On the handling of turbulence equations in RANS adjoint solvers

Recent developments in numerical design tools have made practical the use of gradient-based optimization using high-fidelity computational fluid dynamic simulations. Such has been made possible with the use of adjoint solvers, that can efficiently provide gradients of functions of interest with respect to design variables. However, in the presence of flows modeled by the Reynolds-Averaged Navier–Stokes (RANS) equations, the corresponding adjoint might become too complex to be fully derived or run. This has led to the use of many simplifications in the implementation of such adjoint solvers. In this paper, the constant eddy viscosity (CEV) approximation is explained and its validity tested. Two cases are used, a two-dimensional turbine vane blade and a three-dimensional transonic compressor rotor blade. The gradients computed using both the full RANS and the CEV approximation adjoints are verified against finite-differences. It is shown that the gradients differ slightly but when used in an optimization problem, the optimal solution found is nearly identical. Therefore, the CEV approximation in RANS adjoint solvers proved to be valid for engineering design problems, bringing significant advantages, such as faster implementation and less computational resources needed in terms of CPU and memory size, when compared to the full RANS adjoint solver.