Profile at Incidence

The thin profile is set at incidence a by performing a rotation of angle a in the clockwise direction, about the trailing edge. The exact transformation of a point M (x, z) of the profile into M’ (x’, z’) reads

Fig. 3.1 View of Wright Brothers glider (from NASA, http://wright. nasa. gov/ airplane/kiteQQ. html)

x’ = x + (x — c)(cos a — 1) + z sin a z’ = z — (x — c) sin a + z(cos a — 1)

In thin airfoil theory, we will further assume that the incidence angle (in radians) is small

(3.5)

With this assumption, the above transformation can be simplified to first order in d, c and a as

(3.6)

which corresponds to a shearing transformation. This is a great simplification, because the equation of the profile is not modified by the transformation other than by the addition of a term proportional to a, since the ordinate of the profile at incidence now reads z±(x) = f ±(x) + a(c — x). SeeFig.3.2.

The question arises as to how small these coefficients have to be for the solution to be of practical use? In our experience, good results are obtained with relative cambers of the order of 6%, relative thicknesses of up to 12 % and angles of incidence between —10 and +10°. As will be seen, when a profile is used at its “design condition”, the viscous effects are minimized and the inviscid result will be representative of the real flow as far as the lift and moment coefficients are concerned. In off-design conditions, one should be more cautious with the bounds given above.