Non-linear stability of a perturbed Orr-Sommerfeld solution for the wake of a stationary cylinder at low Reynolds numbers

We derive and investigate the non-linear stability of a perturbed Orr-Sommerfeld solution for the wake of a stationary cylinder at low Reynolds numbers assuming locally parallel flow at a distance greater than 10 diameters downstream. A weakly non-linear asymptotic temporal multiple-scale analysis yields a Stuart-Landau oscillator for the slow time evolution equation of the disturbance which in turn enables a consistent estimation of parameters based on physical measurements. For a streamwise location of 12 diameters behind the cylinder we find: (i) an excellent agreement between the asymptotically computed Strouhal number of this analysis and measurements from several experimental studies; (ii) the spatial variation of the fundamental and second harmonic of the disturbance in this study agree with experimental observations; (iii) the weakly non-linear analysis does not change the location of the critical layer throughout the range of two-dimensional laminar vortex shedding (47