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

TL;DR

This work analyzes the stability of streamwise-rotating plane Poiseuille and plane Couette flows (XPPF/XPCF) by combining linear stability analysis (LSA) with direct numerical simulations (DNS). It identifies two asymptotic regimes in the rotation number $Ro$: a low-$Ro$ regime with $Re_c o ext{const}/Ro$ and a high-$Ro$ regime with $Re_c$ approaching fixed values ($66.45$ for XPPF and $20.66$ for XPCF), accompanied by a constant critical spanwise wavenumber $eta_c$ ($2.459$ for XPPF, $1.558$ for XPCF) and an obliquely inclined most-unstable mode whose angle scales as $Ro$ or $1/Ro$ depending on the regime. The high-$Ro$ limit for XPCF exhibits a Rayleigh–Bénard–convection analogy with $Re_c= frac{ ext{Ra}_c^{1/2}}{2}$, linking rotating shear-flow stability to RB-type instabilities; moreover, the minima of $Re_c$ in the streamwise-rotating cases align with those found for spanwise rotation, explaining a deep equality between these seemingly different configurations. DNS shows subcritical transitions at low $Ro$ with turbulent–laminar patterns, while at larger $Ro$ these subcritical pathways vanish and flows relaminarize for $Re<Re_c$, with a narrow $Ro$ window admitting turbulent patterns under supercritical conditions. Overall, the study clarifies how rotation direction and strength shape the linear and nonlinear fate of these canonical shear flows and provides a coherent bridge between rotating plane Couette/Poiseuille flows and RB-type instabilities.

Abstract

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 ($Ro$): a low-$Ro$ and a high-$Ro$ regime. In the low-$Ro$ regime, the critical Reynolds number $Re_c$ and critical streamwise wavenumber $α_c$ are proportional to $Ro$, while the critical spanwise wavenumber $β_c$ is constant. In the high-$Ro$ regime, as $Ro \rightarrow \infty$, we find $Re_c = 66.45$ and $β_c = 2.459$ for streamwise rotating PPF, and $Re_c = 20.66$ and $β_c = 1.558$ for streamwise rotating PCF, with $α_c\propto 1/Ro$. Our results for streamwise rotating PPF match previous findings by Masuda et al. (2008). Interestingly, the critical values of $β_c$ and $Re_c$ at $Ro \rightarrow \infty$ in streamwise rotating PPF and PCF coincide with the minimum $Re_c$ reported by Lezius & Johnston (1976) and Wall & Nagata (2006) for spanwise rotating PPF at $Ro=0.3366$ and PCF at $Ro=0.5$. We explain this similarity through an analysis of the perturbation equations. Consequently, the linear stability of streamwise rotating PCF at large $Ro$ is closely related to that of spanwise rotating PCF and Rayleigh-Benard convection, with $Re_c = \sqrt{Ra_c}/2$, where $Ra_c$ is the critical Rayleigh number. To explore the potential for subcritical transitions, direct numerical simulations were performed. At low $Ro$, a subcritical transition regime emerges, characterized by large-scale turbulent-laminar patterns in streamwise rotating PPF and PCF. However, at higher $Ro$, subcritical transitions do not occur and the flow relaminarizes for $Re < Re_c$. Furthermore, we identify a narrow $Ro$-range where turbulent-laminar patterns develop under supercritical conditions.

Figures (11)

• Figure 1: (a) XPPF and (b) XPCF configurations.
• Figure 2: Neutral stability curves of three-dimensional modes in XPCF and the two-dimensional $\beta=0$ and three-dimensional modes in XPPF. Horizontal dashed line, $Re=\sqrt{1707.762}/2$; dash-dotted line, $Re=66.45$. Sloped dashed line, $Re=17/Ro$; dash-dotted line, $Re=33.923/Ro$.
• Figure 3: The critical wavenumbers (a) $\alpha_c$ and (b) $\beta_c$ and (c) angle $\theta$ of the wavenumber vector $\hbox{\bf{\em{k}}}_c=(\alpha_c,\beta_c)$ with the $z$-axis as a function of $Ro$ in XPPF and XPCF. In (b) dashed lines, $\beta=1.179$ and $\beta=1.558$; dash-dotted lines, $\beta=1.917$ and $\beta=2.459$. In (c) dashed lines, $\theta=0.5/Ro$ and $\theta=0.8Ro$; dash-dotted lines, $\theta=0.3366/Ro$ and $\theta=1.05Ro$.
• Figure 4: Growth rate $\omega_i$ as a function of $(\alpha,\beta)$ at neutral stability in XPPF. (a) $Re=66.47$ and $Ro=24$, (b) $Re=77.03$ and $Ro=1$, (c) $Re=682.8$ and $Ro=0.05$, (d) $Re=5776$ and $Ro=0.000587$. The neutrally stable modes are indicated by white stars.
• Figure 5: Growth rate $\omega_i$ as a function of $(\alpha,\beta)$ at neutral stability in XPCF. (a) $Re=20.68$ and $Ro=24$, (b) $Re=28.14$ and $Ro=1$, (c) $Re=340.9$ and $Ro=0.05$, (d) $Re=8496$ and $Ro=0.002$. The neutrally stable mode is indicated by a white star.