Invariant Formulation for the Minimum Induced Drag Conditions of Nonplanar Wing Systems

Abstract

Under the hypotheses of linear potential flow and rigid wake aligned with the freestream, a configuration-invariant analytical formulation for the induced drag minimization of single-wing nonplanar systems is presented. Following a variational approach, the resulting Euler–Lagrange integral equation in the unknown circulation distribution is obtained. The kernel presents a singularity of the first order, and an efficient computational method, ideal for the early conceptual phases of the design, is proposed. Munk’s theorem on the normalwash and its relation with the geometry of the wing under optimal conditions is naturally obtained with the present method. Moreover, Munk’s constant of proportionality, not provided in his original work, is demonstrated to be the ratio between the freestream velocity and the optimal aerodynamic efficiency. The augmented Munk’s minimum induced drag theorem is then formulated. Additional induced drag theorems are demonstrated following the derivations of this invariant procedure. Several nonplanar wing systems are proposed and analyzed, and the optimal induced drag and circulation are provided. The conjecture regarding the equality of the optimum induced drag of a quasi-closed C-wing with the induced drag of the corresponding closed system is also verified for several configurations including the box wing.