Fluid and Structural Analysis of Pipes Under Water Hammer Effects

Abstract

The present study contributes to the analysis of pipes under water hammer by developing a hydraulic model, a structural model, and a fluid-pipe interaction model. Under the conventional hydraulic water hammer models based on the classical boundary conditions, the energy is dissipated along the pipe only through friction at the pipe-fluid interface. As a result, the predicted water hammer pressure waves are observed to dissipate in a pattern that differs from that observed in experimental studies. This is particularly the case for water hammer models with unsteady friction based on weighting-functions, in which the predicted pressure wave front keeps its original shape during propagation, as opposed to smoothen and widen as observed in experiments. In a bid to improve the predicted pressure wave front smoothing, the first contribution of the study postulates the presence of a water jet at the pipe inlet throughout water hammer, resulting in a newly proposed boundary expression. The boundary expression introduces an additional source of energy dissipation due to the reflection of the pressure wave at the pipe inlet. The boundary expression was applied in conjunction to three friction models in order to predict the pressure wave history. The model was calibrated based on published experimental data. The proposed boundary expression was shown to improve the predictive capability of the classical water hammer model in replicating of experimentally observed damping patterns in pressure history with respect to peak pressure values, wave front smoothing and phase shifting. While the classical water hammer model omits all inertial effects in the pipe wall, the extended water hammer model captures longitudinal inertial effects. Neither models, however, capture the bending stiffness of the pipe wall, thus predicting an unrealistic discontinuity in the radial displacement of the pipe wall in neighbourhood of the wave front. In order to remedy this limitation, the second contribution of the study develops a finite element formulation for the dynamic structural analysis of pipes subjected to general axisymmetric loading based on thin shell analysis. The validity of the formulation is demonstrated through comparisons with predictions of shell models based on the commercial software ABAQUS for static, natural vibration, and transient dynamic problems. The model is subsequently used to conduct a one-way coupled water hammer analysis, in which the transient pressure histories as predicted from the classical water hammer model is used as input into the structural model. The results suggest that the radial inertial effects in the pipe wall influence the predicted pipe response for fast valve closure scenarios, but the effect becomes negligible when the valve closure time is over eight times larger than the radial period of vibration. Water hammer models based on one-way coupling omit the fluid-pipe interaction effects. In order to capture such effects, the structural finite element model developed herein, was coupled to several hydraulic water hammer models, in a partitioned algorithm. The coupling was based on the Block Gauss-Seidel Algorithm, in which the hydraulic and structural models were iterated sequentially until convergence was attained within a specified tolerance before advancing to a new time step. A linear interpolation technique was adopted to exchange information between the non-matched fluid and structure interfaces. The number of iterations needed for convergence were accelerated by adopting either constant or dynamic relaxation factors. In order to assess the correctness of the implementation of the partitioned approach scheme, the classical and extended water hammer models were solved using the implemented Block Gauss-Seidel Algorithm and the results were compared to those based on the monolithic approach. The close agreement between both predictions demonstrated the validity of the Block Gauss-Seidel Algorithm implementation. The Algorithm was then extended to couple the structural shell finite element model developed herein with the hydraulic water hammer model. The effect of the specified convergence tolerance, the Courant number, the type of relaxation factor and fluid-structure interfaces, were investigated on the stability and computational efficiency for the partitioned models. The results indicate that the Aitken relaxation technique is recommended to accelerate the convergence rate of the two-way coupled analyses.