Weakly nonlinear theory of water hammer induced by slow valve manoeuvres

We develop a weakly nonlinear theory to revisit the water hammer phenomenon resulting from slow valve manoeuvres. The hydraulic head at the valve is known to be nonlinearly coupled with the flow velocity via a relation derived from Bernoulli's principle, so that solutions have been so far obtained only via numerical models. The governing equations and boundary conditions indeed yield a nonlinear boundary-value problem, which is here solved using a perturbation approach, Laplace transform and complex analysis. We obtain space- and time-dependent analytical solutions in all of the pipe and validate our results by comparison with standard numerical methods (i.e. Allievi's method) for determining the exact behaviour of the hydraulic head at the valve. Additionally, we derive algebraic practically relevant closed form expressions for predicting the maximum and minimum hydraulic head values during both valve closure and opening manoeuvres.