A multiscale approach to liquid flows in pipes I: The single pipe

In the present paper we study the propagation of pressure waves in a barotropic flow through a pipe, with a possibly varying cross-sectional area. The basic model is the Saint-Venant system. We derive two multiscale models for the cases of weak and strong damping, respectively, which describe the time evolution of the piezometric head and the velocity. If the damping is weak, then the corresponding first-order hyperbolic system is linear but contains an additional integro-differential equation that takes into account the damping. In the case of strong damping, the system is nonlinear. The full and multiscale models are compared numerically; we also discuss results obtained by a largely used commercial software. The numerical experiments clearly demonstrate the efficiency of the multiscale models and their ability to yield reliable numerical approximations even for coarse grids, that is not the case for the full model.