Finite volume method for simulating water hammer in pumping stations with free-surface flow and optimization of surge chamber

The water hammer simulation for pump station water supply systems with free-surface flow commonly employs the method of characteristics (MOC), often neglecting the influence of unsteady friction and the interaction between pressure and free-surface flow. In complex pipeline network systems, MOC often requires interpolation calculations or adjustments to wave speeds for computation. However, this can lead to issues such as numerical instability and reduced accuracy. This study establishes a second-order finite volume method model that considers unsteady friction for real-time joint calculations of pressure and free-surface flow. The second-order Godunov scheme is utilized to discretize the flow control equations independently for the pressure and free-surface flow. Riemann solvers are employed to compute the discretized fluxes. Simultaneously, the unsteady friction effect is taken into account to accurately simulate energy dissipation during hydraulic transient processes. The model is validated using experimental data. And a variable time-step approach is adopted at the interface between the pressure and free-surface flow. Furthermore, sensitivity analyses are conducted on the dimensions and locations of the throttled surge chamber in practical engineering scenarios. The results indicate that when unsteady friction effects are not considered, the model only matches experimental values at the first peak. However, when unsteady friction effects are taken into account, the model's simulation results are in good agreement with experimental results in terms of pressure peak, decay, and periodicity, and the maximum value of the water hammer pressure error at the valve is only 4.58%. Compared to the characteristics method, the second-order Godunov scheme exhibits higher stability and produces simulation results that are essentially the same when the Courant number is less than 1 as when the Courant number is set to 1.0. The real-time joint calculation method established in this paper can adequately consider the impact of water level fluctuations of the pressure flow outlet pool and the backflow of the free-surface flow. The water hammer pressure and the water levels of the throttled surge chamber obtained from the joint calculation of pressure and free-surface flow exhibit a faster decay compared to the results obtained from pure pressure flow calculations. For power outage conditions of pumps, the engineering scenarios studied in this paper would result in significant water hammer pressures and negative pressures if surge chambers were not installed. Optimization studies on the throttled surge chamber show that as the diameter of the connection tube increases, the maximum pressure head, the highest surge level of the throttled surge chamber, and the maximum reverse rotation speed of the pump increase, and the lowest surge level of the throttled surge chamber decreases. As the diameter of the throttled surge chamber increases, the highest surge level decreases, the lowest surge level increases and the maximum pressure decreases all the time when the connection tube diameter is relatively small, and will decrease first and then increase when the connection tube diameter is relatively large; when the connection tube diameter 3.5 m, throttled surge chamber diameter 15 m can meet the requirements of the system control parameters and also reduce the amount of engineering work. The closer the throttled surge chamber is to the pump station, the larger the maximum pressure, reverse rotation speed, and highest surge level, and the smaller the lowest surge level. When the throttled surge chamber is arranged 1.0 km behind the pump, the maximum surge level will reach 172.35 m, which does not meet the requirements of the control standards. In summary, the established model exhibits high accuracy, stability, and applicability, providing valuable insights for water hammer simulations in pump station water supply systems with free-surface flow.