Bifurcation and stability of near-critical compressible swirling flows

Abstract

The bifurcation and global nonlinear stability of near-critical states of a compressible and axisymmetric swirling flow of a perfect gas in a finite-length straight, circular pipe is studied. This work extends the bifurcation and stability analyses of Wang and Rusak [Phys. Fluids 8, 1007 (1996); Wang and Rusak Phys. Fluids8, 1017 (1996)] to include the influence of Mach number on the flow dynamics. The first- and second-order equations of motion for the evolution of small axially symmetric perturbations on a base columnar state are developed. These equations are reduced to an eigenvalue problem for the perturbation shape function and critical swirl ratio and a model ordinary differential equation for the nonlinear evolution of the perturbations’ amplitude as function of swirl level and Mach number. It is found that noncolumnar equilibrium states bifurcate from the branch of the base columnar equilibrium states at the critical swirl ratio of a compressible vortex flow in the form of a transcritical bifurcation, where both the critical swirl and the bifurcation slope ratio are functions of Mach number. It is also shown that this critical swirl ratio is a point of exchange of global stability for both the columnar and noncolumnar states as the swirl ratio increases across this critical level. When the swirl ratio of the incoming flow is below the critical level the columnar states have an asymptotically decaying mode of perturbation whereas the noncolumnar states are unstable. On the other hand, when the swirl ratio of the incoming flow is greater than the critical level, the columnar states are unstable whereas the noncolumnar states have an asymptotically decaying mode of disturbance. The effect of Mach number on the bifurcation behavior and on the stability characteristics of the various states is presented. The relationship between the present results and the breakdown of a compressible vortex flow is discussed.