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

Abstract

This paper extends an earlier work [Gürcan et al., Proc. Instn. Mech. Engrs, Part C: J. Mech. Eng. Sci., 2003, 217, 353] 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 [J. Fluid Mech., 1964, 18, 1] 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.

On the appearance of Moffatt eddies in viscous cavity flow as the aspect ratio varies

“…We validated the numerical code by comparing to the results of several previous investigators who have performed calculations for the same problem. 1,8,11,16,17 We now briefly recap the structure of the flow solutions ͑full details can be found in the original references͒, and then tabulate computed values for the critical aspect ratios A n , which are the results we shall use to test the asymptotic analysis to follow in Sec. IV.…”

“…8, 9, and 16 and more recently in a series of papers by Gürcan et al 1,11,17 These investigations establish that there exists a discrete sequence of aspect ratios ͕A n ͖ at which bifurcations occur and additional primary eddies are introduced. Depending on the precise nature of the problem considered, the bifurcation may involve corner eddies, or sidewall eddies, or may occur away from the cavity walls.…”

“…Near to y = Ϯ A higher corrections ͉͑n͉ Ͼ 1͒ become comparable in magnitude, but in the bulk of the cavity only n = Ϯ 1 is needed; this explains why the overall form of the flow resembles Moffatt's eigenfunction flow for an infinite channel, the basis for the modeling approximation ͑9͒ of Ref. 17.…”

“…10 used an analytically based superposition method. However, for our present purposes it is convenient to follow the approach used by most other investigators 1,8,9,11,13,17 and seek a solution as a series of eigenfunctions. Smith 20 where n ͑x͒ = s n sin͑s n ͒cos͑s n x͒ − xs n cos͑s n ͒sin͑s n x͒ ͑5͒…”

“…In summary the flow locally looks exactly like the infinite channel eigenfunction, 5 in which each eddy has the same size and shape and is divided from the next by a streamline running between the channel walls. This lead Gür-can et al 17 to model the flow at large A as a simple sequence of eddies from Moffatt's eigenfunction flow. This hypothesis leads to the prediction ͓Eq.…”