On the stability mechanism of a swirling flow confined in an enclosed cylindrical container

The instability mechanism of a swirling flow confined in an enclosed cylindrical container, driven by the rotation of one endwall is studied by DNS and stability analysis. A case study where the cylinder aspect ratio H/R is 4 and Reynolds number $Re=2500$ based on the rotation of the endwall is conducted. A base axisymmetric flow is first established through the DNS by enforcing flow axisymmetry. Then, linear stability analysis of the base flow shows stability to axisymmetric disturbances but instability to spiral disturbances. The fastest linearly growing mode is with an azimuthal wave number $m=3$, which is in an excellent agreement with the DNS results. The physical mechanism of the instability of the $m=3$ mode is revealed through the use of the Reynolds-Orr equation. It pinpoints the exact location of perturbation's kinetic energy production/loss inside of the flow domain. It is also found that the non-axial homogeneity of the base axisymmetric flow plays a major role in the onset of the instability. A similar mechanism was recently identified for the unstable spiral modes found in the solid-body rotation flow in a finite-length, straight, open rotating pipe. The current model has a theoretical advantage since the boundary condition is well defined and used in this study.