Laminar boundary layer separation and reattachment on a rotating sphere

Abstract

A new model of the steady boundary layer flow around a rotating sphere is developed that includes the widely observed collision and subsequent eruption of boundary layers at the equator. This is derived following the Segalini & Garrett (J. Fluid Mech., vol. 818, 2017, pp. 288–318) asymptotic approach for large Reynolds numbers but replacing the Smith & Duck (Q. J. Mech. Appl. Maths, vol. 30, issue 2, 1977, pp. 143–156) correction with a higher-order version of the Stewartson (Grenzschichtforschung/Boundary Layer Research, 1958, pp. 59–71. Springer) model of the equatorial flow. The Stewartson model is then numerically solved, for the first time, via a geometric multigrid method that solves the steady planar Navier–Stokes equations in streamfunction-vorticity form on large rectangular domains in a quick and efficient manner. The results are then compared with a direct numerical simulation of the full unsteady problem using the Semtex software package where it is found that there is broad qualitative agreement, namely the separation and reattachment of the boundary layer at the equator. However, the presence of unobserved behaviour such as a large area of reverse flow seen at lower Reynolds numbers than those observed in other studies, and that the absolute error increases with Reynolds number suggest the model needs improvement to better capture the physical dynamics.