Global stability behaviour for the BEK family of rotating boundary layers

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