Well-balanced and shock-capturing solving of 3D shallow-water equations involving rapid wetting and drying with a local 2D transition approach

Lu, Xinhua; Mao, Bing; Zhang, Xiaofeng; Shi, Rui

doi:10.1016/j.cma.2020.112897

Cited by 9 publications

(9 citation statements)

References 50 publications

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“…In a dry‐bed case with zero velocity in the dry cell, Equation (26) also takes effect. It is worth remarking that in practical simulations in rivers, lakes, and nearshore regions, only a small portion of the simulation region satisfies Equation (26) and thus the overall numerical accuracy is usually not affected by locally switching to a first‐order scheme 7,14 . In Equation (26),

\gamma =1

is a natural choice, but we find that

\gamma =2

still ensures model robustness while maintaining good numerical accuracy.…”

Section: Numerical Modelmentioning

confidence: 92%

“…When a second‐order MUSCL scheme is applied to regions with vanishing water‐depth, nonphysically large velocity may be predicted 11,14 . The main reason is that in these regions, though the reconstructed

{q}_x^{\mathrm{L},\mathrm{R}}

(or

{q}_y^{\mathrm{L},\mathrm{R}}

) and

{h}^{\mathrm{L},\mathrm{R}}

are both small and physically limited, for example,

\mid {\left({q}_x\right)}_k^{\mathrm{L},\mathrm{R}}\mid \leqslant \max \left({\left|{q}_x\right|}_{C_1},\kern0.3em {\left|{q}_x\right|}_{C_2}\right)\sim 0

, and

\mid {h}_k^{\mathrm{L},\mathrm{R}}\mid \leqslant \max \left({h}_{C_1},\kern0.3em {h}_{C_2}\right)\sim 0

, the division of them (e.g.,

{\left({q}_x\right)}_k^{\mathrm{L}}/{h}_k^{\mathrm{L}}

) may be extremely large due to different degrees of infinitesimal values of them.…”

Section: Numerical Modelmentioning

confidence: 99%

“…For instance, Hou et al 11 marked the cells near the wet–dry interface and the cells with relatively small reconstructed cell‐interface water depth as problematic cells and for them, the spatially first‐order scheme is used. Lu et al 14 added an additional criteria in solving the 3D Navier–Stokes equations. As for solving the 2D SWEs, our experience shows that only one criterion as shown in Equation (26) is sufficient to detect potentially problematic cells that needs to be switched from using the second‐order MUSCL scheme to a first‐order scheme,

$$ \left|\frac{{q_x}_k^{\mathrm{L},\mathrm{R}}}{h_k^{\mathrm{L},\mathrm{R}}}\right|\geqslant \gamma \cdotp \max \left(\left|\frac{\underset{C_1}{\overset{\left(I-1\right)}{q_x}}}{\underset{C_1}{\overset{\left(I-1\right)}{h}}}\right|,\left|\frac{\underset{C_2}{\overset{\left(I-1\right)}{q_x}}}{\underset{C_2}{\overset{\left(I-1\right)}{h}}}\right|\right).

…”

Section: Numerical Modelmentioning

confidence: 99%

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Cartesian‐MUSCL‐like face‐value reconstruction algorithm for solving the depth‐averaged 2D shallow‐water equations

2023

Numerical Methods in Fluids

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The aim of this article is to devise a novel second‐order monotone upstream scheme for conservation law (MUSCL) scheme on unstructured quadrilateral meshes, for solving the depth‐averaged 2D nonlinear shallow‐water equations (SWEs) involving rapid wet–dry transitions on uneven bed terrains. The proposed MUSCL scheme elaborately uses the geometry property of the quadrilateral cell and the reconstruction procedure is very similar to the MUSCL scheme on a Cartesian mesh. The edge‐centred value which is to be used for calculating local variable slope in the proposed scheme, is interpolated by both its two adjacent cell‐centred values and its two edge‐node values, where the nodal values are interpolated from surrounding cell‐centred values with a pseudo‐Laplacian formula. To prevent predicting nonphysically extreme velocities near the wet–dry interfaces so as to ensure model robustness, the proposed MUSCL scheme is locally reduced to be first‐order in potentially problematic cells; these cells are detected with a proposed physically‐based criterion. The new scheme does not need to compute variable gradients at cell‐centers, and we proved that the conventional MUSCL scheme on a Cartesian mesh is just a particular case of the newly designed scheme. A 2D SWEs solver is developed based on the new MUSCL scheme. Various numerical tests demonstrate that the developed numerical model is capable of maintaining steady stationary flow at rest and robust for simulation involving large gradients, and captures well the wet–dry front motions. The developed model is also found to adapt to extremely stretched meshes. Though the proposed numerical scheme is designed originally for meshes with purely quadrilateral cells, numerical experiments show that this scheme works well on triangular meshes as well, indicating that the proposed numerical scheme is attractive for simulations on mixed meshes consisting of both quadrilateral and triangular mesh‐cells.

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\gamma =1

is a natural choice, but we find that

\gamma =2

still ensures model robustness while maintaining good numerical accuracy.…”

Section: Numerical Modelmentioning

confidence: 92%

{q}_x^{\mathrm{L},\mathrm{R}}

(or

{q}_y^{\mathrm{L},\mathrm{R}}

) and

{h}^{\mathrm{L},\mathrm{R}}

are both small and physically limited, for example,

\mid {\left({q}_x\right)}_k^{\mathrm{L},\mathrm{R}}\mid \leqslant \max \left({\left|{q}_x\right|}_{C_1},\kern0.3em {\left|{q}_x\right|}_{C_2}\right)\sim 0

, and

\mid {h}_k^{\mathrm{L},\mathrm{R}}\mid \leqslant \max \left({h}_{C_1},\kern0.3em {h}_{C_2}\right)\sim 0

, the division of them (e.g.,

{\left({q}_x\right)}_k^{\mathrm{L}}/{h}_k^{\mathrm{L}}

) may be extremely large due to different degrees of infinitesimal values of them.…”

Section: Numerical Modelmentioning

confidence: 99%

$$ \left|\frac{{q_x}_k^{\mathrm{L},\mathrm{R}}}{h_k^{\mathrm{L},\mathrm{R}}}\right|\geqslant \gamma \cdotp \max \left(\left|\frac{\underset{C_1}{\overset{\left(I-1\right)}{q_x}}}{\underset{C_1}{\overset{\left(I-1\right)}{h}}}\right|,\left|\frac{\underset{C_2}{\overset{\left(I-1\right)}{q_x}}}{\underset{C_2}{\overset{\left(I-1\right)}{h}}}\right|\right).

…”

Section: Numerical Modelmentioning

confidence: 99%

See 1 more Smart Citation

Cartesian‐MUSCL‐like face‐value reconstruction algorithm for solving the depth‐averaged 2D shallow‐water equations

2023

Numerical Methods in Fluids

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show abstract

“…Three-dimensional (3D) non-hydrostatic models are relative to 3D hydrostatic models, which solve the NSE with the hydrostatic pressure assumption and are usually referred to as quasi-3D SWMs (Lu et al, 2020). Quasi-3D SWMs can provide 3D flow patterns with affordable computational expense and are widely applied in river, estuarine, and ocean flow simulations.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional non-hydrostatic model for dam-break flows

Ding

et al. 2022

Physics of Fluids

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A three-dimensional (3D) non-hydrostatic model is presented for the simulation of dam-break flows. The model solves the Reynoldsaveraged Navier-Stokes equations using the projection method. 3D computational grids are constructed from a two-dimensional horizontal unstructured mesh by adding horizontal layers in the vertical direction. Based on the horizontal unstructured grid system, horizontal advection terms are discretized by a momentum conservative scheme. The proposed model is validated with several physical experiments. The agreement between the model results and experimental data is generally good, which demonstrates the capability of the proposed model to resolve dam-break flows over flat and uneven bottoms with complex geometries. Moreover, the efficiency of the model is evaluated with 3D dam-break flow experiments. Comparisons between the non-hydrostatic model and the corresponding quasi-3D shallow water model are also performed, which confirm the role of non-hydrostatic effects in dam-break flows.

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“…In addition, efforts have been made to obtain an accurate solution to the Riemann problems [11,12]. Efforts to deal with wet-dry interfaces were also discussed extensively when developing numerical models for SWE, see [13,14]. Among these methods is the so-called slot-technique applied to the Boussinesq-type model [15], where an artificial porous beach is used to damp the numerical oscillation.…”

Section: Introductionmentioning

confidence: 99%

Staggered Conservative Scheme for 2-Dimensional Shallow Water Flows

Erwina

Adytia

Pudjaprasetya

et al. 2020

Fluids

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Simulating discontinuous phenomena such as shock waves and wave breaking during wave propagation and run-up has been a challenging task for wave modeller. This requires a robust, accurate, and efficient numerical implementation. In this paper, we propose a two-dimensional numerical model for simulating wave propagation and run-up in shallow areas. We implemented numerically the 2-dimensional Shallow Water Equations (SWE) on a staggered grid by applying the momentum conserving approximation in the advection terms. The numerical model is named MCS-2d. For simulations of wet–dry phenomena and wave run-up, a method called thin layer is used, which is essentially a calculation of the momentum deactivated in dry areas, i.e., locations where the water thickness is less than the specified threshold value. Efficiency and robustness of the scheme are demonstrated by simulations of various benchmark shallow flow tests, including those with complex bathymetry and wave run-up. The accuracy of the scheme in the calculation of the moving shoreline was validated using the analytical solutions of Thacker 1981, N-wave by Carrier et al. 2003, and solitary wave in a sloping bay by Zelt 1986. Laboratory benchmarking was performed by simulation of a solitary wave run-up on a conical island, as well as a simulation of the Monai Valley case. Here, the embedded-influxing method is used to generate an appropriate wave influx for these simulations. Simulation results were compared favorably to the analytical and experimental data. Good agreement was reached with regard to wave signals and the calculation of moving shoreline. These observations suggest that the MCS method is appropriate for simulations of varying shallow water flow.

show abstract

Well-balanced and shock-capturing solving of 3D shallow-water equations involving rapid wetting and drying with a local 2D transition approach

Cited by 9 publications

References 50 publications

Cartesian‐MUSCL‐like face‐value reconstruction algorithm for solving the depth‐averaged 2D shallow‐water equations

Cartesian‐MUSCL‐like face‐value reconstruction algorithm for solving the depth‐averaged 2D shallow‐water equations

Three-dimensional non-hydrostatic model for dam-break flows

Staggered Conservative Scheme for 2-Dimensional Shallow Water Flows

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