End-effects in rapidly rotating cylindrical Taylor-Couette flow

Abstract

Abstract. We present numerical simulations of the flow in a rapidly rotating cylindrical annulus. We show that at the rotation rates relevant to the magneto-rotational instability, the flow is strongly constrained by the Taylor-Proudman theorem. As a result, it is controlled almost entirely by the end-plates. We then consider two possible options for minimizing these end-effects, namely (i) simply taking a very long cylinder, and (ii) splitting the end-plates into a series of differentially rotating rings. Regarding option (i), we show that the cylinder would have to be hundreds of times as long as it is wide before end-effects become unimportant in the interior. Since this is clearly not feasible, we turn to option (ii), and show that in order to obtain a smooth angular velocity profile, the end-plates would have to be split into around ten rings. If the end-plates are split into fewer rings, perhaps 3-5, the angular velocity profile will not be smooth, but will instead consist of a series of Stewartson layers at the boundaries from one ring to the next. We suggest therefore that the instabilities one obtains in this system will be the familiar Kelvin-Helmholtz instabilities of these Stewartson layers, rather than the magneto-rotational instability. At best, one might hope to obtain the MRI superimposed on these Kelvin-Helmholtz modes. Any subsequent interpretation of results is thus likely to be quite complicated.

The role of boundaries in the magnetorotational instability

“…See [17] for a more detailed numerical study of this layer. Although one could expect a destabilization of this shear layer to large scale non-axisymmetric modes [18], this has not been observed in our simulations. Note that the layer does not extend very far from the endcaps, suggesting that it is disrupted by small-scale instabilities.…”

Secondary instability and tertiary states in rotating plane Couette flow

“…Despite working on a smaller domain than is used in the experiments, some of the stable states we have attained provide reasonable agreement with the streamwise and spanwise wavelengths of physically observed structures. However, as noted by Tsukahara et al (2010), Ekman layers are likely to form at the lower boundary of their apparatus, causing a change in wavelengths between structures near the lower wall and those away from it, as discussed by Czarny et al (2003), Hollerbach & Fournier (2004), Altmeyer et al (2010), Heise et al (2013). It may therefore be necessary to include the effects of Ekman layers should further agreement between experiment and theory be sought.…”

Stability of quasi-Keplerian shear flow in a laboratory experiment

“…A detailed vertical scan was performed near the ring gap to look for evidence of the Stewartson layer predicted by Hollerbach & Fournier (2004). The results of that scan are plotted in Fig.…”

“…Simulations by Hollerbach & Fournier (2004) Table 2. This leads to a lower q than 2 in the region A), and may explain the small measured Reynolds stress there as shown in Fig.…”

“…The simulations of Kageyama et al (2004) showed that two rings provided an effective trade off between mechanical complexity and reduced Ekman pumping. A simulation of the two-ring design by Hollerbach & Fournier (2004) predicted the formation of Stewartson layers at the boundaries between the components. The ideal Couette profile would then be replaced by two annuli in solid body rotation with the end rings.…”