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Minimum Induced Drag Theorems for Joined Wings, Closed Systems, and Generic Biwings: Applications

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Abstract

An invariant procedure for the minimization of induced drag of generic biwings and closed systems (Joined Wings) was presented in the companion paper (minimum induced drag theorems for Joined Wings, closed systems, and generic biwings: theory) and is now adopted to study several theoretical open questions regarding these configurations. It is numerically verified that a quasi-closed C-wing presents the same optimal induced drag and circulation of the corresponding closed system. It is also verified that when the two wings of a biwing are brought close to each other so that the lifting lines identify a closed path, the minimum induced drag of the biwing is identical to the optimal induced drag of the corresponding closed system. The optimal circulation of this case differs from the quasi-closed C-wing one by an additive constant. The non-uniqueness of the optimal circulation for a closed wing system is also addressed, and it is shown that there are an infinite number of equivalent solutions obtained by adding an arbitrary constant to a reference optimal circulation. This property has direct positive impact in the design of Joined Wings as far as the wing load repartition is concerned: The percentage of aerodynamic lift supported by each wing can be modified to satisfy other design constraints, and without induced drag penalty. Finally, the theoretical open question regarding the asymptotic induced drag behavior of Joined Wings, when the vertical aspect ratio approaches infinity, has been resolved. It has been shown that for equally loaded wings indefinitely distant from each other, the boxwing minimum induced drag tends to zero. In that condition, the upper and lower wings present a constant aerodynamic load. Prandtl’s approximated formula for the minimum induced drag of a boxwing (Best Wing System) cannot be used to describe the asymptotic behavior. This work also shows that the optimal distribution over the equally loaded horizontal wings of a boxwing is not the superposition of a constant and an elliptical functions. This is an acceptable approximation only for small vertical aspect ratios (of aeronautical interest).

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Notes

  1. Closed System’s C-Wing Limit and Closed System’s Biwing Limit Theorems have not been completely proven in mathematical terms (see [13, 14]); however, they have been numerically verified. Therefore, from a mathematical point of view they are still conjectures. In the following, “Theorem” will mean a conjectured theorem when referred to CSCWLT and CSBLT.

  2. The induced drag minimization is carried out by projecting the lifting system on a single plane perpendicular to the freestram (Munk’s stagger theorem).

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Authors and Affiliations

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

    Luciano Demasi & Rauno Cavallaro

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

    Giovanni Monegato

  3. SkyBox Engineering S.r.l., Pisa, Italy

    Emanuele Rizzo

  4. Department of Structural Engineering, University of California San Diego, La Jolla, CA, USA

    Rauno Cavallaro

  5. Dipartimento di Ingegneria Aerospaziale, Università di Pisa, Pisa, Italy

    Antonio Dipace

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  1. Luciano Demasi
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  2. Giovanni Monegato
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  3. Emanuele Rizzo
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  4. Rauno Cavallaro
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  5. Antonio Dipace
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Correspondence to Luciano Demasi.

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Demasi, L., Monegato, G., Rizzo, E. et al. Minimum Induced Drag Theorems for Joined Wings, Closed Systems, and Generic Biwings: Applications. J Optim Theory Appl 169, 236–261 (2016). https://doi.org/10.1007/s10957-015-0850-5

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  • DOI: https://doi.org/10.1007/s10957-015-0850-5

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