The critical swirl of a subsonic compressible swirling flow of a perfect gas in a finite-length straight circular pipe

Functional analysis techniques are used to rigorously determine the range of flow Mach number $Ma_0$ for the existence of the critical swirl ratio $\omega_1$ for exchange of stability of a base columnar compressible swirling flow of a perfect gas in a finite-length straight circular pipe. For swirling flows with a monotonic circulation profile, it is first established that $\omega_1$ definitely exists in the range $0< Ma_0 <2\sqrt{\gamma-1}/\gamma$, where $\gamma>1$ is the ratio of specific heats of the gas. Then, the existence of a limit Mach number $Ma_{0l}$ between $2\sqrt{\gamma-1}/\gamma$ and $1$ is proven for a subclass of swirling flows; i.e. $\omega_1$ does not exist and the flow is stable for all swirl level when $Ma_{0l}< Ma_0 <1$. For example, $0.903