Page 255 - Aerodynamics for Engineering Students

238 Aerodynamics for Engineering Students

- I t J 4 J 4 J c J.1

4 4
w =zero J. J t i 4 4 4 w=2wcp
WCP
I
Fig. 5.27 Variation in magnitude of downwash in front of and behind wing

the resultant velocity VR and is, therefore, tilted back against the actual direction of
motion of the wing V. The two-dimensional lift L, is resolved into the aerodynamic
forces L and D, respectively, normal to and against the direction of the forward
velocity of the wing. Thus the second important consequence of downwash emerges.
This is the generation of a drag force D,. This is so important that the above
sequence will be explained in an alternative way.
A section of a wing generates a circulation of strength I?. This circulation super-
imposed on an apparent oncoming flow velocity V produces a lift force L, = pVF
according to the Kutta-Zhukovsky theorem (4.10), which is normal to the apparent
oncoming flow direction. The apparent oncoming flow felt by the wing section is the
resultant of the forward velocity and the downward induced velocity arising from the
trailing vortices. Thus the aerodynamic force L, produced by the combination of I?
and Y appears as a lift force L normal to the forward motion and a drag force D,
against the normal motion. This drag force is called trailing vortex drug, abbreviated
to vortex drag or more commonly induced drug (see Section 1.5.7).
Considering for a moment the wing as a whole moving through air at rest at
infinity, two-dimensional wing theory suggests that, taking air as being of small to
negligible viscosity, the static pressure of the free stream ahead is recovered behind
the wing. This means roughly that the kinetic energy induced in the flow is converted
back to pressure energy and zero drag results. The existence of a thin boundary layer
and narrow wake is ignored but this does not really modify the argument.
In addition to this motion of the airstream, a finite wing spins the airflow near the
tips into what eventually becomes two trailing vortices of considerable core size. The
generation of these vortices requires a quantity of kinetic energy that is not recovered

Fig. 5.28 The influence of downwash on wing velocities and forces: w = downwash; V = forward
speed of wing; V, = resultant oncoming flow at wing; a = incidence; E = downwash angle = w/V;
am = (g .- E) = equivalent two-dimensional incidence; L, = two-dimensional lift; L = wing lift;
D, =trailing vortex drag