To determine the circulation about the airfoil we need an additional
condition on the flow field. If we think of the total flow as being composed
of a uniform contribution (with no circulation) plus a circulatory
contribution, then the circulation will adjust itself until the
total flow leaving the trailing edge of the airfoil is smooth. This is
called the *Kutta-Joukowsky condition* , and uniquely determines the circulation, and therefore the lift,
on the airfoil. The origin of this condition can be seen from Fig. 4.3.

**Figure 4.3:** The development of circulation about an airfoil.

We initially have flow without circulation, with two stagnation points on
the upper and lower surfaces of the airfoil. The fluid on the lower surface
of the airfoil must accelerate around the sharp trailing edge in order to
reach the rear stagnation point on the upper surface. This actually requires
that the fluid velocity at the trailing edge be infinite--an unlikely
circumstance in a viscous fluid. This flow configuration is unstable, and
the rear stagnation point gets washed downstream, until it coincides with
the trailing edge of the airfoil. At this point there is a net circulation
about the airfoil, with the zero velocity stagnation point ``cancelling''
the infinite velocity at the trailing edge, resulting in smooth flow from
the trailing edge. Now in a nonviscous fluid circulation is conserved--this
is known as the *Kelvin circulation theorem* . Since at high Reynolds numbers the viscosity is only really
important in the boundary layer (again assuming that it remains attached),
then the circulation should be approximately conserved for flow of a viscous
fluid about an airfoil at high Re. To compensate for the circulation which
is established about the airfoil, a *starting vortex* with the opposite circulation is produced downstream of the airfoil.
This is demonstrated in a dramatic way in the classic flow visualizations of
Prandtl--see the accompanying figures.

This starting vortex (often called the *downwash* )
places limitations the spacing of take-offs. The downwash usually becomes
turbulent, and for a large jet the downwash can actually flip small planes.
How long does it take before the vortex decays away due to viscosity?
Dimensional analysis tells us that , with *L* the typical
size of the vortex. If we take *L*=20 cm, and m
s , then s. This is an overestimate, but it does indicate
that vortices in air can be long-lived entities. The typical spacing between
jets is about 90 s, apparently enough time for the vorticity to decay away
to a manageable level.

With these considerations it is possible to calculate the lift on an airfoil
in terms of its geometry and angle of attack. The circulation about a
symmetric airfoil at an angle of attack and chord *c* is
approximately , so that the lifting force
per unit length is . This information is
often expressed in terms of the coefficient of lift (per unit length) for
the airfoil, defined as

For more complicated airfoils the lift coefficient must be computed using
numerical methods; typical plots of coefficient of lift as a function of
angle of attack are shown in Fig. 4.4.
It is seen that, for small angles of attack, Eq. (4.2) works very well,
and even better is the simple linear approximation
*c _{L}* = 2pa with a in
radians, or

**Figure 4.4:** Lift coefficient versus angle of attack
for the symmetric airfoil NACA 0009 and the cambered airfoil NACA 2408.

A cambered airfoil can generate lift at zero angle of attack; cambered
airfoils are the norm on modern airplanes. Notice that once the angle of
attack exceeds a critical value the flow separates from the top surface of
the airfoil, resulting in a precipitous reduction in lift and an increase in
the drag (due to pressure drag); this is called the *stall angle*, and
beyond this angle the airfoil is said to be *stalled*.

Tue Oct 21 21:23:27 EDT 1997