On the laminar–turbulent transition of the rotating-disk flow: the role of absolute instability

Abstract

This paper describes a detailed experimental study using hot-wire anemometry of the laminar–turbulent transition region of a rotating-disk boundary-layer flow without any imposed excitation of the boundary layer. The measured data are separated into stationary and unsteady disturbance fields in order to elaborate on the roles that the stationary and the travelling modes have in the transition process. We show the onset of nonlinearity consistently at Reynolds numbers, $R$, of $\sim $510, i.e. at the onset of Lingwood’s (J. Fluid Mech., vol. 299, 1995, pp. 17–33) local absolute instability, and the growth of stationary vortices saturates at a Reynolds number of $\sim $550. The nonlinear saturation and subsequent turbulent breakdown of individual stationary vortices independently of their amplitudes, which vary azimuthally, seem to be determined by well-defined Reynolds numbers. We identify unstable travelling disturbances in our power spectra, which continue to grow, saturating at around $R=585$, whereupon turbulent breakdown of the boundary layer ensues. The nonlinear saturation amplitude of the total disturbance field is approximately constant for all considered cases, i.e. different rotation rates and edge Reynolds numbers. We also identify a travelling secondary instability. Our results suggest that it is the travelling disturbances that are fundamentally important to the transition to turbulence for a clean disk, rather than the stationary vortices. Here, the results appear to show a primary nonlinear steep-fronted (travelling) global mode at the boundary between the local convectively and absolutely unstable regions, which develops nonlinearly interacting with the stationary vortices and which saturates and is unstable to a secondary instability. This leads to a rapid transition to turbulence outward of the primary front from approximately $R=565$ to 590 and to a fully turbulent boundary layer above 650.

Global linear instability of the rotating-disk flow investigated through simulations

“…Malik (1986), Lingwood (1995a), Pier (2003), and experimental studies, e.g. Wilkinson & Malik (1985), Lingwood (1996), Othman & Corke (2006), Siddiqui et al (2013), Imayama, Alfredsson & Lingwood (2014a), have focused on various aspects of the instability characteristics and laminar-turbulent transition. There have also been several investigations of the turbulent rotating-disk flow, e.g.…”

Experimental study of rotating-disk boundary-layer flow with surface roughness

“…Figure 1 in Imayama, Alfredsson & Lingwood (2014) shows typical examples of laminar velocity profiles from rotating-disk boundary-layer flow. Here, quantities U * , V * , W * , r * , z * , θ and the non-dimensional quantities, U, V, W, R, z are defined as † Email address for correspondence: lingwood@mech.kth.se in Imayama et al (2014), where * denotes a dimensional value. The inflection point of the radial velocity profile makes the flow inviscidly unstable, so-called cross-flow (or Type-I) instability.…”

Quantifying non-Newtonian effects in rotating boundary-layer flows

“…To the best of the authors' knowledge no such experiments have yet taken place. Certainly in the Newtonian regime a wealth of literature exists, and this is currently a topic of particular interest, see Imayama et al [22][23][24], for example. Personal communication with these authors has revealed the difficulty of obtaining consistently accurate experimental results; therefore we can only envisage that the introduction of non-Newtonian fluids would serve to significantly complicate any experimental procedure.…”