A discrete adjoint harmonic balance method for turbomachinery shape optimization

The harmonic balance method has seen an increasing popularity in the solution of time-periodic problems because of its computational efficiency and its ability to model dynamically nonlinear fluid phenomena. In addition, the mathematically steady nature of this technique makes it ideal for adjoint sensitivity analysis of unsteady problems. In this work, a novel optimization framework consisting of three components: a harmonic balance based unsteady cascade flow solver, an accompanying adjoint solver and a quasi-Newton optimization solver; have been developed. The discrete adjoint solver is obtained with the aid of an automatic differentiation tool, TAPENADE. To demonstrate the efficiency and accuracy of the method, we present shape optimization and adjoint sensitivity computations for a two-dimensional compressor cascade. Steady inverse design of this cascade is performed to investigate the effects of two shape parameterization methods, namely Hicks-Henne bump functions and mesh points. Shape optimization is performed to improve the aerodynamic damping characteristics of a vibrating cascade row. In addition, the unsteady adjoint technique is used to determine the frequency of vibration that would drive the system to limit-cycle, which defines the stability limit of the cascade.