

Article
Transition to turbulence in the rotatingdisk boundarylayer flow with stationary vortices
AbstractThis paper proposes a resolution to the conundrum of the roles of convective and absolute instability in transition of the rotatingdisk boundary layer. It also draws some comparison with sweptwing flows. Direct numerical simulations based on the incompressible Navier–Stokes equations of the flow over the surface of a rotating disk with modelled roughness elements are presented. The rotatingdisk flow has been of particular interest for stability and transition research since the work by Lingwood (J. Fluid Mech., vol. 299, 1995, pp. 17–33) where an absolute instability was found. Here stationary disturbances develop from roughness elements on the disk and are followed from the linear stage, growing to saturation and finally transitioning to turbulence. Several simulations are presented with varying disturbance amplitudes. The lowest amplitude corresponds approximately to the experiment by Imayama et al. (J. Fluid Mech., vol. 745, 2014a, pp. 132–163). For all cases, the primary instability was found to be convectively unstable, and secondary modes were found to be triggered spontaneously while the flow was developing. The secondary modes further stayed within the domain, and an explanation for this is a proposed globally unstable secondary instability. For the lowamplitude roughness cases, the disturbances propagate beyond the threshold for secondary global instability before becoming turbulent, and for the highamplitude roughness cases the transition scenario gives a turbulent flow directly at the critical Reynolds number for the secondary global instability. These results correspond to the theory of Pier (J. Engng Maths, vol. 57, 2007, pp. 237–251) predicting a secondary absolute instability. In our simulations, high temporal frequencies were found to grow with a large amplification rate where the secondary global instability occurred. For smaller radial positions, lowfrequency secondary instabilities were observed, tripped by the global instability.

