A new matched asymptotic expansion for the intermediate and far flow behind a finite body

An approximated Navier–Stokes steady solution is here presented for the two dimensional bluff body wake region that is intermediate between the field on the body scale LD, which includes the two symmetric counter-rotating eddies, and the ultimate far wake. The nonparallelism of the streamlines in the intermediate wake cannot yet be considered negligible. The R is of the order of the critical value for the onset of the first instability and the limiting behavior for large R is not considered. The solution is obtained by matching an inner solution—a Navier–Stokes expansion in powers of the inverse of the longitudinal coordinate—and an outer solution, which is a Navier–Stokes asymptotic expansion in powers of the inverse of the distance from the body. The matching is built on the criteria that, where the two solutions meet, the longitunal pressure gradients and the vorticities must be equal and the flow toward the inner layer must be equal to the outflow from the external stream. At high orders in the inner expansion solution, the lateral decay turns out to be algebraic. This approximate solution is here examined in relation to the class of asymptotic solutions that, in the past, were obtained by adopting the rapid decay principle, which implies an irrotational outer flow. The theme running through this paper is the necessity of the addition of this criterion to the equations of motion to build a solution that describes the intermediate wake. The present solution has been obtained by relaxing the imposition of the rapid decay principle. It can be concluded that, at Reynolds numbers as low as the first critical value and where the nonparallelism of the streamlines is not yet negligible, the division of the field into two basic parts—an inner vortical boundary layer flow and an outer potential flow—is spontaneously shown up to the second order of accuracy: at higher orders in the expansion solution the vorticity is first convected and then diffused in the outer field. If exploited to represent the basic flow of bluff body wakes, the analytical simplicity of this asymptotic expansion could be useful for the nonparallel analysis of the instability of two-dimensional wakes.