A Simple Model for Minimum Induced Drag of Multiplanes: Could Prandtl Do the Same?

Abstract

In a famous report about the induced drag of multiplanes, L. Prandtl conceived the Best Wing System: the lifting system having the minimum induced drag among all the other ones, for assigned wingspan and lift. The resulting shape was a boxwing, asymptotically equivalent to a multiplane having infinite wings. At the end of the report, he gave the performance of this optimal lifting system (in terms of dimensionless induced drag and vertical gap) through a curve, without presenting the method he used. Nowadays, modern computational aerodynamics show an excellent agreement between the numerical results and Prandtl's prevision. The first part of the paper aims at clarifying which method Prandtl used for the Best Wing System only. In the second part of the paper, starting from Prandtl's different approach for the biplane and triplane problem, we propose a simple model for the evaluation of the induced drag of multiplanes, using notions an aerodynamicist of 1920s likely had. We show that the limit multiplane having infinitely many wings, under the ansatz of elliptic lift distribution on the outermost wings and constant lift distribution on the innermost ones, matches the Best Wing System curve. This leads to a simple but interesting interpretation of the actual results.

An Alternative Approach to Induced Drag Reduction

“…The other option is to reduce induced drag by increasing wingspan efficiency factor. Towards this end several concepts and schemes, such as winglets (Whitcomb, 1976;Chattot, 2006;Demasi et al, 2019) and non-planar lifting surfaces in the form of the C-wing (McMasters & Kroo, 1998), the joined wings (Wolkovich, 1985) and the box wing (Prandtl, 1924;Scardaoni, 2020), have been proposed. Among them, only winglets have been used in operational aircraft.…”

“…Another group of publications focuses on minimizing induced drag under structural constraints (Jones, 1950;Pate & German, 2013;Phillips et al, 2019;Taylor & Hunsaker, 2020a). Lately, there is a renewed interest in Prandtl's box wing (Prandtl, 1924) as the lifting system with minimum induced drag (Frediani & Montanari, 2009;Cavallaro & Demasi, 2016;Scardaoni, 2020).…”