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

Abstract

The global linear stability of the family of infinite rotating-cone boundary layers in an otherwise still fluid is investigated using a velocity-vorticity form of the linearized Navier-Stokes equations. The formulation is separable with respect to the azimuthal direction. Thus, disturbance development is simulated for a single azimuthal mode number. Numerical simulations are conducted for an extensive range of cone half-angles (ψ ∈ [20 • : 80 • ]) and stability parameters (Reynolds number, azimuthal mode number), where conditions are taken to be near those specifications necessary for the onset of absolute instability. A localized impulsive wall forcing is implemented that excites disturbances that form wave packets. This allows the disturbance evolution to be traced in the spatial-temporal plane. When a homogeneous flow approximation is utilised that neglects the spatial variation of the basic state, linear perturbations display characteristics consistent with local stability theory. For disturbances to the genuine spatially dependent inhomogeneous flow, global linear instability characterized by a faster than exponential temporal growth arises for azimuthal mode numbers greater than the conditions for critical absolute instability. Furthermore, a reasonable prediction for the azimuthal mode number needed to bring about a change in global behavior is achieved by coupling solutions of the Ginzburg-Landau equation with local stability properties. Thus, the local-global stability behavior is qualitatively similar to that found in the infinite rotating-disk boundary layer and many other globally unstable flows.

Boundary-layer transition over a rotating broad cone

“…Also for broad cones stability theory has shown that an absolute instability exists (see, for instance, Refs. [10,11]), similar to the one for the disk as suggested by Lingwood [4]. However, since the stationary corotating vortices always exist in a physical experiment the role or importance of this absolute instability with respect to transition is not clear [12].…”

Effects of partial slip on the local-global linear stability of the infinite rotating disk boundary layer

“…Numerical computations of the genuine inhomogeneous flow display global linear stability characteristics comparable with that found by Thomas and Davies 15,16 for the infinite rotating disc and family of rotating cones. A form of global linear instability, characterised by a faster than exponential growth, arises when the azimuthal mode number n exceeds a threshold value n g .…”

“…Thomas and Davies 15 also speculate that global linear instability exists for other related rotating flows. Their follow-up study 16 confirms this for the rotating cone boundary layer.…”

“…Similar to the recent studies undertaken by Thomas and Davies 15,16 we will now attempt to predict the azimuthal mode number n g necessary for globally unstable behaviour to occur.…”

“…Using the modeling developed by Thomas and Davies [14][15][16]20 the azimuthal mode number n g for the onset of globally unstable disturbances was predicted accurately by coupling solutions to the Ginzburg-Landau equation 18 with local disturbance characteristics from the homogeneous flow simulations. For all cases modeled, predictions for n g were consistent with the numerical simulations of the inhomogeneous flow.…”