Figure 3 presents the time variation for the piston motion, where the displacement z_p and the velocity U₀ (= dz_p/dt) are plotted. The motion time T₀ is 0.748s. The velocity profile is trapezoid, and the maximum velocity U_m (=154 mm/s) is maintained for 0.64T₀.

When a fluid is discharged from a cylinder into the still fluid by a piston, a laminar vortex ring or a turbulent one is launched depending on the condition for the discharge. Glezer [18] identified the conditions that lead to the generation of the laminar and turbulent vortex rings as shown in Figure 4. The identification is based on the cylinder inner diameter D₀, the piston stroke L₀ and the circulation of vortex ring Γ₀. Γ₀ is computed by the following equation:

(1)

where T₀ and U₀ are the motion time and velocity of the piston respectively.

The circulation Γ₀ calculated from Equation (1) results in Γ₀/ν = 7252. According to the identification of Glezer in Figure 4, it is found that a laminar vortex ring is launched into still water in this experiment.

2.3. Launch of Vortex Ring

Small hydrogen bubbles are released by the electrolysis of water. A copper wire of diameter 0.1 mm is employed for the cathode. It is wound around the cylinder of the vortex ring launcher, as illustrated in Figure 2. The height is 5 mm below the cylinder outlet.

The hydrogen bubbles released from the circular cathode are spherical, and their diameter is less than 0.5 mm. The voltage between the electrodes varies the amount of the bubbles released per unit time. This study sets the voltage at 60 V. The bubble volume release rate, measured by downward displacement of water, is 4.1 mm³/s, while the bubble mean diameter is 0.2 mm.

When the terminal velocity v_t for a single bubble with diameter 0.2 mm rising in still water is estimated from the numerical calculation, it is 16.3 mm/s.

3. Experimental Results and Discussions

3.1. Behavior of Vortex Ring Launched into Still Water

First, the behavior of the vortex ring in still water is investigated to examine the characteristics of the vortex ring in this experimental set-up. A visualized image for the central vertical cross-section of the vortex ring is presented in Figure 5. It is acquired by adding water paint to the launched water in the cylinder. The mean diameter and volume fraction of the paint are about 50 μm and 0.006 respectively. A vortex pattern is clearly observed. A shear layer originates at the boundary between the water discharged from the cylinder and the still water in the tank, and the paint is entrained into the vortex core with the roll up of the shear layer. A laminar distribution of the paint is found behind the vortex ring. There are no active mixings. The visualized image indicates the characteristics of a laminar vortex ring. This is parallel to the Glezer’s identification [18] demonstrated in Figure 4.

On the visualized image of the central vertical crosssection of the vortex ring shown in Figure 5, the z coordinate of the center for vortex core is regarded as the displacement of vortex ring z_v, while the distance between the centers of the vortex cores is defined as the diameter of vortex ring D. 

The displacement of vortex ring z_v changes with the

lapse of time as shown in Figure 6, where the piston maximum velocity U_m and the cylinder inner diameter D₀ are used for the nondimensional expressions. The time when the piston reaches the top dead center is set at t = 0. The change of z_v is almost linear, indicating that the vortex ring rises with a constant velocity in still water. The rising velocity, dz_v/dt, is 0.41 U_m. The rise almost ceases at z_v/D₀ ≒ 5.8. Because the vortex ring is affected by the water surface at z/D₀ = 6. 

The change in the diameter D is plotted against the displacement z_v in Figure 7. The diameter remains almost unaltered at z_v/D₀ ≦ 5.8. The marked increment at z_v/D₀ > 5.8 is attributable to the effect of the water surface. 

This study measures the water velocity distribution on the vertical central cross-section of the vortex ring at four displacements of z_v/D₀ = 0.78, 2.77, 4.1, 5.01, as mentioned later. The measurement section includes the vortex core. The radial and vertical widths are 2.23D₀ and 1.59D₀ respectively. When the vorticity is calculated at each displacement, the circulation Γ can be estimated from the distribution. The nondimensional circulation Γ/ν is plotted by the open symbols in Figure 8. The circulation remains almost unaltered in the region investigated by this study. When the piston reaches the top dead  

center (t = 0), the displacement z_v/D₀ is 0.78. The circulation Γ/ν at this displacement is 8063. It is slightly larger than the value, Γ/ν = 7252, estimated from Equation (1). Similar result was also reported by Glezer [18] and Gharib et al. [19]. Equation (1) is based on a slug model assuming that the water velocity is uniform within the cylinder cross-section. The difference from the measurement is attributable to this assumption. 

3.2. Characteristics of Bubble Plume

Figure 9 illustrates the bubble distribution for the temporally developed bubble plume, where the distribution on the vertical plane passing through the plume centerline is depicted. The bubbles, generated from the cathode wound around the cylinder outlet, rise due to the buoyant force. They hardly move straight in the vertical direction. They shift toward the plume centerline near the cathode (z/D₀ ≦ 1), and meander slightly with their rise at z/D₀ > 1. The shift is caused by the fact that the water outside of the plume flows toward the centerline to satisfy the mass conservation low of the water. Because the rising bubbles induce an upward water flow around the centerline. 

When the time-averaged water velocity is measured, the axial (vertical) component changes in the radial direction as plotted in Figure 10, where the distributions at four elevations from the cylinder outlet are presented. At z/D₀ = 0.78, takes its maximum value at r/D₀ = 0.26. This is because the bubbles released from the cylinder tip shift toward the centerline and they concentrate at this radial position as seen in Figure 9, and accordingly an upward water flow with higher vertical velocity is induced there. At z/D₀ = 2.77 and 4.1, reaches the maximum on the centerline of the plume. The velocity increases with the elevation z. One can confirm the entrainment of the water into the plume. A spatially developing buoyant jet occurs. The velocity lowers at z/D₀ = 5.01 owing to the effect of the water surface.

The time-averaged bubble velocity is measured at the radial positions of r/D₀ = 0.25. It changes in the vertical (z) direction as shown in Figure 11, where the

bubble terminal velocity v_t is used for the nondimensional expression. The radial positions correspond to the regions where the bubbles concentrate as seen in Figure 9. The axial (vertical) velocity heightens with increasing the elevation, demonstrating that the plume is spatially developing in the measured region. But it shows a tendency to be saturated. The velocity is higher than the bubble terminal velocity at z/D₀ ≧ 2. This is owing to the fact that the bubbles rise relative to the vertically upward water flow induced by them. The radial (horizontal) velocity is much lower than, and the change in the vertical direction is small. 

3.3. Behavior of Vortex Ring Launched into Bubble Plume and Bubble Distribution

When the vortex ring is vertically launched into the bubble plume along the centerline, the bubbles distribute as shown in Figure 12. The distributions on the central vertical cross-section at four displacements of vortex ring z_v are presented. At z_v/D₀ = 0.78, the bubbles above the vortex ring tend to move to the rear of the vortex ring

along the vortex core, and they are being spirally entrained into the vortex core. The bubble entrainment is caused by the roll up of the shear layer originating from the cylinder tip. This displacement is at the time tU_m/D₀ = 0 when the piston reaches the top dead center. At z_v/D₀ = 2.77, the spiral entrainment further proceeds. The bubble volume fraction within the cross-section reaches its maximum value at this displacement as mentioned later. At z_v/D₀ = 4.1, the number of bubbles in the vortex ring around the center axis (r/D₀ = 0) is smaller than that at z_v/D₀ = 2.77, and a lot of bubbles distribute behind the vortex ring. Because the bubbles moving to the rear of vortex ring are hardly entrained into the vortex ring due to the cease of the roll up for the shear layer, and accordingly they are left behind the vortex ring. When the vortex ring rises more (z_v/D₀ = 5.01), the amount of bubbles inside the vortex ring increases again due to the collision between the vortex ring and the bubbles accumulated near the water surface. A marked increment occurs around the center axis. 

On the visualized image of the bubble distribution within the central vertical cross-section of the vortex ring, the z coordinate of the center for vortex core is regarded as the displacement z_v, while the distance between the centers of the vortex cores is defined as the diameter D. Figure 13 indicates the image at z_v/D₀ = 1.22. It is confirmed that z_v and D are clearly identified.

Figure 14 shows the time variation for z_v. The displacement in the bubble plume is almost the same as that in still water at tU_m/D₀ ≦ 2.42. But it is larger at tU_m/D₀ > 2.42, indicating that the rising velocity of the vortex ring in the bubble plume is higher. Because the vortex ring is affected with the vertically upward water flow induced by the rising bubbles as explained later. When tU_m/D₀ > 2.42, the time variation of z_v is linear. The vortex ring rises at a constant velocity.

The diameter of vortex ring D changes as a function of z_v as presented in Figure 15. The diameter D in the bubble plume decreases with the rise of vortex ring. Thus, the diameter is smaller than that in still water. A buoyant jet is generated as shown in Figure 9, and there exists an upward water flow around the centerline (r/D₀ = 0). Therefore, the water is entrained into the bubble plume, and the water flowing toward the centerline reduces the D value.

3.4. Void Fraction in Central Vertical Cross-Section of Vortex Ring

To grasp the change in the amount of bubbles inside the vortex ring with the convection of the vortex ring, the void fraction α in a rectangular region surrounding the vortex core is calculated. Binarized images for the bubble distribution are used for the calculation. The region is depicted in Figure 13. The width and the height are 1.25D₀, and the centroid is fixed at the center of the vortex core. 

Figure 16 shows the relation between the void fraction and the displacement of vortex ring z_v. At 0.78 ≦ z_v/D₀ ≦ 2.77, α increases monotonically. This is because the bubbles are being spirally entrained into the vortex ring with the roll up of the velocity shear layer, as seen in Figures 12(a) and (b). But α decreases at 2.77 ≦ z_v/D₀ ≦ 4.1 where the strength of the shear layer lessens. Because the bubbles are left behind the vortex ring, as seen in Figure 12(c). At z_v/D₀ > 4.1, α increases again (Figure 12(d)). This is attributable to the fact that the vortex ring collides with the bubbles accumulated near the water surface. The maximum value of α at z_v/D₀ = 2.77 is 2.43 times as large as that near the cylinder outlet (z_v/D₀ = 0.78).

3.5. Distributions for Velocity and Vorticity of Water

The distribution of water velocity on the central vertical cross-section of the vortex ring is shown in Figure 17,

where the ensemble-averaged velocity at four displacements of vortex ring is presented. The distribution in still water and that in the bubble plume are compared, and the bubble distribution is also presented. When the vortex ring is launched into still water, the vortex core remains nearly circular. The water velocity scarcely depends on the displacement of the vortex ring. Even when the vortex ring is launched into the bubble plume, the vortex core exists. The vortex core at z_v/D₀ = 0.78 and 2.77 is not circular, but deforms with the convection of vortex ring. The deformation is reconfirmed from the contour lines for the magnitude of velocity shown in Figure 18. The deformation is mainly caused by the shear flow of water induced by the rising bubbles. It may also be attributable to the fact that the time variation of the bubble distribution around the vortex core changes markedly due to the active bubble entrainment. Sridhar and Katz [17] made clear that a vortex ring deforms even when a few bubbles are entrained. The present authors elucidate the variation of the water velocity in the vortex core with the convection of vortex ring. They also grasp the change of the shape for vortex core. 

Figure 19 presents the axial (vertical) component of the ensemble-averaged water velocity, where the distribution on the horizontal line passing through the

centers of the vortex cores shown in Figure 17 is plotted. The velocity remains unaltered between the vortex cores. The velocity in the bubble plume is higher than that in still water at z_v/D₀ = 2.77 and 4.1. This is due to the upward water flow of the plume. The velocity gradient around the center of the vortex core in the bubble plume is smaller than that in still water. 

The distribution for the ensemble-averaged vorticity on the central vertical cross-section of vortex ring is shown in Figure 20, where the bubbles are superimposed on the result in the bubble plume. The absolute value of the vorticity around the vortex core in the bubble plume is lower than that in still water at every displacement of the vortex ring. Because the friction acts on the interface between the bubbles and the water. Ruetsch and Meiburg [20] conducted a numerical simulation of a plane shear layer laden with bubbles. They highlighted that the bubbles concentrate around the vortex center and that the concentration reduces the strength of the vortex strength.

The present experiment grasps the variation in the vorticity distribution with the convection of vortex ring, and therefore it elucidates the dynamic interaction between the vortex ring and the entrained bubbles.

The circulation Γ around the vortex core in the bubble plume is calculated from the vorticity field. The nondi-

mensional circulation Γ/ν is plotted by the painted symbols in Figure 8. At z_v/D₀ ≦ 4, the circulation in the bubble plume is higher than that in still water. Because the effect of the upward water flow is superimposed on Γ. The circulation in the bubble plume lessens with the rise of vortex ring. Because the absolute value of the vorticity in the vortex ring decreases owing to the friction between the water and the entrained bubbles.

The vorticity on the horizontal line passing through the

centers of the vortex cores is plotted in Figure 21. At z_v/D₀ = 0.78, the maximum value for the vorticity in the water is markedly high. One can find the occurrence of a velocity shear layer from the cylinder tip. But the maximum value in the bubble plume at the displacement is not so high. This is attributable to the fact that the velocity gradient of the shear layer is reduced by the bubbles. The absolute value of lessens greatly around the vortex core, indicating that the vorticity distribution is flattened by the entrained bubbles. 

4. Conclusions

The interactions of bubbles with a vortex ring launched

into a bubble plume were investigated. Laminar vortex ring was vertically launched into bubble plume. The mean bubble diameter is 0.2 mm. The results are summarized as follows: 

1) The bubbles are spirally entrained into the vortex ring just after the launch of vortex ring. When the vortex ring reaches a certain height, the amount of the bubbles around the center axis of vortex ring lessens, and the bubbles moving to the rear of the vortex ring are left behind the vortex ring. 

2) The rising velocity of vortex ring in the bubble plume is higher than that in still water. This is because the vortex ring is affected by the vertically upward water flow induced by the bubbles. The velocity of vortex ring is almost constant. 

3) The diameter of vortex ring reduces with the rise of the vortex ring. The bubble plume is generated by the rising bubbles, and there exists the vertically upward water flow around the centerline. Therefore the water is entrained into the plume, and the water flowing toward the centerline reduces the diameter.

  4) The void fraction in the central vertical cross-section of vortex ring increases until a certain displacement

of vortex ring, where the reduction occurs. It heightens again when the vortex ring approaches the water surface. Because the vortex ring collides with the bubbles accumulated near the water surface.

5) The vortex core deforms with the convection of vortex ring. The entrained bubbles cause the decrement of the absolute value for the vorticity around the vortex core, reducing the strength of vortex ring.

REFERENCES