Abstract
We propose a one-dimensional Saint-Venant (open-channel) model for overland flows, including a water input–output source term modeling recharge via rainfall and infiltration (or exfiltration). We derive the model via asymptotic reduction from the two-dimensional Navier–Stokes equations under the shallow water assumption, with boundary conditions including recharge via ground infiltration and runoff. This new model recovers existing models as special cases, and adds more scope by adding water-mixing friction terms that depend on the rate of water recharge. We propose a novel entropy function and its flux, which are useful in validating the model’s conservation or dissipation properties. Based on this entropy function, we propose a finite volume scheme extending a class of kinetic schemes and provide numerical comparisons with respect to the newly introduced mixing friction coefficient. We also provide a comparison with experimental data.
Highlights
In quantifying the dynamics of a watercourse, the most important components of the hydrologic recharge and loss are the precipitation and infiltration processes, respectively
For the specific problem of modeling flooding caused by precipitation, the inclusion of a source term corresponding in published maps and institutional to the recharge or infiltration in the Saint-Venant system turns it from a conservation affiliations
Our aim is to construct a mathematical model for overland flows that is consistent with the physical phenomena that can affect the motion of such water
Summary
In quantifying the dynamics of a watercourse, the most important components of the hydrologic recharge and loss are the precipitation and infiltration processes, respectively These are important today in understanding and forecasting the impact of climate variability on the human and natural environment. Our goal in this paper is to derive a model akin to (1) via vertical averaging under the shallow water assumption, starting from the Navier–Stokes equations with a permeable Navier boundary condition to account for the infiltration and a kinematic boundary condition to consider the precipitation. We will see below that these friction terms are necessary to avoid paradoxical outcomes, such as perpetual motion, and, for simplicity, we will assume in this paper the most basic constitutive relations for this friction: linear in R for k+(R) and piecewise linear in I for k−(I), in agreement with approximations based on experimental results [3,17]. A C and C++ implementation of this code written by Matthieu Besson, Omar Lakkis, and Philip Townsend is freely available on request (an older version is given by [20])
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
