Secondary instabilities in the shear layer of a compressible jet over a convex wall

The compressible jet over a convex wall is numerically studied using the delayed detached-eddy simulation method based on the two-equation shear-stress transport model. In particular, the current study focuses on the secondary instabilities in the shear layer. The results show that the alternating high-speed and low-speed streaks in the shear layer induced by the streamwise vortices exhibit secondary instabilities similar to those found in other wall-bounded flows. The varicose mode secondary instability induces the thinning and thickening motions of the low-speed regions, whereas the sinuous mode leads to the side-to-side sway motions. The hairpin vortices and the counter-rotating roll-mode, which are often associated with the symmetric (varicose-type) mode, are dominated in the transient stage of the shear layer. However, as the shear layer evolves downstream, there still appear one-sided roll-modes and quasistreamwise vortices with vorticity of alternate sign produced by the spanwise overlapping of the hairpin vortex legs, which are well known to be related to the antisymmetric (sinuous-type) mode. Further, the dynamic mode decomposition analysis performed on the spanwise velocity fluctuation of the instantaneous snapshots reveals the presence of both sinuous- and varicose-type secondary instabilities, as well as their competitions, in the shear layer. Quantitatively, from θ=25∘ to θ=45∘, the proportion of the varicose mode decreases from 84.4% to 50.6%, and accordingly, the sinuous mode increases from 15.6% to 49.4%. The analysis of the turbulent kinetic energy production terms reveals that both varicose and sinuous modes are influenced by the radial and spanwise shear components of the turbulent energy production term, and they are in the same order of magnitude. Additionally, the mechanism of the instability is due to the velocity fluctuations and Reynolds stress parallel to the local mean flow gradient. The local inflection point instability caused by the mean flow gradient is also the source of turbulent energy that sustains the instabilities.