SUPERSONIC TRANSITIONAL AIRFOIL SHAPES OF MINIMUM DRAG

Feasibility of a procedure incorporating transition considerations in optimizing total drag has been demonstrated, The Schittkowski algorith m 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 airfoi l was transitional, generally with the forward portion Laminar and the aft turbulent, The transition locus was calculated using the fast tra nsition prediction module, which provides rapid computation of the Tol lmien-Schlichting wave amplification factor N and estimates transition point by the e(N) method, The laminar friction drag was evaluated usi ng self-similar solutions of the boundary-layer equations, The frictio n drag for the turbulent portion was computed assuming a one-seventh v elocity profile, With this framework, the total drag was a function of the functional x(tr), the streamwise location of transition. As a val idation of the method, the algorithm gave the correct global optimum f or the inviscid case, which is the parabolic are profile. For the visc ous case, significantly different locations of the maximum thickness l ed to only small differences in the minimum total drag. In addition, t he drag reductions from the optimal inviscid parabolic profile were ab out 10%. Convergence with respect to the number of spline knots was ac hieved, Important challenges were met to reduce the effect of truncati on errors in the numerical approximation of differentiations such as t hose used in the evaluation of the Hessian matrix, Despite these diffi culties, generalization of the approach to treat infinite yawed and sw ept wings accounting for suction, crossflow instabilities, and vortex drag appears feasible.