Formulation of consistent finite volume schemes for hydraulic transients

Abstract

ABSTRACTIntentionally generated transient waves are used in the detection of defects in pipelines. Experience to date in these applications shows that the wider the band of injected wave frequencies the better is the detection effectiveness. Thus, as these technologies develop further, models that can handle high frequency waves will be required. Finite volume (FV) methods are well suited for high frequency wave phenomena, but their use in the water hammer field has been limited to date compared with other fields such as open channel flow. A major reason for this is that FV methods are formulated for simple boundary conditions such as no-flux or no-slip, but have not been well developed for the typically more complex boundary conditions found in pressurized pipeline systems (e.g. junctions, control valves, orifices, tanks and reservoirs). In those few instances in the literature where FV has been applied to water hammer problems, the approach has been to use FV for internal sections and method of characteristics (MOC) at the boundaries. The global order of accuracy of FV–MOC is governed by the MOC solution. To our knowledge, this paper constitutes a first attempt at handling boundary conditions within the FV framework. The approach places the boundary element within a FV to enforce mass and momentum conservation within this volume. The fluxes between the FV and the adjacent elements are then formulated in the usual manner. The approach is illustrated for the case of a valve, a reservoir and a junction. The finite volume method used is the Boltzmann-type scheme. The accuracy and efficiency of all schemes with the proposed non-iterative FV formulation of the boundary conditions are demonstrated through the following test cases: (i) water hammer wave interactions with a junction boundary characterized by a geometric discontinuity, (ii) water hammer wave interactions with a junction boundary characterized by a discontinuity in the value of wave speed, and (iii) water hammer wave interactions with a junction characterized by a flow rate discontinuity. The pure FV formulation guarantees mass and momentum conservation. The Boltzmann-based FV schemes capture discontinuity as well as wave interactions with boundary elements accurately. The stability of the proposed FV schemes is satisfied when .