Page 213 - Aerodynamics for Engineering Students

1 96 Aerodynamics for Engineering Students

In Section 4.10 (Fig. 4.26), the predictions of thin-aerofoil theory, as embodied in Eqns
(4.91) and (4.92), are compared with accurate numerical solutions and experimental data. It
can be seen that the predictions of thin-aerofoil theory are in satisfactory agreement with the
accurate numerical results, especially bearing in mind the considerable discrepancy between the
latter and the experimental data.

4.9 Thickness problem for thin-aerofoil theory

Before extending the theory to take account of the thickness of aerofoil sections, it is
useful to review the parts of the method. Briefly, in thin-aerofoil theory, above, the
two-dimensional thin wing is replaced by the vortex sheet which occupies the camber
surface or, to the first approximation, the chordal plane. Vortex filaments comprising
the sheet extend to infinity in both directions normal to the plane, and all velocities
are confined to the xy plane. In such a situation, as shown in Fig. 4.12, the sheet
supports a pressure difference producing a normal (upward) increment of force of
(p1 - p2)Ss per unit spanwise length. Suffices 1 and 2 refer to under and upper sides of
the sheet respectively. But from Bernoulli’s equation:

1 2 2 u2 + u1 (4.93)
2
P1 -p2 = -P(U, - u1) = p(u2 - 241)- 2
Writing (242 + u1)/2 U the free-stream velocity, and u2 - u1 = k, the local loading
on the wing becomes

(PI - p2)S~ = PUkSS (4.94)
The lift may then be obtained by integrating the normal component and similarly the
pitching moment. It remains now to relate the local vorticity to the thin shape of the
aerofoil and this is done by introducing the solid boundary condition of zero velocity
normal to the surface. For the vortex sheet to simulate the aerofoil completely, the
velocity component induced locally by the distributed vorticity must be sufficient to
make the resultant velocity be tangential to the surface. In other words, the compon-
ent of the free-stream velocity that is normal to the surface at a point on the aerofoil
must be completely nullified by the normal-velocity component induced by the
distributed vorticity. This condition, which is satisfied completely by replacing the
surface line by a streamline, results in an integral equation that relates the strength of
the vortex distribution to the shape of the aerofoil.
So far in this review no assumptions or approximations have been made, but thin-
aerofoil theory utilizes, in addition to the thin assumption of zero thickness and small
camber, the following assumptions:
(a) That the magnitude of total velocity at any point on the aerofoil is that of the
local chordwise velocity U + u’.
(b) That chordwise perturbation velocities u’ are small in relation to the chordwise
component of the free stream U.
(c) That the vertical perturbation velocity v anywhere on the aerofoil may be taken
as that (locally) at the chord.
Making use of these restrictions gives
v=s,- ‘k dx
27T x - XI