Examining the water hammer number

This paper presents a dimensionless analysis of the water hammer phenomenon, focusing on the derived continuity and motion equations in their dimensionless form. The study establishes the water hammer number ( Wh ) as a critical dimensionless criterion for ensuring consistency between laminar flows in similar piping systems. Through both numerical (method of characteristics) and analytical solutions (Rich model with quasi-steady friction; Mei-Jing and Muto-Takahashi models with unsteady friction), the significance of Wh is demonstrated, alongside its relationship with other dimensionless numbers such as the Ghidaoui, Womersley, and Strouhal numbers. The paper identifies key thresholds for Wh , including critical values for maximal "line packing" ( Wh ≈ 0.02), cavitation-free reservoir-pipe-valve systems ( Wh ≥ 0.072), and overdamped water hammer flows ( Wh ≥ 0.44). Additionally, the study highlights the unidirectional nature of the dispersion process, with increasing water hammer periods as Wh increases, particularly for larger values ( Wh ≥ 0.01). These findings emphasize the importance of the Reynolds number ( Re ) on water hammer dynamics, particularly for high Re values ( Re ≥ 1·10 5 ), where the effects of turbulence and unsteady shear stress become less significant. This research underscores the need for dimensionless analyses in water hammer studies and calls for future experimental investigations that account for varying Wh values to avoid redundant experimental practices. The present analysis is limited by the assumption of laminar flow, the idealization of instantaneous valve closure, and the neglect of cavitation, thereby precluding pressures from approaching saturated vapor pressure.