Abstract
| There is not much computational work on travelling wave computations in pipe
flows at very large $R$ in existing literature apart from those reported in
Ozcakir (2016). In this talk, we present results that extend reliable traveling
wave computations through greater efficiency to a far greater Reynolds number
(up to $R =5 \times 10^5$) regime than previously reported. Firstly, we confirm
that travelling waves states which are referred to as C1 and C2 in Ozcakir
(2016) are indeed finite $R$ realization of Nonlinear Viscous Core states
because of much closer agreement of numerical results with asymptotics. The
second part of the talk concerns determination of a new branch of solution (WK2)
which connects to Wedin-Kerswell (WK) when continued to sufficiently large $R$
which ascertains that it is a finite $R$ realisation of asymptotic VWI states,
with peak roll, wave, and stream amplitudes scaling as $R^{-1}$, $R^{-5/6}$ and
$O(1)$ respectively. In the last part of the talk linear stability of traveling
waves are discussed. We extend linear stability calculations to large enough $R$
so that asymptotics of unstable eigenvalues are apparent. These scalings are in
agreement with the $R^{-1/2}$, $R^{-1}$ asymptotics for edge and meandering
modes predicted by Deguchi \& Hall (2016) for uni-directional shear flow. We do
not, however, find any $R^0$ unstable eigenvalue within the class of
pressure-preserving two-fold azimuthally symmetric disturbances. |