Large-scale and periodic modes of rectangular cell flow

Abstract

Linear stability of the rectangular cell flow: Ψ=cos kx cos y (0<k<1), is studied, both numerically and analytically. Owing to its spatial periodicity, the disturbances are characterized by the Floquet exponents (α,β). Based on numerical results, it is found that two types of the critical modes with vanishingly small exponents exist. One type (large-scale mode) has an almost uniform spatial structure. The other type (periodic mode) has a structure with the same periodicity as the main flow. The large-scale mode gives the critical Reynolds number in a more isotropic case (i.e., k≳0.6), while the periodic mode does so in the less isotropic case (i.e., k<0.6). Asymptotic expansions from (α,β)=(0,0) agree with the numerical results. Using the periodic mode, a possible explanation is given for the merging process of a pair of counter-rotating vortices observed in the experiments of a linear array of vortices by Tabeling et al. [J. Fluid Mech. 213, 511 (1990)].

Stability of a pair of planar counter-rotating vortices in a rectangular box

“…One drawback in using the model ow is that the realistic viscous boundary condition is not satisÿed along the walls but an artiÿcial stress-free one is. Nevertheless, topological changes of streamlines brought about by a computed critical mode agree with those of the experimental observation in the four-vortex system (Pleshanova, 1982) and the linear array of vortices qualitatively (Gotoh et al, 1995;Fukuta and Murakami, 1998). Furthermore, it is satisfactory to give the range of the aspect ratio of the vessel where the oscillation occurs in the four-vortex system.…”

“…It is found that vortex merging or self-oscillation is observed with stronger forcing in some cases. Some experimental results have been explained theoretically (see Pleshanova, 1982;Tabeling et al, 1990;Thess, 1992;Gotoh et al, 1995;Fukuta and Murakami, 1998). In previous theoretical works, simple stream functions such as = sin kx sin y have been used to represent the basic arrangements of the vortices because the eigenvalue problem is solved with relative ease in numerical computation.…”

Stability of periodic arrays of vortices

“…This explains why the system does not One possible reason that our simulation, unlike the experiment of Tabeling et al [9], does not reach a final equilibrium state could be that we considered the flow to be perfectly twodimensional. In the experiment, two-dimensionality is enforced by using a shallow fluid layer: a frictional force proportional to the velocity could capture this bottom-friction effect [32,33].…”