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Optimum Induced Drag of Wingtip Devices: The Concept of Best Winglet Design

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Abstract

Sustainable air transportation requires aerodynamically efficient airplanes. Thus, reduction of drag is of paramount importance. From a pure induced drag perspective, this goal can be achieved by the adoption of nonplanar configurations such as C-Wings, Joined Wings or with other design options such as wingtip devices (winglets). Under the assumption of inviscid flow with wake aligned with the freestream velocity, several winglet designs are investigated and general properties are demonstrated. In particular, under optimal conditions, given a closed simply connected wingtip region bounded by a curve, any winglet design geometrically included in that region will be less efficient than the winglet whose lifting line is represented by the bounding curve. Moreover, closed winglets are characterized by undetermined optimal aerodynamic load but unique and global minimum for the induced drag. Finally the Box Winglet and several variations of it are proposed as effective forms to reduce induced drag.

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Authors and Affiliations

  1. Department of Aerospace Engineering, San Diego State University, San Diego, USA

    Luciano Demasi & Rachel Rybarczyk

  2. Dipartimento di Scienze Matematiche, Politecnico di Torino, Turin, Italy

    Giovanni Monegato

  3. Departamento de Bioingeniería e Ingeniería Aeroespacial, Universidad Carlos III de Madrid, Madrid, Spain

    Rauno Cavallaro

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  1. Luciano Demasi
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  2. Giovanni Monegato
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Correspondence to Luciano Demasi.

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The original online version of this article was revised: there was an error in equation 2.

Appendix

Appendix

This appendix reports the validations of Cone’s “closed-arc tips” winglets. We have already seen Form A. Forms B, C, D, E, and F are presented in Figs. 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63. Form B, configurations shown in Figs. 44 and 46, has a circle starting at \(0.5b_w\) outboard of the wing with a radius of \(0.25b_w\). This is the largest circular radius examined in the group. Form C, configurations shown in Figs. 48 and 50, has an ellipse starting at \(0.5b_w\) outboard of the wing with a major axis of length \(0.5b_w\) and a minor axis of length \(0.3b_w\). Form C has a major axis aligned with the horizontal. Form D, configurations shown in Figs. 52 and 55, has an ellipse starting at \(0.7b_w\) outboard of the wing with a major axis of length \(0.5b_w\) and a minor axis of length \(0.3b_w\). Form D has a major axis aligned with the vertical. Form E, configurations shown in Figs. 56 and 58, has an ellipse starting at \(0.85b_w\) outboard of the wing with a major axis of length \(0.4b_w\) and a minor axis of length \(0.15b_w\). Form E has a major axis aligned with the vertical. Form F, configurations shown in Figs. 44 and 62, has an ellipse starting at \(0.6b_w\) outboard of the wing with a major axis of length \(0.4b_w\) and a minor axis of length \(0.15b_w\). Form F has a major axis aligned with the horizontal.

Figures 45, 47, 49, 51, 53, 54, 57, 59, 61, 63 show the reproduced configurations and the calculated aerodynamic efficiencies. A notable increase in aerodynamic efficiency is realized when adding circles to the wingtips as seen in Form A and Form B. Form B has a larger radius than Form A, and therefore a higher vertical aspect ratio, resulting in a aerodynamic efficiency. Form C, Form D, Form E, and Form F all explore how the aerodynamic efficiency changes with varying elliptical orientations. When the major axis is aligned with the vertical axis rather than the horizontal, the vertical aspect ratio and thus the aerodynamic efficiency is higher. This is seen when comparing Form C and Form D as well as Form E and Form F. Additionally, when the major axis is aligned with the vertical, the ellipses with a higher eccentricity have a higher aerodynamic efficiency. This is seen when comparing Form D and Form E. Conversely, when a major axis is aligned with the horizontal, the ellipses with a higher eccentricity have a lower aerodynamic efficiency. This is seen when comparing Form C and Form F.

While Cone’s results show the correct trend, it is found that Cone slightly over estimated the optimal aerodynamic efficiency ratios when examining Form A, Form E, and Form F. Cone was very close, however, in the cases of Form B, Form C, and Form D. Cone’s closed-arc tip results are compared to the calculated results and presented in Table 5.

Table 5 Cone’s closed-arc tip forms
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Fig. 44
figure 44

Cone’s closed winglet Form B, strategy 1

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Fig. 45
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Cone’s closed winglet Form B, strategy 1

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Fig. 46
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Cone’s closed winglet Form B, strategy 2

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Fig. 47
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Cone’s closed winglet Form B, strategy 2

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Cone’s closed winglet Form C, strategy 1

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Fig. 49
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Cone’s closed winglet Form C, strategy 1

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Cone’s closed winglet Form C, strategy 2

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Cone’s closed winglet Form C, strategy 2

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Cone’s closed winglet Form D, strategy 1

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Cone’s closed winglet Form D, strategy 1

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Fig. 54
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Cone’s closed winglet Form D, strategy 2

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Cone’s closed winglet Form D, strategy 2

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Fig. 56
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Cone’s closed winglet Form E, strategy 1

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Fig. 57
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Cone’s closed winglet Form E, strategy 1

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Fig. 58
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Cone’s closed winglet Form E, strategy 2

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Fig. 59
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Cone’s closed winglet Form E, strategy 2

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Fig. 60
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Cone’s closed winglet Form F, strategy 1

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Fig. 61
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Cone’s closed winglet Form F, strategy 1

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Fig. 62
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Cone’s closed winglet Form F, strategy 2

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Fig. 63
figure 63

Cone’s closed winglet Form F, strategy 2

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Demasi, L., Monegato, G., Cavallaro, R. et al. Optimum Induced Drag of Wingtip Devices: The Concept of Best Winglet Design. Aerotec. Missili Spaz. 101, 61–93 (2022). https://doi.org/10.1007/s42496-022-00110-z

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