An experimental study of absolute instability of the rotating-disk boundary-layer flow

Abstract

In this paper, the results of experiments on unsteady disturbances in the boundary-layer flow over a disk rotating in otherwise still air are presented. The flow was perturbed impulsively at a point corresponding to a Reynolds numberRbelow the value at which transition from laminar to turbulent flow is observed. Among the frequencies excited are convectively unstable modes, which form a three-dimensional wave packet that initially convects away from the source. The wave packet consists of two families of travelling convectively unstable waves that propagate together as one packet. These two families are predicted by linear-stability theory: branch-2 modes dominate close to the source but, as the packet moves outwards into regions with higher Reynolds numbers, branch-1 modes grow preferentially and this behaviour was found in the experiment. However, the radial propagation of the trailing edge of the wave packet was observed to tend towards zero as it approaches the critical Reynolds number (about 510) for the onset of radial absolute instability. The wave packet remains convectively unstable in the circumferential direction up to this critical Reynolds number, but it is suggested that the accumulation of energy at a well-defined radius, due to the flow becoming radially absolutely unstable, causes the onset of laminar–turbulent transition. The onset of transition has been consistently observed by previous authors at an average value of 513, with only a small scatter around this value. Here, transition is also observed at about this average value, with and without artificial excitation of the boundary layer. This lack of sensitivity to the exact form of the disturbance environment is characteristic of an absolutely unstable flow, because absolute growth of disturbances can start from either noise or artificial sources to reach the same final state, which is determined by nonlinear effects.

The absolute instability of the boundary layer on a rotating cone

“…This experimental observation leads us to expect that the underlying transition mechanism here is an absolute instability, and indeed there is reasonable agreement between the experimental transition location and our predicted absolute-instability boundary for ψ > 50 • . Note, however, that Kobayashi and Izumi measure a local Reynolds number of 566 for transition on the rotating disk (ψ = 90 • ), which differs from Lingwood's [2] experimental measurement of between R = 502 and R = 514 (which in turn is itself close to results from a number of other experimental papers, see [1,2] for details). This discrepancy is presumably due to different experimental definitions of transition location.…”

“…Interestingly, this is reasonably close to the value calculated for the rotating-sphere boundary layer in still fluid [3], in circumstances in which the onset is close to the pole (latitude Kobayashi and Izumi (1982). The dashed line shows the transition value on the rotating disk as measured by [2]. …”

“…There has been considerable recent interest in the role of absolute instability in the transition to turbulence on a rotating disk, following the important discovery by Lingwood [1,2] of the presence of absolute instability for local Reynolds numbers in excess of a value very close to the experimentally measured transition location. Lingwood's analysis has been extended to the case of a rotating sphere by the present authors, both in still fluid, [3], and in uniform axial flow [4].…”

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

“…Spatio-temporal development of |ω θw | for an impulsively excited disturbance with n = 67 and r e = 311. ×, + denote experimental data points for the leading and trailing edges of the wavepacket taken from figure 15(b) of Lingwood (1996). (Contours drawn using a logarithmic scale with levels separated by a factor of two.…”

“…Comparison with Lingwood's experimental data It is not possible to carry out a simulation that exactly mimics the experiment described in Lingwood (1996). Both the numerical and physical experiments generate the initial disturbances with excitations that roughly approximate a point impulse.…”