**8. Conclusions**

In the current work, a numerical study of an underwater piezoelectric energy harvester has been carried out for the extraction of kinetic energy from the water flow. The mechanism, which is planned to be used inside water pipelines of 2–5 inches of diameter, contains a piezoelectric beam assembled to an oscillating body. Two different geometries for the oscillating body have been considered. The first one is the traditional circular cylinder, and the second one is a novel U-shaped geometry. Both geometries have been studied for two different diameters: *D* = 10 and 20 mm. Thus, 2D numerical simulations have been performed around each proposed geometry at Reynolds numbers Re = 3000, 6000, 9000, and 12,000. Simulations in unsteady-state conditions were made during a period of time of 20 seconds in order to evaluate the vortex shedding generated in the region behind the oscillating bodies. The lift coefficient of the oscillating bodies obtained in the simulations has been used as an input variable in the control system.

Furthermore, a multivariable JADE-based optimization algorithm has been designed to optimize the design process of the harvester and maximize the power extracted from it. The two parameters optimized with the JADE algorithm are the structural spring of the harvester and the constant gain associated to its control algorithm. According to the obtained results, the power generated by the U-shape-based energy harvester is always larger than the one obtained by the circular cylinder for all the Reynolds numbers studied except for Re = 3000 and *D* = 10 mm. The maximum power extracted from the harvester is 5321.7 μW and corresponds to the case with Re = 12,000 and *D* = 10 mm. The results show that thanks to the U-shaped geometry of the oscillating body and to the JADE optimization algorithm, the power output of the harvester has significantly improved.

The power generated by the underwater piezoelectric energy harvester follows an exponential law for all the cases investigated, including the U-shaped geometry. Additionally, the proportional gain of the control law maintains approximately constant at the water speeds studied in the current work.

**Author Contributions:** I.A. and U.F.-G. conceived and performed the CFD simulations; A.S.-A. and E.Z. developed the new control algorithm, and A.B. analyzed the results and provided constructive instructions in the process of preparing the paper.

**Funding:** This research was funded by the Government of the Basque Country SAIOTEK (S-PE11UN112) and the University of the Basque Country UPV/EHU EHU12/26 research programs.

**Acknowledgments:** The authors are grateful to the Government of the Basque Country and the University of the Basque Country UPV/EHU through the SAIOTEK (S-PE11UN112) and EHU12/26 research programs, respectively. The funding of Vital Fundazioa is also acknowledged. Development Agency of the Basque Country (SPRI) is gratefully acknowledged for economic support through the research project "Refrigeración de dispositivos de alto flujo térmico mediante impacto de chorro" (AIRJET), KK-2018/00109, Programa ELKARTEK.

**Conflicts of Interest:** The authors declare no conflict of interest.
