Shallow Flows over Curved Beds: Application of the Serre–Green–Naghdi Theory to Weir Flow

Abstract

The Serre–Green–Naghdi (SGN) theory is used widely in maritime hydraulics for the computation of water waves from shallow to intermediate water depths. However, it has been largely ignored in problems of open-channel hydraulics. This research applies the SGN theory to compute shallow flows over curved beds, focusing on round-crested control structures. A systematic and complete analysis is presented, including the solution of both the unsteady and steady versions of the SGN theory. A new steady SGN solver is presented, allowing for automatic computation of the rating curves of the structures, thereby avoiding tedious trial and error iterations. The SGN theory is applied to a wide portfolio of obstacle shapes used in practice, resulting in agreement with experimental evidence within the shallowness limit determined. A higher-order approach is discussed, pointing to a possible path to increase the validity of SGN-type solvers in high overflow problems.