Best Practices for Adjoint-based Shape Optimization of Turbomachinery Stages

Abstract

In the pursuit of a new ultra-efficient generation of turbomachines, novel and more sophisticated flow simulation and optimization methods have to be adopted. With the development of harmonic balance (HB) methods and adjoint optimization, questions arise on the concrete benefits of using one method over another. This research investigates the difference in predictive capability and computational cost between unsteady harmonic balance and steady-state flow solving methods in particular. The research is performed using the SU2 software suite. The relevant set of scaling parameters is established based on a dimensional analysis. Variation of several of these nondimensional parameters - namely the flow and work coefficient, specific heat ratio gamma, isentropic pressure-volume exponent gamma-Pv and volumetric flow ratio - constitute the different setups. Given the excessive computational times associated with full optimizations, this large set of case studies is subjected to one flow evaluation. Trends and conclusions are made based on the resulting data. It is found that the accuracy difference between steady and HB solvers decreases slightly with an increase in flow coefficient as reduced frequency goes down, and increases significantly with the stage work coefficient as Mach effects increase. Varying working fluids under the ideal gas law is found not to impact the stage and solver performance given the volumetric flow ratio as a similarity parameter. In cases where gamma-Pv is significantly larger than gamma, a major increase in stage Mach numbers is observed. Consequently, the difference between results found by steady-state and harmonic balance solvers is considerably larger. Upon increasing the volumetric flow ratio over the stage, stage efficiencies have been found to increase and solver performance differences to decrease. A pair of complete optimizations then aims to bring additional insights and to verify the validity of the simulation results. The optimizations illustrate the effects of the behavior observed in flow simulations on a full optimization process and reiterate the difference in computational time between both methods. In both cases, the HB-optimized turbine stages are more efficient than the equivalent steady-optimized cascades, namely by 0.58% and 0.12%. The predictive capability of steady-state solvers in comparison with unsteady solvers is found to be dependent on reduced frequency, work coefficient and stage maximum Mach number. A new parameter consisting of those three quantities, the adjusted reduced frequency, forms a linear relationship with the deviation in steady versus unsteady results. It can thus be used as an indicative parameter in preliminary design to trade off accuracy and computational time and select the right solver for a given case.