Flow induced by a rotating cone: Base flow and convective stability analysis

The steady flow over a rotating cone is investigated theoretically and numerically in order to improve the traditional von Kármán solution by proposing a self-similar correction which is an explicit function of the cone angle. The effect of the correction on the linear stability analysis of the rotating-cone flow is successively investigated through a weakly divergent approach. Both the base flow correction and the results of the stability analysis are validated against dedicated numerical simulations. As for the base flow, the comparison shows a clear improvement obtained by using the proposed correction in comparison with the classical von Kármán solution. As for the stability properties of the flow, the com-parison against the reference simulations shows a good agreement among all the approaches for large azimuthal wave numbers, but a better performance is obtained with the weakly divergent approach for lower wave numbers. The latter approach provides a lower critical Reynolds number than that predicted by parallel theory and, most importantly, changes the interplay between modes I and II with respect to what predicted by the parallel stability calculations. Finally, it is observed that the proposed correction of base flow has a slight effect on the stability analysis of the considered cases, but it may have important effects for low cone angles. Thus, while the classical Kármán solution is appropriate for large cone angles, the proposed correction is recommended for future stability analyses of slender cones.