Transition to turbulence in the rotating-disk boundary-layer flow with stationary vortices

Abstract

This paper proposes a resolution to the conundrum of the roles of convective and absolute instability in transition of the rotating-disk boundary layer. It also draws some comparison with swept-wing ows. Direct numerical simulations based on the incompressible Navier{Stokes equations of the ow over the surface of a rotating disk with modelled roughness elements are presented. The rotating-disk ow 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 nally transition 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, 2014, 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 ow was developing. The secondary modes further stayed within the domain, and an explanation for this is a proposed globally unstable secondary instability. For the low-amplitude roughness cases, the disturbances propagate beyond the threshold for secondary global instability before becoming turbulent, and for the high-amplitude roughness cases the transition scenario gives a turbulent ow directly at the critical Reynolds number for the secondary global instability. These results correspond to the theory of Pier (J. Eng. Math, vol. 57, 2007, pp. 237-251) predicting a secondary absolute instability. In our simulations, high temporal frequencies were found to grow with a large ampli cation rate where the secondary global instability occurred. For smaller radial positions, low-frequency secondary instabilities were observed, tripped by the global instability.