On the uniform vorticity in a high Reynolds number flow

Abstract

A method is proposed for determining the value of the uniform vorticity (ω₀) in the inviscid region of a high Reynolds number (Re) flow with closed streamlines. An asymptotic treatment of the area integral of the Navier-Stokes equations over the enclosed region leads to a constraint involving the core vorticity; this requires the solution of the momentum equations at O(1) and O(Re ^{−% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac+% cacaaIYaaaaa!3816!\[1/2\]}) both in the core and in the surrounding boundary layers, although we are subsequently able to show that, under the assumption that the core vorticity at O(δ) is also constant, the value of ω₀ depends only on the flow at O(1). The analysis is verified numerically for the case where the boundary is an ellipse, and is also shown to be in agreement with the only case for which an analytic solution is available, namely when the enclosing boundary is circular. The validity of the above-mentioned assumption is also discussed.