High-speed photography technology is adopted in this paper to systematically investigate the entire process of water jet breakup into droplets in the air after ejection from the water splitter tray of low-pressure rotary sprinklers under varying operating pressures and nozzle diameters. The experimental results revealed significant dependencies of jet breakup length, characteristic jet velocity, droplet characteristics, including their size and velocity, on the operating pressure and nozzle diameter. There are three types of spray water jet breakup, which are the split jet breakup, column jet breakup, and liquid-film jet breakup. A statistical linear regression model links the jet breakup length to the working pressure and nozzle diameter, and the breakup length of the water jet increases with decreasing working pressure or increasing nozzle diameter. Meanwhile, an increase in working pressure or nozzle diameter leads to a high flow rate, which in turn results in a power-law growth of the characteristic jet velocity. A quantitative nonlinear model of the breakup droplet velocity concerning its size and operating parameters is proposed through parametric analyses, and it is revealed that the breakup droplet velocity increases in a power-law manner with increasing pressure or nozzle diameter, and exhibits a logarithmic increase with increasing droplet size. This paper quantifies the relationships between jet breakup dynamics and operating parameters through experimental investigations and analytical modeling, which offers a valuable reference and experimental basis for performance-driven optimized design of industrial spray systems.

Pumping stations serve as the foundation platform for large-scale vertical fluid machinery, and their structural dynamics directly govern the vibration levels and long-term reliability of the installed pump units. In low-head vertical pumping stations, the interaction among the massive underwater substructure, flexible above-ground powerhouse, and surrounding backfill soil creates a complex dynamic system whose behavior remains insufficiently characterized. This study presents a comprehensive dynamic analysis of a large-scale vertical pumping station using a high-fidelity three-dimensional finite element model that incorporates the powerhouse superstructure, submerged concrete substructure, and backfill soil. Modal analysis under four boundary condition scenarios—varying in soil participation and interface contact conditions—systematically quantifies the influence of soil–structure interaction on natural frequencies and mode shapes. Resonance verification against three primary excitation sources—rotational frequency (4.917 Hz), blade passage frequency (24.583 Hz), and rotor–stator interaction frequency (196.667 Hz)—is extended from the first 50 modes to the 400th mode to assess potential high-order resonance risks. Results show that the roof slab, with its large span and low stiffness, exhibits the highest vibration susceptibility. For the rotational frequency, modes 4–12 fall below the 20% code-specified safety margin but rapidly exceed the threshold thereafter. For the blade passage frequency, the separation ratio decreases progressively with increasing mode order within the first 50 modes, and the extended analysis up to the 400th mode shows that the separation ratio remains well above 20% throughout modes 51–400. Consequently, no substantial resonance risk exists for the blade passage frequency within the entire computed range. The rotor–stator interaction frequency remains safely separated with margins exceeding 95%. These findings demonstrate the profound influence of soil–structure interaction and confirm that, despite a decreasing trend in frequency separation at higher orders, the blade passage frequency poses no substantial resonance risk up to the 400th mode. This work provides a rigorous analytical framework for vibration-informed design and optimization of pump foundation systems, with direct implications for the reliability and operational safety of large-scale vertical fluid machinery.

A three-dimensional droplet motion model considering the rotational effect of the sprinkler tray was established in a fixed reference frame for low-pressure rotary sprinklers to reveal the effects of tray rotation of a rotary sprinkler on the flight trajectories of water droplets and spray uniformity. Conventional ballistic models for fixed sprinklers only incorporate radial and vertical ejection velocities and neglect the circumferential velocity generated by tray rotation, which cannot describe the circumferential deflection behavior of water droplets. The circumferential velocity induced by tray rotation was introduced as the initial boundary condition of the droplets, and an initial velocity model of the droplets leaving the tray, including radial, circumferential and vertical components, was constructed. Meanwhile, influences of air drag, gravity, buoyancy, ambient wind speed and evaporation on the droplet flight process were comprehensively considered. The fitting performance of Rosin–Rammler, Upper-Limit Log-Normal and Gamma distributions for droplet size was evaluated based on laser disdrometer measured data, showing that the Gamma distribution performed best and can serve as the droplet size input for trajectory and spray uniformity calculations. With droplet sizes and initial velocities defined, the Runge–Kutta fourth-order (RK4) method was adopted to solve three-dimensional trajectories, obtaining droplet distributions and uniformity coefficient through landing-point interpolation and grid statistics. Model validation results show that the simulated ranges of single water droplets agree well with the measured values, with relative errors all controlled within 6%. Further comparison of the spray characteristics between fixed and rotary sprinklers reveals that the circumferential initial velocity imparted to droplets by the rotational effect can significantly alter their spatial flight trajectories and landing point distributions, enabling droplets to complement each other in the circumferential direction and thereby enhancing the continuity of spray coverage. The single-sprinkler application uniformity coefficient was increased from 54.6% in the fixed state to 76.9% in the rotating state under the identical operating conditions. The findings can provide a theoretical basis and computational tools for analyzing the spraying mechanism, structural optimization and application uniformity improvement of low-pressure rotary sprinklers. Experiments in this study had no external wind speed criteria, but model development includes wind speed if relevant.