An Efficient Finite-Volume Scheme for Modeling Water Hammer Flows
The study of water hammer flows has great significance in a wide range of industrial and municipal applications including power plants, petroleum industries, water distribution systems, etc. The understanding of water hammer phenomena is also important in hydraulic conveyance systems such as stormwater and sanitary sewer systems. Although the latter two systems are generally designed based on gravity flow, in practice large variations in the inflow and outflow from these systems may result in the pressurization of the systems that, in turn, may produce water hammer phenomena. For modeling this type of flows, several numerical approaches have been proposed. The efficiency of a model is a critical factor for Real-Time Control (RTC), since several simulations are required within a control loop in order to optimize the control strategy, and small simulation time steps are needed to reproduce the rapidly varying hydraulics (León et al. 2006). RTC is becoming increasingly indispensable for industrial and municipal applications in general. For instance, in the case of water distribution systems, RTC facilitates delivery of safe, clean and high-quality water in the most expedient and economical manner.
Among the approaches proposed to solve the water hammer equations are the Method of Characteristics (MOC), Finite Differences (FD), Wave Characteristic Method (WCM), Finite Elements (FE), and Finite Volume (FV). In-depth discussions of these methods can be found in Chaudhry and Hussaini 1985, Ghidaoui and Karney 1994, Szymkiewicz and Mitosek 2004, Zhao and Ghidaoui 2004, and Wood et al. 2005. Among these methods, MOC-based schemes are most popular because these schemes provide the desirable attributes of accuracy, numerical efficiency, programming simplicity, and ability to handle complex boundary conditions (e.g., Wylie and Streeter 1993; Ghidaoui et al. 2005; Zhao and Ghidaoui 2004). In fact, in a review of commercially available water hammer software packages, it is found that eleven out of fourteen software packages examined use MOC schemes (Ghidaoui et al. 2005).
Recently, FV Godunov-Type Schemes (GTS), which belong to the family of shock-capturing schemes, have been applied to water hammer problems with good success. The underlying idea of GTS is the Riemann problem that must be solved to provide fluxes between adjacent computational cells. The first application of GTS to water hammer problems is due to Guinot (2000), who presented first and second-order schemes based on Taylor series expansions of the Riemann invariants. He showed that his second-order scheme is largely superior to his first-order scheme, although the Taylor series development introduces an inevitable inaccuracy in the estimated pressure, especially in the case of low pressure-wave celerities. A second application is due to Hwang and Chung (2002), whose second-order accuracy scheme is based on the conservative form of the compressible flow equations. Although this scheme requires an iterative process to solve the Riemann problem, these authors state that their scheme requires a little more arithmetic operation and CPU time than the so-called Roe's scheme, but is able to get more accurate computational results than the latter scheme. Later, Zhao and Ghidaoui (2004) presented first and second-order schemes for solution of the non-conservative water hammer equations. They show that, for a given level of accuracy, their second-order GTS requires much less memory storage and execution time than either their first-order GTS or the fixed-grid MOC scheme with space-line interpolation. Although highly efficient, this chapter shows that the second-order scheme of Zhao and Ghidaoui (2004) has an overall accuracy of at most first-order.
The reason why the second-order scheme of Zhao and Ghidaoui (2004) has at most only first-order rate of convergence is because this scheme uses a second-order formulation solely at the internal cells, but not at the boundaries for which only a first-order formulation is used. This chapter focuses on the formulation and assessment of a second-order accurate (internal and boundary nodes) FV scheme for modeling water hammer flows. Unlike the scheme of Zhao and Ghidaoui (2004), the proposed approach uses the conservative form of the water hammer equations. To achieve second-order accuracy at the internal cells in the proposed approach, the Monotone Upstream-centred Scheme for Conservation Laws (MUSCL) - Hancock method is used, which also was used by Zhao and Ghidaoui (2004). Thus, the only difference between the second-order scheme of Zhao and Ghidaoui and the proposed approach is in the order of accuracy used at the boundaries (first-order in the scheme of Zhao and Ghidaoui and second order in the proposed approach).
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