Successive bifurcations from steadiness to unsteadiness in a diffuser flow

Based on the diffuser flow configuration of Yu et al. (2010), we provide an answer to the debated question of the criticality of the first bifurcation, which we show to be a supercritical pitchfork bifurcation. We also show that this first bifurcation is quickly followed by a second, saddle–node type bifurcation, leading to the coexistence of two stable solution branches. Quite unexpectedly, the symmetric base flow undergoes a secondary supercritical pitchfork bifurcation which coincides with the saddle–node bifurcation mentioned above. The coincidence of local bifurcations, although not generic in dynamical systems theory, nevertheless seems to occur in fluid mechanics, as shown by Deng et al. (2021) in the wake of the fluidic pinball. Several steady-state solutions, both symmetric and asymmetric, stable or unstable, coexist over a finite range of Reynolds numbers, before one of them destabilizes, leading to periodic oscillations of the jet at the nozzle outlet. Inspired by the pioneering work of Deng et al. (2020), we propose a least-order model of the first bifurcation involving only two degrees of freedom and show that the model reproduces well the transient dynamics toward the asymmetric flow state.