Coupled numerical and theoretical study of the flow transition between a rotating and a stationary disk

Abstract

Both direct numerical simulation and theoretical stability analysis are performed together in order to study the transition process to turbulence in a flow between a rotating and a stationary disk. This linear stability analysis considers the complete rotor-stator flow and then extends the results of Lingwood [J. Fluid Mech. 299, 17 (1995); 314, 373 (1996)] obtained in a single disk case. The present linear analysis also extends the former two-disk computations of Itoh [ASME FED 114, 83 (1991)], only limited to a hydrodynamic spatial instability analysis. Moreover, in the present work, this approach is completed by discussing the effects of buoyancy-driven convection on the flow stability and by absolute/convective analysis of the flow. Coupled with accurate numerical computations based on an efficient pseudo-spectral Chebyshev–Fourier method, this study brings new insight on the spatio-temporal characteristics of this flow during the first stages of transition. For instance, an exchange of stability from a steady to a periodic flow with spiral structures is observed for the first time numerically in such cavity of large aspect ratio. The nature of the first bifurcation is discussed as well as the influence on it of disturbances coming from the end-wall boundary layer. Annular and spiral patterns are observed in the unstable stationary disk layer with characteristic parameters agreeing very well with the present theoretical results. Then, these structures are interpreted in terms of type I and type II generic instabilities. Moreover, the absolute instability regions which are supposed to be strongly connected with the turbulent breakdown process are also identified and the critical Reynolds numbers of the convective/absolute transition in both Ekman and Bödewadt layers are given.