Continuous Adjoint Approach for the Spalart–Allmaras Model in Aerodynamic Optimization

In this paper, the continuous adjointmethod to compute shape sensitivities in aerodynamic designwith turbulence modeling is described and developed. The focus is on compressible flows described by the Reynolds-averaged Navier–Stokes equations and the classical Spalart–Allmaras model for turbulence. Turbulence modeling usually requires, in particular, computation of the distance to the surface. Here, this distance is incorporated to the system as a new variable, solving the Eikonal equation. The accuracy of the sensitivity derivatives obtained with the complete turbulent approach is assessed by comparison with finite difference computations and the classical continuous adjoint with frozen viscosity, showing substantial improvements in the convergence properties of the method and in the quality of the obtained gradients. The validity of the overall methodology is illustrated with several design examples, including the optimization of three-dimensional geometries in combination with advanced freeform techniques for mesh deformation.