On the origin of the inflectional instability of a laminar separation bubble

Abstract

This is an experimental and theoretical study of a laminar separation bubble and the associated linear stability mechanisms. Experiments were performed over a flat plate kept in a wind tunnel, with an imposed pressure gradient typical of an aerofoil that would involve a laminar separation bubble. The separation bubble was characterized by measurement of surface-pressure distribution and streamwise velocity using hot-wire anemometry. Single component hot-wire anemometry was also used for a detailed study of the transition dynamics. It was found that the so-called dead-air region in the front portion of the bubble corresponded to a region of small disturbance amplitudes, with the amplitude reaching a maximum value close to the reattachment point. An exponential growth rate of the disturbance was seen in the region upstream of the mean maximum height of the bubble, and this was indicative of a linear instability mechanism at work. An infinitesimal disturbance was impulsively introduced into the boundary layer upstream of separation location, and the wave packet was tracked (in an ensemble-averaged sense) while it was getting advected downstream. The disturbance was found to be convective in nature. Linear stability analyses (both the Orr–Sommerfeld and Rayleigh calculations) were performed for mean velocity profiles, starting from an attached adverse-pressure-gradient boundary layer all the way up to the front portion of the separation-bubble region (i.e. up to the end of the dead-air region in which linear evolution of the disturbance could be expected). The conclusion from the present work is that the primary instability mechanism in a separation bubble is inflectional in nature, and its origin can be traced back to upstream of the separation location. In other words, the inviscid inflectional instability of the separated shear layer should be logically seen as an extension of the instability of the upstream attached adverse-pressure-gradient boundary layer. This modifies the traditional view that pegs the origin of the instability in a separation bubble to the detached shear layer outside the bubble, with its associated Kelvin–Helmholtz mechanism. We contend that only when the separated shear layer has moved considerably away from the wall (and this happens near the maximum-height location of the mean bubble), a description by the Kelvin–Helmholtz instability paradigm, with its associated scaling principles, could become relevant. We also propose a new scaling for the most amplified frequency for a wall-bounded shear layer in terms of the inflection-point height and the vorticity thickness and show it to be universal.

Destabilisation and modification of Tollmien–Schlichting disturbances by a three-dimensional surface indentation

“…These and other works on boundary layers with separation bubbles thus indicate, the dominant peaks of the disturbances are located at the critical layer, these peaks increase with distance from the separation point and achieve maximum values which can be attributed to the critical layer. Häggmark et al (2000) experimentally studied a 2-D separation bubble on a flat plate by means of hot-wire anemometry and found that the streamwise velocity disturbances for both natural and forced cases exhibited three maxima; Diwan & Ramesh (2009) experimental results confirmed this observation.…”

“…The more recent research demonstrated that the primary instability mechanism in a separation bubble is inflectional in nature (Diwan & Ramesh 2009). In figure 12, normalised streamwise components of the base flow and the perturbation in the symmetric plane for λ = 81 mm are shown.…”

Response of a laminar separation bubble to impulsive forcing

“…Gaster & Grant (1975) were the first to introduce an impulsive disturbance, exciting the full spectrum of instability modes, thus generating a wave packet through selective amplification. This method was subsequently used by Watmuff (1999) and Diwan & Ramesh (2009). It must be noted that impulsive disturbances have, so far, been locally introduced at a single point, usually a pressure tap.…”

“…It is generally accepted that disturbances developing in a LSB are initially convectively amplified via a Kelvin-Helmholtz instability mechanism, similar to free shear layers (Ho & Huerre 1984;Watmuff 1999;Marxen et al 2015). Inviscid inflectional and Göertler type instability mechanisms have been proposed by Diwan & Ramesh (2009) and Marxen et al (2009) respectively. Furthermore, Theofilis et al (2000) conjectured a self-exciting, globally unstable process in a disturbancefree numerical environment.…”

“…It should be noted, however, that bursting and elongation do not necessarily imply that the LSB acquires long bubble characteristics. Although determining appropriate criteria for bubble bursting has been a recurrent topic (Gaster 1967;Diwan & Ramesh 2009;Marxen & Henningson 2011;Serna & Lázaro 2015), the development of a universal criterion has proven difficult, mainly due to the lack of sufficient understanding of the bursting process.…”