Optimum Lift Distributions

The elliptic lift distribution has long been known to be the distribution that minimizes induced drag for a given amount of lift and wingspan. However, once structural effects are included, other optimum lift distributions exist (like Prandtl's Bell-Shaped distribution), depending on the design constraints.

Problem Formulation

We wish to find the lift distribution that minimizes induced drag for a given set of design constraints.
From lifting-line theory, a dimensionless lift distribution can be written as

where Bn are dimensionless Fourier coefficients. The resulting induced-drag can be expressed as

where W is the total weight, and b is the wingspan. We seek solutions that minimize the induced drag Di, recognizing that the total weight is the sum of structural weight Ws and non-structural weight Wn within the wing (W = Ws + Wn), and that the required structural weight depends on the lift distribution, i.e. Ws = f(Bn).

Well-Known Solutions

Elliptic Lift Distribution

Constraints

Solution

Bn = 0 for all n

References


Prandtl's 1933 Bell-Shaped Lift Distribution

Constraints

Assumptions

Note that by assuming the proportionality constant is independent of spanwise location, Prandtl implied that the chord is constant. In fact, in his paper he states, “Admittedly, this would be exactly true only if the spar had the same height everywhere and the web weight were negligible compared with the weight of the flanges.”

Solution

B3 = -1/3, Bn = 0 for all n≠3. This solution allows an increase in wingspan of 22.5% and a decrease in induced drag of 11.1% relative to that of the elliptic lift distribution. Because the span is allowed to vary with no change in chord or weight, the wing area increases by 22.5% and the wing loading decreases by 18.4%. This can be problematic, since the aircraft design speeds are a strong function of wing loading.

References


Other Solutions

A family of optimum solutions exists, which vary only in the value of B3 with Bn=0 for all n≠3. For example, the elliptic lift distribution is B3=0, and the bell-shaped lift distribution is B3=-1/3. Other optimum solutions exist within this family of lift distributions, depending on the set of design constraints.

In order to obtain analytic solutions based on realistic design parameters, the following assumptions were made:

Case 1

Constraints

Solution

B3 = -0.1356. This solution allows an increase in wingspan of 4.98% and a decrease in induced drag of 4.25% relative to that of the elliptic lift distribution. The wing area and wing loading are the same as the case for the elliptic lift distribution.

Case 2

Constraints

Solution

B3 = -0.0597. This solution allows an increase in wingspan of 1.03% and a decrease in induced drag of 0.98% relative to that of the elliptic lift distribution. The wing area and wing loading are the same as the case for the elliptic lift distribution.

Case 3

Constraints

Solution

B3 = -1/3. This solution allows an increase in wingspan of 25.99%, a decrease in induced drag of 16.01%, and an increase in wing area of 33.33% relative to that of the elliptic lift distribution.

Case 4

Constraints

Solution

B3 = -0.1771. This solution allows an increase in wingspan of 9.07%, a decrease in induced drag of 8.03%, and an increase in wing area of 17.71% relative to that of the elliptic lift distribution.

References

Any of the lift distributions in this family can be created by twisting the wing. The theoretical twist distribution, twist amount, and root angle of attack required to produce any of these lift distributions can be expressed as

References