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

Abstract

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.

Appendix: Two Useful Mathematical Identities

In the following two propositions, the functions ${{\varGamma }}^{\mathrm {opt}}(\eta )$ and $\delta (\eta )$ are those defined in the paper for the closed curve and for the biwing system.

1. Lifting line represented by a closed curve

Proposition 10.1

Under the assumption ${{\varGamma }}^{\mathrm {opt}}, \delta \in W$ (see (5)), the following relation holds:

(88)

where $\delta '(\xi )=\frac{\mathrm { d}\delta (\xi )}{\mathrm { d}\xi }$.

Proof

To prove this relation, the left-hand side of identity (88) is elaborated taking into account relation (3):

(89)

The inner integral on the right-hand side of relationship (89) can be integrated by parts. Recalling that $\mathbf {r}\left( 0\right) = \mathbf {r}\left( b\right)$, we obtain:

(90)

This new expression can now be used to modify the right-hand side of (89):

(91)

where the last double integral has been obtained from the preceding one by exchanging $\xi$ with $\eta$. By applying now integration by parts to the outer integral in the last expression in (91) we obtain:

(92)

Due to a result of Tricomi (see [44], p.171), in the last integral we can exchange the order of integration, that is:

(93)

from which identity (88) follows. $\square$

2. Biwing System case.

Proposition 10.2

Under the assumption ${{\varGamma }}^{\mathrm {opt}}_i, \delta _i\in V_i, i=1,2$ (see (38)), for $i,j=1,2$ the following relations hold:

(94)

and, when $j\not =i$,

$$\begin{aligned}&\int \limits _{a_j}^{b_j} \,\delta _j\left( \eta _j\right) \int \limits _{a_i}^{b_i}\,\,\frac{\mathrm { d}{\varGamma }^\mathrm {opt}_i\left( \xi _i\right) }{\mathrm { d}\xi _i} \,\,Y_{ji}\left( \eta _j,\xi _i\right) \,\,\mathrm { d}\xi _i \,\mathrm { d}\eta _j \nonumber \\&\quad = \int \limits _{a_i}^{b_i}\,{\varGamma }^\mathrm {opt}_i\left( \eta _i\right) \int \limits _{a_j}^{b_j}\,\,\delta '_j(\xi _j) \,Y_{ij}\left( \eta _i,\xi _j\right) \,\,\mathrm { d}\xi _j\,\mathrm { d}\eta _i. \end{aligned}$$

(95)

To prove relationship (94), it is sufficient to repeat the same steps we have performed for deriving the integral exchange property (88). The proof of (95) is similar, but simpler, since the above-mentioned Tricomi result is replaced by the well-known Fubini theorem.