Multi-objective Optimization of S-shaped Hydrofoil and Bidirectional Axial-flow Pump Based on NSGA-II

To meet the urban demand for drought response and two-way water transfer, as well as improve the hydraulic efficiency of bidirectional axial-flow pumps, this study focuses on optimizing the design of high-specific-speed bidirectional pumps with a specific speed (forward operation) of around 1200. Based on the validation of the accuracy of numerical calculations, a reversible hydrofoil (S-shaped hydrofoil) near the middle span of the impeller was selected and optimized using NSGA-II (Non-dominated sorting genetic algorithm-II) for various attack angles (+2°, +4°, and +6°). The optimization aims to maximize the weighted cavitation performance and lift-to-drag ratio of the hydrofoil, while maintaining the lift coefficient at the original hydrofoil design angle under the constraint of unchanged bidirectional pump head. The maximum thickness and its position of the S-shaped hydrofoil were kept constant, and the camber line shape was described using a fourth-order Bezier curve, while the thickness distribution was described by a combination of arcs and fourth-order Bezier curves. Through optimization of 12 design parameters related to the camber line and thickness distribution, a total of 1680 iterations were performed using NSGA-II, resulting in a Pareto solution set for cavitation performance and lift-to-drag ratio. The emergence of the Pareto frontier indicates a trade-off relationship between the lift-drag performance and cavitation performance of the hydrofoil. Among the 115 Pareto solutions obtained, the maximum improvement in the lift-to-drag ratio was approximately 27.7%, and the maximum improvement in cavitation performance was approximately 1.7%. The optimal Sc1 for lift-to-drag ratio, the optimal Sc3 for cavitation performance, and the compromise solution Sc2 were selected for further comparison of their shape and pressure coefficient distribution. The lift coefficients of the three hydrofoil schemes before and after optimization were basically consistent at different attack angles, and the minimum pressure coefficient on the hydrofoil surface gradually increased from Sc1 to Sc3. Compared with the original Ori scheme, the maximum camber of the Sc1 and Sc2 hydrofoils decreased, resulting in a more uniform distribution of pressure coefficients on the hydrofoil surface. However, the maximum camber of the Sc3 hydrofoil increased, indicating that reducing the maximum camber within a certain range is beneficial for improving the lift-to-drag ratio, while increasing the camber is beneficial for improving cavitation performance. The Sc2 scheme was selected as the preferred model for the impeller of the bidirectional pump, and the impeller performance before and after optimization was compared. The results showed that the head of the optimized bidirectional pump remained basically unchanged, while the efficiency improved significantly. At 0.8, 1.0, and 1.2 times the forward and reverse design flow rates, the forward efficiency increased by approximately 0.6, 0.5, and 2.2 percentage points, and the reverse efficiency increased by approximately 1.1, 0.2, and 0.4 percentage points. The optimized impeller exhibited lower shaft power within the calculated flow range. The flow separation at the leading edge of the blade and near the trailing edge of the hub was significantly suppressed, resulting in a more uniform velocity distribution within the blade passage and significant improvement in the internal flow field of the impeller. Additionally, under both forward and reverse design flow conditions, the pump's critical cavitation number undergoes relatively minor changes (increasing by 0.013 and 0.006 respectively), whereas the incipient cavitation number decreases significantly (decreasing by approximately 0.5 in both cases). Based on the combination of numerical simulation and NSGA-II algorithm, this study proposes a multi-objective optimization design method for S-shaped hydrofoils of bidirectional pumps that considers both lift-to-drag ratio and cavitation performance. The results can provide a reference for the optimization design of reversible rotating machinery.