The effect of compressibility on the critical swirl of vortex flows in a pipe

The effect of compressibility on the critical swirl level for breakdown of subsonic vortex flows in a straight circular pipe of finite length is studied. This work extends the critical-state concept of Benjamin (1962) to include the influence of Mach number on the flow behaviour. The analysis is based on a linearized version of the equations for the motion of a steady, axisymmetric, inviscid and compressible swirling flow of a perfect gas. The relationship between the velocity, density, temperature and pressure perturbations to a base columnar flow state are derived. An eigenvalue problem is formulated to determine the first critical level of swirl at which a special mode of a non-columnar small disturbance may appear on the base flow. It is found that when the characteristic Mach number of the base flow tends to zero the eigenvalue problem and the critical swirl are the same as defined by Wang & Rusak (1996a, 1997a) in their study of incompressible swirling flows in pipes. As the characteristic Mach number is increased, the critical swirl level increases and the flow perturbation expands in the radial direction. As the Mach number is increased toward a certain limit value related to the core size of the vortex, the critical swirl reaches very large values and becomes singular. The present results indicate that the axisymmetric breakdown of high-Reynolds-number compressible vortex flows may be delayed with the increase of the flow Mach number.