The Centrifugal Instability of a Slender Rotating Cone

Abstract

In this study, we provide a mathematical description of the onset of counterrotating circular vortices observed for a family of slender rotating cones (of half-angles 15°or less) in quiescent fluid. In particular, we apply appropriate scalings in order to simplify the basic-flow profiles, which are subsequently perturbed, accounting for the effects of streamline curvature. A combined large Reynolds number and large vortex wavenumber analysis is used to obtain an estimate for the asymptotic right-hand branch of neutral stability for a slender rotating cone. Our results confirm our earlier predictions pertaining to the existence of the new Görtler mode and capture the effects of the governing centrifugal instability mechanism. Meanwhile, favourable comparisons are drawn with existing numerical neutral stability curve results.

The centrifugal instability of the boundary-layer flow over slender rotating cones

“…These values correspond to the theoretical values of φ presented in Kobayashi & Izumi (1983) and hence facilitate a useful comparison with their results, as shown in Hussain et al (2012) for ψ = 15…”

“…As discussed above, these are not considered here and the interested reader is referred to Hussain (2010), Hussain et al (2012) for the associated stability analyses.…”

“…Furthermore, discussion of the formulation, analysis and results in this parameter regime where circular vortices are observed is given in Hussain et al (2012).…”

The centrifugal instability of the boundary-layer flow over a slender rotating cone in an enforced axial free stream

“…Indeed, physically, the departure of a vortex from the wall suggests that vorticity within the boundary layer is no longer fixed on the cone surface, but is instead propagating or travelling in the effective velocityx-direction. Ultimately, in order to confirm whether travelling instabilities may harbour the most unstable modes for the slender rotating-cone problem, a further investigation would be required, taking account of time-dependent terms within the governing disturbance equations (11)- (14).…”

“…These arise due to the fact that the basic flow quantitiesŨ andṼ are now functions of the logarithmic spiral coordinatesx andȳ, as well as η. We manipulate the disturbance equations (11)- (14) and subsequently express the basic flow terms in terms of η 1 by making use of the coordinate stretching (3). The analysis involves neglecting Coriolis terms and viscous streamline-curvature effects.…”