Three-Dimensional Airfoils

Suppose that air circulation is set up around the wing parallel to the - plane in such a manner as to produce an upward lift. It follows that the average pressure on the lower surface of the wing must exceed that on its upper surface. Consider Figure 9.9, which shows a back view of the airfoil shown in Figure 9.8. As we go from the median plane ( ) to a wing tip, , whether along the upper or the lower surface of the wing, we must arrive at the same pressure at . It follows that there is a drop in pressure as we move outward, away from the median plane, along the wing's bottom surface, and a further drop in pressure as we move inward, toward the median plane, along the upper surface. Because air is pushed in the direction of decreasing pressure, it follows that the air that impinges on the wing's leading edge, and then passes over its upper surface, deviates sideways toward the median plane. Likewise, the air that passes over the wing's lower surface deviates sideways away from the median plane. (See Figure 9.8.)

The air that leaves the trailing edge of the wing at some point must have impinged on the leading edge at the different points and , depending on whether it travelled over the wing's upper or lower surfaces, respectively. Moreover, air that travels to via the wing's upper surface acquires a small sideways velocity directed towards the median plane, whereas that which travels to via the lower surface acquires a small sideways velocity directed away from the median plane. On the other hand, the air speed at must be the same, irrespective of whether the air arrives from the wing's upper or lower surface, because the pressure (which, according to Bernoulli's theorem, depends on the air speed) must be continuous at . Thus, we conclude that there is a discontinuity in the direction of the air emitted by the trailing edge of a wing. This implies that the interface, , between the two streams of air that travel over the upper and lower surfaces of the wing is a vortex sheet. (See Section 9.5.) Of course, this vortex sheet constitutes the wake that trails behind the airfoil. Moreover, we would generally expect the wake to be convected by the incident wind. It follows that the vorticity per unit length in the wake can be written

(9.63) |

where , and is tangential velocity discontinuity across the wake. [See Equation (9.56).]

As we saw previously, the boundary layer that covers the airfoil is such that the tangential velocity
just outside the
layer is sharply reduced to zero at the airfoil surface. Actually, the nature of the substance enclosed by the surface is
irrelevant to our argument, and nothing is changed in our analysis if we suppose that this region contains air at rest. Thus, we can replace
the airfoil by air at rest, and the boundary layer by a vortex sheet,
, with a vortex intensity per unit length
**
**
that is determined by the velocity
discontinuity
between the air just outside the boundary layer and that at rest in the region where the airfoil was previously located. In fact, Equation (9.56) yields

where is an outward unit normal to the airfoil surface.

We conclude that a stationary airfoil situated in a uniform wind of constant velocity is equivalent to a vortex sheet , located at the airfoil surface, and a wake that trails behind the airfoil, the airfoil itself being replaced by air at rest. The vorticity within is largely parallel to the -axis [because and are both essentially parallel to the - plane--see Equation (9.64)], whereas that in is parallel to the incident wind direction. (See Figure 9.11.) The vortex filaments within are generally termed bound filaments (because they cannot move off the airfoil surface). Conversely, the vortex filaments within are generally termed free filaments. The air velocity both inside and outside can be written

(9.65) |

where is the external wind velocity, the velocity field induced by the free vortex filaments that constitute , and the velocity field induced by the bound filaments that constitute .

Consider some point that lies on . Let and be two neighboring points that are equidistant from , where lies just outside , and lies just inside , and the line is normal to . We can write

However, , where is the tangential air velocity just above point on the airfoil surface, and , as the air within is stationary. Moreover, , because we expect to be continuous across . On the other hand, we expect the tangential component of to be discontinuous across . Let us define

(9.68) |

This quantity can be identified as the velocity induced at point by the bound vortices on , excluding the contribution from the local bound vortex at (because this vortex induces equal and opposite velocities at and ). Finally, taking half the sum of Equations (9.66) and (9.67), we obtain