This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number Re < 1150 and bend opening ratio (ratio of bend width to inlet height) 0.2 ≤ β ≤ 5. This range of Re and β captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For 0.2 ≤ β ≤ 1, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as β increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for β = 0.2 and a spanwise synchronous mode for β ≥ 0.3. The critical Reynolds number and the spanwise wavelength of perturbations increase as β increases. For 1 < β ≤ 2 both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as β increases. Finally, for 2 < β ≤ 5, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.
- channel flow
- separated flows