Spin-up & Spin-down

Spin-up & Spin-down:

The axisymmetric nonlinear evolution of a homogeneous fluid in a cylinder during spin-up may lead to shock formation in the azimuthal velocity field. This shows the critical case where shock formation is just not occurring. In the linearized regime, the presence of a free-surface may slow down this spin-up process. Spin-up of a fluid in a rectangle with a sloping bottom is in first instance described well by assuming conservation of absolute vorticity, leading to a single anticyclonic cell (with centre over the deeper part). Fig. Schematic of rectangular geometry with weakly sloping bottom.
Fig. Top views of theoretical (upper) and experimental (bottom) streamline patterns of initial flow in such a rectangular basin upon impulsive start of steady basin rotation. The vorticity generated in the boundary layers organises itself in a sequence of is cellular patterns of horizontal aspect ratio close to one. When the bottom is steep, this pattern is unstable, and smaller and smaller scales of motion appear. When the bottom slopes up to the surface, and the remaining flow domain is thus of triangular cross-sectional shape, the initial vorticity-conserving, inviscid start-up flow can be computed exactly and is shown here in side (left) and plan (right) views for three different container shapes. For a channel that is as wide (B) as it is high (H=1), it shows results in terms of along-channel length L for a long L=3 (top), moderate L=1 (middle) and short L=1/3 (bottom) channel.

Publications:
Heijst, G.J.F. van, Maas, L.R.M., Williams, C.W.M. (1994) The spin-up of fluid in a rectangular container with a sloping bottom. Journal of Fluid Mechanics, 265: 125-159. Maas, L.R.M. (1993) Nonlinear and free-surface effects on the spin-down of barotropic axisymmetric vortices. Journal of Fluid Mechanics, 246: 117-141.