Modelling of unsteady pipe flow with "liquid column separation" (water hammer induced transient cavitation)

Abstract

Understanding of waterhammer pressure transients in liquid-filled pipelines and their computational (numerical) modelling as unsteady one-dimensional flow is well established in the literature and engineering practice. However, for the particular issue of the potential for low transient pressure to initiate a change of phase or release of dissolved gases, leading to the phenomenon of localised transient cavitation known as liquid column separation, there is not yet a consensus on the reliability of the various numerical models that have been proposed in the literature. To contribute to further progress on this, therefore, this present work builds primarily on two previous studies, by Bergant & Simpson (1999) at Adelaide University and Arfaie (1989) at Newcastle University. The aim of this work is to repeat and extend the Bergant & Simpson (1999) comparison of the Discrete Vapour Cavity Method (DVCM) and Discrete Gas Cavity Method (DGCM) while also taking into account the contribution of Arfaie (1989) in respect of:  his suggestion for an improved transient internal boundary condition at the moving liquid column separation interface;  his recommendation that the greater damping associated with unsteady pipe friction models (as opposed to conventional quasi-steady friction) may improve overall model performance; and  his observation that the mode of column separation behaviour (particularly when column separation causes a pressure spike that may exceed the widely regarded Joukowsky pressure maximum) may influence the choice of best model. The basic DVCM and DGCM models tested (with the gas release physical parameters for the latter) are those established in the literature. In these the transient internal boundary condition at the moving column separation interface can be either the conventional Wylie & Streeter (1993) formulation as in Bergant & Simpson (1999) or the Arfaie (1989) improvement. There are many models available for unsteady friction, but previous work by Bughazem (1997) at Newcastle University had established that a Brunone-type Instantaneous Acceleration Based model is not only simple to implement but also works well on the specific experimental apparatus used ii in this study. Bughazem & Anderson (1996) had outlined (but not implemented or tested) a possible alternative integration of this into a fully Method of Characteristics approach. This is developed and applied, but its additional implementation complexity for no obvious gain in performance led to its being set aside. The very simple Arfaie (1989) experimental apparatus used is intended to eliminate any modelling issues (especially for external boundary conditions) not associated with column separation as well as to attempt to restrict column separation to a single location (to support its visualisation). Flow visualisation on this apparatus did not show the conventional full-bore vapour cavity suggested by the term “column separation”. Rather scattered vapour or released gas bubbles appeared along the pipe soffit during the transient column separation event. To support clarification of Arfaie’s views on the different modes of column separation behaviour, an extensive series of experimental runs were recorded to facilitate development of a map for the occurrence of these, with the intention of helping analysts and designers to determine if pressure higher than Joukowsky might occur. It was determined that these may occur for PM =~1.2~2 where PM is the Martin ratio: PM = ρ.a.VoPR−Pv Initially the comparison of computed against experimental results followed the conventional qualitative approach as in Arfaie (1989) and others. However, this proves problematic where a large number of experimental runs (with scatter due to uncertainties) have been taken, as well as when there are more than one factor for comparison. This process, though, did highlight an issue with predicting the data value for vapour pressure, where the actual value on the experimental traces is different from the Steam Tables value used for prediction and thus appearing on the computed traces. This introduced a further factor to the investigation. Following Arfaie (1989) and others, initially qualitative comparison taken over a period including up to five pressure peaks were made (overall shape of peaks and ability to maintain phase of solution features). However, for consistent comparison across a number of experimental runs, two specific quantifiable criteria are defined:  the time duration of the first column separation event; and iii  the maximum pressure peak amplitude occurring as a result of that. Graphs can be compiled to attempt to explore the behaviour of different model options, but with a large amount of data showing scatter due to uncertainty these do not lead to clear outcomes. Consequently, following previous work on CFD modelling at Newcastle University by Ahmeid (1997), a statistical approach using Design of Experiments (DOE) with Analysis of Variance (ANOVA) was adopted which demanded quantified criteria. ANOVA indicates whether significant differences can be detected from the data and the DOE approach (as compared with “one factor at a time” testing) can indicate if there are interactions present between the factors. With this recourse to statistical methodology, Normal Probability plots indicated that the data for first cavity duration are better than data for maximum pressure peak amplitude, giving more significant ANOVA outcomes for the former than the latter. Though this first attempt at using these techniques has not produced clear or comprehensive outcomes, the methodology is promising for future studies. The present outcomes are that:  For basic method, DGCM, as suggested by Bergant & Simpson (1999), performs best for cavity duration, but it is not yet possible to say this for maximum pressure amplitude.  Similarly, with quasi-steady friction at least, the Arfaie (1989) internal boundary condition is a small improvement over the conventional Wylie & Streeter (1993), certainly for cavity duration.  Unsteady friction does reduce error magnitude and scatter, but the greater damping may lead to non-conservative (under-estimation) prediction of maximum pressure amplitude.  There is evidence that the mode of column separation behaviour does interact with the other factors, but it is not yet clear exactly what, if any, real effect it has.  Finally, though the data value for vapour pressure is significant (certainly for cavity duration), in practice small variations in its value seem to make little difference to computed predictions. There is sufficient evidence that with better quality data and further consideration of quantifiable criteria for comparison that the statistical methodology demonstrated can be an effective tool for computational model testing. iv Unfortunately for this present study, it exposed the limitation of the apparatus used in producing repeatable results with controlled uncertainties, especially for peak pressure. A clear conclusion is that better experimental data from an improved experimental apparatus are required.