The effects of mass transfer on the global stability of the rotating-disk boundary layer

Abstract

Numerical simulations were conducted to investigate the effects of surface suction and injection on the global behaviour of linear disturbances in the rotating-disk boundary layer. This extends earlier work, which considered the case with no mass transfer. For disturbances in the genuine base flow, where radially inhomogeneity is retained, mass injection at the disk surface led to behaviour that remained qualitatively similar to that which was found when there was no mass transfer. The initial development of disturbances within the absolutely unstable region involved temporal growth and upstream propagation, as should be anticipated for an absolute instability. However, this did not persist indefinitely. Just as for the case without mass transfer, the simulation results suggested that convective behaviour would eventually dominate, for all the Reynolds numbers investigated. In marked contrast, the results obtained for flows with mass suction indicate a destabilization due to the effects of the base-flow radial inhomogeneity. It was possible to identify disturbances excited within the absolutely unstable region that grew continually, with a temporal growth rate that increased as the disturbance evolved. The strong locally stabilizing effect of suction on the absolute instability, which gives rise to large increases in critical Reynolds numbers, appears to be obtainable only at the expense of introducing a new form of global instability. Analogous forms of global behaviour can be found in impulse solutions of the linearized complex Ginzburg–Landau equation. These solutions were deployed to interpret and make comparisons with the numerical simulation results. They illustrate how the long-term behaviour of a disturbance can be determined by the precise balance between radial increases in temporal growth rates, corresponding shifts in temporal frequencies and diffusion/dispersion effects. This balance provides some insight into why disturbances that are absolutely unstable, for the homogenized version of the rotating-disk boundary-layer flow, may become, in the genuine radially inhomogeneous flow, either globally stable or globally unstable, depending on the level of mass transfer that is applied at the disk surface.