SIMPLY AND MULTIPLY CONNECTED REGIONS

The region exterior to a two-dimensional airfoil and that exterior to a three-dimensional wing or body are fundamentally different in a mathematical sense and lead to velocity potentials with different properties. To point out the difference in these regions, we need to introduce a few basic definitions.

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. Any curve surrounding the body can be translated away from the body and then contracted.) 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. The degree of connectivity of a region is n +1 where n is the minimum number of barriers needed to make the remaining region simply connected. For example, consider the region in Fig. 2.6 exterior to an airfoil. Draw a barrier from the trailing edge to downstream infinity. The original region minus the barrier is now simply connected (note that curves in the region can no longer surround the airfoil). Therefore n = 1 and the original region is doubly connected.

Consider irrotational motion in a simply connected region. The circula­tion around any curve is given by

r = <J>q-<fl = <J>V<&-dl = (2.36)

Barrier

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With the use of Eqs. (2.4) and with £ = 0 the circulation is seen to be zero. Also, since the integral of гіФ around any curve is zero (Eq. (2.36)), the velocity potential is single-valued.

Now consider irrotational motion in the doubly connected region exterior to an airfoil as shown in Fig. 2.7. For any curve not surrounding the airfoil, the above results for the simply connected region apply and the circulation is zero. Now insert a barrier as shown in the figure. Consider the curve that consists of Ci and C2, which surround the airfoil, and the two sides of the barrier. Since the region excluding the barrier is simply connected, the circulation around this curve is zero. This leads to the following equation:

Note that the first term is the circulation around C, and the second is minus the circulation around C2. Also, the contributions from the barrier cancel for steady flow (since the barrier cannot be along a vortex sheet). The circulation around curves Cx and C2 (and any other curves surrounding the airfoil once) are the same and may be nonzero. From Eq. (2.36) the velocity potential is not single-valued if there is a nonzero circulation.