On the evolution and circulation dynamics of the counter rotating vortex rings in the starting phases of under-expanded jets

Abstract

In the starting phases of laminar, axisymmetric, continuously blowing under-expanded jets, the evolution of a counter rotating vortex ring (CRVR) over the periphery of the primary vortex ring (PVR) is numerically investigated. During CRVR evolution, the circulation dynamics of both these vortex rings is also studied. The effect of jet Reynolds number (Rej), jet Mach number (Mj), and pressure ratio (pjpa) on these phenomena is also shown. The CRVR evolution involves the rolling of the CRVR over the PVR periphery followed by the shock–vortex interaction between the CRVR and vortex induced shock (VIS) of the PVR. The PVR and CRVR circulations attain quasi-constancy as the CRVR starts to roll over the PVR periphery. The PVR circulation is shown to attain quasi-constancy even in the absence of its pinch-off (detachment) from the shear layer, which is a manifestation of Kelvin's theorem. The slip-stream strength is found to govern the magnitude of CRVR circulation. New type of discontinuities (induced shocks and vortex sheet) are observed inside the rolling CRVRs, which are generated from medium/high strength slip-streams. Due to the vorticity imparted from such slip-streams, there is self-roll-up of the CRVR leading to the generation of transient spiral stagnation point and vortex sheet which causes the formation of the transient saddle stagnation point. The formation of induced shocks inside CRVRs is due to the combination of two different constraints, i.e., (a) cyclic distribution of fluid properties inside the CRVR and (b) quasi-constancy of the CRVR circulation. The shock–vortex interaction between CRVR and VIS is a weak shock-strong vortex interaction, which occurs due to the constraint of quasi-constancy of net-circulation enclosing both vortex rings (i.e., PVR and CRVR) and causes the weakening of VIS.