Three-dimensional linear stability analysis of the flow around a sharp 180° bend

This study seeks to characterise the stability of a two-dimensional channel flow involving a 180-degree sharp bend, to infinitesimal three-dimensional disturbances by way of a linear stability analysis. A highly accurate global linear stability analysis of the flow is presented via the Reynolds number Re varies in the range 100 ≤ Re ≤ 700, this Re range produces steady-state two-dimensional flow solutions for bend opening ratio (ratio of bend width on inlet height) β = 1. The two-dimensional base flow solutions demonstrate that as β decreases, the transition from steady to unsteady occurs at lower Reynolds number. The stability analysis shows that the flow first becomes unstable to a synchronous three-dimensional instability mode with span-wise wavenumber k = 2 at approximately Re = 400, whereas the two-dimensional solution branch undergoes transition to unsteady flow somewhere near Re ≈ 800. Instability mode structures associated with the leading eigenvalues are localized at the re-attachment point of the first separation bubble and the separation point of the second separation bubble. The stability analysis is used to produce neutral stability curves, and visualisations of the global modes of the system for typical Reynolds number are also presented.