Purpose This study aims to provide discussions of the numerical method and the bubbly flow characteristics of an annular bubble plume. Design/methodology/approach The bubbles, released from the annulus located at the bottom of the domain, rise owing to buoyant force. These released bubbles have diameters of 0.15–0.25 mm and satisfy the bubble flow rate of 4.1 mm3/s. The evolution of the three-dimensional annular bubble plume is numerically simulated using the semi-Lagrangian–Lagrangian (semi-L–L) approach. The approach is composed of a vortex-in-cell method for the liquid phase and a Lagrangian description of the gas phase. Findings First, a new phenomenon of fluid dynamics was discovered. The bubbly flow enters a transition state with the meandering motion of the bubble plume after the early stable stage. A vortex structure in the form of vortex rings is formed because of the inhomogeneous bubble distribution and the fluid-surface effects. The vortex structure of the flow deforms as three-dimensionality appears in the flow before the flow fully develops. Second, the superior abilities of the semi-L–L approach to analyze the vortex structure of the flow and supply physical details of bubble dynamics were demonstrated in this investigation. Originality/value The semi-L–L approach is applied to the simulation of the gas–liquid two-phase flows.

Purpose This paper aims to provide discussions of a numerical method for bubbly flows and the interaction between a vortex ring and a bubble plume. Design/methodology/approach Small bubbles are released into quiescent water from a cylinder tip. They rise under the buoyant force, forming a plume. A vortex ring is launched vertically upward into the bubble plume. The interactions between the vortex ring and the bubble plume are numerically simulated using a semi-Lagrangian–Lagrangian approach composed of a vortex-in-cell method for the fluid phase and a Lagrangian description of the gas phase. Findings A vortex ring can transport the bubbles surrounding it over a distance significantly depending on the correlative initial position between the bubbles and the core center. The motion of some bubbles is nearly periodic and gradually extinguishes with time. These bubble trajectories are similar to two-dimensional-helix shapes. The vortex is fragmented into multiple regions with high values of Q, the second invariant of velocity gradient tensor, settling at these regional centers. The entrained bubbles excite a growth rate of the vortex ring's azimuthal instability with a formation of the second- and third-harmonic oscillations of modes of 16 and 24, respectively. Originality/value A semi-Lagrangian–Lagrangian approach is applied to simulate the interactions between a vortex ring and a bubble plume. The simulations provide the detail features of the interactions.

Collision of two vortex rings (VR) initially arranged in axis-offset and orthogonal configurations at Reynolds numbers (ReΓ) in the range of 5000–200 000 was simulated to investigate turbulent energy cascade associated with their reconnection. Two elliptical VRs are generated by joining each part of the first VR with another part of the second VR for the axis-offset collision, while two VRs associate to form a double U-shaped vortex, and this vortex reconnects itself at two points to form three elliptical VRs linked by the vortex filaments for the orthogonal collision. Many vortex structures in various scales and shapes, including small-scale VRs and horseshoe vortices, are observed in connection regions for both cases. As ReΓ increases, the energy of formed small vortices raises and their wavenumber (k) range enlarges. The flow energy spectrum approaches a k−5/3 slope of the Kolmogorov hypotheses at low wavenumbers. For the axis-offset collision, the energy spectrum at medium wavenumbers continuously changes from k−3.0 at ReΓ= 5000 to k−1.8 at ReΓ= 200 000, and the exponent (α) of the wavenumber is determined by a function as α=0.3304 ln(ReΓ)−5.6538. Meanwhile, the energy spectrum at two medium-wavenumber subranges for the orthogonal collision with ReΓ≥ 20 000 approaches the slopes of k−3.0 and k−2.6. Turbulent mixing performance due to the axis-offset collision of two vortex rings is better than that with the orthogonal one.

The reconnection of a vortex ring and a vortex tube in a viscous fluid with the effects of two vortex core sizes (σ0=0.12r0 and 0.24r0, where r0 are initial ring radius) and three initial flow configurations (left-offset, center, and right-offset) at Reynolds number (ReΓ) of 10 000 was investigated using a high-order vortex-in-cell method combined with a large-eddy simulation model. For the left-offset case, a large part of the ring, slipping over the tube, associates with a small part of the tube to establish a new vortex ring, whereas the rest of the tube is reconnected by another part of the ring. For the center case, half of the ring joins with a part of the tube to construct an elliptical vortex ring while the rest connects because of viscosity. The reconnected ring and tube become more stable and are like the initial ones in the ultimate stage. For the right-offset case, both the ring and tube's reconnection occurs, and the reconnected elliptical vortex ring is rapidly distorted. The proportion of reconnected ring increases, and then this ring section loses its integrity, decaying into a complex cluster of various-scales vortex structures in different shapes. At σ0=0.12r0, the secondary vortex structures surrounding the tube and ring appear in three cases, while they are only observed for the center case at σ0=0.24r0. For three flow configurations and two vortex core sizes, after the reconnection, the energy cascade of the flow approaches a k−5/3 slope of Kolmogorov's similarity hypotheses and a k−3 slope in the ranges of wavenumbers (k) from 3 to 10 and from 10 to 40, respectively. The highest population of small-scale coherent vortex structures is observed for the right-offset, followed by the center and left-offset. In addition, a larger number of these structures was observed for a smaller core size. This validates that the mixing performance is the best at a small vortex core and in the right-offset configuration.

The deformation of a vortex ring caused by its impingement on a sphere was numerically investigated using a proposed vortex-in-cell method. The method was validated by simulation of the collision of a vortex ring with a rigid planar surface and proved to be most satisfactory in the analysis of the dynamics of a vortex structure. In a coaxial collision, the behavior of the vortex structure is similar to that in the case of a planar surface. A secondary vortex ring is formed owing to the separation of the boundary layers on the sphere, caused by the effect of the primary vortex ring. The interaction between the secondary and primary vortex rings plays an important role in the dynamics of the vortex structure when the secondary vortex ring is completely formed. In a noncoaxial collision, the structure of the secondary vortex is moderately different from that in the coaxial collision. Moreover, the vortex structure in the coaxial collision is two-dimensional, in which the vorticity field is dominated by two transverse components, whereas that in the noncoaxial collision is three-dimensional. The total kinetic energy in both the cases decreases gradually during the entire period of evolution, whereas the enstrophy reduces in the early stage, then increases considerably, before a gradual reduction in the final stage. The enstrophy reaches a peak when the secondary vortex ring is completely formed, at which stage the effects of vortex stretching and vorticity production at the solid boundary are higher than that of vortex diffusion.