Adjoint-Based Sensitivity Analysis for Unsteady Bladerow Interaction Using Space–Time Gradient Method

Temporal variation of components' performance is becoming a crucial parameter in turbomachinery design process. The main physical mechanism driving the time-dependent behavior is the unsteady bladerow interaction as stator–rotor relative motion due to rotating frame of reference. However, so far unsteady effects have been ignored in design processes in common engineering practice. In fact, steady approach has been generally employed for computational fluid dynamics (CFD)-based turbomachinery design. Moreover, conventional blade design has been based on single operating point considerations. Taking into account multiple time-dependent phenomena, as the unsteady performance parameters variation, might be beneficial in making a further improvement on component performance. In quantitative terms, first of all it is important to investigate the relative effect of unsteady variation, compared to the standard steady approach, and to create a capability for calculating temporal sensitivity variation, while keeping a reasonable computing cost. This work investigates the unsteady variation of turbomachinery performance on quasi-three-dimensional (3D) geometries: single-stage turbine and single-stage compressor. Steady flow solutions using mixing plane method are compared to the unsteady flow solutions using a direct unsteady calculation, while assessing the introduction of the space–time gradient (STG) method. The results clearly show how the unsteady variation is a non-negligible effect in performance prediction and blade design. Then, a new computational technique to quantify temporal sensitivity variation is introduced, based on the STG method, with an extension to adjoint-based sensitivity analysis. The relation between time and space in multipassage-multirow domain, the fundamental assumption of the STG method, is applied within the adjoint operator formulation, which gives unsteady sensitivity information on a broad range of design parameters, at the cost of a single computation. Finally, the unsteady sensitivities are compared to the ones resulting from steady solution in the two quasi-3D cases. This work introduces a coherent and effective mathematical formulation for accounting deterministic unsteadiness on component design, while reducing computational cost compared to standard unsteady optimization techniques.