The onset of absolute instability of rotating Hagen–Poiseuille flow: A spatial stability analysis

Abstract

A spatial, viscous stability analysis of Poiseuille pipe flow with superimposed solid body rotation is considered. For each value of the swirl parameter (inverse Rossby number) L>0, there exists a critical Reynolds number Rec(L) above which the flow first becomes convectively unstable to nonaxisymmetric disturbances with azimuthal wave number n=−1. This neutral stability curve confirms previous temporal stability analyses. From this spatial stability analysis, we propose here a relatively simple procedure to look for the onset of absolute instability that satisfies the so-called Briggs–Bers criterion. We find that, for perturbations with n=−1, the flow first becomes absolutely unstable above another critical Reynolds number Ret(L)>Rec(L), provided that L>0.38, with Ret→Rec as L→∞. Other values of the azimuthal wave number n are also considered. For Re>Ret(L), the disturbances grow both upstream and downstream of the source, and the spatial stability analysis becomes inappropriate. However, for Re<Ret, the spatial analysis provides a useful description on how convectively unstable perturbations become absolutely unstable in this kind of flow.