Evolution of an elliptic vortex ring in a viscous fluid

Abstract

The evolution of a viscous elliptic vortex ring in an initially quiescent fluid or a linear shear flow is numerically simulated using a lattice Boltzmann method. A wide range of parameters are considered, namely, aspect ratios (AR) (1 ≤ AR ≤ 8), core radius to ring radius ratios (σ0) (0.1 ≤ σ0 ≤ 0.3), Reynolds number (Re) (500 ≤ Re ≤ 3000), and shear rate (K) (0 ≤ K ≤ 0.12). The study aims to fill the gap in the current knowledge of the dynamics of an elliptic vortex ring in a viscous fluid and also to address the issue of whether an elliptic ring undergoes vortex stretching and compression during axis-switching. In a quiescent fluid, results show that for fixed Re and σ0, there exists a critical aspect ratio (ARc), below which an elliptic ring undergoes oscillatory deformation with the period that increases with increasing AR. Above ARc, the vortex ring breaks up into two or three sub-rings after the first half-cycle of oscillation. While higher Reynolds number enhances vortex ring breakup, larger core size has the opposite effect. Contrary to an inviscid theory, an elliptic ring does undergo vortex stretching and compression during oscillatory deformation. In the presence of a linear shear flow, the vortex ring undergoes not only oscillatory deformation and stretching but also tilting as it propagates downstream. The tilting angle increases with the shear rate K and is responsible for inducing a “tail” that consists of a counter-rotating vortex pair (CVP) near the upstream end of the initial major axis after the first half-cycle of oscillation. For a high shear rate, the CVP wraps around the ring and transforms its topological structure from a simple elliptic geometry to a complicated structure that eventually leads to the generation of turbulence.