Thickness to Chord Ratio of Symmetrical Aerofoil

The maximum thickness of the aerofoil occurs where dn/dd = 0. Therefore, differentiating

П = 2be(1 + cos в) sin в with respect to в, and equating to zero, we get:

— = 2be(1 + cos в) cos в — 2be sin в sin в = 0 d0 v ‘

cos2 в — sin2 в + cos в = 0 2 cos2 в + cos в — 1 = 0 (2 cos в — 1) (cos в + 1) = 0.

Thus, either:

(2 cos в — 1) = 0

(cos в + 1) = 0.

From the above relation, we get the following conditions corresponding to the maximum and minimum thickness of the aerofoil:

• cos в = 1/2, giving в = n/3 = 60°, at the maximum thickness location.

• cos в = —1, giving в = n, at the minimum thickness location.

Therefore, the maximum thickness is at the chord location, given by:

n

9 = 2b cos — = b.

9 3

This point (b, 0), from the leading edge of the aerofoil, in Figure 4.6(b), is the quarter chord point. Thus, the maximum thickness tmax is at the quarter chord point. The maximum thickness tmax is given by 2ц, with в = n/3 in Equation (4.4b).

tmax = 2ц = 2x [2be(1 + cos в) sin в]

= 2x [2be(1 + cos 60°) sin 60°]

= 2 x 2be x (1 +—^ x.

I 2 2

That is:

Thus, the thickness to chord ratio of the aerofoil becomes:

From the above relation for maximum thickness and thickness-to-chord ratio, it is seen that the thickness is dictated by the shift of the center of the circle or eccentricity e. The eccentricity serves to fix the fineness ratio (t/c ratio) of the profile. For example, a 20% thick aerofoil section would require an eccentricity of:

e = 0.2/1.3 = 0.154.

Thus, at the trailing edge, both upper and lower surface are tangential to the f-axis, and therefore, to each other. In other words, the trailing edge is cusped. This kind of trailing edge would ensure that the flow will leave the trailing edge without separation. But this is possible only when the trailing edge is cusped with zero thickness. Thus, this is only a mathematical model. For actual aerofoils, the trailing edge will have a finite thickness, and hence, there is bound to be some separation, even for the thinnest possible trailing edge.

Note: Transformation of a circle with its center at a distance be on the negative side of the x-axis, in the physical plane, will result in a symmetrical aerofoil, with its leading edge on the negative side of the f-axis (mirror image of the aerofoil profile about the n-axis, in Figure 4.6), in the transformed plane. Similarly, positioning the center of the circle, with an offset, on the у-axis, will get transformed to a symmetrical aerofoil, with its leading and trailing edges on the n-axis, in the transformed plane.