Submitted to: Journal of Hydraulic Research IAHR
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 6/7/2007
Publication Date: 1/20/2008
Citation: Ying, X., Wang, S.S. 2008. Improved implementation of the HLL approximate Riemann solver for one-dimensional open channel flows. Journal of Hydraulic Research IAHR. Vol. 46, No. 1 (2008), pp. 21-34.
Interpretive Summary: Modeling open channel flows with discontinuities and mixed flow regimes has been a great challenge to hydraulic researchers and engineers. Recently, applications of approximate Riemann solvers to one-dimensional open channel flows have frequently been reported. These schemes are accurate in some situations, but problems such as numerical imbalance and non-convergent solution of discharge arise when the channel has complex geometry and the flow includes a hydraulic jump. In the conventional formulation, the momentum equation includes three terms that respectively represent the hydrostatic pressure force, the pressure force due to cross-sectional variations, and the gravity effect due to bed slope. The numerical imbalance is created when these terms are calculated using different methods, which leads to unrealistic physical flow in a still water test case. These limitations have existed for more than 10 years and restrain the applications of the approximate Riemann solvers to real-life open channel flows. In this paper, several new techniques are proposed to overcome these deficiencies. It is shown by various test problems that the new schemes can totally eliminate the problems of numerical imbalance and non-convergent solution of discharge, and are capable of satisfactorily reproducing various complicated open channel flows.
Technical Abstract: Several new techniques are proposed to overcome the deficiencies in the conventional formulation of the approximate Riemann solvers for one-dimensional open channel flows, which include numerical imbalance and inaccuracy in the solution of discharge. The former arises in the case of irregular geometry and the latter in the presence of a hydraulic jump. These new techniques include: (1) adopting the form of the Saint Venant equations that include both gravity and pressure in one source term; (2) using water surface level as one of the primitive variables, in stead of cross-sectional area; (3) defining discharge at interface and evaluating it according to the flux obtained by the HLL Riemann solver; (4) estimating water surface gradient based on piecewise linearly reconstructed variables in the 2nd-order scheme. The performance of the resulting schemes is evaluated by means of theoretical analysis and various test examples, including ideal dam-break flows with dry and wet bed, hydraulic jump, steady flow over bump with hydraulic jump, wave interactions, tidal flow in an open channel, and wave propagation. It is demonstrated that the schemes have excellent numerical balance and mass conservation property and are capable of satisfactorily reproducing various open channel flows.