Three-dimensional instabilities in a rapidly counter-rotating split cylinder

The three-dimensional flow in a counter-rotating cylinder that is split at its mid-plane is studied numerically via spectral methods. The cylinder of radius $a$ and length $h$ is completely filled with fluid of kinematic viscosity $\nu$. The top half rotates with angular speed $\omega$ and the bottom half with angular speed $-\omega$. There are two nondimensional parameters governing the flow, $Re=\omega a^2/\nu$ and $\Gamma=h/a$. For small values of $Re$ and $\Gamma$ the flow is steady, axisymmmetric and reflection symmetric about the mid-height (with appropriate changes of sign for some flow components). In this regime the interior flow in each half of the cylinder rotate as solid-body rotation of opposite senses. Apart from the boundary layers on the cylinder walls, there is also an internal shear layer separating the two counter-rotating halves. Above a critical $Re$ that depends on $\Gamma$, this internal shear layer becomes unstable to low frequency instabilities that break both the axisymmetry and the reflection symmetry. For these cases there exist rotating waves associated with the shear-layer instability. The variation of the critical $Re$ and the azimuthal wavenumbers of the instability as a function of $\Gamma$ is studied, along with the nonlinear dynamics.