On the stability of an axisymmetric rotating flow in a pipe

Abstract

The linear stability of an inviscid, axisymmetric and rotating columnar flow in a finite length pipe is studied. A well posed model of the unsteady motion of swirling flows with compatible boundary conditions that may reflect the physical situation is formulated. A linearized set of equations for the development of infinitesimal axially-symmetric disturbances imposed on a base rotating columnar flow is derived. Then, a general mode of axisymmetric disturbances, that is not limited to the axial-Fourier mode, is introduced and an eigenvalue problem is obtained. Benjamin’s critical state concept is extended to the case of a rotating flow in a finite length pipe. It is found that a neutral mode of disturbance exists at the critical state. In the case of a solid body rotating flow with a uniform axial velocity component, analytical solution of the eigenvalue problem is found. It is demonstrated that the flow changes its stability characteristics as the swirl changes around the critical level. When the flow is supercritical an asymptotically stable mode is found, and when the flow is subcritical, an unstable mode of disturbance may develop. This result cannot be predicted by Rayleigh’s classical stability criterion. In the case of a general columnar swirling flow in a pipe, the asymptotic solution of the eigenvalue problem around the critical state is also studied. It is shown that the critical swirl ratio is a point of exchange of stability for any swirling flow in a finite length pipe. This result reveals an unknown instability mechanism of swirling flows that cannot be detected by previous stability analyses and sheds new light on the relation between stability of vortex flows and the vortex breakdown phenomenon.