Pakravan Khouzani Mahdi
Numerical modeling of Water Hammer using Roe’s Scheme
Mohammad Reza Chamani
A Scheme to model water hammer in pipeline systems is presented. The scheme is an explicit first-order accurate and it is stable for Courant numbers up to unity. The scheme solves mass and momentum equations using a Godunov-type finite volume (FV) method named Roe's approximate Riemann solver. Godunov methods provide fluxes at interfaces by the exact solution of Riemann problem that ensures conserving properties of scheme using FV discretization. The well-known Method of characteristics (MOC) with linear space-line interpolation is used for comparison. Both complete and approximate water hammer equations are used. Multi-step, multi-stage and iterative approaches are 'used to compute fluxes and to integrate the time terms. The first numerical model includes a horizontal pipe, a valve and a reservoir upstream (without friction). Other models include series pipes and real valves. The first numerical model includes a horizontal pipe, a valve and a reservoir upstream (without friction). Other models include series pipes and real valves.
For the first model, the effects of Courant number (Cr) and spatial steps are investigated. For Cr<l, numerical dissipation occurs. Increasing the number of spatial steps increases the accuracy. For the second model, the results show that the application of one-step approach for fluxes computation and multi-stage approach for time integration are more suitable in terms of convergence. It is then decided to use one-step and multi-stage approaches for the remaining model. For the third model, the results show that water hammer analysis is very sensitive to Courant numbers. For series pipes, the optimum number of spatial steps is obtained. The approximate water hammer equations are more convergence than complete water equations. Roe's scheme using approximate water hammer equations is more convergence than that of MOC. The multi-stage Roe's scheme using approximate water hammer equations has the most convergence. Convergence at a section where maximum pressure occurs is more than that of a section with minimum pressure. It is shown that spatial variations of pressure are more than time variations. It is also shown that the use of series pipes instead of single pipe is very efficient to control extreme water hammer pressures.