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

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.Abstract: This paper extends an earlier work [1] on the development of eddies in rectangularcavities driven by two moving lids. The streamfunction describing Stokes flow in such cavitiesis 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 aspectratio of the fully developed eddies is found computationally to be 1.38+0.05, which is inclose agreement with that obtained from Moffatt’s [2] analysis of the decay of a disturbancebetween infinite stationary parallel plates. Extended control space diagrams for both negativeand positive lid speed ratios are presented, and show that the pattern of bifurcation curvesseen previously in the single-eddy cavity is repeated at higher aspect ratios, but with a shift inthe speed ratio. Several special speed ratios are also identified for which the flow in one ormore eddies becomes locally symmetric, resulting in locally symmetric bifurcation curves.By superposing two semi-infinite cavities and using the constant velocity damping factorfound by Moffatt, a simple model of a finite multiple-eddy cavity is constructed and used toexplain both the repetition of bifurcation patterns and the local symmetries. The speed ratiosproducing partial symmetry in the cavity are shown to be integer powers of Moffatt’s velocitydamping factor.