The critical layer in pipe flow at high Reynolds number

Abstract

We report the computation of a family of travelling wave solutions of pipe flow up to Re =75 000. As in all lower branch solutions, streaks and rolls feature prominently in these solutions. For large Re , these solutions develop a critical layer away from the wall. Although the solutions are linearly unstable, the two unstable eigenvalues approach 0 as Re →∞ at rates given by Re −0.41 and Re −0.87 ; surprisingly, the solutions become more stable as the flow becomes less viscous. The formation of the critical layer and other aspects of the Re →∞ limit could be universal to lower branch solutions of shear flows. We give implementation details of the GMRES-hookstep and Arnoldi iterations used for computing these solutions and their spectra, while pointing out the new aspects of our method.