Subcritical axisymmetric solutions in rotor-stator flow

Abstract

Rotor-stator cavity flows are known to exhibit unsteady flow structures in the form of circular and spiral rolls. While the origin of the spirals is well understood, that of the circular rolls is not. In the present study the axisymmetric flow in an aspect ratio $R/H=10$ cavity is revisited numerically using recent concepts and tools from bifurcation theory. It is confirmed that a linear instability takes place at a finite critical Reynolds number $\text{Re}={\text{Re}}_c$ and that there exists a subcritical branch of large amplitude chaotic solutions. This motivates the search for subcritical finite-amplitude solutions. The branch of periodic states born in a Hopf bifurcation at $\text{Re}={\text{Re}}_c$, identified using a self-consistent method (SCM) and arclength continuation, is found to be supercritical. The associated solutions only exist, however, in a very narrow range of $\text{Re}$ and do not explain the subcritical chaotic rolls. Another subcritical branch of periodic solutions is found using the harmonic balance method with an initial guess obtained by SCM. In addition, edge states separating the steady laminar and chaotic regimes are identified using a bisection algorithm. These edge states are biperiodic in time for most values of $\text{Re}$, where their dynamics is analyzed in detail. Both solution branches fold around at approximately the same value of $\text{Re}$, which is lower than ${\text{Re}}_c$ yet still larger than the values reported in experiments. This suggests that, at least in the absence of external forcing, sustained chaotic rolls have their origin in the bifurcations from these unstable solutions.