Continuous Unsteady Adjoint Variational Multiscale Method for airfoils applications

Airfoil optimization using the adjoint-based method has been widely used thanks to its efficiency in computing sensitivities. In this research, the finite element method based on the Variational MultiScale (VMS) method is used to solve the flow and adjoint equations. The derivation of the VMS terms for the resolution of the continuous adjoint problem is shown. The method has been implemented for airfoil design with the aim of improving time-averaged aerodynamic performance. In the implementation Class Shape Transformation (CST) is used to parameterize the control surface to ensure that the obtained geometries are always airfoil-like. In the test case, from a NACA0012 at Re = 1000, AoA = 2.5°, CL = 0.35 is targeted and achieved. The effect of the time window for time averaging has been investigated, which shows a crucial role for the convergence of the algorithm.