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

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 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. 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.

Boundary-Layer Transition on Broad Cones rotating in an Imposed Axial Flow

“…The question of the global-linear stability of the rotating disk was first investigated by Davies & Carpenter,32 who use a numerical simulation to demonstrate that the global mode associated with the local absolute instability is damped. This has also been theoretically verified in the preliminary study of Garrett & Peake 22 for the boundary layers on disks and cones rotating in otherwise still fluids.…”

The absolute instability of the boundary layer on a rotating cone

“…The results of this paper are restricted to local linear stability. The question of the global-linear stability of the rotating disk has been investigated by Davies and Carpenter [22], who show that the global mode associated with the local absolute instability is damped. Given the close relationship between the boundary layers on rotating disks and rotating cones, one may also expect this to hold for the rotating cone (at least for large half angle).…”

The neutral curve for stationary disturbances in rotating disk flow for power-law fluids

“…The value of the critical Reynolds number agrees exceptionally well with experimental data, leading to Lingwood's hypothesis that absolute instability plays a part in turbulent transition on the rotating disk. Subsequently, Davies and Carpenter [6] investigated the global behaviour of the absolute instability of the rotating disk boundary-layer. By direct numerical simulations of the linearised governing equations they were able to show that the local absolute instability does not produce a linear global instability, instead suggesting that convective behaviour eventually dominates at all the Reynolds numbers.…”