Mechanics of Fluids

408 The flow of an inviscid fluid
Fig. 9.35
Fig. 9.36
Elliptical lift distribution
downwash velocity vi is combined with the velocity of the approaching fluid
the effective angle of attack is altered to αe as shown in Fig. 9.36. If the
wing is now treated as one of infinite span but set at the effective angle of
attack αe, the lift force divided by length in the spanwise direction is given
by Le =−ϱUeƔ, and is perpendicular to Ue. The force Le may be resolved
into two perpendicular components: the useful lift L, normal to U, and a
component Di, parallel to U in the rearward direction, called the vortex
drag, formerly known as the induced drag.
In a real fluid, the drag force acting on a wing can be subdivided into
various components (see Section 8.3.3). Some are directly attributable to the
viscous nature of real fluids and affect wings of finite and infinite span. The
vortex drag experienced by a wing of finite span, however, is additional to
these and, as it depends entirely on the downwash velocity induced by the tip
vortices, it would occur in an inviscid fluid. That drag can occur in an inviscid
fluid may seem paradoxical. However, it can be explained by the fact that
the work done against the vortex drag appears as the kinetic energy of the
fluid in the tip vortices which are continuously shed behind the aircraft wing.
From similar triangles in Fig. 9.36
Di/Le = vi/Ue
(9.42)
For a wing producing an elliptical distribution of lift (this is a useful approximation
for aircraft wings) (Fig. 9.35), the downwash velocity vi is constant
along the span and equal in magnitude to
−Ɣ0/2b (9.43)