Görtler-number-based scaling of boundary-layer transition on rotating cones in axial inflow

Abstract

This paper reports on the efficacy of the Görtler number in scaling the laminar-turbulent boundary-layer transition on rotating cones facing axial inflow. Depending on the half-cone angle $\psi$ and axial flow strength, the competing centrifugal and cross-flow instabilities dominate the transition. Traditionally, the flow is evaluated by using two parameters: the local meridional Reynolds number $Re_l$ comparing the inertial versus viscous effects and the local rotational speed ratio $S$ accounting for the boundary-layer skew. We focus on the centrifugal effects, and evaluate the flow fields and reported transition points using Görtler number based on the azimuthal momentum thickness of the similarity solution and local cone radius. The results show that Görtler number alone dominates the late stages of transition (maximum amplification and turbulence onset phases) for a wide range of investigated $S$ and half-cone angle ( $15^{\circ } \leq \psi \leq 50^{\circ }$ ), although the early stage (critical phase) seems to be not determined by the Görtler number alone on the broader cones ( $\psi =30^{\circ }$ and $50^{\circ }$ ) where the primary cross-flow instability dominates the flow. Overall, this indicates that the centrifugal effects play an important role in the boundary-layer transition on rotating cones in axial inflow.