Page 254 - Aerodynamics for Engineering Students

Finite wing theoly 237

Fig. 5.26 Prandtl's lifting line model

is the total circulation due to all the vortex filaments passing through the wing section
at z. Physically the approximate theoretical model implicit in Eqn (5.32) and (5.33)
corresponds to replacing the wing by a single bound vortex having variable strength
I', the so-called Zijting Zine (Fig. 5.26). This model, together with Eqns (5.32) and
(5.33), is the basis of Prandtl's general wing theory which is described in Section 5.6.
The more involved theories based on the full version of Eqn (5.31) are usually
referred to as lifting surface theories.
Equation (5.32) can also be deduced directly from the simple, less general, theor-
etical model illustrated in Fig. 5.21. Consider now the influence of the trailing vortex
filaments of strength ST shed from the wing section at z in Fig. 5.21. At some other
point z1 along the span, according to Eqn (5.1 l), an induced velocity equal to

will be felt in the downwards direction in the usual case of positive vortex strength.
All elements of shed vorticity along the span add their contribution to the induced
velocity at z1 so that the total influence of the trailing system at z1 is given by Eqn
(5.32).

5.5.2 The consequences of downwash - trailing vortex drag
The induced velocity at z1 is, in general, in a downwards direction and is sometimes
called downwash. It has two very important consequences that modify the flow
about the wing and alter its aerodynamic characteristics.
Firstly, the downwash that has been obtained for the particular point z1 is felt to
a lesser extent ahead of z1 and to a greater extent behind (see Fig. 5.27), and has the
effect of tilting the resultant oncoming flow at the wing (or anywhere else within its
influence) through an angle

where w is the local downwash. This reduces the effective incidence so that for the
same lift as the equivalent infinite wing or two-dimensional wing at incidence ax an
incidence a = am + E is required at that section on the finite wing. This is illustrated
in Fig. 5.28, which in addition shows how the two-dimensional lift L, is normal to