Wing Geometry Required for Elliptical Loading, LiftingLine Theory
It is shown in the previous section that ellipticalspanwise loading corresponds to minimum induced drag. There are three ways to design a wing to achieve elliptical – spanwise loading: (1) using a suitable wing planform, or (2) using wing twist, and it can be approximated (3) using a simple tapered planform.
where K = (T0/qsCt), which is a constant. Thus, the variation of the wing chord, c, with span, y, is given by:
K2 (b / 2)2
ro 
f 1 } 
f 2 CLVsS ^ 
f 4S Ї 
f Cl ^ 
4S 

qC 
1 q~C1J 
1 nb J 
tp Vsnb J 
v CL J 
pVjnh 
which is the equation for an ellipse. The constant K is independent of lift because from Eq. 6.39: 
K= 
Eq. 6.48 states that if a wing is untwisted and of constant section with constant liftcurve slope, then an ellipticallift distribution is generated if the wing has an elliptical planform. Furthermore, the ellipticallift distribution occurs at any value of winglift coefficient. Thus, ellipticallift distribution can be achieved in practice but at the cost of complexity of manufacture.
2. Twist
Assume (for simplicity) only that m0 = constant for any wing section. A geometric twist (varies with span) or an aerodynamic twist (aLo varies with span) or both are permitted. The induced angle of attack, a;, is a constant because of the ellipticalloading requirement. Then, from Eq. 6.45:
CL= mo(« «i “L0 ),
and using Eqs. 6.36 and 6.42:
The local variation of wing chord as expressed by c(y) and wing twist as expressed by the absolute angle of attack aa(y) = [a(y) – aLo(y)] must be such as to satisfy Eq. 6.49 for all values of y. In particular, for a rectangular wing of constant section, Eq. 6.49 shows that an elliptical spanwise loading can be achieved provided that the geometric twist of the wing also varies elliptically with span. However, because CL does not cancel out in Eq. 6.49, this twist provides elliptical spanwise loading at only one (design) value of winglift coefficient. Again, as in the case of an elliptical planform, spanwise elliptical loading can be achieved at the cost of fabrication complexity.
3. Tapered Wing
Recall from Eqs. 6.31 and 6.44 that a wing of arbitrary planform with arbitrary spanwise loading has an induced drag that is a factor (1 + 5) larger than that for elliptical loading. A straighttapered wing with a taper ratio of about 0.4 has a value of 5 or about 0.01. Thus, this wing—which is relatively straightforward to manufacture—has an induceddrag coefficient that is only about 1 percent above the minimum. We verify this conclusion by running Program PRANDTL for various taper ratios and examining the values of induced drag in the neighborhood of a taper ratio of 0.4.
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