Consistent section-averaged shallow water equations with bottom friction

In this paper, we present a general framework to construct section-averaged models when the flow is constrained – e.g. by topography – to be almost one-dimensional. These models are consistent with the two-dimensional shallow water equations. After rewriting the two-dimensional shallow water equations in a suitable set of coordinates allowing to take care of a meandering configuration, we consider the quasi one-dimensional regime. Then, we expand the water elevation and velocity field in the spirit of the diffusive wave equations and establish a set of one-dimensional equations made of a mass, momentum and energy equations, which are close to the ones usually used in hydraulic engineering. Our model reduces to classical shallow water models with variable sections found in the literature. Out of these configurations, there is an O(1) deviation of our model from the classical ones. Finally, we present the main mathematical properties of our model and carry out numerical simulations to validate our approach by comparing the results to the full two-dimensional shallow water equations.