Direct Methods for Airfoils

Aerodynamic analysis methods coupled with numerical optimization techniques offer several advantages from the software point of view, since this approach allows for modularity to a large scale. It ows its existence to the avaibility of powerful computers, because, as explained below, the computational requirements are their major drawback. In general, the problem can be formulated as follows: minimize an objective function f subject to the aerodynamic constraints and the side constraints fs. Aerodynamic constraints are set on global aerodynamic quantities, such CL, drag CD, pitching moment CM, aerodynamic efficiency CL/CD, that on turns depend upon the angle of attack and the free stream Mach number M. Side constraints are given as thickness ratio t/c, trailing edge gap , and leading edge radius r.

where N is the number of design points; a and b are the matrices of the influence coefficients. Determination of these coefficients is not obvious. Usually not all the above requirements appear in the formulation of the objective function, and the number of design points seldom exceeds one, for reasons that will be evident further down. A useful simplification of Eq. 5 is the following

with , . Eq. 7 can be effectively used for drag minimization, while keeping the lift to a fixed value, and the airfoil thickness within specified bounds. The airfoil/wing is then described in terms of perturbations over a base geometry. The perturbation is a linear combination of shape functions

The design variables are the coefficients of the linear combination, . The choice of the shape functions, their number and the base airfoil is of fundamental importance, and will discussed in further details. A noteworthy point is that all the three-dimensional methods described herein assume a fixed wing planform. One of the first successful attempts in numerical optimization is attributed to Hicks-Murman and Vanderplaats, by using Vanderplaats’ feasible directions/gradient optimization method.