Efficient Quasi-Two-Dimensional Model for Water Hammer Problems

Quasi-two-dimensional models for turbulent flows in water hammer are necessary for advancing the understanding of flow behavior in pipe transient; conducting detailed investigation of the fate of transient-induced contamination; and validating onedimensional water hammer models. An existing quasi-two dimensional numerical model for turbulent water hammer flows has the attributes of being robust, consistent with the physics of wave motion and turbulent diffusion, and free from the inconsistency associated with the enforcement of the no slip condition while neglecting the radial velocity at boundary elements, such as valves and reservoirs. However, this scheme is computationally intensive making it unsuitable for practical pipe systems or for conducting numerical experiments. This paper addresses the efficiency and stability of this existing scheme. In particular, algebraic manipulations show that the original scheme can be decoupled into two tridiagonal systems, one for piezometric head and radial flux and another for axial velocity. This decoupling is the reason for the high efficiency of the modified scheme. The original and proposed schemes are applied to a pipe–reservoir–valve system. It is found that, for the same spatial and temporal discretization, both schemes are of equal accuracy. However, significant saving in computer execution time is achieved by using the modified scheme. Application of the modified scheme to pipes of realistic dimensions and wavespeeds (length 35.2 km, diameter 200 mm, and wave speed 1000 m/s) takes only a few minutes to execute. This small execution time requirement makes the current quasi-two-dimensional model suitable for application to practical water hammer problems. The stability domain of the proposed scheme is established using the Von Neumann method.