Examples of Camber and Thickness Distributions

A symmetric profile has zero camber, i. e. d (x) = 0. The simplest camber distribution is the parabolic camber line:

Fig. 3.2 Rotation and shearing transformation

A cubic distribution can represent a small windmill if an axis is placed at the mid-chord x x x

d(x) = 6V3dX (1 – 2^1 – -) (3.8)

see the camber distributions in Fig. 3.3, where the z-coordinate has been stretched. A simple thickness distribution is represented by a semi-cubic

(3.9)

which gives a finite trailing edge angle. The maximum thickness is at x = 3.

The Joukowski profile described in Sect. 2.5.1 can be linearized by expanding its analytic expression to first order in є and S, where (e, S) is related to the center of the cylinder image of the profile and also to thickness and camber. SeeFig.2.11. For є ^ 1 and S ^ 1, to first order the radius reads

r(ff) = —(є cos ff — S sin ff) + a ^1 + -) (3.10)

Substituting this into the Cartesian coordinates of the profile and expanding in terms of the small coefficients gives

x(ff) = (a(1 + aa) — (- cos ff — S sin ff) + a(1

z(ff) = (a(1 + a) — (- cos ff — S sin ff) — a(1

which simplifies to give

x (ff) = 2a cos ff

z(ff) = 2є(1 — cos ff) sin ff + 2S sin2 ff

This is a parametric representation of a Quasi-Joukowski airfoil. In z(ff), the term proportional to є represents the thickness distribution and the term proportional to S, the camber. It is easy to see that the mean camber line is parabolic by elimination of ff in terms of x. The thickness distribution can also be expressed in terms of x as

16 lx x 2

) = 3^75 eh (‘- c) t3J3)

The Quasi-Joukowski airfoil has a cusped trailing edge. The maximum thickness is at x = 4.

The following thickness distribution has a maximum at x = | .As will be dis­cussed with viscous effects, having the maximum thickness far forward helps the boundary layer recovery as the flow slows down towards the trailing edge. This thickness distribution may be interesting for that reason:

See Fig. 3.4 for the different thickness distributions, where the scale has been stretched in the z-direction.

A Quasi-Joukowski airfoil at incidence, with the chord and camber lines is shown in Fig. 3.5.