Vortex formation in starting buoyant jets at moderate Richardson numbers

In this paper, the formation process of the leading vortex ring in positively and negatively buoyant starting jets with moderate Richardson
number in the range of 0.06 < |Ri| ≤ 0.321 has been investigated numerically and theoretically. Using the similarity variables |Ri| and |Ri|2/5 for
the positively and negatively buoyant starting jets, respectively, fitting equations can be obtained to predict the buoyant jet penetration rate.
Based on these fitting equations, a revised circulation model is proposed by incorporating the effects of both over-pressure and buoyancy.
The revised model is well consistent with the numerical results for all positively buoyant starting jets. However, for the negatively buoyant
starting jets, this model can predict well only during the initial period. The over-pressure has little influence on the vortex ring characteristics
of the starting jets with positive buoyancy, whereas it can significantly affect the vortex ring formation of the negatively buoyant starting jets
at moderate Richardson numbers. As the positive Richardson number increases, the instabilities of the trailing shear layer occur earlier. At
moderately negative Richardson numbers, a “double plume” structure (an inner sinking circular forced plume and an outer rising annular
plume) can be observed. The outer negative vorticity layers develop gradually due to the baroclinic effect. Consequently, the size and strength
of the leading vortex progressively decrease. As the negative Richardson number decreases, the negative vorticity layers occur earlier and grow
faster.