Eddy genesis and transformation of Stokes flow in a double-lid-driven cavity. Part 2: deep cavities

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Gurcan F., Wilson M. C. T., Savage M. D.

PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART C-JOURNAL OF MECHANICAL ENGINEERING SCIENCE, vol.220, no.12, pp.1765-1774, 2006 (SCI-Expanded) identifier identifier

Abstract

Abstract: This paper extends an earlier work [1] on the development of eddies in rectangular
cavities driven by two moving lids. The streamfunction describing Stokes flow in such cavities
is expressed as a series of Papkovich-Faddle eigenfunctions. The focus here is deep cavities,
i.e. those with large height-to-width aspect ratios, where multiple eddies arise. The aspect
ratio of the fully developed eddies is found computationally to be 1.38+0.05, which is in
close agreement with that obtained from Moffatt’s [2] analysis of the decay of a disturbance
between infinite stationary parallel plates. Extended control space diagrams for both negative
and positive lid speed ratios are presented, and show that the pattern of bifurcation curves
seen previously in the single-eddy cavity is repeated at higher aspect ratios, but with a shift in
the speed ratio. Several special speed ratios are also identified for which the flow in one or
more eddies becomes locally symmetric, resulting in locally symmetric bifurcation curves.
By superposing two semi-infinite cavities and using the constant velocity damping factor
found by Moffatt, a simple model of a finite multiple-eddy cavity is constructed and used to
explain both the repetition of bifurcation patterns and the local symmetries. The speed ratios
producing partial symmetry in the cavity are shown to be integer powers of Moffatt’s velocity
damping factor.

This paper extends an earlier work [1] on the development of eddies in rectangular cavities driven by two moving lids. The streamfunction describing Stokes flow in such cavities is expressed as a series of Papkovich-Faddle eigenfunctions. The focus here is deep cavities, i.e. those with large height-to-width aspect ratios, where multiple eddies arise. The aspect ratio of the fully developed eddies is found computationally to be 1.38 +/- 0.05, which is in close agreement with that obtained from Moffatt's [2] analysis of the decay of a disturbance between infinite stationary parallel plates. Extended control space diagrams for both negative and positive lid speed ratios are presented, and show that the pattern of bifurcation curves seen previously in the single-eddy cavity is repeated at higher aspect ratios, but with a shift in the speed ratio. Several special speed ratios are also identified for which the flow in one or more eddies becomes locally symmetric, resulting in locally symmetric bifurcation curves. By superposing two semi-infinite cavities and using the constant velocity damping factor found by Moffatt, a simple model of a finite multiple-eddy cavity is constructed and used to explain both the repetition of bifurcation patterns and the local symmetries. The speed ratios producing partial symmetry in the cavity are shown to be integer powers of Moffatt's velocity damping factor.