Multi-objective optimization of S-shaped hydrofoil and bidirectional axial-flow pumps based on NSGA-II

Bidirectional axial-flow pumps can be used to fully meet the urban demand for drought response and two-way water transfer in recent years. This study aims to optimize the high-specific-speed bidirectional pumps with a specific speed (forward operation) of around 1 200, in order to improve the hydraulic efficiency. The numerical calculations were also validated using NSGA-II (Non-dominated sorting genetic -II). A reversible hydrofoil (S-shaped hydrofoil) near the middle span of the impeller was selected and then optimized for various attack angles (+2°, +4°, and +6°). A series of optimizations were implemented to maximize the weighted cavitation and lift-to-drag ratio of the hydrofoil while maintaining the lift coefficient at the original hydrofoil angle under the constraint of an unchanged bidirectional pump head. The maximum thickness and position of the S-shaped hydrofoil remained constant during this time. The camber line shape was described using a fourth-order Bezier curve, while the distribution of thickness was represented by combining the arcs and fourth-order Bezier curves. 12 parameters were optimized to consider the camber line and thickness distribution. A total of 1 680 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 indicated a trade-off relationship between the lift-drag and cavitation performance of the hydrofoil. Among the 115 Pareto solutions, the maximum lift-to-drag ratio and cavitation performance were approximately 27.7% and 1.7%, respectively. The optimal Sc1 for lift-to-drag ratio, the optimal Sc3 for cavitation performance, and the compromise solution Sc2 were selected to further compare their shape and pressure coefficient distribution. The lift coefficients of three hydrofoil schemes before and after optimization were basically consistent at different attack angles. The minimum pressure coefficient on the hydrofoil surface gradually increased from Sc1 to Sc3. Compared with the original scheme, the maximum camber of 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 the reducing maximum camber was beneficial for the lift-to-drag ratio, while the increasing camber for the cavitation performance. The Sc2 scheme was selected as the preferred model for the impeller of the bidirectional pump. The impeller performance was compared before and after optimization. The results showed that the optimal head of the bidirectional pump remained constant, while the efficiency was 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, respectively, while the reverse efficiency increased by approximately 1.1, 0.2, and 0.4 percentage points, respectively, the internal flow field of the impeller was significantly improved, under forward and reverse design flow conditions, the critical cavitation number of the pump changed relatively little (increased by 0.013 and 0.006 respectively), while the initial cavitation number decreased significantly (both decreased by about 0.5). The optimal impeller exhibited the lower shaft power within the flow range. The flow separation was suppressed significantly at the leading edge of the blade and near the trailing edge of the hub, resulting in a more uniform distribution of velocity within the blade passage and significant improvement in the internal flow field of the impeller. Additionally, the critical cavitation number of the pump changed slightly, while the incipient cavitation number decreased significantly. Numerical simulation and NSGA-II algorithm were combined to realize the multi-objective optimization on S-shaped hydrofoils of bidirectional pumps, in terms of the lift-to-drag ratio and cavitation performance. The findings can also provide a strong reference to optimize the reversible rotating machinery.