Numerical simulations were conducted to investigate the linear global
stability behaviour of the B¨odewadt, Ekman, von K´arm´an (BEK) family of flows,
for cases where a disk rotates beneath an incompressible fluid that is also rotating.
This extends the work reported in recent studies that only considered the
rotating-disk boundary layer with a von K´arm´an configuration, where the fluid
that lies above the boundary layer remains stationary. When a homogeneous flow
approximation is made, neglecting the radial variation of the basic state, it can
be shown that linearized disturbances are susceptible to absolute instability. We
shall demonstrate that, despite this prediction of absolute instability, the disturbance
development exhibits globally stable behaviour in the BEK boundary layers
with a genuine radial inhomogeneity. For configurations where the disk rotation
rate is greater than that of the overlying fluid, disturbances propagate radially
outwards and there is only a convective form of instability. This replicates the
behaviour that had previously been documented when the fluid did not rotate
beyond the boundary layer. However, if the fluid rotation rate is taken to exceed
that of the disk, then the propagation direction reverses and disturbances grow
while convecting radially inwards. Eventually, as they approach regions of smaller
radii, where stability is predicted according to the homogeneous flow approximation,
the growth rates reduce until decay takes over. Given sufficient time, such
disturbances can begin to diminish at every radial location, even those which are
positioned outwards from the radius associated with the onset of absolute instability.
This leads to the confinement of the disturbance development within a finitely
bounded region of the spatial-temporal plane.