Cylinders
The flow on a circular cylinder may be computed from a uniform stream and
a doublet. (See previous section.)
Some interesting conclusions and generalizations follow from the expressions
for the velocity and the potential on a circular cylinder shown above.
Note that on the surface of the cylinder, the tangential velocity is: V
= 2U sin q,
so the maximum velocity is twice the freestream value.
The more general forms of these results hold for all ellipsoids:
Vmax = V (1 + t/c) and V at surface = n x (n x Vmax))
Notice that this holds exactly in incompressible potential flow, even if
the ellipse has a t/c much larger than 1. Of course, in such a case, the
real flow will probably look quite different from the potential flow solution.
The force on a general 2-D cylinder can be computed by calculating the velocities,
using Bernoulli's law to compute pressures, then integrating the surface
pressures. However, the total forces and moments can be derived directly
from the complex potential. The result is called the Blasius theorem.
It is not derived here, but the result follows from the theory of residues,
the complex potential, and the incompressible Bernoulli equation. (Or one
might just use the momentum equation and compute the net force by far field
integrals.)
where G is the total circulation and S is the net source strength. In
the case of no net source strength, the net force exerted on a collection
of sources and vortices in a flow with freestream velocity U is perpendicular
to the freestream and proportional to U and the total circulation.