The connection between centrifugal instability and Tollmien–Schlichting-like instability for spiral Poiseuille flow

Abstract

For spiral Poiseuille flow with radius ratio $\eta\,{\equiv}\,R_i\slash R_o\,{=}\,0.5$, we have computed complete linear stability boundaries for several values of the rotation rate ratio $\mu\equiv{\it\Omega}_o\slash{\it\Omega}_i$}, where $R_i$ and $R_o$ are the inner and outer cylinder radii, respectively, and ${\it \Omega}_i$ and ${\it \Omega}_o$ are the corresponding (signed) angular speeds. The analysis extends the previous range of Reynolds number $Re$ studied computationally by more than eightyfold, and accounts for arbitrary disturbances of infinitesimal amplitude over the entire range of $Re$ for which spiral Poiseuille flow is stable for some range of the Taylor number $Ta$. We show how the centrifugally driven instability (beginning with steady or azimuthally travelling-wave bifurcation of circular Couette flow at $Re\,{=}\,0$ when $\mu\,{<}\,\eta^2$) connects, as conjectured by Reid (1961) in the narrow-gap limit, to a non-axisymmetric Tollmien–Schlichting-like instability of non-rotating annular Poiseuille flow at $Ta\,{=}\,0$. For $\mu\,{>}\,\eta^2$, we show that there is no instability for $0\,{\leq}\,Re\,{\leq}\,Re_{min}$. For $\mu\,{=}\,0.5$, $Re_{min}$ corresponds to a turning point, beyond which exists a range of $Re$ for which there are two critical values of $Ta$, with spiral Poiseuille flow being stable below the lower one and above the upper one, and unstable in between. For the special case $\mu\,{=}\,1$, with the two cylinders having the same angular velocity, $Re_{min}$ corresponds to a vertical asymptote smaller than found by Meseguer & Marques (2002), whose results for $\mu\,{>}\,\eta^2$ fail to account for disturbances with a sufficiently wide range of azimuthal wavenumbers.