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Abstract
A theoretical and numerical investigation of the streamwise-oriented Dean vortices in a curved channel is presented. With increasing Reynolds number we find a sequence of bifurcations similar to the bifurcations observed in nonturbulent Taylor-Couette flow between concentric rotating cylinders. We examine the transition from laminar curved channel Poiseuille flow to a flow containing axisymmetric Dean vortices using linear and weakly nonlinear analyses. These results are compared to pseudo-spectral simulations based on the incompressible Navier-Stokes equations. With the code, we find a second transition (at higher Reynolds number) in which the axisymmetric vortices develop a long wavelength streamwise waviness which travels in the streamwise direction. These wavy vortices are very similar to those in wavy Taylor vortex flow, and produce a flow we call "undulating Dean vortex flow". At slightly higher Reynolds numbers, we find that axisymmetric Dean vortex flow can undergo transition to a second type of nonaxisymmetric flow, which we call "twisting Dean vortex flow". This type of flow also corresponds to a travelling wave, but it is of shorter streamwise wavelength and has a considerably different appearance than undulating Dean vortex flow. To our knowledge, twisting vortex flow is not observed in the Taylor-Couette problem. Linear stability analysis of axisymmetric Dean vortex flow to nonaxisymmetric perturbations provides insight into the nature of these two instabilities. Further insight is gained from nonlinear numerical solutions. At high enough Reynolds numbers, linear growth rates associated with twisting vortices far exceed those associated with undulating vortices. For the channel curvatures studied, angular speeds of the travelling waves are only weakly dependent on Reynolds number and wavenumber. Analysis of the development of these flows from small amplitude disturbances indicates that full development of undulating vortices may require a streamwise distance greater than one circumference, whereas the twisting vortices reach steady state within half this distance at sufficiently large Reynolds numbers. We suggest that twisting Dean vortex flow is due to a shear instability of axisymmetric Dean vortex flow. By calculating temporal power spectra in a moving reference frame we also identify a transient, quasi-periodic flow which contains a nonpropagating mode superimposed on our travelling wave vortex flows.
