Adjoint-based unsteady optimization of turbomachinery operating with nonideal compressible flows

The lack of established optimal design guidelines for turbomachinery operating in the nonideal flow regime (e.g., organic Rankine cycle turbines, CO₂ compressors, compressors for refrigeration systems) demands for effective and efficient automated design methods. Past research work focused on gradient-free methods applied to computational fluid-dynamic simulations. The application of the adjoint method is a cost-effective alternative as it enables gradient-based optimization irrespective of the number of design variables. This paper presents the application of a fully turbulent unsteady adjoint method for the automated design of multirow turbomachinery partly operating in the nonideal flow regime. The method therefore allows for the solution of constrained unsteady fluid-dynamic optimization problems, in which the thermodynamic properties of the working fluid need to be modeled by means of complex equations of state. The optimal designs computed with unsteady simulations obtained with the harmonic balance method are then compared with optimal design resulting from mixing-plane simulations. The method is applied to the optimization of 1) a two-dimensional turbine cascade subject to time-varying inlet conditions, and 2) a two-dimensional turbine stage of an organic Rankine cycle power system. The results demonstrate the importance of computing fluid properties using accurate thermodynamic models and of using unsteady simulations for shape optimization of these machines.