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# Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer

 Davies, Christopher and Carpenter, Peter W. 2003. Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. Journal of fluid mechanics 486 , pp. 287-329. 10.1017/S0022112003004701

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## Abstract

A study is carried out of the linear global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. It is based on direct numerical simulations of the complete linearized Navierâ€“Stokes equations obtained with the novel velocityâ€“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. Impulse-like 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 so-called parallel-flow 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 short-term 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 rotating-disc 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 rigid-walled 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: Article Publication Published Mathematics Publisher’s copyright requirements: “All contributors retain the right to post the definitive version of the contribution as published at Cambridge Journals Online (in PDF or HTML form) in the Institutional Repository of the institution in which they worked at the time the paper was first submitted, or (for appropriate journals) in PubMed Central or UK PubMed Central, no sooner than one year after first publication of the paper in the journal, subject to file availability and provided the posting includes a prominent statement of the full bibliographical details, a copyright notice in the name of the copyright holder (Cambridge University Press or the sponsoring Society, as appropriate), and a link to the online edition of the journal at Cambridge Journals Online. Inclusion of this definitive version after one year in Institutional Repositories outside of the institution in which the contributor worked at the time the paper was first submitted will be subject to the additional permission of Cambridge University Press (not to be unreasonably withheld). See: http://journals.cambridge.org/action/forAuthors?page=copyright Cambridge University Press 14697645 04 Jun 2017 01:41 http://orca.cf.ac.uk/id/eprint/1708