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

An analytical formulation for the induced drag minimization of closed wing systems is presented. The method is based on a variational approach, which leads to the Euler–Lagrange integral equation in the unknown circulation distribution. It is shown for the first time that the augmented Munk’s minimum induced drag theorem, formulated in the past for open single-wing systems, is also applicable to closed systems, joined wings and generic biwings. The quasi-closed C-wing minimum induced drag conjecture discussed in the literature is addressed. Using the variational procedure presented in this work, it is also shown that in a general biwing, under optimal conditions, the aerodynamic efficiency of each wing is equal to the aerodynamic efficiency of the entire wing system (biwing’s minimum induced drag theorem). This theorem holds even if the two wings are not identical and present different shapes and wingspans; an interesting direct consequence of the theorem is discussed. It is then verified (but yet not demonstrated) that in a closed path, the minimum induced drag of the biwing is identical to the optimal induced drag of the corresponding closed system (closed system’s biwing limit theorem). Finally, the nonuniqueness of the optimal circulation for a closed wing system is rigorously addressed, and direct implications in the design of joined wings are discussed.

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