Interaction between Ekman pumping and the centrifugal instability in Taylor–Couette flow

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The endwalls in a Taylor–Couette cell introduce adjacent boundary layers that interact with the centrifugal instability. We investigate the interaction between the endwall Ekman layers and the Taylor vortices near transition from nonvortical to vortical flow via direct numerical simulation using a spectral method. We consider a radius ratio of η=0.75 in a short annulus having a length-to-gap ratio of Γ=6. To analyze the nature of the interaction between the vortices and the endwall layers, three endwall boundary conditions were considered: fixed endwalls, endwalls rotating with the inner cylinder, and stress-free endwalls. Below the critical Taylor number, endwall vortices for rotating endwalls are more than twice the strength of the vortices for fixed endwalls. This trend continues well above the transition to vortical flow, consistent with a simple force balance analysis near the endwalls. Stress-free endwalls result in endwall vortices that are similar in strength to those for rotating endwalls above the critical Taylor number. The endwall conditions significantly change the bifurcation diagram based on the radial velocity near the center of the annulus. For stress-free endwall conditions, the bifurcation is quite sharp, although only one fork of the bifurcation results unless the initial conditions are specifically set to favor the other fork. For rotating and fixed endwalls, there is a continuous transition from a featureless flow to a vortical flow due to the endwall vortices.