On Stewartson's collision problem: the laminar stationary flow around a rotating sphere at asymptotically large Reynolds number

The collision and thus separation of two symmetrical, laminar, steady planar wall jets of a Newtonian fluid with uniform properties at arbitrarily large Reynolds number (Re) poses a long-standing problem in boundary layer theory and beyond. In its to date most celebrated form, it was posed first by Stewartson (Boundary Layer Research: IUTAM Symposium Freiburg/Br., 1958) as the equatorial collision of the meridional flow past a sphere (body of revolution, generally) due to its rotation in an unbounded, otherwise quiescent environment. The asymptotic flow structure put forward originally consists essentially of an Euler region governing the separation of the viscous sublayer underneath. Even though indicated intuitively back then, this structure can be shown rigorously as inconsistent with the branching of the incident wall jet, which is inevitably controlled by viscous-inviscid interaction of double-deck type. This mechanism was proposed to be at play already by Smith & Duck (Q. J. Mech. Appl. Math., 30 (2), 1977). In turn, they predicted much larger recirculation eddies and the absence of the Euler region. Despite its rationally founded preference, this self-consistent structure was never detected in the numerical solutions of the full Navier-Stokes equations (NSE), which raised a quite recent controversy. We aim at resolving this by demonstrating that the flow detaches in an eddy-free manner for large but finite values of Re, whereas it is either intrinsically unsteady or unstable for even larger ones and presumably undergoes quickly laminar-turbulent transition. To this end, we obtained accurate transient numerical solutions of the NSE and performed an asymptotic analysis, amongst others devoted to the stability of the structure by Smith & Duck. These preliminary findings also tie in with well-understood analogous situations in the theory of external high-Re flows. Remarkably, the recirculation eddies indeed appear for a fraction of time, which points to a too short transient period in the recent numerical studies and might explain their partial favouritism of Stewartson's structure.