High Reynolds number asymptotic theories are used to describe nonlinear equilibrium solutions of the Navier Stokes equations. In the first case we describe vortex-wave interaction states involving inviscid wave systems propagating on streaky flows. We give explicit forms for the back effect of the wave on the roll with the latter then acting on the streak to make the system self-sustained. The asymptotic theory describes with remarkable accuracy Navier Stokes solutions of equilibrium solutions of channel flows. The first type of vortex-wave interaction (VWI) appears at quite low Reynolds number and defines explicitly the edge states visited by DNS simulations of sufficiently large initial disturbances to shear flows leading to turbulent flows. At low streamwise wavenumbers a new asymptotic structure emerges and it describes localized solutions closely resembling turbulent spots observed experimentally or calculated numerically. The VWI results imply the size of the minimal seed needed for a flow to become turbulent and the minimum box size needed for sustained turbulence to exist.
Another class of VWI states connects with the first branch and represent flows with inflected critical layers. These solutions have much higher drag than the first branch and are visited regularly by DNS simulations of turbulent flows. In the long wavelength limit the two branches merge in a complex manner and the nature of that merger has consequences for the existence of sustained turbulence in channels or pipes of infinite length.
The work is extended to external boundary layers and the corresponding equilibrium states are discussed. In the first case a parallel boundary layer is investigated we find that in addition to states corresponding to internal flows, in external flows a new kind of asymptotic state emerges. The new state is generated in a layer at the edge of the boundary layer and drives streaky flows there and in the main part of the boundary layer. We refer to the new state as an exact freestream coherent structure. The work is extended to growing boundary layers and we see nonparallel effects now dominate the process and restrict the structure to only persist over finite distances downstream. The relationship of the structures to experiments is discussed.
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