Solution, Simulation and Optimization of the Equations Governing the Water Hammer Problem in the Case of a Valve Closing

This study provides a rigorous numerical investigation into transient hydrodynamic regimes (water hammer) induced by slow valve closure configurations within a Reservoir-Pipe-Valve Supply System (RPVSS). Utilizing the Method of Characteristics (MOC) to solve the fundamental hyperbolic Saint-Venant equations, this research evaluates and optimizes the transient response of four widely implemented piping materials: steel, ductile cast iron, glass-reinforced plastic (GRP), and high-density polyethylene (HDPE), under both standardized and optimized non-linear boundary conditions. The multi-parametric optimization framework reveals that each material exhibits an intrinsic, periodic distribution of the optimal closure shape exponent (m), systematically confined between converging upper (m_max) and lower (m_min) envelope curves. Cross-analysis of the transient dynamics demonstrates that while rigid metallic conduits (cast iron and steel) suffer severe initial overpressures up to 28.5 bar, they achieve rapid wave attenuation; conversely, visco-elastic HDPE inherently cushions the initial pressure shock to 20.5 bar but demands a broader stabilization window (up to 105 s) due to its lower acoustic wave celerity. Furthermore, the overpressure profiles exhibit distinct non-monotonic localized bumps resulting from constructive acoustic wave interference, initiating at a threshold of 1.4t₄ and fading beyond 4t₄ (where t₄ represents the transient reflection period). The resulting parametric predictive models demonstrate that optimal closure profiles systematically converge toward a convex configuration (1.20 ≤ m ≤ 1.30). These analytical envelopes provide a powerful, high-fidelity engineering tool, enabling operators to determine optimal valve closing schedules that simultaneously mitigate extreme overpressures below the material’s nominal pressure capacity (PN) and prevent transient depressions from dropping beneath the liquid’s vaporization threshold, thereby effectively eliminating cavitation risks.