Bifurcation and stability analysis of a jet in cross-flow: onset of global instability at a low velocity ratio

Abstract

AbstractWe study direct numerical simulations (DNS) of a jet in cross-flow at low values of the jet-to-cross-flow velocity ratio$R$. We observe that, as the ratio$R$increases, the flow evolves from simple periodic vortex shedding (a limit cycle) to more complicated quasi-periodic behaviour, before finally becoming turbulent, as seen in the simulation of Bagheriet al. (J. Fluid. Mech., vol. 624, 2009b, pp. 33–44). The value of$R$at which the first bifurcation occurs for our numerical set-up is found, and shedding of hairpin vortices characteristic of a shear layer instability is observed. We focus on this first bifurcation, and find that a global linear stability analysis predicts well the frequency and initial growth rate of the nonlinear DNS at the critical value of$R$and that good qualitative predictions about the dynamics can still be made at slightly higher values of$R$where multiple unstable eigenmodes are present. In addition, we compute the adjoint global eigenmodes, and find that the overlap of the direct and the adjoint eigenmode, also known as a ‘wavemaker’, provides evidence that the source of the first instability lies in the shear layer just downstream of the jet.