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In this paper, we propose an extension of the Harmonic Balance method for threedimensional, unsteady, multi-stage turbomachinery problems modeled by the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations. This time-domain algorithm simulates the true geometry of the turbomachine (with the exact blade counts) using only one blade passage per blade row, thus leading to drastic savings in both CPU and memory requirements. Modified periodic boundary conditions are applied on the upper and lower boundaries of the single passage in order to account for the lack of a common periodic interval for each blade row. The solution algorithm allows each blade row to resolve a specified set of frequencies in order to obtain the desired computation accuracy; typically, a blade row resolves only the blade passing frequencies of its neighbors. Since every blade row is setup to resolve different frequencies the actual Harmonic Balance solution in each of these blade rows is obtained at different instances in time or time levels. The interaction between blade rows occurs through sliding mesh interfaces in physical time. Space and time interpolation are carried out at these interfaces and can, if not properly treated, introduce aliasing errors that can lead to instabilities. With appropriate resolution of the time interpolation, all instabilities are eliminated. This new procedure is demonstrated using both two-and three dimensional test cases and can be shown to significantly reduce the cost of multi-stage simulations while capturing the dominant unsteadiness in the problem.

The authors would like to thank the many stakeholders with whom we have engaged to elicit feedback on the opportunities and challenges associated with developing and deploying standard modular hydropower technologies. Ongoing engagement with these technology innovators, commercial service/equipment providers, project developers, and environmental stewards has yielded insight into the myriad perspectives and experiences that shape the current hydropower landscape and inform standard modular hydropower research.