Development of an upwinding kernel in SPH-SWEs model for 1D trans-critical open channel flows

In this study, an upwinding SPH model with a non-symmetric kernel function is proposed to predict one-dimensional open channel flows. Due to the application of non-symmetric kernel function biased in favor of the upstream side, numerical diffusion is intrinsically added into the discretized momentum equation using SPH. The proposed model thus has shown to have good potential to resolve steep gradient or discontinuous solutions without the need of exactly adding artificial viscosity to the discretized equation. Furthermore, an upwinding coefficient for the determination of the degree of upwinding is derived to accommodate the dispersion-relation-preserving (DRP) property. In wave number space, the error between the discretized SPH equations and the original partial differential equations is minimized, thereby yielding the optimized upwinding coefficient. The proposed model has been validated by solving four benchmark problems involving non-rectangular cross section, varying channel width, non-uniform bed slope and hydraulic jump. Comparison of the numerical and exact solutions shows that the proposed model has the ability of accurately predicting various open channel flows involving complicated transcritical flows. The consistency condition of the proposed model is also analyzed theoretically for the sake of completeness.