Journal of Fluid Mechanics



On the boundary layer arising in the spin-up of a stratified fluid in a container with sloping walls


P. W. DUCK a1, M. R. FOSTER a2 and R. E. HEWITT a1
a1 Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
a2 Department of Aerospace Engineering, Applied Mechanics and Aviation, The Ohio State University, Columbus, OH, 43210, USA

Abstract

In this paper we consider the boundary layer that forms on the sloping walls of a rotating container (notably a conical container), filled with a stratified fluid, when flow conditions are changed abruptly from some initial (uniform) state. The structure of the solution valid away from the cone apex is derived, and it is shown that a similarity-type solution is appropriate. This system, which is inherently nonlinear in nature, is solved numerically for several flow regimes, and the results reveal a number of interesting and diverse features.

In one case, a steady state is attained at large times inside the boundary layer. In a second case, a finite-time singularity occurs, which is fully analysed. A third scenario involves a double boundary-layer structure developing at large times, most significantly including an outer region that grows in thickness as the square-root of time.

We also consider directly the nonlinear fully steady solutions to the problem, and map out in parameter space the likely ultimate flow behaviour. Intriguingly, we find cases where, when the rotation rate of the container is equal to that of the main body of the fluid, an alternative nonlinear state is preferred, rather than the trivial (uniform) solution.

Finally, utilizing Laplace transforms, we re-investigate the linear initial-value problem for small differential spin-up studied by MacCready & Rhines (1991), recovering the growing-layer solution they found. However, in contrast to earlier work, we find a critical value of the buoyancy parameter beyond which the solution grows exponentially in time, consistent with our nonlinear results.

(Received December 13 1995)
(Revised October 24 1996)


Correspondence:
Present address: Department of Civil Engineering, The University, Dundee DD1 4HN, UK.


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