Design sensitivities are computed using the adjoint approach and verified against finite differences. The compressor maps are first computed and adjoint solutions for both design and typical off-design conditions are calculated. Combining an already validated Newton–Krylov flow solver with the adjoint solver developed in this work, robust and efficient residual convergence is demonstrated for representative turbomachinery cases, including an axial and a centrifugal compressor. Consequently, the development of an adjoint solver is significantly simplified, as reverse differentiation is not needed. The developed parallel adjoint solver reuses the Jacobian matrix computed by the flow solver and obtains the adjoint matrix–vector product via an accumulative parallel communication. This work attempts to alleviate such problems by using the Newton–Krylov method to solve both the flow and adjoint equations. However, for industrial applications, the degradation of robustness and efficiency of adjoint solvers for edge-of-the-envelope conditions still poses a challenge to the successful deployment of adjoint methods in the industry. Adjoint methods are widely used for turbomachinery aerodynamic shape optimization.
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