Minimum Induced Drag Theorems for Multiwing Systems

Abstract

Under the assumption of a rigid wake aligned with the freestream velocity, a computationally efficient induced drag minimization procedure, tailored for the preliminary design phases of generic multiwing systems, is presented. The method is based on a variational approach, which leads to the formulation of a system of coupled integral equations in the unknown circulation distributions. It is shown that the augmented Munk’s minimum induced drag theorem has general validity for any given number and types of wings. Under optimal conditions, the aerodynamic efficiency of each wing is shown to coincide with the aerodynamic efficiency of the entire system, regardless of the number of wings and their shapes/wingspans; moreover, the aerodynamic load on each wing cannot be negative. In the past, it was demonstrated that changing the sign of curvature did not affect the optimal circulation and minimum induced drag for single nonplanar wings. This has been proven to be a far more general theorem in which, given an arbitrary number of generic nonplanar wings, the optimal circulations, load repartitions, and induced drag do not change if all lifting lines are replaced with mirrored (with respect to the horizontal plane) counterparts. It has also been verified that any wing system internal to a closed wing should be unloaded under optimal conditions, demonstrating that, if canard/tail are added to Box Wings, these additional lifting surfaces should be placed outside the region identified by the closed wing to avoid penalty on the induced drag. Several other properties of multiwings are presented, with emphasis on the relationship between closed and open systems conceptually filling the space identified by the closed wing. Results present several multiwing systems including nonplanar wings and closed systems. It is verified that the optimal conditions of a closed system part of a more complex multiwing can be studied as a limit case of a C-wing whose tips are brought close to each other or as a limit case of a biwing, confirming the generality of previous findings. The classical straight lifting line has been observed to have the minimum optimal aerodynamic efficiency, in terms of induced drag, for a given wingspan and lift if compared to any optimally loaded generic nonplanar open or closed systems.