Mode selection in concentric jets: the steady–steady 1 : 2 resonant mode interaction with O(2) symmetry

Abstract

The linear and nonlinear stability of two concentric jets separated by a duct wall is analysed by means of global linear stability and weakly nonlinear analysis. Three governing parameters are considered, the Reynolds number based on the inner jet, the inner-to-outer jet velocity ratio ( $\delta _u$ ) and the length of the duct wall ( $L$ ) separating the jet streams. Global linear stability analysis demonstrates the existence of unsteady modes of inherent convective nature, and symmetry-breaking modes that lead to a new non-axisymmetric steady state with a single or double helix. Additionally, we highlight the existence of multiple steady states, as a result of a series of saddle-node bifurcations and its connection to the changes in the topology of the flow. The neutral lines of stability have been computed for inner-to-outer velocity ratios within the range $0 < \delta _u < 2$ and duct wall distances in the interval $0.5 < L < 4$ . They reveal the existence of hysteresis, and mode switching between two symmetry-breaking modes with azimuthal wavenumbers $1:2$ . Finally, the mode interaction is analysed, highlighting the presence of travelling waves emerging from the resonant interaction of the two steady states, and the existence of robust heteroclinic cycles that are asymptotically stable.