Unsteady Continuous Adjoint Approach for Aerodynamic Design on Dynamic Meshes

This paper presents the development and application of a new unsteady continuous adjoint formulation for optimal shape design of aerodynamic surfaces in motion, such as rotating or pitching applications. The arbitrary Lagrangian–Eulerian form of the unsteady, compressible Reynolds-averaged Navier–Stokes equations with a generic source term is considered, and from these governing flow equations, an adjoint formulation centered around finding surface sensitivities using shape calculus is derived. This surface formulation provides the gradient information necessary for performing gradient-based aerodynamic shape optimization at a computational cost equivalent to solving the flow equations and with minimal memory overhead. To verify the methodology, gradients provided by the continuous adjoint and finite-differencing approaches are compared. Optimal shape design is demonstrated in both two and three dimensions for pitching airfoil and wing test cases.