Finite-amplitude crossflow vortices, secondary instability and transition in the rotating-disk boundary layer

Abstract

In the three-dimensional boundary layer produced by a rotating disk, the experimentally well-documented sharp transition from laminar to turbulent flow is shown to coincide with secondary absolute instability of the naturally selected primary nonlinear crossflow vortices. Fully saturated primary finite-amplitude waves and the associated nonlinear dispersion relation are first numerically computed using a local parallel flow approximation. Exploiting the slow radial development of the basic flow, the naturally selected primary self-sustained flow structure is then derived by asymptotic analysis. In this state, outward-spiralling nonlinear vortices are initiated at the critical radius where primary absolute instability first occurs. A subsequent secondary stability analysis reveals that as soon as the primary nonlinear waves come into existence they are absolutely unstable with respect to secondary perturbations. Secondary disturbances growing in time at fixed radial locations continuously perturb the primary vortices, thus triggering the direct route to turbulence prevailing in this configuration.

The absolute instability of the boundary layer on a rotating cone

“…Pierre and Huerre [23] have shown that when non-linear effects are included a self-sustained non-linear oscillator will always be generated when there is a region of local linear absolute instability. On the rotating disk, Pier [24] has shown that this oscillator will undergo secondary instability close to the absolute instability boundary, providing a possible route to transition. The global behaviour needs to be investigated for the cone boundary layer, perhaps with a view to understanding the apparent change in behaviour when the cone angle is reduced.…”

Transition to turbulence of the Batchelor flow in a rotor/stator device

“…Many authors have then analyzed the stability of 1 these flows and two types of waves (type I and II) have been revealed in both boundary layers. For instance, Faller [7], Pithkov & Smirnov [8], Lingwood [9] and Pier [10] performed linear stability analyses of the different velocity profiles that occur in the different regions of the flow. More recently, and following the pioneer work of Lingwood [9], Serre et al [11] studied theoretically and numerically the transition from convective to absolute instability of these flows.…”

A new way to describe the transition characteristics of a rotating-disk boundary-layer flow

“…The transition process once triggered probably takes place through a secondary instability. 2,7 It is the purpose of the present paper to introduce a probability density function (PDF) map of the azimuthal velocity fluctuations to elucidate the changing flow characteristics through the stable and unstable laminar-flow regions, through the transitional region into the fully turbulent region for the rotating-disk flow.…”

“…Furthermore, as discussed by Viaud et al, 11 in a strongly nonlinear and weakly non-parallel regime, the presence of a finite region of absolute instability has been shown theoretically to be a sufficient condition for a nonlinear global mode with a steep front (a so-called "elephant mode"), located at the upstream boundary (primary front) between local convective and absolute instability, and, further, when the global mode is itself absolutely unstable to local secondary perturbations the transition process is likely to be via the secondary instabilities a short distance downstream of the primary front. 2,7,11 To summarize, the present work shows a new way to describe the characteristics of a rotatingdisk flow by introducing the PDF contour plot of the normalized fluctuation velocity (where the PDF at each z-position is normalized by its maximum value). The map is shown in Fig.…”

“…Moreover, the evidence presented herein may represent the first experimental validation of the suggested secondary absolute instability of the primary nonlinear (steep-fronted) global mode. 7,11 As mentioned above, this PDF method may be useful not only for rotating-disk flows (the so called BEK system of rotating boundary-layer flows 9 ), but also for other transitional flows, for example other three-dimensional boundary-layers, such as those over swept wings, or even more conventional two-dimensional boundary-layer configurations.…”