In this article calculus of variations has been used to define five optimum rotors. Two of them correspond to the well-known ITR and OR. The third rotor analysed in this paper is the rotor which minimizes the total power coefficient, MPR. An interesting result is that the MPR does not produce a constant induced velocity along the blade span. Moreover, the induced velocity radial distribution is a linear one. As far as the authors know, the MPR obtained in the present paper has not been reported in the literature before. Also, the derivation from a calculus of variation approach of the wellknown ITR and OR has not been reported before. The results presented herein are valuable and interesting in themselves for two reasons. First, from an educational point of view, the use of calculus of variations is an excellent tool to show students a rigorous mathematical demonstration of the solidity, twist, and induced velocity distributions of the ITR, OR, and MPR. Second, calculus of variations can be used with more complex simulation models and the final optimization problems will also be more complicated to deal with. Evidently, in this case, the problems should be tackled with numerical methods in order to obtain the corresponding solution. The MPR tends to the OR when the aerodynamic efficiency is high enough. This behaviour opens an important practical application for the MPR owing to the low values of aerodynamic efficiency that rotary wing MAV exhibit.When both root and tip losses are included in the optimization process, the optimum solidity is significantly changed at the blade root and tip, leading to more realizable rotors. For maximum aerodynamic efficiency of about 10, the ratio between the FM of the MPRL and that of the ORL can be about 3 per cent for high values of thrust coefficient, about 0.01, and of 11 per cent for thrust coefficient about 0.002. Finally, and considering the blade shape of the MPRL obtained, its implementation to MAV could lead to new designs in this leading technological field. Of course, a trade-off between aerodynamic performance and simple manufacturing of MAV rotor blades would be encountered. The MPRL obtained in this paper should stimulate micro air vehicles manufacturing processes that permit the production of more complex geometry rotor blades.
Use of calculus of variations to determine the shape of hovering rotors of minimum power and its application to micro air vehicles
Anwendung der Variationsrechnung zur Bestimmung der Form von Rotoren im Schwebeflug bei minimaler Leistung und ihre Anwendung auf miniaturisierte Luftfahrzeuge
2012
15 Seiten, 15 Bilder, 1 Tabelle, 23 Quellen
Article (Journal)
English
CFD-based aerodynamic shape optimization of hovering rotors
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