• variational
• propeller
• optimization
• goldstein
• aerodynamics

The aim of this paper is to evaluate the theoretical efficiency of propellers with non-planar blade, optimally shaped.
It is well known that non-planar wing configurations can significantly re- duce the induced drag [5], hence this can be of interest also for propeller design. Furthermore the adoption of a curvilinear blade system can be justi- fied not only for an efficiency improvement, but also for reason that concerns the structure and the noise reduction [12], [1].
A solution to the optimum rotor problem, in the context of propeller vortex theory, was given by Goldstein [7]. He considered straight blade pro- pellers and expressed the optimum circulation function via a trigonometrical series of Bessel functions. However, such were the difficulties of computa- tion, even after the solution was found, that Theodorsen resorted to the use of rheoelectrical analogy to evaluate the circulation function, unfortunately without great success [14]. Accurate tabulated values of the Goldstein func- tion covering a wide range of parameters became available with an extensive mathematical effort by Tibery and Wrench [15]. Although this work is based on a completely different approach, Goldstein results are fundamental to validate the procedure for the case of straight blade.
In this dissertation, a variational formulation1 of the optimum rotor prob- lem is proposed in order to support the optimization of more complex blade configurations, such as the non-planar ones. The first step of the formulation consists into finding a class of functions (representing the circulation distri- bution along the blade) for which the thrust and the aerodynamic resisting moment functionals are 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 exists and is unique.
Some of the configurations analysed are: - Classical straight blade
- Parabolic blade
- Elliptical blade
- Superelliptic blade
Configurations with the same value of maximum dimensions and perfor- mances required are compared in the case of single and multiple blade pro- pellers.
This formulation can be seen as the extension of the one proposed in [11].
The main difficulty in the functionals’ evaluation, is the fact that an an- alytical expression of the velocity induced by a semi-infinite helical vortex filament do not exist2. For this reason the Euler-Lagrange equation asso- ciated with the variational problem is not obtained and a direct method is used. In particular the Ritz Method is adopted.
Another task to deal with, is the evaluation of the singular integral rep- resenting the induced velocity. A two-dimensional quadrature rule, based on Legendre polynomials, is used [9].
This procedure is implemented in a MATLAB program that, given the parametric expression of the curve representing the blade, allows the eval- uation of the momentum in the required condition and plots the optimal circulation along the curve.