Instability and transition in curved channel flow

Abstract

A theoretical and numerical investigation of streamwise-oriented Dean vortices in curved channel flow is presented. The principal results are obtained from three-dimensional pseudospectral simulations of the incompressible time-dependent Navier-Stokes equations. With increasing Reynolds number, a sequence of transitions similar to that observed in non-turbulent Taylor-Couette flow is found. The transition from laminar curved channel Poiseuille flow to axisymmetric Dean vortex flow is studied using linear and weakly nonlinear analyses; these results are compared to the full simulations. Using the code, two transitions that cause the axisymmetric vortices to develop waves travelling in the streamwise direction at higher Reynolds numbers are discovered. The linear stability of axisymmetric Dean vortex flow to non-axisymmetric perturbations is examined. Associated with the two transitions are two different non-axisymmetric flows: undulating and twisting Dean vortex flow. Undulating vortices are similar to wavy Taylor vortices. Twisting vortices, with a much shorter streamwise wavelength, are dissimilar; to our knowledge, they have no counterpart in the Taylor-Couette problem. At sufficiently high Reynolds numbers, linear growth rates associated with twisting vortices far exceed those associated with undulating vortices. For the channel curvatures studied, angular speeds of both kinds of travelling waves are only weakly dependent on Reynolds number and wavenumber. A bifurcation limits the vortex spacings that can be examined and suggests an Eckhaus stability boundary. The development of wavy vortex flows from small-amplitude disturbances shows that full development of undulating vortices may require a streamwise distance greater than one circumference, whereas for sufficiently large Reynolds numbers, twisting vortices reach equilibrium amplitude within half this distance and are therefore more likely to be observed experimentally. We suggest twisting vortices are due to a shear instability.