ON PRANDTL-BATCHELOR THEORY OF A CYLINDRICAL EDDY - ASYMPTOTIC STUDY

The inviscid limit of vorticity in steady Navier-Stokes flows in two d imensions containing regions of closed streamlines (eddies) is studied . Prandtl [International Mathematical Congress, Heidelberg, Teubner, L eipzig, 1904, pp. 484-491; see also Gesammelte Abhandlungen zur angewa ndten Mechanik, Hydro- und Aerodynamik, vol. 2, Springer-Verlag, Berli n, 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 vortic ity in such a region becomes constant. For the circular domain, Batche lor and Wood [J. Fluid Mech., 2 (1957), pp. 77-87] gave a formula whic h computes the constant vorticity. To study the effect of finite Reyno lds number R on this formula, a matched asymptotic expansion of the ve locity in two parameters, epsilon (the perturbation parameter) and R, is calculated up to order of O(epsilon(2) = root R) and compared with the numerical results. For the resulting vorticity, including the corr ection for finite R, omega(full), we obtain values in reasonably good agreement with the numerical results.