Meshless computation of water hammer

Water hammer concerns the generation, propagation, reflection and damping of pressure waves in liquidfilled pipe systems. The numerical simulation of water-hammer events is usually based on one-dimensional mathematical models. Two equations representing the conservation of mass and momentum govern the unsteady flow in the pipes. Boundary and/or intermediate conditions model the behaviour of hydraulic machinery and the additional complication of column separation. The method of characteristics is the preferred method of solution and the conventional approach is to define a fixed grid in the distance-time plane. On this grid, the unknown pressures and velocities (or heads and discharges) are numerically computed in a time-marching procedure that starts from a given initial condition. Rectangular or diamond grids are used that in general do not exactly match the given pipe lengths or the chosen time step. Therefore numerical interpolations and/or adjustments of wave speeds or pipe lengths are necessary, and these necessities introduce error. This paper presents a different way of water-hammer computation. The procedure is based on the method of characteristics, but a numerical grid is not required. Any point in the distance-time plane can be selected to compute the local solution without explicitly using stored previous solutions. The computation is based on back-tracking waves by means of a very simple recursion, so that the programming effort is small. Exact solutions are thus obtained for frictionless water-hammer. Approximate solutions are obtained when the distributed friction in individual pipes is concentrated at the pipe boundaries. The new algorithm is fully described and the new method is evaluated and compared with conventional water-hammer calculations. The following boundary conditions are tested: reservoir, instantaneous and gradual valve closure, and pipe junction. The algorithm is easy to implement and its exact solutions can be used to asses the numerical error in more conventional methods for the test cases of frictionless waterhammer and lumped friction.