Variational Approach to the Problem of the Minimum Induced Drag of Wings

Abstract

A closed form solution of the problem of the minimum induced drag of a finite span straight wing was given by Prandtl. In this chapter, a mathematical theory, based on a variational approach, is proposed in order to revise such a problem and provide one with a support for optimizing more complex wing configurations, which are becoming of interest for future aircraft. The first step of the theory consists in finding a class of functions (lift distributions) for which the induced drag functional is well defined. Then, in this class, the functional to be minimized is proved to be strictly convex; taking into account this result, it is proved that the global minimum solution exists and is unique. Subsequently, we introduce the image space analysis associated with a constrained extremum problem; this allows us to define the Lagrangian dual of the problem of the minimum induced drag and show how such a dual problem can supply a new approach to the design. After having obtained the Prandtl exact solution in the context of a variational formulation, a numerical algorithm, based on the Ritz method, is presented, and its convergence is proved.