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

• Published in: Journal of fluid mechanics Vol. 696; pp. 94 - 121
• Main Authors: Ilak, Miloš, Schlatter, Philipp, Bagheri, Shervin, Henningson, Dan S.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 10.04.2012
• Subjects: absolute/convective instability; bifurcation; Bifurcations; Computational fluid dynamics; Exact sciences and technology; Flow control; Flow velocity; Fluid dynamics; Fluid flow; Fluid mechanics; Fundamental areas of phenomenology (including applications); Hydrodynamic stability; Instability; Jets; Nonlinearity (including bifurcation theory); Physics; Shear layers; Simulation; Stability; Stability analysis; Turbulence; Turbulent flow; Turbulent flows, convection, and heat transfer; Velocity; bifurcation; absolute/convective instability; jets; Digital simulation; Vortices; Bifurcation; Flow control; Jets; Crossflow; Modelling; Interactions
• ISSN: 0022-1120; 1469-7645; 1469-7645
• DOI: 10.1017/jfm.2012.10

We 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 Bagheri et 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.