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Global behaviour corresponding to the absolute instability of the rotatingdisc boundary layer
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Davies, Christopher and Carpenter, P. W.. (2003) Global behaviour corresponding to the absolute instability of the rotatingdisc boundary layer. Journal of Fluid Mechanics, Volume 486 . pp. 287329. ISSN 00221120

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Official URL: http://dx.doi.org/10.1017/S0022112003004701
Abstract
A study is carried out of the linear global behaviour corresponding to the absolute instability of the rotatingdisc boundary layer. It is based on direct numerical simulations of the complete linearized Navier–Stokes equations obtained with the novel velocity–vorticity method described in Davies & Carpenter (2001). As the equations are linear, they become separable with respect to the azimuthal coordinate, $\theta$. This permits us to simulate a single azimuthal mode. Impulselike excitation is used throughout. This creates disturbances that take the form of wavepackets, initially containing a wide range of frequencies. When the real spatially inhomogeneous flow is approximated by a spatially homogeneous flow (the socalled parallelflow approximation) the results ofthe simulations are fully in accordance with the theory of Lingwood (1995). If the flow parameters are such that her theory indicates convective behaviour the simulations clearly exhibit the same behaviour. And behaviour fully consistent with absolute instability is always found when the flow parameters lie within the theoretical absolutely unstable region. The numerical simulations of the actual inhomogeneous flow reproduce the behaviour seen in the experimental study of Lingwood (1996). In particular, there is close agreement between simulation and experiment for the ray paths traced out by the leading and trailing edges of the wavepackets. In absolutely unstable regions the shortterm behaviour of the simulated disturbances exhibits strong temporal growth and upstream propagation. This is not sustained for longer times, however. The study suggests that convective behaviour eventually dominates at all the Reynolds numbers investigated, even for strongly absolutely unstable regions. Thus the absolute instability of the rotatingdisc boundary layer does not produce a linear amplified global mode as observed in many other flows. Instead the absolute instability seems to be associated with transient temporal growth, much like an algebraically growing disturbance. There is no evidence of the absolute instability giving rise to a global oscillator. The maximum growth rates found for the simulated disturbances in the spatially inhomogeneous flow are determined by the convective components and are little different in the absolutely unstable cases from the purely convectively unstable ones. In addition to the study of the global behaviour for the usual rigidwalled rotating disc, we also investigated the effect of replacing an annular region of the disc surface with a compliant wall. It was found that the compliant annulus had the effect of suppressing the transient temporal growth in the inboard (i.e. upstream) absolutely unstable region. As time progressed the upstream influence of the compliant region became more extensive.
Item Type:  Journal Article 

Subjects:  Q Science > QC Physics 
Divisions:  Faculty of Science > Engineering 
Library of Congress Subject Headings (LCSH):  NavierStokes equations, Boundary layer, Wave packets, Reynolds number, Fluid mechanics 
Journal or Publication Title:  Journal of Fluid Mechanics 
Publisher:  Cambridge University Press 
ISSN:  00221120 
Official Date:  24 June 2003 
Volume:  Volume 486 
Page Range:  pp. 287329 
Identification Number:  10.1017/S0022112003004701 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  Briggs, R. J. 1964 ElectronStream Interaction with Plasmas. MIT Press. 
URI:  http://wrap.warwick.ac.uk/id/eprint/766 
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