Three-Dimensional Unsteady Aerodynamic Optimization of a Transonic Compressor Stage Using an Implicit Discrete Adjoint Harmonic Balance Method

Abstract The design of high-loading and compact modern turbomachines results in strong unsteady interactions between two adjacent blade rows, which has been found to have a noticeable impact on aerodynamic performances of turbomachines. Therefore, it is necessary to consider unsteady effects arising from blade row interactions in multirow turbomachinery aerodynamic analyses and design optimizations. In this work, three-dimensional multirow unsteady aerodynamic design optimizations of turbomachinery blades are performed using a full-viscosity discrete adjoint solver developed by using the source code transformation automatic differentiation tool—Tapenade. An efficient time domain harmonic balance (HB) method with a complete rotor–stator interface coupling treatment is used to analyze multirow unsteady flow and adjoint fields. To stabilize the solution, the one-step Jacobi iteration combined with the lower–upper symmetric Gauss-Seidel (LU-SGS/one-step Jacobi) method is used for an implicit solution of the HB equation systems. For an efficient sensitivity evaluation, the effect of an adjoint solver’s root-mean-square (RMS) residual convergence levels on adjoint sensitivity accuracy is thoroughly studied to find an adjoint solver’s convergence criterion. The results from a single-stage transonic compressor-NASA Stage 35 reveal that when the adjoint RMS residual is reduced by three orders, accurate sensitivity information can be obtained, leading to a 41% reduction in computational cost compared with a fully converged one. The fully implicit LU-SGS/one-step Jacobi method can stabilize the solution of a multirow discrete adjoint solver while the solution of the semi-implicit LU-SGS equation system diverges quickly. Furthermore, compared with a steady one, the unsteady optimization can achieve more performance gains.