A critical Reynolds number for global instability of rotating disc flow

Lecturer: Prof. Jonathan Healey, Keele University, Anglie
Date: May 15, 2012 (Tuesday),10:00-11:00
Location: Institute of Thermomechanics AS CR, v. v. i., Dolejškova 5, Prague, director's meeting room (1st floor)

The flow set up by rotating a disc at constant angular velocity about its axis, in otherwise still fluid, is a classical problem in fluid mechanics. The resulting three-dimensional boundary layer is a useful model for the flow over swept wings and turbine blades. Experiments show a transition from laminar to turbulent flow at radii in a range corresponding to 500 < Re < 560, where $Re$ is a suitably defined local Reynolds number. Is this transition the result of spatial amplification produced by a convective instability? Or is it the result of a global instability that sets in at a well-defined critical Re? In convectively unstable flows, like the Blasius boundary layer, transition depends sensitively on the background disturbance level, leading to strong dependence on the experimental facility. On the other hand, for flows with a global instability, like that which produces periodic vortex-shedding in the flow around a cylinder placed in a uniform stream, the critical Re is highly repeatable and independent of facility. In this seminar we develop a global instability theory for the rotating disc, and show that the critical Re for global instability in fact depends not only on the flow over the disc, but also on the flow beyond the edge of the disc, which depends on the facility. We suggest that this may account for the variability in transitional Re seen in experiments. The theory also predicts that the lowest critical Re for global instability is Re = 507, and the largest critical Re for global instability is Re = 566, which is remarkably close to the range of transitional Re seen in experiments.

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