Geometric optimization of a vertical-shaft spiral spillway tunnel based on a Kriging surrogate model

Abstract

Vertical-shaft spiral spillways are widely used in high-head reservoirs because of their compact configuration and high energy-dissipation efficiency. The complex three-dimensional swirling flow makes it difficult for the structure to be optimized using empirical design methods. To address this problem, a numerical multi-objective optimization framework was developed, which integrates computational fluid dynamics (CFD), a Kriging surrogate model, and a multi-objective genetic algorithm (MOGA). The contraction ratio (n), the ratio of the stilling-well depth to the shaft-section diameter (H / d), and the ratio of the pressure-slope-section length to the shaft-section diameter (L / d) were defined as the design variables, while the energy-dissipation rate (η) and the velocity-uniformity coefficient (C_V) served as optimization objectives. A Kriging surrogate model was constructed and validated using CFD simulation results, with determination coefficients (R²) exceeding 0.94 and Root Mean Square Errors (RMSE) below 0.40. The MOGA was applied to obtain a Pareto-optimal solution set reflecting the trade-offs between η and C_V. The optimal configuration was characterized by an increased H / d of 1.03, an extended L / d of about 4.13, and a reduced n of 14.24. Compared with the original design, η increased by 9.97% and C_V decreased by 64.77%, indicating significant improvements in energy-dissipation efficiency and flow uniformity. The proposed framework provides a reliable and efficient tool for the hydraulic optimization of vertical-shaft spiral spillway tunnels.