Page 191 - Aerodynamics for Engineering Students

174 Aerodynamics for Engineering Students

where k is the distribution of vorticity over the element of camber line 6s and
circulation is taken as positive in the clockwise direction. The problem now becomes
one of determining the function k(x) such that the boundary condition
dYc
I
v=U--Ua at y=O, O<x<l (4.14)
dx
is satisfied as well as the Kutta condition (see Section 4.1.1).
There should be no difficulty in accepting this idealized concept. A lifting wing
may be replaced by, and produces forces and disturbances identical to, a vortex
system, and Chapter 5 presents the classical theory of finite wings in which the idea of
a bound vortex system is fully exploited. A wing replaced by a sheet of spanwise
vortex elements (Fig. 5.21), say, will have a section that is essentially that of the
replaced camber line above.
The leading edge is taken as the origin of a pair of coordinate axes x and y;
Ox along the chord, and Oy normal to it. The basic assumptions of the theory permit
the variation of vorticity along the camber line to be assumed the same as the
variation along the Ox axis, i.e. Ss differs negligibly from Sx, so that Eqn (4.13)
becomes

I? = LCkdx (4.15)

Hence from Eqn (4.10) for unit span of this section the lift is given by

(4.16)

Alternatively Eqn (4.16) could be written with pUk = p:

I = L'pUkdx = (4.17)

Now considering unit spanwise length, p has the dimensions of force per unit area
or pressure and the moment of these chordwise pressure forces about the leading
edge or origin of the system is simply

(4.18)

Note that pitching 'nose up' is positive.
The thin wing section has thus been replaced for analytical purposes by a line
discontinuity in the flow in the form of a vorticity distribution. This gives rise to an
overall circulation, as does the aerofoil, and produces a chordwise pressure variation.
For the aerofoil in a flow of undisturbed velocity U and pressure PO, the insert
to Fig. 4.12 shows the static pressures p1 and p2 above and below the element 6s
where the local velocities are U + u1 and U + 242, respectively. The overall pressure
difference p is p2 - p1. By Bernoulli:
1 1
p1 +p(U+u1)2 =po +-pu2
2
1 1
p2 + 5 p( u + u2)2 = Po + - pu2
2