Circulation, Vorticity, and Stokes Theorem
Stokes' theorem is an integral identity that may be written:
When the vector function F is taken to be the velocity field, V, then
this relation in 2-D may be restated as:
This result implies that the circulation around a contour that contains
a group of vortices is just equal to the sum of the enclosed vortex strengths.
This allows application of the Blasius theorem to find the force acting
on a group of vortices. The result is sometimes called the Kutta-Joukowski
law:
We can also treat the flow field far from a group of vortices as if it were
created by a single vortex with a strength equal to the sum of the individual
vortices. Such far field solutions can be especially simple and useful as
a check of more complex results. Far field solutions can also be used as
boundary conditions for the more complex near field solution, reducing the
required extent of computational grids.
We should note here that just because we find a superposition of singularities
that satisfies the boundary conditions and the differential equation, it
does not mean that we have found the only solution to the problem. For example,
we could add a vortex to the doublet that was used to model the circular
cylinder, and we would still find that the flow went around the cylinder.
These non-unique solutions are problemsome and we appeal to additional considerations
to find the one(s) that actually will appear in nature. Just such an auxiliary
condition, the Kutta condition, is provided by viscous effects which then
determine the value of circulation.