The recently understood relationship between high Reynolds number vortex-wave interaction
theory and computationally-generated self-sustaining processes provides a possible
route to an understanding of some of the underlying structures of fully turbulent flows.
Here vortex-wave interaction theory, which we now refer to as VWI, is used in the long
streamwise wavelength limit to continue the development found at order one wavelengths
by Hall and Sherwin (2010). The asymptotic description given reduces the Navier-Stokes
equations to the so-called boundary-region equations for which we find equilibrium states
describing the change in the VWI as the wavelength of the wave increases from O(h) to
O(Rh) where R is the Reynolds number and 2h is the depth of the channel. The reduced
equations do not include the streamwise pressure gradient of the perturbation or the
effect of streamwise diffusion of the wave-vortex states. The solutions we calculate have
an asymptotic error proportional to R−2 when compared to the full Navier-Stokes equations.
The results found correspond to the minimum drag configuration for VWI states
and might therefore be of relevance to the control of turbulent flows. The key feature of
the new states discussed here is the thickening of the critical layer structure associated
with the wave part of the flow to completely fill the channel so that the roll part of
the flow is driven throughout the flow rather than as in Hall and Sherwin (2010) as a
stress discontinuity across the critical layer. We identify a critical streamwise wavenumber
scaling which when approached causes the flow to localise and take on similarities
with computationally-generated or experimentally-observed turbulent spots. In effect the
identification of this critical wavenumber for a given value of the assumed high Reynolds
number fixes a minimum box length necessary for the emergence of localized structures.
Whereas nonlinear equilibrium states of the Navier-Stokes equations are thought to form
a backbone on which turbulent flows hang, our results suggest that the localized states
found here might play a related role for turbulent spots.