Numerical Aerodynamic Optimization Incorporating Laminar-Turbulent Transition Prediction

A two-dimensional Newton–Krylov aerodynamic shape optimization algorithm is applied to several optimization problems in which the location of laminar-turbulent transition is free. The coupled Euler and boundary-layer solver MSES is used to obtain transition locations through the e method, which are then used in Optima2D, a Newton–Krylov discrete-adjoint optimization algorithm based on the compressible Reynolds-averaged Navier– Stokes equations. The algorithm is applied to the design of airfoils with maximum lift-to-drag ratio, endurance factor, and lift coefficient. The design examples demonstrate that the optimizer is able to control the transitionpoint locations to provide optimum performance, often producing pressure distributions with laminar rooftops followed by concave pressure recovery. In particular, the optimization algorithm is able to design an airfoil that is very similar, in terms of both shape and performance, to one of the high-lift airfoils designed by Liebeck (Liebeck, R. H., “A Class of Airfoils Designed for High Lift in Incompressible Flow,” Journal of Aircraft, Vol. 10, No. 10, 1973, pp. 610–617) in the 1970s. The results provide a striking demonstration of the capability of the Newton– Krylov aerodynamic optimization algorithm to design airfoils with characteristics that previously required a great deal of expertise to achieve.