Stability of periodically modulated rotating disk boundary layers

The linear stability properties of the boundary layer generated above a disk of infinite extent which rotates around its azimuth are explored for a novel configuration. The rotation rate is taken to be temporally periodic, motivated by findings from Thomas et. al. (Proc. Royal Soc. A, 2011) that the addition of an oscillatory component to an otherwise steady flow has stabilising effects. The vorticity-based methods that were first adopted by Davies and Carpenter (J. Comput. Phys., 2001) are utilised in a novel way for the solution of steady and temporally periodic eigenvalue dispersion relations. Validation of this method is provided by archetypal flow configurations such as the steady Blasius boundary layer and the temporally periodic Stokes layer, where Floquet theory is incorporated. Floquet stability theory is applied to the periodically modulated rotating disk for fixed wavenumber and fixed frequency disturbances, where it is shown that the addition of a modulated rotation rate has a stabilising effect on the boundary layer across a range of modulation frequencies. Confirmation is provided by frozen profile analyses and direct numerical simulations of the subsequent flow development. An energy analysis of the perturbation quantities is conducted to provide insights into the physical mechanisms for the stabilisation. The flow response to impulsive excitations in the periodically modulated rotating disk boundary layer is explored. Direct numerical simulations of radially homogeneous and inhomogeneous configurations are conducted and global stability behaviour is investigated.