On the linear instability of a finite Stokes layer: Instantaneous versus Floquet modes

The linear instability of a finite Stokes layer is investigated numerically. The aim is to clarify the relationship between the instantaneous instability and the recently discovered Floquet global instability [P. J. Blennerhassett and A. P. Bassom, J. Fluid Mech. 464, 394 (2002); P. J. Blennerhassett and A. P. Bassom, J. Fluid Mech. 556, 1 (2006)], and their role in causing transition to turbulence. For the first time, a numerical approach based on initial-value calculations is able to confirm the global instability, which occurs as the Reynolds number $R$ (based on the Stokes-layer thickness) exceeds the critical value $Rc=708$⁠. In both supercritical $(R>Rc)$ and subcritical $(R<Rc)$ regimes, strong instantaneous growth occurs during the deceleration phase. By projecting the disturbance at an arbitrary instant to instantaneous eigenfunctions, it is shown that the energy of the disturbance is primarily carried by the most unstable instantaneous mode. During the main growing phase, both the spatial structure and the growth rate agree well with those of the instantaneous mode; during the decaying phase part of the energy is continuously transferred to the precursor (i.e., the backward continuation in time) of the most unstable mode existing in the next cycle. The most unstable mode including its continuation therefore constitutes the crucial part of a Floquet mode. Unstable Floquet modes exist despite that none of instantaneous modes is periodic, implying that the commonly held premise that a Floquet mode is equivalent to a periodic instantaneous mode is erroneous. It is also found that in the subcritical regime, the flow responds very sensitively to small surface roughness, modeled by a wavy wall. The sensitivity can be measured by a response function, and our calculations show that significant disturbances can be generated by surface imperfections less than $0.1μm$ in typical experimental setups. The response becomes stronger as $R$ increases, and tends to infinity as the wavenumber of the wavy wall approaches that of the critical Floquet mode at $Rc$⁠, indicating that roughness with a suitable wavelength can excite a global mode through resonance, and the large response may be attributed to a detuned resonance with a global mode.