On the stability of a solid-body-rotation flow in a finite-length pip

The three-dimensional, inviscid and viscous flow instability modes that appear on a solid-body rotation flow in a finite-length, straight, circular pipe are analyzed. This study is a direct extension of the Wang \& Rusak (1996) analysis of axisymmetric instabilities on inviscid swirling flows in a pipe. We study a general mode of perturbation that satisfies the inlet, outlet and wall conditions of a flow in a finite-length pipe with a fixed-in-time and in-space vortex generator ahead of it. The eigenvalue problem for the growth rate and the shape of the perturbations for any azimuthal wave number $m$ is solved numerically for all azimuthal wave number $m$. In the inviscid flow case, the $m=1$ modes are the first to become unstable as the swirl ratio is increased and dominate the perturbation's growth in a certain range of swirl levels. In the viscous flow case, the neutral stability line is presented in a Reynolds number ($Re$) versus swirl ratio ($\omega$) diagram and can be used to predict the first appearance of of axisymmetric or spiral instabilities as a function of $Re$ and $L$. We will discuss and demonstrate the physical mechanism and evidences of the onset of the instability.