The linear stability of a Stokes layer subjected to high-frequency perturbations

Abstract

Quantitative results for the linear stability of planar Stokes layers subject to small, high frequency perturbations are obtained for both a narrow channel and a flow approximating the classical semi-infinite Stokes layer. Previous theoretical and experimental predictions of the critical Reynolds number for the classical flat Stokes layer have differed widely with the former exceeding the latter by a factor of two or three. Here it is demonstrated that only a one percent perturbation, at an appropriate frequency, to the nominal sinusoidal wall motion is enough to result in a reduction of the theoretical critical Reynolds number of as much as 60%, bringing the theoretical conditions much more in line with the experimentally reported values. Furthermore, within the various experimental observations there is a wide variation in reported critical conditions and the results presented here may provide a new explanation for this behaviour.

The linear stability of an acceleration-skewed oscillatory Stokes layer

“…Several explanations for the large variations between theory and experiments have been postulated, including geometry (Blennerhassett & Bassom 2006;Thomas, Bassom & Blennerhassett 2012), quasi-steady instabilities (Cowley 1987;Hall 2003;Luo & Wu 2010) and wall roughness (Vittori & Verzicco 1998). More recently, Thomas et al (2015) suggested that imperfections in the experimental apparatus could introduce some low level noise that establishes the premature onset of unstable behaviour. It was shown that high-frequency modulation at amplitudes near 1 % of the wall motion could reduce the critical Reynolds number for linear instability by more than a half, bringing the theoretical observations in line with those values reported experimentally.…”

Linear and nonlinear dynamics of pulsatile channel flow

“…More recently, in a series of theoretical and numerical papers, Blennerhassett and Bassom, with Thomas and Davies, have used Floquet analysis and linear simulation to address the stability of a range of related time-periodic flows due to an oscillating plate (Blennerhassett & Bassom 2002;Thomas et al 2010Thomas et al , 2014Thomas et al , 2015, a streamwise oscillating channel (Blennerhassett & Bassom 2006;Thomas et al 2011) or pipe (Blennerhassett & Bassom 2006;Thomas et al 2011Thomas et al , 2012, as well as a torsionally oscillating pipe (Blennerhassett & Bassom 2007;Thomas et al 2012), thereby resolving some of the inconsistencies of previous linear stability analyses and establishing, among others, curves of marginal linear instability for this family of flows. The spatio-temporal impulse response of the Stokes layer is studied by Thomas et al (2014), and the fate of some disturbances when they become nonlinear is also considered.…”

Transient growth of perturbations in Stokes oscillatory flows

“…Consequently, the idealised analytical flow has quite different stability properties from those seen in practice. For example, the perturbation growth can be amplified by wall roughness (Blondeaux & Vittori 1994;Vittori & Verzicco 1998) or a base flow variation (Thomas et al 2015).…”