Unsteady turbulent round jets and vortex motion

Abstract

A new model to predict the velocity distribution in round jets with time-varying injection profiles has been formulated as an extension of steady jet theory. The approach introduces an effective injection velocity within the jet based on a representative response time. It is assumed that the instantaneous injection velocity affects the velocity within the jet with an exponential response function and that the response time is related to the fluid particle’s residence time within the jet, consistent with the theory of translation of jet vortex rings from Helmholtz’s vortex motion analysis [P. G. Tait, London Edinburgh Dublin Philos. Mag. J. Sci. 33, 485 (1867)]. The Helmholtz theory is also shown to reduce to the well-known velocity decay rate in the case of steady turbulent gas jets. A Duhamel superposition integral is used to determine the effective injection velocity for time-varying injection rates. The model is tested with different injection profiles and different ambient densities. The results are also compared with numerical results from a computational fluid dynamics code. The comparisons agree very well and the new model is shown to offer an efficient method to predict jet tip penetrations for unsteady jets.