Numerical treatment of a discontinuous top surface in 2D shallow water mixed flow modeling

Abstract

Summary This paper presents a numerical strategy based on shallow water equations (SWE) coupled with the 2D Preissmann slot model to handle a ceiling step discontinuity in finite volume schemes for mixed flow modeling. In practice, a typical situation would be a closed structure, such as a bridge or culvert, which induces a sudden vertical flow constriction and may even run partly or totally full in high flow conditions. In such case, both the inlet and outlet of the structure involve a discontinuity in the top elevation. This special singularity is topologically represented by inserting a fictitious cell between 2 adjacent computational cells characterized by sharply different ceiling elevation. The 2D SWE are solved by means of a well‐balanced quasi‐conservative Godunov‐type numerical scheme based on the Slope Limiter Centered (SLIC) scheme. The flow variables at each boundary of the fictitious cell are reconstructed by adopting the cross‐sectional shape of the adjoining cell. Accordingly, the dynamic effect of the structure deck on the flow is suitably modeled, and the C‐property for a stationary solution is rigorously satisfied, even when the closed structure is partially full. The capability of the numerical scheme is verified by comparison with both novel analytical solutions of 1D Riemann problems with a ceiling step discontinuity and experimental data of steady and unsteady mixed flows available in literature. Finally, a real‐scale application to a multiple arch bridge is presented. The results show that the method is robust and effective in predicting the 2D features induced by a crossing structure on the flow dynamics.

Extension of the two‐component pressure approach for modeling mixed free‐surface‐pressurized flows with the two‐dimensional shallow water equations

“…The numerical model gives a good prediction of the height of the shock front but with a time lag of approximately 1.5 s. This is because shallow water models do not provide the correct wave celerity when the hydrostatic pressure assumption is not fulfilled. A typical situation when this happens is when a surface wave or a supercritical flow impacts a closed wall generating high dynamic pressures, as reported in previous studies 12,55,56 …”

“…Observed results from Reference 12. Maranzoni results refer to Reference 55.…”

“…The case modeled here corresponds to the Deck 1 geometry from Reference 12, and its geometry is presented in Figure 12. This case was also studied numerically in Reference 55. The experiments were carried out in a smooth flat channel, 6 m long, with a 24 cm × 16 cm rectangular cross‐section.…”

“…Two different inlet discharges and outlet conditions were tested, corresponding to cases in which the bridge is partially submerged (Table 2). Regarding the numerical discretization, the same conditions used in Reference 55 are used, namely, m, a Courant number of 0.7, and a Manning's roughness coefficient of 0.011 s/m 1/3 . The initial water level in the whole domain is set equal to the water level at the downstream boundary, and the simulation is run until the steady state is reached.…”

Suppress Numerical Oscillations in Transient Mixed Flow Simulations with a Modified HLL Solver

“…To simplify this question, we can reduce the number of governing equations by adopting one set of governing equations for both flow conditions. Based on this idea, the Preissmann slot model (PSM) is proposed by Cunge [11] and it has been adopted by many researchers and commercial software packages for its simplicity and accuracy [12][13][14][15][16][17][18]. The gradient of piezometric head at the interface between two flow regimes are very steep which forms a discontinuity in flow, therefore the governing equations are normally discretized using the state-of-art finite volume method for it admits discontinuous solutions [19].…”