Centrifugal instability over a rotating cone

In this study, we provide a mathematical description of the on-set of counter-rotating circular vortices observed for a family of slender rotating cones (of half-angles 15◦, 30◦ and 45◦) in quiescent fluid. In particular, we apply appropriate scalings and apply a change of coordinates, accounting for the effects of streamline curvature. A combined large Reynolds number and large vortex wave number analysis is used to obtain an estimate for the asymptotic right-hand branch of neutral stability for the family of slender rotating cones. Existing experimental and theoretical studies are discussed which lead to the clear hypothesis of a hitherto unidentified convective instability mode that dominates within the boundary-layer flow over slender rotating cones. The mode manifests as G ̈ortler-type counter-rotating spiral vortices, indicative of a centrifugal mechanism. Although a formulation consistent with the classic rotating-disk problem has been successful in predicting the stability characteristics over broad cones, it is unable to identify such a centrifugal mode as the half-angle is reduced. An alternative formulation is developed and the governing equations solved using both short-wavelength asymptotic and numerical approaches to independently identify the centrifugal mode. Our results confirm our earlier predictions pertaining to the existence of the new G ̈ortler mode and capture the effects of the governing centrifugal instability mechanism. Meanwhile, favourable comparisons are drawn between numerical and asymptotic neutral stability curve predictions.