The onset of steady vortices in Taylor-Couette flow: The role of approximate symmetry

The onset of steady cellular motion in Taylor-Couette flow between a pair of finite length cylinders is studied. This is most often portrayed in the literature as an example of a simple pitchfork bifurcation where the trivial state of rotary Couette flow is replaced by cellular motion above a critical Reynolds number. However, numerous experiments and simulations of the Navier-Stokes equations, as well as the detailed numerical bifurcation study reported here, all lead to the following, seemingly paradoxical, conclusion: On the one hand, no matter how long the apparatus, finite-length effects greatly perturb the disconnected branch of the pitchfork of the periodic model. This corresponds to anomalous-mode flows which are observed to exist above a range of Reynolds number that is at least a factor of two greater than the value corresponding to the onset of cells. On the other hand, in long cylinders these effects appear to change the connected branch of normal-mode flows only minimally. We propose a resolution of this paradox in terms of a symmetry breaking bifurcation. The relevant symmetry, which is only approximate, is between two normal-mode flows with large, and nearly equal, numbers of cells. Additionally, our numerical calculations establish a scaling law that quantifies the magnitude of finite-length effects on normal-mode flows at large lengths.