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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
Fixed W
Fixed b
Solution
Bn = 0 for all n
References
Phillips, W. F., Fugal, S. R., and Spall, R. E., "Minimizing Induced Drag with Wing Twist, Computational-Fluid-Dynamics Validation," Journal of Aircraft, Vol. 43, No. 2, 2006, DOI: 10.2514/1.15089
Phillips, W. F., Hunsaker, D. F., and Joo, J. J., "Minimizing Induced Drag with Lift Distribution and Wingspan," Journal of Aircraft, Vol. 56, No. 2, pp. 431-441, 2019, DOI: 10.2514/1.C035027
Prandtl's 1933 Bell-Shaped Lift Distribution
Constraints
Fixed W
Assumptions
The weight at any spanwise location is proportional to the wing bending moment
The proportionality constant between the bending moment and wing weight is independent of spanwise location
Wingspan is allowed to vary
The bending moment at any spanwise location is a function only of the lift distribution
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
Hunsaker, D. F., and Phillips, W. F., “Ludwig Prandtl’s 1933 Paper Concerning Wings for Minimum Induced Drag, Translation and Commentary,” AIAA Scitech Forum, Orlando, Florida, January 2020, AIAA-2020-0644 , DOI: 10.2514/6.2020-0644 Also available here
Phillips, W. F., Hunsaker, D. F., and Joo, J. J., "Minimizing Induced Drag with Lift Distribution and Wingspan," Journal of Aircraft, Vol. 56, No. 2, pp. 431-441, 2019, DOI: 10.2514/1.C035027
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:
The bending moment at any spanwise location is a function of lift, non-structural weight, and structural weight distributions
The wing is sized by both a maximum maneuvering-flight load factor and a hard-landing load factor
Optimum non-structural weight distributions within the wing and at the wing root were used
All planforms are rectangular (this allowed an analytic solution to be found)
Case 1
Constraints
Fixed total weight
Fixed maximum stress
Fixed wing loading
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
Fixed total weight
Fixed maximum deflection
Fixed wing loading
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
Fixed non-structural weight
Fixed maximum stress
Fixed stall speed
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
Fixed non-structural weight
Fixed maximum deflection
Fixed stall speed
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
Phillips, W. F., Hunsaker, D. F., and Joo, J. J., "Minimizing Induced Drag with Lift Distribution and Wingspan," Journal of Aircraft, Vol. 56, No. 2, pp. 431-441, 2019, DOI: 10.2514/1.C035027
Phillips, W. F., Hunsaker, D. F., and Taylor, J. D., "Minimising induced drag with weight distribution, lift distribution, wingspan, and wing-structure weight," The Aeronautical Journal, Vol. 124, No. 1278, pp. 1208-1235, 2020, DOI: 10.1017/aer.2020.24
Taylor, J., Hunsaker, D. F.,and Joo, J.,“Numerical Algorithm for Wing-Structure Design,” AIAA Aerospace Sciences Meeting, Kissimmee, Florida, January 2018, AIAA-2018-1050, DOI: 10.2514/6.2018-1050
Taylor, J. D., and Hunsaker, D. F., "Minimum Induced Drag for Tapered Wings Including Structural Constraints," Journal of Aircraft, Vol. 57, No. 4, pp. 782-786, 2020, DOI: 10.2514/1.C035757
Taylor, J. D., and Hunsaker, D. F., “Low-Fidelity Method for Rapid Aerostructural Optimisation and Design-Space Exploration of Planar Wings,” The Aeronautical Journal, July 2021, Vol. 125, No. 1289, pp. 1209–1230, DOI: 10.1017/aer.2021.14
Taylor, J. D., and Hunsaker, D. F., “Comparison of Theoretical and Multifidelity Optimum Aerostructural Solutions for Wing Design,” Journal of Aircraft, Vol. 59, No. 1, Jan-Feb 2022, pp. 103-116, DOI: 10.2514/1.C036374
Taylor, J. and Hunsaker, D. F., “Effects of Active Wing-Morphing on Aircraft Fuel Burn along Fuel-Optimal Trajectories” AIAA SciTech 2023 Forum, January 2023, AIAA-2023-0038, DOI: 10.2514/6.2023-0038
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
Phillips, W. F. and Hunsaker, D. F., "Designing Wing Twist or Planform Distributions for Specified Lift Distributions," Journal of Aircraft, Vol. 56, No. 2, pp. 847-849, 2019, DOI: 10.2514/1.C035206
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