Toward an asymptotic description of Prandtl–Batchelor flows with corners

Abstract

The Prandtl–Batchelor theorem states that the vorticity in a steady laminar high Reynolds (Re) number flow containing closed streamlines should be constant; however, apart from the simple case of circular streamlines, very little is known about how to determine this constant (ω0). This paper revisits earlier work for flow driven by a surrounding smooth moving boundary, with a view to extending it to the case where the enclosing boundary has corners; for this purpose, a benchmark example from the literature for flow inside a semi-circle is considered. However, the subsequent asymptotic analysis for Re≫1 and numerical experimentation lead to an inconsistency: the asymptotic approach predicts boundary-layer separation, whereas a linearized asymptotic theory and computations of the full Navier–Stokes equations for Re≫1 do not. Nevertheless, by considering a slightly modified problem instead, which does not suffer from this inconsistency, it is found that, when extrapolating the results of such high-Re computations to infinite Re, the agreement for ω0 is around 5%, which is roughly in line with previous comparisons of this type. Possible future improvements of the asymptotic method are also discussed.