Canonical Nonlinear Viscous Core Solution in pipe and elliptical geometry

In an earlier paper (Ozcakir et. al.(2016)), two new nonlinear traveling wave solutions were found with collapsing structure towards the center of the pipe as Reynolds number $R \rightarrow \infty$, which were called Nonlinear Viscous Core (NVC) states. Asymptotic scaling arguments suggested that the NVC state collapse rate scales as $R^{-1/4}$ where axial, radial and azimuthal velocity perturbations from Hagen-Poiseuille flow scale as $R^{-1/2}$, $R^{-3/4}$ and $R^{-3/4}$ respectively, while $(1-c)=O(R^{-1/2})$ where $c$ is the traveling wave speed. The theoretical scaling results were roughly consistent with full Navier-Stokes numerical computations in the range $10^{5}