Structure from the critical layer framework in turbulent flow

We extend the critical layer framework for turbulent pipe flow proposed by McKeon \& Sharma (\textit{J. Fluid Mech, 2010}) to investigate vortical structure generated at particular streamwise/azimuthal wavenumber and frequency combinations, $(k,n,\omega)$. This framework utilizes an input-output formulation of the Navier-Stokes equations in a divergence-free basis to analyze the transfer function (the ``resolvent'') and identify the dominant forcing and response mode shapes at each $(k,n,\omega)$ combination relevant to experimental spectra. It is shown that the hairpin vortex is a natural constituent of the velocity field associated with so-called wall modes, such that our model gives important predictive information about both the statistical and structural make-up of wall turbulence. Thus the dominant response mode shapes form a suitable basis by which to decompose the full turbulent velocity field. \textbf{Acknowledgements:} This research is sponsored by an Imperial College Junior Research Fellowship and the AFOSR (program manager J. Schmisseur).