Publication Date:
2017-03-30
Description:
A comprehensive study of the classical linear spin-down of a constant-density viscous fluid (kinematic viscosity ) rotating rapidly (angular velocity ) inside an axisymmetric cylindrical container (radius , height ) with rigid boundaries, which follows the instantaneous small change in the boundary angular velocity at small Ekman number , was provided by Greenspan & Howard (J.Fluid Mech., vol.17, 1963, pp.385-404). For that problem Ekman layers form quickly, triggering inertial waves together with the dominant spin-down of the quasi-geostrophic (QG) interior flow on the time scale. On the longer lateral viscous diffusion time scale , the QG flow responds to the sidewall shear layers. In our variant, the sidewall and top boundaries are stress-free, a set-up motivated by the study of isolated atmospheric structures such as tropical cyclones or tornadoes. Relative to the unbounded plane layer case, spin-down is reduced (enhanced) by the presence of a slippery (rigid) sidewall. This is evidenced by the QG angular velocity, , evolution on the time scale: spatially, increases (decreases) outwards from the axis for a slippery (rigid) sidewall; temporally, the long-time behaviour is dominated by an eigensolution with a decay rate slightly slower (faster) than that for an unbounded layer. In our slippery sidewall case, the corner region that forms at the sidewall intersection with the rigid base is responsible for a singularity within the layer, causing our asymptotics to apply only at values of far smaller than can be reached by our direct numerical simulation (DNS) of the linear equations governing the entire spin-down process. Instead, we solve the boundary layer equations for given numerically. Our hybrid asymptotic-numerical approach yields results in excellent agreement with our DNS. © 2017 Cambridge University Press.
Print ISSN:
0022-1120
Electronic ISSN:
1469-7645
Topics:
Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
,
Physics
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