A unified theory for the envelope radiation of ring-mode coherent structures in the very-near-nozzle and developed regions of a circular jet

Large-scale coherent structures (CS) are present on a turbulent circular jet and are known to constitute an important source of jet noise. These structures may be treated as wavepackets of instability modes supported by the time-averaged mean flow. The nonlinear evolution and acoustic radiation of such CS in the fully developed region of a jet have been described in our recent work [Zhang and Wu, “Nonlinear evolution and low-frequency acoustic radiation of ring-mode coherent structures on subsonic turbulent circular jets,” J. Fluid Mech. 940(A39), 1–56 (2022)], where the mechanism of envelope radiation, or low-frequency radiation, was identified. The present paper focuses on the distinguished very-near-nozzle region, where the nozzle diameter is comparable with the length scale of the envelope of the CS. The CS resides in the thin shear layer developing from the lip line, and its nonlinear dynamics resembles that on a planar shear layer, only slightly affected by circular geometry, or circularity. However, circularity causes a leading-order effect on the acoustic radiation. One of significant new features is the presence of an inner acoustic field in the potential core. The outer and inner acoustic fields interact with each other, and the equivalent sources of sound have to be determined along with the sound fields. The waves in the inner acoustic field propagate mainly upstream. It is shown for the first time that nonlinear interactions of CS in the thin shear layer excite via receptivity the so-called discrete “trapped acoustic modes” as well as low-frequency Kelvin–Helmholtz modes, which may evolve into preferred modes in the developed region. The envelope radiation provides a key mechanism coupling the dynamics in the thin shear layer with the dynamics in the jet column regions. As the jet develops downstream, the inner acoustic field narrows gradually and disappears eventually. It is demonstrated that the theory developed here for nonlinear dynamics and acoustic radiation is valid from the very-near-nozzle to the developed regions, and thus stands as a unified theory.