Aerodynamic Shape Optimization for Natural Laminar Flow Using a Discrete-Adjoint Approach

A framework for the design of natural-laminar-flow airfoils is developed based on multipoint aerodynamic shape optimization capable of efficiently incorporating and exploiting laminar-turbulent transition. A two-dimensional Reynolds-averaged Navier-Stokes flow solver making use of the Spalart-Allmaras turbulence model is extended to incorporate an iterative laminar-turbulent transition prediction methodology. The natural transition locations due to Tollmien-Schlichting instabilities are predicted using a simplified e(N) method or the compressible form of the Arnal-Habiballah-Delcourt criterion. The Reynolds-averaged Navier-Stokes solver is subsequently used in a gradient-based sequential quadratic programming shape optimization framework. The transition criteria are tightly coupled into the objective and gradient evaluations. The gradients are obtained using an augmented discrete-adjoint formulation for nonlocal transition criteria. Robust design over a range of cruise flight conditions is demonstrated through multipoint optimization. Finally, a technique is proposed and demonstrated to enable the design of natural-laminar-flow airfoils with robust performance over a range of critical N factors: the optimizer is seen to produce transition ramps similar to those used by experienced designers.