Vortex ring formation in starting forced plumes with negative and positive buoyancy

Abstract

The limiting process of vortex ring formation in starting forced plumes, with Richardson number in the range of −0.06 ≤ Ri ≤ 0.06, was studied numerically under the Boussinesq approximation. The examination of the dynamics of the starting flow evolution reveals that the plume-ambient density difference affects the vortex ring pinch-off mainly through three mechanisms, i.e., the baroclinic production of vorticity, the buoyancy acceleration (or deceleration) on the vortical structures, and its effect on the trailing shear layer instability. As Ri increases from negative to positive values, three regimes can be identified in terms of the vortex interaction patterns during the pinch-off process, i.e., the weak-interaction regime (−0.06 < Ri < − 0.02), the transition regime (−0.02 ≤ Ri < 0), and the strong-interaction regime (0 ≤ Ri < 0.06). By eliminating the influence of the baroclinic vorticity production, the circulation method proposed for the starting jets is revised to determine the buoyant formation number F in the starting forced plumes. Besides the formation number F, another dimensionless time scale (dubbed as the separation number S), which corresponds to the end of the pinch-off process, is identified by the time of vanishment of the vorticity flux feeding the leading vortex ring. The numerical results show that the variation trends of formation number and separation number against Ri change near the critical value of Ric ≈ − 0.02. In the weak-interaction regime, both formation number and separation number increase rapidly against Ri. While in the transition and strong-interaction regimes alike, the formation number increases at a much slower rate than in the weak-interaction regime, and the separation number declines dramatically as Ri increases. Finally, a qualitative explanation on the variation patterns of formation number and separation number is proposed based on the buoyancy effects on the dynamic properties of the leading vortex ring and the vortex interaction patterns.