A numerical scheme based on discontinuous Galerkin method is proposed for the two-dimensional shallow water flows. The scheme is applied to model flows with shock waves. The form of shallow water equations that can eliminate numerical imbalance between flux term and source term and simplify computation is adopted here. The HLL approximate Riemann solver is employed to calculate the mass and momentum flux. A slope limiting procedure that is suitable for incompressible two-dimensional flows is presented. A simple method is adapted for flow over initially dry bed. A new formulation is introduced for modeling the net pressure force and gravity terms in discontinuous Galerkin method. To validate the scheme, numerical tests are performed to model steady and unsteady shock waves. Applications include circular dam break with shock, shock waves in channel contraction, and dam break in channel with

Free surface flows that take place in rivers, oceans, and estuaries are of great importance to human activities. Numerical models for the open channel flows have been of great interests to hydraulic researchers and engineers. In practice, numerical models for flows with shock waves present difficulties in capturing shock wave and preserving conservative properties of the flow equations. Many shock-capturing methods have been developed in previous studies [

In recent years, discontinuous Galerkin (DG) finite element method has gained popularity in modeling shallow water flows [

Various upwind schemes can be applied to solve for the numerical flux across the boundaries of an element with discontinuities. These include Roe’s flux function [

In this paper, a numerical model based on DG method is presented for two-dimensional (2D) shallow water flows. In contrast to the previous studies based on the DG scheme, the net pressure force and the gravity terms are combined in this paper. A discretization scheme is adapted for the combined term that eliminates numerical imbalance between the pressure force term and gravity term and simplifies computation. The flux terms are approximated using HLL flux function. A modified HLL flux function is used to handle flow over dry bed. It is well known that when higher-order numerical schemes are used, nonphysical oscillations are generated around discontinuities and steep gradients. TVD slope limiters are widely used to minimize these oscillations and stabilize numerical schemes. To achieve oscillation-free solutions, a four-step slope-limiting procedure in DG method is adapted for the two-dimensional incompressible flows. The governing equations are first introduced, and then the DG formulation is briefly described. The implementation of HLL flux function and treatment of the source term are discussed. The Total Variation Diminishing (TVD) method for time integration scheme and slope limiter is presented. Finally, several numerical examples are presented to test the present numerical scheme for shallow water flows with shock waves and flow over dry bed.

The vector form of the depth-averaged, two-dimensional, shallow water flow equations can be written as

The two-dimensional shallow water equations are derived by integrating the Navier-Stokes equations along the depth of the fluid body. Several assumptions are made such as hydrostatic pressure distribution and uniform velocity profile in the vertical direction. The advantage is that free surface location is determined as part of the solution. The two-dimensional shallow water flow equations can be applied in situations where vertical acceleration may be neglected, and the horizontal extent is much greater than the depth of flow [

For

Configuration of triangular elements in DG method.

The Discontinuous Galerkin formulation is written for each element. The variation of variables within an element is represented by the values of the variables at the vertices and shape functions

Since the discontinuous elements are allowed in discontinuous Galerkin method, a generalized local Riemann problem can be solved for the numerical flux. The numerical flux in (

For

The numerical flux,

The water level slopes in the source term can be determined with Green’s theorem as follows [

The Total Variation Diminishing (TVD) scheme is chosen to eliminate numerically generated oscillations near shocks and steep gradients. The TVD Runge-Kutta time integration and the slope limiter are used here to achieve the TVD property. Former studies have shown that, to conserve the TVD property, the Runge-Kutta time integration scheme should be one order higher than the shape functions [

A slope-limiting procedure developed by Tu and Aliabadi [

Tests are performed in this section to examine the accuracy of the proposed numerical scheme to model shallow water flows with shock waves. Numerical tests include circular dam break, shock wave in circular dam break, steady shock wave in channel contraction, and dam break in channel with 45° bend. Numerical results are compared with an exact solution or measured experimental data, if available.

To test the symmetric shock-capturing capability of the scheme, the idealized circular dam break problem is used [^{3}, and the water volume at 0.8 seconds is 5910 m^{3}, showing the mass is well conserved.

Configuration of circular dam break test.

Computed water surface at 0.8 s after dam removal.

Computed water surface contour at 0.8 s after dam removal.

Computed velocity field at 0.8 s after dam removal.

The same domain as used in the previous test is adopted here with different initial conditions. The initial water depth is 1 m inside the dam and 10 m outside the dam. After removing the dam, the circular shock moves inwards, passes through the singularity, and then expands outwards. The shock at 2 seconds is shown below in Figures ^{3}, and the water volume at 2 seconds is 21590 m^{3}, showing the mass conservation is well preserved in this model.

Computed water surface at 2 s.

Computed water surface contour at 2 s.

Computed velocity field at 2 s.

The steady supercritical shock wave due to channel contraction is simulated to test the numerical scheme. The plan view of shock wave in a symmetric channel contraction is illustrated in Figure

Plan view of shock wave in a symmetric channel contraction.

Computation domain and mesh for the symmetric channel contraction.

Steady flow solutions for water depth are shown in Figures

Accuracy evaluation for supercritical flow in channel contraction.

Type | NSE | PBIAS | RSR |
---|---|---|---|

Dash line, coarse mesh | 0.95 | −0.40% | 0.22 |

Dash line, fine mesh | 0.98 | −0.24% | 0.15 |

Solid line, coarse mesh | 0.94 | −0.17% | 0.24 |

Solid line, fine mesh | 0.97 | 0.01% | 0.17 |

Water surface profile for in the symmetric channel contraction.

Water depth contour in the symmetric channel contraction

Comparison of water depths along the dash line.

Comparison of water depths along the solid line.

Physical models were built in the Civil Engineering Department Laboratory, Université Catholique de Louvain (UCL, Belgium) to model dam break and strong transient flows in sharp bends. Experimental data were collected and used to validate numerical models developed by the CADAM group [

The plan view of the channel with horizontal bed and 45° bend is shown in Figure ^{1/3} for bottom and 0.0195 s/m^{1/3} for wall, as suggested by Frazão et al. [

Gauge points location and accuracy of simulated results.

Gauge point | y (m) | NSE | PBIAS | RSR | |
---|---|---|---|---|---|

G1 | 1.59 | 0.69 | 0.99 | 0.57% | 0.074 |

G2 | 2.74 | 0.69 | 0.34 | −13.67% | 0.81 |

G3 | 4.24 | 0.69 | 0.83 | −1.39% | 0.42 |

G4 | 5.74 | 0.69 | 0.82 | 1.87% | 0.43 |

G5 | 6.74 | 0.72 | 0.81 | −7.89% | 0.44 |

G6 | 6.65 | 0.80 | 0.87 | −6.38% | 0.36 |

G7 | 6.56 | 0.89 | 0.80 | −5.46% | 0.45 |

G8 | 7.07 | 1.22 | 0.84 | −4.55% | 0.40 |

G9 | 8.13 | 2.28 | 0.84 | 6.43% | 0.40 |

Plan view of channel with 45° bend.

After the removal of the gate, water flows rapidly into the channel and reaches the bend. The water reflects against the wall, and a shock forms and moves upstream. The velocity field at 3 seconds is shown in Figure

Velocity field at 3 s after dam break.

Water surface at 10 s after dam break.

Comparison of simulated and measured hydrographs at G1.

Comparison of simulated and measured hydrographs at G2.

Comparison of simulated and measured hydrographs at G3.

Comparison of simulated and measured hydrographs at G4

Comparison of simulated and measured hydrographs at G5.

Comparison of simulated and measured hydrographs at G6.

Comparison of simulated and measured hydrographs at G7.

Comparison of simulated and measured hydrographs at G8.

Comparison of simulated and measured hydrographs at G9.

A new numerical model is developed for two-dimensional shallow water flows with shock waves. In this model, the two-dimensional shallow water equations are solved by discontinuous Galerkin (DG) finite element method. The form of shallow water equations that simplifies computation and eliminate numerical imbalance between flux term and source term is used here. In this formulation, the net pressure force term is combined with the source term due to gravity. A proper treatment of the new source term is given. The HLL approximate Riemann solver is employed to calculate the mass and momentum flux. A slope-limiting procedure that is suitable for incompressible two-dimensional flows in DG solver is presented. The performance of the numerical scheme is tested by simulating circular dam break, shock in channel contraction, and dam break in channel with 45° bend. These tests include steady and unsteady shock waves, flow over initially dry bed, and subcritical and supercritical flow. The circular dam break tests show the shock capturing and symmetry preservation capabilities of the proposed scheme. The comparisons of the numerical results with analytical solutions and measured data demonstrate that the scheme is capable of simulating two-dimensional shallow water flows with shock waves as well as flow over dry bed.