Adjoint methods are widely used for turbomachinery aerodynamic shape optimization. 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. This work attempts to alleviate such problems by using the Newton–Krylov method to solve both the flow and adjoint equations. 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. Consequently, the development of an adjoint solver is significantly simplified, as reverse differentiation is not needed. 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. The compressor maps are first computed and adjoint solutions for both design and typical off-design conditions are calculated. Design sensitivities are computed using the adjoint approach and verified against finite differences. Compared with a representative implicit scheme, the Newton–Krylov approach allows the flow, adjoint, and sensitivities to be stably computed over a wider operating range, which facilitates whole-map adjoint aerodynamic shape optimization for turbomachinery components.

伴随方法广泛用于涡轮机械空气动力学形状优化。然而，对于工业应用，伴随求解器在包络边缘条件下的鲁棒性和效率的下降仍然对工业中成功部署伴随方法构成挑战。这项工作试图通过使用 Newton-Krylov 方法来求解流动方程和伴随方程来缓解此类问题。开发的并行伴随求解器重用了由流求解器计算的雅可比矩阵，并通过累积并行通信获得伴随矩阵-向量乘积。因此，由于不需要反向微分，因此显着简化了伴随求解器的开发。将已经验证过的 Newton-Krylov 流求解器与本工作中开发的伴随求解器相结合，对于具有代表性的涡轮机械案例，包括轴流式和离心式压缩机，证明了稳健且有效的残差收敛。首先计算压缩机图，并计算设计和典型非设计条件的伴随解。使用伴随方法计算设计灵敏度并针对有限差异进行验证。与具有代表性的隐式方案相比，Newton-Krylov 方法允许在更宽的操作范围内稳定计算流量、伴随和灵敏度，这有助于涡轮机械部件的全图伴随气动形状优化。首先计算压缩机图，并计算设计和典型非设计条件的伴随解。使用伴随方法计算设计灵敏度并针对有限差异进行验证。与具有代表性的隐式方案相比，Newton-Krylov 方法允许在更宽的操作范围内稳定计算流量、伴随和灵敏度，这有助于涡轮机械部件的全图伴随气动形状优化。首先计算压缩机图，并计算设计和典型非设计条件的伴随解。使用伴随方法计算设计灵敏度并针对有限差异进行验证。与具有代表性的隐式方案相比，Newton-Krylov 方法允许在更宽的操作范围内稳定计算流量、伴随和灵敏度，这有助于涡轮机械部件的全图伴随气动形状优化。