On the stability of a compressible axisymmetric rotating flow in a pipe

The linear stability of a compressible inviscid axisymmetric and rotating columnar flow of a prefect gas in a finite-length straight circular pipe is investigated. This work extends a previous analysis to include the influence of Mach number on the flow dynamics. A well-posed model of the unsteady motion of a swirling flow, with inlet and outlet conditions that may reflect the physical situation, is formulated. The linearized equations of motion for the evolution of infinitesimal axially symmetric disturbances are derived. A general mode of disturbance, that is not limited to the axial-Fourier mode, is introduced and an eigenvalue problem is developed. It is found that a neutral mode of disturbance exists at 'the critical swirl ratio for a compressible vortex flow'. The flow changes its stability characteristics as the swirl ratio increases across this critical level. When the swirl ratio is below the critical level (supercritical flow), an asymptotically stable mode is found and, when it is above the critical level (subcritical flow), an unstable mode of disturbance develops. This result cannot be predicted by any of the previous stability criteria. When the characteristic Mach number of the base flow tends to zero, the results are the same as found for incompressible swirling flows in pipes. The growth rate ratio is positive for all Mach numbers, but decreases as Mach number is increased. This ratio vanishes at the limit Mach number at which the critical swirl tends to infinity. The present results also demonstrate that the axisymmetric breakdown of high-Reynolds-number compressible vortex flows may be delayed to higher swirl ratios with the increase of the incoming flow Mach number.