Stability of plane Couette and Poiseuille flows rotating about the streamwise axis

We study the stability of plane Poiseuille flow (PPF) and plane Couette flow (PCF) subject to streamwise system rotation using linear stability analysis and direct numerical simulations. The linear stability analysis reveals two asymptotic regimes depending on the non-dimensional rotation rate ( $\textit{Ro}$ ): a low- $\textit{Ro}$ and a high- $\textit{Ro}$ regime. In the low- $\textit{Ro}$ regime, the critical Reynolds number $\textit{Re}_c$ and critical streamwise wavenumber $\alpha _c$ are proportional to $\textit{Ro}$ , while the critical spanwise wavenumber $\beta _c$ is constant. In the high- $\textit{Ro}$ regime, as $\textit{Ro} \rightarrow \infty$ , we find $\textit{Re}_c = 66.45$ and $\beta _c = 2.459$ for streamwise-rotating PPF, and $\textit{Re}_c = 20.66$ and $\beta _c = 1.558$ for streamwise-rotating PCF, with $\alpha _c\propto 1/Ro$ . Our results for streamwise-rotating PPF match previous findings by Masuda et al. ( J. Fluid Mech. , vol. 603, 2008, pp. 189–206). Interestingly, the critical values of $\beta _c$ and $\textit{Re}_c$ at $\textit{Ro} \rightarrow \infty$ in streamwise-rotating PPF and PCF coincide with the minimum $\textit{Re}_c$ reported by Lezius & Johnston ( J. Fluid Mech. , vol. 77, 1976, pp. 153–176) and Wall & Nagata ( J. Fluid Mech. , vol. 564, 2006, pp. 25–55) for spanwise-rotating PPF at $\textit{Ro}=0.3366$ and PCF at $\textit{Ro}=0.5$