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Astron. Astrophys. 347, 734-738 (1999)

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3. Are keplerian flows unstable?

More precisely, the question we address is whether the Couette flow is unstable when the angular velocity decreases outwards and the angular momentum increases outwards, as in keplerian flow.

There is no definite answer yet, because this regime has not been explored at high enough Reynolds number. But some information can be gleaned from Wendt's study. He reports the results of experiments where the two cylinders rotate such that [FORMULA]. At low Reynolds number, this setup enforces a laminar flow of constant angular momentum (neutral flow), but at high Reynolds number this flow becomes turbulent for two of the three gaps used by Wendt.

The angular velocity and angular momentum profiles for one of these turbulent flows are reproduced in Fig. 3. (According to Wendt's data, [FORMULA] actually exceeds [FORMULA] by a half a percent.) The profiles clearly demonstrate the flow instability, with angular momentum being transported down the angular velocity gradient, which becomes somewhat flater far enough from the boundaries, whereas the angular momentum profile steepens substantially.

[FIGURE] Fig. 3. Angular velocity and momentum profiles in the case of decreasing angular velocity and initially constant angular momentum - dotted line (experimental data from Wendt 1933).

In the turbulent bulk of the flow [FORMULA], compared to the initial [FORMULA]. We recall that [FORMULA] in keplerian flow, and that in the numerical simulations performed by Balbus et al. (1996, 1998), the instability is lost already at about [FORMULA]. A crude estimate of the parameter [FORMULA] in the viscosity formulation (7) indicates that the size of the turbulent eddies is much smaller than the gap width: [FORMULA].

The corresponding values of [FORMULA] for the three experiments are reported in Fig. 4 together with the critical line of Fig. 1. They are located respectively above and below this line for the unstable and stable flows. The data are too scarce to locate precisely the critical line of these "neutral" flows, but we can conclude that the critical gradient Reynolds number [FORMULA] then lies between [FORMULA] and [FORMULA].

[FIGURE] Fig. 4. Reynolds number vs. aspect ratio for the three "neutral" experiments from Wendt: the two filled circles correspond to the unstable flows, the open circle to the stable flow. The stability curve obtained for the inner cylinder at rest (Fig. 1) is displayed for comparison.

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© European Southern Observatory (ESO) 1999

Online publication: June 30, 1999