Turbulence cascade model for viscous vortex ring-tube reconnection

Abstract

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.