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

Abstract

In this study, a new centrifugal instability mode, which dominates within the boundary-layer flow over a slender rotating cone in still fluid, is used for the first time to model the problem within an enforced oncoming axial flow. The resulting problem necessitates an updated similarity solution to represent the basic flow more accurately than previous studies in the literature. The new mean flow field is subsequently perturbed leading to disturbance equations that are solved via numerical and short-wavelength asymptotic approaches, importantly yielding favourable comparison with existing experiments. Essentially, the boundary-layer flow undergoes competition between the streamwise flow component, due to the oncoming flow, and the rotational flow component, due to effect of the spinning cone surface, which can be described mathematically in terms of a control parameter, namely the ratio of streamwise to axial flow. For a slender cone rotating in sufficiently strong axial flow rates, the instability mode breaks down to Görtler-type counter-rotating spiral vortices, governed by an underlying centrifugal mechanism, which is consistent with experimental and theoretical studies for a slender rotating cone in otherwise-still fluid.

RETRACTED: Competing instabilities of rotating boundary-layer flows in an axial free-stream

“…In general, we observe close agreement between our asymptotic and OS numerical stability results, as well as with the numerical calculations of [2] (see [9] for comparisons of various slender cones with ψ < 40 • ). Importantly, while we have used the asymptotic results to provide an envelope for the right-hand branch of the numerical neutral stability curve, they are unable to predict the effect of varying axial flow on the critical Reynolds numbers.…”

“…In contrast, the OS numerical stability results complement the asymptotics in confirming the existence of the neutral stability curve for the centrifugal mode. Furthermore, we observe a reduction in the critical Reynolds number Re c as well as an increase in the critical amplification rate α 1,c with increasing s (see [9] for various slender cones with ψ < 40 • ).…”

Global linear instability of rotating-cone boundary layers in a quiescent medium

“…However, results were unchanged for those boundary layers with a zero axial flow. Further investigations on the family of rotating-cone boundary layers were undertaken by the Garrett group [37][38][39][40], who studied the type of convective instabilities that develop for variable cone half-angles. For broad rotating-cones, the crossflow instability that forms the corotating vortices on the rotating-disk was found to dominate the boundary layer stability process (at least until the conditions required…”

Boundary layer instability over a rotating slender cone under non-axial inflow

“…Recently, Hussain et al. (2016) revisited this topic and developed a distinct theoretical analysis based on the centrifugal instability mode. They highlighted that, even though cross-flow and Tollmien–Schlichting instabilities are present in the flow field, the centrifugal instability is prevalent.…”