Analysis of inviscid shear instability of axisymmetric flows

TL;DR

This work addresses the inviscid stability of axisymmetric base flows in annuli and pipes by deriving simple analytic criteria that separate stable and unstable parameter regimes. It extends the classical KA-I condition with a KA-II criterion and introduces a hurdle-theorem–type instability mechanism based on the key quantity $W_{\alpha,N}(r)=\frac{rQ'(r)}{U(r)-\alpha}$ and geometry-aware thresholds, enabling rapid stability assessment without full eigenvalue computations. The authors prove the corresponding theorems, apply them to annular and pipe model flows, and demonstrate close alignment with neutral points obtained from eigenvalue analyses and with viscous linear stability results at high $Re$. The approach offers a practical framework for inviscid stability screening in complex shear flows and highlights both the utility and limitations of inviscid theory in predicting real-fluid behavior, including potential extensions to jet-like configurations.

Abstract

Simple analytical criteria are derived to determine whether axisymmetric base flows in annuli and pipes are stable or unstable. Both axisymmetric and non-axisymmetric inviscid disturbances are considered. Our sufficient condition for stability improves upon the classical result of \cite{Batchelor_Gill_1962}, following the idea of the second Kelvin-Arnol'd stability theorem. A novel sufficient condition for instability is also derived by extending the recently proposed hurdle theorem for parallel flows \citep{Deguchi_Hirota_Dowling_2024}. These analytical criteria are applied to annular and pipe model flows and are shown to effectively predict the neutral parameters obtained from eigenvalue computations of the stability problem.

Figures (11)

• Figure 1: The annular model flow used in section \ref{['sec:4.1']}. (a) Schematic of the model flow in the dimensional cylindrical coordinates $(r_*,\varphi,z_*)$. Here, the radii $r_{i*}$ and $r_{o*}$ satisfy $r_{o*}-r_{i*}=2d_*$, $r_{i*}/r_{o*}=\eta$. (b) The base flow $U(r)$ given in (\ref{['AnnularU']}) for $\chi=-7,-1,1,7$.
• Figure 2: Stability diagram of the annular model flow (figure \ref{['fig:fig1']}-(a)) in the $\chi$--$\eta$ plane. All physically possible wavenumbers are covered. The black solid line represents the neutral curve of the inviscid problem (\ref{['Geq']}). Stability is guaranteed by Theorem \ref{['KA']} in the blue region, while unstable modes are found by Theorem \ref{['Hannulus']} in the red region.
• Figure 3: Stability diagram of the annular model flow at the narrow-gap limit $\eta\rightarrow 1$. The eigenvalue problem (\ref{['GeqNG']}) indicates the presence of unstable modes for $\chi <-6$ and $\chi>3.81$. The grey line shows that the profile of $W_{\alpha,N}$, given in (\ref{['narrowW']}), becomes singular when $\chi\leq -6$.
• Figure 4: Profiles of $W_{\alpha,N}$ with $N=10$ for the annular model flow at $\eta=0.7$. The constant $\alpha$ is set equal to $U(r_c)$. (a) $\chi=7$; (b) $\chi=1$; (c) $\chi=-1$; (d) $\chi=-7$. In panel (a), the red line shows $h$ from (\ref{['hthm2']}). In panel (b), the blue line shows $H$ defined in (\ref{['HNHN']}).
• Figure 5: Inviscid stability result for the annular model flow at $(\eta,\chi)=(0.7,7)$. (a) Imaginary part of the phase speed $c_i$ for $N=10$. The neutral point is at $k=k_0=0.502$. The dashed red line indicates the result using (\ref{['dcidk']}). (b) Eigenfunction of the neutral mode found at $N=1/k$, $k=0.9475$ (i.e. $n=1$).

Theorems & Definitions (2)

• Theorem 1
• Theorem 2