Characterisation of the secondary instability in a 1 : 3 sudden expansion through inviscid structural sensitivity

The inviscid mechanism, driving flow instabilities in a $1:3$ , planar and symmetric sudden expansion, is discerned through a sensitivity-based protocol, also referred to as inviscid structural sensitivity analysis, with a specific focus on the onset and nature of the secondary instability. The fundamental idea of this methodology is to change the contribution of viscosity solely in the global stability equations, while freezing the base-flow field at the critical conditions. This is practically implemented by decoupling the Reynolds number that serves as the control parameter for determining the steady base flow from that governing the disturbance evolution, in order to repeat the structural sensitivity analysis while progressively increasing the Reynolds number in the linearised equations only. Accordingly, the sequence of structural sensitivity maps enables us to highlight the flow regions where the inviscid instability mechanism acts. The numerical results reveal that the classical structural sensitivity analysis accurately locates the wavemaker region within the primary recirculation zone, but only its inviscid limit can unveil that the core of the instability coincides with the centre of the primary vortex: a hallmark of an elliptic instability. To validate the global findings, the results of the inviscid structural sensitivity analysis are compared with those obtained from geometric optics. The agreement of the two approaches confirms the inviscid character of the instability, thereby providing a complete picture of the nature of the secondary bifurcation.