Modeling positive surge propagation in open channels using the Serre-Green-Naghdi equations

Abstract

Numerical modeling of upstream propagation of a positive surge wave in an open-channel flow caused by a sudden rise of downstream water level, either by gate operation or by tidal influence, remains an important challenge from many aspects. A perusal of literature suggests that the Serre-Green-Naghdi (SGN) models have not been methodically explored for a positive surge flow modeling. In this study, the SGN equations for variable bed profiles are solved for simulating the positive surge waves. The governing equations are reformulated in the form of a conservation law, which is solved using a hybrid finite volume-finite difference method. The eddy-viscosity type wave breaking model is coupled with the governing equations for simulating a breaking surge. The performance of second- and fourth-order time stepping schemes and second- and fourth-order accurate schemes for determining the numerical flux is assessed. The results indicate that using an overall fourth-order accurate scheme is preferred for an accurate depiction of the positive surge waves. A comparison is also made with laboratory observations from a wide range of experiments conducted under different discharge, Froude number and bed slope conditions, which demonstrates that the proposed model can accurately reproduce the key free surface characteristics corresponding to positive surge waves. The importance of computing an accurate initial gradually varied flow profile before applying the transient downstream condition, such as a gate closure, is highlighted. Further, the significance of considering full non-linearity is demonstrated through numerical tests as the weakly non-linear Boussinesq-type equations are found to perform poorly in predicting the secondary undulations for an undular surge. The non-hydrostatic pressure field occurring during the propagation of an undular surge has rarely been reported from laboratory experiments. In this study, the bed pressure beneath a propagating undular surge is measured and compared with that computed according to the SGN equations. Finally, the results obtained from the proposed SGN model are compared with those obtained from Large Eddy Simulation results available in the literature.