Eddies induced in cylindrical containers by a rotating end wall

Abstract

The flow generated in a viscous liquid contained in a cylindrical geometry by a rotating end wall is considered. Recent numerical and experimental work has established several distinct phases of the motion when fluid inertia plays a significant role. The current paper, however, establishes the nature of the flow in the thus far neglected low Reynolds number regime. Explicitly, by employing biorthogonality relations appropriate to the current geometry, it is shown that a sequence of exponentially decaying eddies extends outward from the rotating end wall. The cellular structure is a manifestation of the dominance of complex eigensolutions to the homogeneous problem and arises as the result of nonlinear forcing associated with an inertial correction to the Stokes flow.

Nonlinear vortex development in rotating flows

“…This is particularly the case if the bubble appears or disappears near a boundary, since we recognize that recirculating eddies can always occur when sufficiently close to such corner regions; these local eddy cascades were described in the work of Moffatt (1964) with a rotational analogue described by Hills (2001). The approach we take here is one that was successfully employed by Mullin et al (1998); the criterion for the existence of a bubble is the presence of a recognizably enclosed region in the meridional streamline plot when viewed over the full computational domain.…”

“…The appearance of multiple regions of recirculation in the meridional plane is not purely a finite Reynolds number phenomenon; toroidal vortices may also develop at low Reynolds numbers. In particular, an analysis of the slow-flow regime by Hills (2001) has shown that a cascade of toroidal vortices can be found at large aspect ratios. In this work, we briefly comment on some new features observed in the experimental visualization of such lid-driven cavity flows.…”

Bifurcations of a creeping air–water flow in a conical container

“…At the interface, v and the swirl shear stresses are continuous: v w = v a and v w /z =  r v a /z at z = H w . Thus the problem for the swirl velocity becomes separated from the problem for the meridional motion in the limiting case as Re  0, similar to that in the one-fluid problem studied by Hills [24]. We first solve this linear problem for swirl.…”

Flow Patterns in a Two-Roll Mill

“…To minimise end effects the light sheet was placed between 20 and 30cm above the base of the container. (Near the base eddies can be introduced associated with the homogeneous eigensolutions for the geometry that will destroy planar flow -see Hills [30]). Figure 11 illustrates the flow patterns we obtained for co-and counter-rotation of the cylinders for a variety of Reynolds numbers.…”