On Prandtl--Batchelor Theory of a Cylindrical Eddy: Asymptotic Study

Abstract

The inviscid limit of vorticity in steady Navier--Stokes flows in two dimensions containing regions of closed streamlines (eddies) is studied. Prandtl [International Mathematical Congress, Heidelberg, Teubner, Leipzig, 1904, pp. 484--491; see also Gesammelte Abhandlungen zur angewandten Mechanik, Hydro- und Aerodynamik, vol. 2, Springer-Verlag, Berlin, 1961, pp. 575--584] and Batchelor [J. Fluid Mech., 1 (1956), pp. 177--190] showed that, in the limit of large Reynolds number R, the vorticity in such a region becomes constant. For the circular domain, Batchelor and Wood [J. Fluid Mech., 2 (1957), pp. 77--87] gave a formula which computes the constant vorticity. To study the effect of finite Reynolds number R on this formula, a matched asymptotic expansion of the velocity in two parameters, $ \epsilon $ (the perturbation parameter) and R, is calculated up to order of $O(\eps^2 /\sqrt{R})$ and compared with the numerical results. For the resulting vorticity, including the correction for finite R, $\omega_{full}$, we obtain values in reasonably good agreement with the numerical results.