The continuous adjoint to the incompressible (D)DES Spalart-Allmaras turbulence models

This article formulates the continuous adjoint method for the gradient-based shape optimization of fluid flows governed by the incompressible Detached Eddy Simulation (DES) and Delayed-DES (DDES) models, based on the Spalart-Allmaras turbulence model. As both flow models are inherently unsteady, challenges arise regarding the availability of flow fields during the backward in time integration of the unsteady adjoint equations. To minimize both the computational cost and the memory demands, the computed flow fields are compressed using the iPGDZ${}^{+}$ lossy compression technique, recently developed by the authors. Its application in the context of turbulence-resolving flows, where the compression of flow fields poses an intricate challenge, is a second original contribution of this article. Everything is implemented as an extension to the publicly available adjointOptimisation library of OpenFOAM, which is used to solve the flow and adjoint equations and conduct the optimization. Using two shape optimization problems in external aerodynamics, it is demonstrated that including the adjoint to the turbulence model equation is crucial for the computation of accurate sensitivity derivatives. In contrast to sensitivities computed under the “frozen turbulence” assumption, which neglects variations in turbulent viscosity due to changes in the design variables, the proposed adjoint method yields sensitivities that align with those obtained using Finite Differences. This is due to the Think-Discrete Do-Continuous adjoint method which, inspired by hand-differentiated discrete adjoint, gives rise to consistent discretization schemes of the terms involved in the equations derived by continuous adjoint. Furthermore, it is demonstrated that the proposed adjoint method can significantly benefit from the iPGDZ${}^{+}$ algorithm, by reducing memory requirements by more than two orders of magnitude, eliminating the need for flow recomputations, while maintaining the accuracy of the computed derivatives. Ways to handle large integration windows of the objective function with this type of flow models are beyond the scope of this article.