Friction and geometric source terms in a 1D augmented shallow water equations system

Abstract

This paper deals with source terms due to flow resistance and geometric variability in a new formulation of the one-dimensional augmented Shallow Water Equations (SWE) for open channels and rivers with arbitrarily shaped cross sections. In the classical treatment of the Shallow Water Equations, source terms are due to geometric irregularities on the one hand and to friction on the other. In the present approach, geometrical irregularities are incorporated in the convective term, while a specific numerical treatment of the friction source term is introduced, which is able to face stiff problems.

The robustness of the augmented inviscid model is maintained when the cross section presents high irregularities; the focused treatment of the stiffness allows to preserve the accuracy when the water depth assumes very low values, as in the case of wave propagation over dry bed.

The additional variable introduced to obtain the augmented SWE depends on the section considered and the type of geometric irregularity encountered, but the formulation is general and designed for an extended variety of practical cases.

The numerical method used to integrate the system of hyperbolic balance laws with source terms is a Strong Stability Preserving Implicit–Explicit (IMEX) Runge–Kutta method, which is embedded on a path-conservative Dumbser Osher Toro (DOT) Finite Volume Method (FVM) method, which is second order accurate in space and time. This accuracy is maintained in the stiff limit, which is reached when a very small depth occurs.

After checking the order of accuracy of the numerical scheme on two smooth test cases – wet bed and dry bed, respectively – the mathematical model and its numerical implementation are validated on very different examples: i) the computation of quasi uniform flow in an uneven trapezoidal channel, which allows to generalize the concept of bed slope when several generatrice lines of different slope are used to reconstruct the wetted perimeter of the channel; ii) the simulation of dam break flows on a dry bed including friction for different power-law cross section channels, which is specifically dedicated to show the robustness of the method on the wave front where the water depth approaches zero, in very different narrowness configuration of the channel geometry. Very good results are obtained in all cases, demonstrating the wide applicability of the method.