Feasibility of a procedure incorporating transition considerations in optimizing total drag has been demonstrated, The Schittkowski algorithm was modified to demonstrate the approach on two-dimensional airfoils, Cubic spline basis functions were used to describe the airfoils, and total drag (wave + friction) was minimized under the constraint of a fixed airfoil area, Reynolds numbers were assumed such that the airfoil was transitional, generally with the forward portion Laminar and the aft turbulent, The transition locus was calculated using the fast transition prediction module, which provides rapid computation of the Tollmien-Schlichting wave amplification factor N and estimates transition point by the e(N) method, The laminar friction drag was evaluated using self-similar solutions of the boundary-layer equations, The friction drag for the turbulent portion was computed assuming a one-seventh velocity profile, With this framework, the total drag was a function of the functional x(tr), the streamwise location of transition. As a validation of the method, the algorithm gave the correct global optimum for the inviscid case, which is the parabolic are profile. For the viscous case, significantly different locations of the maximum thickness led to only small differences in the minimum total drag. In addition, the drag reductions from the optimal inviscid parabolic profile were about 10%. Convergence with respect to the number of spline knots was achieved, Important challenges were met to reduce the effect of truncation errors in the numerical approximation of differentiations such as those used in the evaluation of the Hessian matrix, Despite these difficulties, generalization of the approach to treat infinite yawed and swept wings accounting for suction, crossflow instabilities, and vortex drag appears feasible.
