Flow instability in weakly eccentric annuli

A temporal linear stability analysis of laminar flow in weakly eccentric annular channels has been performed. It has been shown that, even for eccentricities ε and Reynolds numbers that were much smaller than those considered in previous studies, flow instability occurred in the form of travelling waves having characteristics that are very different from those of Tollmien-Schlichting waves and which were triggered at mid-gap by an inviscid mechanism that is associated with the presence of inflection points in azimuthal profiles of the base velocity. The critical stability conditions have been determined for 0 ≤ ε ≤ 0.1 and for diameter ratios 0 < γ < 1. The critical Reynolds number Rec decreased with increasing γ for 0 < γ ≲ 0.13, reached a minimum at γ ≈ 0.13, and increased with further increase in γ. The lowest observed Rec was 529 and occurred for ε = 0.1 and γ ≈ 0.13. As ε → 0, Rec ∝ ε−2. The critical wave number and the critical frequency of the disturbances decreased with increasing γ and approached zero as γ → 1, whilst their ratio was nearly constant in the range of parameters considered in this study. The most unstable regions were found to be at roughly mid-gap on the two flanks of the annulus, and the phase speed of the disturbances was close to the base flow velocity at these regions.