Automatic Differentiation Adjoint of the Reynolds-Averaged Navier-Stokes Equations with a Turbulence Model

Abstract

This paper presents an approach for the rapid implementation of an adjoint solver for the ReynoldsAveraged Navier–Stokes equations with a Spalart–Allmaras turbulence model. Automatic differentiation is used to construct the partial derivatives required in the adjoint formulation. The resulting adjoint implementation is computationally efficient and highly accurate. The assembly of each partial derivative in the adjoint formulation is discussed. In addition, a coloring acceleration technique is presented to improve the adjoint efficiency. The RANS adjoint is verified with complex-step method using a flow over a bump case. The RANS-based aerodynamic shape optimization of an ONERA M6 wing is also presented to demonstrate the aerodynamic shape optimization capability. The drag coefficient is reduced by 19% when subject to a lift coefficient constraint. The results are compared with Euler-based aerodynamic shape optimization and previous work. Finally, the effects of the frozenturbulence assumption on the accuracy and computational cost are assessed.