Aerodynamic Shape Optimization of a Turbine Blade Considering Geometric Uncertainty Using an Adjoint Method

Abstract

Abstract Aerodynamic shape optimization of turbomachinery blades taking the performance impact of geometric uncertainty into account is presented in the paper. A continuous adjoint method for the steady-state Euler equations is used to calculate the first order (FO) sensitivities of performance functions to the design parameters. For each performance function, all the sensitivities can be obtained by about two equivalent flow computations, regardless of the number of design parameters. The performance changes due to the geometric variations are evaluated on use of the second order (SO) sensitivities. A direct-ad joint method is used to calculate the FO and SO sensitivities of performance functions to the basis modes of geometric variations, which are extracted from a covariance function using principal component analysis method. Firstly, the sensitivities of the mass-averaged adiabatic efficiency and mass flow rate to the basis modes of geometric variations for a three-dimensional turbine blade are calculated. The validations of sensitivity-based performance impact are then illustrated. For the optimization maximizing the adiabatic efficiency with mass flow rate constrained, a series of shape functions are imposed on the turbine blade to perturb the aerodynamic shape. For each deformed blade with respect to each design parameter, the statistic mean values of performance changes are evaluated by using the method of moment and then contribute to the calculation of the corresponding FO sensitivity of performance functions. The design optimization without and with geometric uncertainty are finally presented. The results are compared in detail to demonstrate the effects of geometric variations to the optimization, especially to the statistical performance of the optimized blades. Through design optimization, the shock loss can be significantly reduced, compared with the original blade. The optimized blade is more robust in respect of geometric uncertainty.