The relationship between numerical finite-amplitude equilibrium solutions of the full
Navier-Stokes equations and nonlinear solutions arising from a high Reynolds number
asymptotic analysis is discussed for a Tollmien-Schlichting wave type two-dimensional
vortical flow structure. The specific flow chosen for this purpose is that which arises from
the mutual axial sliding of co-axial cylinders for which nonlinear axisymmetric travellingwave
solutions have been discovered recently by Deguchi & Nagata (J. Fluid Mech., vol.
678, 2011, pp. 156–178). We continue this solution branch to a Reynolds number R = 108
and confirm that the behaviour of its so-called lower branch solutions, which typically
produce a smaller modification to the laminar state than the other solution branches,
quantitatively agrees with the axisymmetric asymptotic theory developed in this paper.
We further find that this asymptotic structure breaks down when the disturbance wavelength
is comparable with R. The new structure which replaces it is investigated and the
governing equations are derived and solved. The flow visualization of the resultant solutions
reveals that they possess a streamwise localized structure, with the trend agreeing
qualitatively with full Navier-Stokes solutions for relatively long wavelength disturbances.