An energy analysis of convective instabilities of the Bödewadt and Ekman boundary layers over rough surfaces

An energy balance equation for the three-dimensional Bödewadt and Ekman layers of the so called “BEK family" of rotating boundary-layer flows is derived. A Chebyshev discretisation method is used to solve the equations and investigate the effect of surface roughness on the physical mechanisms of transition. All roughness types lead to a stabilization of the Type I (cross-flow) instability mode for both flows, with the exception of azimuthally-anisotropic roughness (radial grooves) within the Bödewadt layer which is destabilising. In the case of the viscous Type II instability mode, the results predict a destabilisation effect of radially-anisotropic roughness (concentric grooves) on both flows, whereas both azimuthally-anisotropic roughness and isotropic roughness have a stabilisation effect. The results presented here confirm the results of our prior linear stability analyses.