Linear stability of confined flow around a 180-degree sharp bend

TL;DR

This paper performs a linear stability analysis of flow around a 180-degree sharp bend, identifying how 3D instabilities develop depending on the bend ratio and Reynolds number, revealing different instability modes.

Contribution

It provides a detailed global linear stability analysis of confined flow around a sharp bend, characterizing the onset of 3D instabilities across various flow parameters.

Findings

3D transition occurs via synchronous instability of steady flows.

Critical Reynolds number and spanwise wavelength depend on the bend opening ratio.

Different 3D modes are localized near the reattachment point depending on the opening ratio.

Abstract

This study seeks to characterise the breakdown of the steady 2D solution in the flow around a 180-degree sharp bend to infinitesimal 3D disturbances using a linear stability analysis. The stability analysis predicts that 3D 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) $0.2\leq\beta\leq5$. This range of $Re$ and $\beta$ captures both steady-state 2D flow solutions as well as the inception of unsteady 2D flow. For $0.2\leq\beta\leq1$, the 2D base flow transitions from steady to unsteady at higher Reynolds number as $\beta$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely spanwise oscillating mode for $\beta=0.2$, and spanwise synchronous mode for $\beta \geq 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\beta$ increases. For $1<\beta\leq2$ both the critical Reynolds for onset of unsteadiness and the spanwise wavelength decrease as $\beta$ increases. Finally, for $2<\beta\leq5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different 3D modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.