The continuous adjoint method to the γ − R ˜ e θ t $$ \gamma -\tilde{R}{e}_{\theta t} $$ transition model coupled with the Spalart–Allmaras model for compressible flows

The continuous adjoint method for transitional flows of compressible fluids is developed and assessed, for the first time in the literature. The gradient of aerodynamic objective functions (aerodynamic forces) with respect to design variables, in problems governed by the compressible Navier–Stokes equations coupled with the Spalart–Allmaras turbulence model and the $\gamma -\tilde{R}{e}_{\theta t}$ transition model (in three, non-smooth and smooth, variants of it), is computed based on the continuous adjoint method. The development of the adjoint to the smooth transition model variant proved to be beneficial. The accuracy of the computed sensitivity derivatives is verified against finite differences. Programming is performed in an in-house, vertex-centered finite-volume code, efficiently running on GPUs. The proposed continuous adjoint method is used in 2D and 3D aerodynamic shape optimization problems, namely the constrained optimization of the NLF(1)–0416 isolated airfoil and that of the ONERA M6 wing. The impact of “frozen transition” (assumption according to which the adjoint to the transition model equations are not solved) or “frozen turbulence” (by additionally ignoring the adjoint to the turbulence model) are evaluated; it is shown that both lead to inaccurate sensitivities.