Figure 1 shows a schematic of the experimental setup. The water tank is made of transparent acrylic resin to enable flow visualization. The width and depth of the tank are both 14.7D, and the height is 19.7D. The water level is 16.7D. The top of the tank is open to the atmosphere. A cylinder with diameter D of 30 mm is horizontally placed in the tank, and five carbon rods are placed at the bottom of the tank. The rod diameter is 0.16D. The origin of the coordinates is at the center of the carbon rods. The x-y plane is horizontal and the z-axis is vertical.

Figure 2 shows the carbon rods and their positions relative to the cylinder. The length of each rod is 3.3D. The rods are arranged at intervals of 0.7D along the x-axis on the tank bottom. Two rods are used as anodes, and the others are used as cathodes. Applying a DC voltage between the electrodes generates small oxygen and hydrogen bubbles at the anodes and cathodes, respectively. The generated bubbles rise in the water because of buoyancy with inducing water flow around them. Inside the generated bubble plume, a circular cylinder is horizontally placed. The cylinder length is 6.5D and its axis is orthogonal to the carbon rods. Its distance from the tank bottom is 6.5D.

The volume of hydrogen and oxygen bubbles generated per unit time at the carbon rods is controlled by the electric current. In this study, 30 V is applied, and the experiments are performed using electric current I of 0.1 and 0.3 A. The bubble volume Q_g is measured by the downward displacement of water. The Q_g values at I = 0.1 and 0.3 A are 16 and 53 mm³/s, respectively.

The bubbles are visualized using a laser light sheet with 0.5 W power, 532 nm wavelength, and 1 mm thickness. The bubble images in the central section (x-z section) perpendicular to the cylinder axis and those in the central section (y-z section) parallel to the cylinder axis are captured using a video camera to investigate the bubble motion.

The bubble diameter is measured in the upper regions of the cylinder using the bubble images acquired at 23 positions in the central section (x-z section) perpendicular to the cylinder axis. At each position, a small rectangular section is arranged, as shown in Figure 3. The side lengths in the horizontal (x) and vertical (z) directions are 0.53D and 0.33D, respectively. The diameter of the bubbles inside each section is measured to calculate the average diameter.

The bubble diameter is smaller than 0.06 mm at all measurement sections, as discussed below. The bubble velocity in the x-z section is measured with a particle image velocimetry (PIV) system by treating the bubbles as tracer particles. The frame rate and shutter speed of the camera are 200 fps and 1/1000 s, respectively.

The water flow around the cylinder is visualized using fluorescent paint (Rhodamine B). First, the lower-half surface of the cylinder is thinly coated with transparent glue mixed with the fluorescent paint, as shown in Figure 4. The axial length of the coated part is 1.7D. After the glue dries, the cylinder is placed in the water tank. Because the water flow dissolves the paint coated on the cylinder surface, the paint can be used to visualize the water flow. The trajectories of the paint correspond to the water streaklines.

First, the fundamental characteristics of the bubble plume are investigated for this particular experimental setup. The electrolysis of water is started before placing the cylinder in the tank. The bubbles generated at the cathode and anode on the tank bottom rise in the tank due to buoyancy. The bubble plume forms after the bubbles reach the water surface. Figure 5 shows the visualized bubble images at the central vertical section (x-z section) of the tank. The images at 5.5 ? z/D ? 9 are presented. In the case of the bubble flow rate of Q_g = 16 mm³/s, a number of bubbles concentrate at −0.5 ? x/D ? 0.5 near the central axis of the tank. The bubble distribution is mostly non-uniform and resembles a belt. The bubbles form cluster locally, especially at the periphery of the bubble plume (x/D ≃ ∓0.5). This is because the large-scale eddies are induced by the water velocity shear layers at the periphery, and accordingly the bubbles are entrained in the eddies. When Q_g = 53 mm³/s, the horizontal width of the region in which the bubbles are distributed is larger than when Q_g = 16 mm³/s. Bubble entrainment in the large-scale eddies occurs at the periphery of the bubble plume x/D ≃ ∓1, similar to the case of Q_g = 16 mm³/s.

Figure 6 shows the distribution of the bubble diameter d_b. For Q_g = 16 mm³/s, the variations of d_b in the horizontal (x) and vertical (z) directions are not remarkable. There are no data at z/D ? 9.5 at a horizontal distance of 2D from the cylinder. This is because no bubbles pass through this region. For Q_g = 53 mm³/s, the variation of d_b is also small. The average bubble diameters at Q_g of 16 and 53 mm³/s are 0.043 mm and 0.047 mm, respectively.

The terminal velocity of a small bubble is given by the following equation [13] :

where g is the gravitational constant, Δρ is the density difference between the bubble and the ambient fluid, and μ is the viscosity of the ambient fluid. Parkinson et al. [14] measured the terminal velocity of bubbles in water and confirmed the validity of Equation (1). d_b ranged from 10 to 100 μm. According to Figure 6, the value of d_b at position (x/D, z/D) = (0, 0.5), where the circular cylinder would be placed, is 0.045 mm irrespective of the value of Q_g. The terminal velocity of the bubble is calculated at 1.7 mm/s from Equation (1).

The bubble velocity u_g is measured by the PIV system in the central vertical section (x-z section) of the tank, and the time-averaged velocity during 60 s is calculated. Figure 7 shows the distribution of around the plume center (−0.75 ? x/D ? 0.75), where the velocity is expressed in a non-dimensional form using the bubble terminal velocity u_t (= 1.7 mm/s). For Q_g = 16 mm³/s, the bubbles flow nearly vertically upward. The magnitude of is much larger than that of u_t, indicating that high-velocity vertical water flow is induced by the rising bubbles. is maximum at the plume center (x = 0) and decreases with increasing horizontal distance from the center. This is attributed to the bubble concentration around the plume center (Figure 5). Note that the velocity variation in the vertically upward direction (z-direction) is small. This suggests that the plume develops spatially in this direction. The velocity for Q_g = 53 mm³/s is higher than that for Q_g = 16 mm³/s. The decrease with increasing is lower because the bubble distribution is broader in the horizontal direction owing to the larger bubble flow rate (Figure 5), and therefore, the induced water flow has higher velocity in the broader region. The variation in the vertically upward direction is small, as in the case of Q_g = 16 mm³/s. Because the bubble diameter is smaller than 0.047 mm (Figure 6), the bubble velocity is assumed to be the same as that of water. Thus, Figure 7 shows the approximate water velocity distribution.

Figure 8 shows images of the bubbles in the central vertical section (x-z section) perpendicular to the cylinder axis. For Q_g = 16 mm³/s, the rising bubbles collide with the front surface of the cylinder and form a layer along the lower-half surface of the cylinder. The bubble layer is separated from both sides of the cylinder (x/D = ∓0.5, z/D = 6.5) and the separated bubbles flow almost vertically. This is attributed to the separation and convection of water shear layers, as discussed below. The separated layers of the bubbles roll up downstream of the cylinder. The region where bubbles are hardly entrained is just behind the cylinder. For Q_g = 53 mm³/s, the separated layers are observed at the cylinder sides. They convect more vertically when compared with the results for Q_g = 16 mm³/s. Presumably, the rising velocity of the plume increases.

The bubble distribution in the vertical section (y-z section) passing through the cylinder axis is shown in Figure 9. Irrespective of the Q_g value, the bubbles colliding with the cylinder move in the horizontal (y) direction along the lower-half surface of the cylinder. Thus, the bubble distribution is broader in the horizontal direction downstream of the cylinder. Bubble clusters appear at the plume edges downstream of the cylinder, and a bubble plume with three-dimensional features is seen.

Figure 10 shows the bubbles and Rhodamine B around the cylinder for Q_g = 16 mm³/sat every 1/3 s after the reference time t₀. The Rhodamine B images show that the water separated shear layers originating at both sides

of the cylinder convect with rolling up. Several bubbles are distributed along the shear layers. The water and bubbles separate at the cylinder surface and convect together. Large-scale eddies induced by the roll-up of the separated shear layers are observed downstream of the cylinder, and some bubbles are entrained in the eddies.

Bubbles and Rhodamine B images at Q_g = 53 mm³/s are shown in Figure 11. In comparison with the images at Q_g = 16 mm³/s, the water separated shear layers convect more vertically and the position where the roll-up occurs shifts downstream.

The bubble diameter d_b distributes as shown in Figure 12. Compared with the result for the cylinder-free condition in Figure 6, no effect of the cylinder on d_b is observed. The average diameters at Q_g = 16 and 53 mm³/s are 0.047 and 0.045 mm, respectively.

The distribution of the time-averaged bubble velocity in the central vertical section (x-z section) is shown in Figure 13, where the distribution of the same region as in Figure 7 is presented. At Q_g = 16 mm³/s, the velocity decreases along the plume centerline while approaching the leading point of the cylinder. A stagnant flow is seen at the leading point. The bubbles colliding with the leading point flow along the cylinder surface with being accelerated, and their velocity reaches the maximum at the cylinder sides. This is attributed to the fact that the water is accelerated owing to the curvature of the streamline along the cylinder surface. The separated layers

originating at both sides of the cylinder are reconfirmed. Note that the velocity is lower near the central region just behind the cylinder and the wake ranges in the vertical direction. At Q_g = 53 mm³/s, the velocity distribution is similar to that at Q_g = 16 mm³/s. Nevertheless, the wake region is narrower and the velocity recovers promptly. On the cross-section at z/D = 5, upstream of the cylinder, the averaged velocity u_g0 is calculated. The values of u_g0 at Q_g = 16 and 53 mm³/s are 23.4 and 32.7 mm/s, respectively.

The horizontal bubble velocity u_gx is measured at a position in the central vertical section (x-z section) perpendicular to the cylinder axis. The coordinates of the position (x, z) are (0.63D, 7.7D). The luminance of the bubble L is assumed to be proportional to the bubble volume fraction and is measured at (1.3D, 7.7D). Figure 14 shows the two positions. When measuring L, a square region of 11 × 11 pixels is set around the measurement position to obtain the luminance at 121 pixels by the 256 gradations and calculate the average.

Figure 15 shows the time variation of u_gx at Q_g = 16 mm³/s. When the cylinder is not placed, no marked variations are seen. However, variations with almost constant period and amplitude appear when the cylinder is placed. The cylinder induces a bubbly flow with periodic oscillation in the horizontal direction. The velocity is measured at the location where the water separated shear layer with bubbles passes periodically (Figure 10).

The time variation of u_gx at Q_g = 53 mm³/s is shown in Figure 16. Though u_gx varies even in the cylinder-free condition, the variation induced by the cylinder is nearly periodic and the amplitude increases. The amplitude is larger than that at Q_g = 16 mm³/s because of the higher velocity of the separated layers.

The frequency analysis of u_gx shows the dominant frequency f_B, which is 0.14 and 0.11 Hz at Q_g = 16 and 53 mm³/s, respectively. The non-dimensional frequency f_BD/u_g0 is calculated, where u_g0 is the velocity at z/D = 5 upstream of the cylinder. The values at Q_g = 16 and 53 mm³/s are 0.18 and 0.1, respectively. The Reynolds number of the bubble plume is defined as Re = u_g0D/ν, where ν is the kinematic viscosity of water. The Re values at Q_g = 16 and 53 mm³/s are 702 and 981, respectively.

The vortex shedding frequency f behind a circular cylinder for uniform flow with velocity U is expressed by the Strouhal number St (= fD/U). Figure 17 shows the relation between Re (= UD/ν) and Stinvestigated by Roshko [15] . The solid line in the figure is the Roshko’s correlation, which is given as

The values of St = f_BD/u_g0 are also shown by painted symbols in Figure 17. They are much lower than the

results for uniform flow. This may be attributed to the non-uniform distribution of the fluid density due to the bubbles.

Figure 18 shows the time variation of L at Q_g = 53 mm³/s, where L is normalized by the maximum value (255). L varies irregularly in the cylinder-free condition. Nevertheless, a regular variation is produced when the cylinder is placed. The time-averaged L and amplitude increase. The variation of L also demonstrates the periodicity of the water separated shear layer with bubbles.

The frequency analysis of L shows that the dominant frequency f_L is 0.14 Hz. This is slightly larger than that of u_gx (0.11 Hz). Apparently, the positions where L and u_gx are minimum or maximum do not always correspond.