By Kuethe A.M., Chow C.-Y.
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A reducible curve in a region can be contracted to a point without leaving the region. For example, in the region exterior to an airfoil, any curve surrounding the airfoil is not reducible and any curve not surrounding it is reducible. A simply connected region is one where all closed curves are reducible. (The region exterior to a finite three-dimensional body is simply connected. ) A barrier is a curve that is inserted into a region but is not a part of the resulting modified region. The insertion of barriers into a region can change it from being multiply connected to being simply connected.
30), with the second term on the left-hand side replaced by the right-hand side of Eq. 5). 7) along with the incompressible continuity equation and the fact that the vorticity is divergence free (note that for any vector A, ∇ · ∇ × A ≡ 0). 9) For a flow that is two-dimensional, the vorticity is perpendicular to the flow direction and Eq. 11) Dt and the vorticity of each fluid element is seen to remain constant. The vorticity equation (Eq. 8)) strongly resembles the Navier–Stokes equation and for very high values of the Reynolds number we see that the vorticity that is created at the solid boundary is convected along with the flow at a much faster rate than it can be diffused out across the flow and so it remains in the confines of the boundary layer and trailing wake.
In Fig. 3 two examples are shown to illustrate the concept of circulation. The curve C (dashed lines) is taken to be a circle in each case. In Fig. 3a the flowfield consists of concentric circular streamlines in the counterclockwise direction. It is clear that along the circular integration path C (Fig. 3a) q and dl in Eq. 3) are positive for all dl and therefore C has a positive circulation. In Fig. 3b the flowfield is the symmetric flow of a uniform stream past a circular cylinder. It is clear from the symmetry that the circulation is zero for this case.