for bubble plumes, which demonstrates the intrinsic similarities of these two bubbly flows. We found that the radial profiles of normalized mean vertical velocity for bubble-in-chain induced flows and bubble plumes collapse well using the center-line velocity and the half-width of the velocity profile. We found the half-width correlated with the characteristic length scales of the bubble wakes. The normalized Reynolds stresses using the local velocity square show the same shape of profiles for both bubbly flows, but they do not collapse onto universal curves. The normalized turbulent kinetic energy is a function of normalized gas flow rate and approaches an asymptotic value of 0.05. The dynamic length scale, a function of initial kinematic buoyancy flux, was found to govern turbulent dissipation rates in both bubbly flows. The normalized one-dimensional velocity energy spectra collapse in the regime of $k_v\eta >0.1$ where $k_v$ is the wave number and $\eta$ is the Kolmogorov length scale, showing a $-3$ slope as the result of energy balance between production from bubble motion and dissipation in the bubble wakes. The wake length scale was found to define the lower-end wave number of the $-3$ slope.

Figure 1

Schematic picture of the experimental setup (not drawn to scale). The coordinate system is shown as $x$ and $y$ in the laser sheet.

Figure 2

Mean velocity profile of bubble induced flow at different release frequencies: (a) in physical scale; (b) in normalized scale.

Figure 3

(a) Relationship between $b_g$ and flow rate $Q$ in normalized formats, using the bubble slip velocity $U_s$ and the bubble diameter $D$; (b) relationship between $b_g$ and the wake characteristics length $l_c$. The error bars for $b_g$ represent the standard deviation of the measured $b_g$ from all runs for each flow rate; the error bars for $b_g/D$ represent the propagated error from $b_g$ and $D$; the error bars for $l_c$ represent the propagated error due to measurement errors of $V_B$ and $D$.

Figure 4

(a) Relationship between normalized bubble rise velocity and flow rate; (b) bubble slip velocities from measurements (blue circle) and from empirical equations (red triangle) [23].

Figure 5

Profiles of (a) Reynolds normal stress $\overline{u^{\prime }{u}^{\prime }}$, (b) Reynolds normal stress $\overline{v^{\prime }{v}^{\prime }}$, (c) Reynolds shear stress $\overline{u^{\prime }{v}^{\prime }}$.

Figure 6

Normalized profiles of (a) Reynolds normal stress $\overline{u^{\prime }{u}^{\prime }}$, (b) Reynolds normal stress $\overline{v^{\prime }{v}^{\prime }}$, (c) Reynolds shear stress $\overline{u^{\prime }{v}^{\prime }}$.

Figure 7

Normalized profiles for bubble-in-chain induced flows and bubble plumes: (a) Reynolds normal stress $\overline{u^{\prime }{u}^{\prime }}$, (b) Reynolds normal stress $\overline{v^{\prime }{v}^{\prime }}$, (c) Reynolds shear stress $\overline{u^{\prime }{v}^{\prime }}$.

Figure 8

(a) TKE profiles normalized using the profiles of velocity square ($V^2$) for bubble-in-chain induced flows and bubble plumes. Same data key as in Fig. 7. (b) The TKE in the center line normalized using the center-line velocity square ($V_c^2$) as a function of flow rate ($Q$) normalized with bubble slip velocity ($V_s$) and bubbly flow length scale ($D_b$).

Figure 9

(a) The profiles of turbulent dissipation rates; (b) normalized turbulent dissipation using the streamwise velocity ($V$) and the bubbly flow length scale ($D_b$).

Figure 10

Normalized 1D center line velocity energy spectrum as a function of the wave number in the vertical direction.