# Development of a Three-Dimensional Hydrodynamic Model Based on the Discontinuous Galerkin Method

^{*}

## Abstract

**:**

## 1. Introduction

Grid Type | Numerical Method | ||
---|---|---|---|

Finite Difference Method | Finite Volume Method | Finite Element Method | |

Structured grid | ROMS [6], POM [7], MoM [8], GETM [9], TRIM [10], DELFT3D [11], EFDC [12], ECOMSED [13], COHERENS [14] | MITgcm [15], MOHID [16] | |

Unstructured grid | FVCOM [17], UnTRIM [18], HydroInfo [19] | FESOM [20], ICOM, SELFE [21], ADCIRC [22], SCHISM [23], TELEMAC [24] |

## 2. Governing Equations

## 3. Numerical Implementation

#### 3.1. Domain Partition and Polynomial Space

#### 3.2. Numerical Discretization of Momentum Equations

#### 3.2.1. Convective and Bottom Topography Terms

#### 3.2.2. Horizontal Eddy Viscosity Terms

#### 3.2.3. Vertical Eddy Viscosity and Coriolis Acceleration Terms

#### 3.3. Numerical Discretization of the Primitive Continuity Equation

#### 3.4. Calculation of Vertical Velocity

#### 3.5. Time Stepping

- Calculate the explicit terms ${K}_{1}^{expl}$ and ${K}_{D,1}^{expl}$ for the three-dimensional momentum equations and primitive continuity equation according to the variables at ${t}_{n}$.
- Calculate the intermediate water depth as$${D}_{i}={D}_{n}+\Delta t{\displaystyle \sum _{j=1}^{i}{\widehat{a}}_{i+1,j}{K}_{D,j}^{expl}},i\in \left[1,2\right]$$Likewise, the intermediate conservative variables ${U}_{i}$ and ${K}_{i}^{impl}$ are$${U}_{i}={U}_{n}+\Delta t{\displaystyle \sum _{j=1}^{i}{\widehat{a}}_{i+1,j}{K}_{j}^{expl}}+\Delta t{\displaystyle \sum _{j=1}^{i}{a}_{i,j}{K}_{j}^{impl}}.i\in \left[1,2\right]$$With ${U}_{i}$ available, the intermediate depth-averaged momenta ${\left(DU\right)}_{i}$ and ${\left(DV\right)}_{i}$ are calculated through the depth-integration of ${U}_{i}$, followed by the calculation of the intermediate vertical velocity ${\omega}_{i}$. Later, the explicit terms ${K}_{i+1}^{expl}$ and ${K}_{D,i+1}^{expl}$ for the three-dimensional momentum equations and primitive continuity equation, are calculated according to the intermediate variables.
- Calculate the water depth ${D}_{n+1}$ and the conservative variables ${U}_{n+1}$ at ${t}_{n+1}$ as$$\begin{array}{l}{U}_{n+1}={U}_{n}+\Delta t{\displaystyle \sum _{j=1}^{3}{\widehat{b}}_{j}{K}_{j}^{expl}}+\Delta t{\displaystyle \sum _{j=1}^{2}{b}_{j}{K}_{j}^{impl}},\\ {D}_{n+1}={D}_{n}+\Delta t{\displaystyle \sum _{j=1}^{3}{\widehat{b}}_{j}{K}_{D,j}^{expl}}.\end{array}$$
- Integrate ${U}_{n+1}$ along the water depth to obtain the final depth-averaged momenta ${\left(DU\right)}_{n+1}$ and ${\left(DV\right)}_{n+1}$, followed by the calculation of the final vertical velocity ${\omega}_{n+1}$.

## 4. Numerical Tests

^{2}. In addition, all the experiments are run on an Intel Xeon E5-2620 processor with 16 GB of internal memory. Our program is parallelized using OpenMP to run on six cores.

#### 4.1. Manufactured Solution

^{2}s

^{−1}, the vertical eddy viscosity coefficient ${K}_{m}$ is taken as 0.01 m

^{2}s

^{−1}, and the analytical solution is given as follows:

#### 4.2. Tide Wave Propagation in a Semi-Closed Bay

#### 4.3. Wind-Induced Water Circulation

^{2}s

^{−1}; the whole simulation lasts for 3600 s.

#### 4.4. Generation of the Ekman Profile

^{2}/s, ${\tau}_{sx}=0.1$ Pa, and $\theta $ is the geophysical latitude (${45}^{\xb0}$). Theoretically, the flow direction at the surface would rotate counterclockwise 45 degrees to that of the wind stress.

#### 4.5. Tidal Flow in Bohai Bay

_{1}, P

_{1}, O

_{1}, K

_{1}, N

_{2}, M

_{2}, S

_{2}, K

_{2}, and Sa. The Coriolis parameter was taken as $9.1557\times {10}^{-5}{\mathrm{s}}^{-1}$, and the horizontal eddy viscosity coefficient was set to 70 m

^{2}s

^{−1}. The vertical eddy viscosity coefficient was modelled using the $k-\u03f5$ turbulence closures provided by GOTM [47], and the coupling between GOTM and the hydrodynamic model is established following the same philosophy presented by Tuomas et al. [32]. The bottom roughness length is set as 0.1 mm. Field measurements from 12/08/2018 to 12/09/2018 are used to verify the developed model.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic view of a column of three-dimensional computational elements. (

**a**) quadrangular prisms; (

**b**) triangular prisms.

**Figure 2.**L

^{2}error versus DOFs and normalized CPU time for water depth D, vertical velocity ω, and horizontal momenta Du and Dv at different approximation orders (N

_{h}= N

_{v}= 1(black), N

_{h}= N

_{v}= 2(red)) on a sequence of meshes. (

**a**) L

^{2}error versus DOFs; (

**b**) L

^{2}error versus normalized CPU time.

**Figure 4.**Comparison between the numerical result (black line) and the analytical solution (red circle) for surface elevation $\eta $, horizontal velocity $u$, and vertical velocity $w$ at Gauges (

**A**–

**C**).

**Figure 5.**Comparison between the numerical results and the analytical solution for the vertical distribution of horizontal velocity at the center of the basin. (

**a**) Homogeneous Dirichlet boundary condition; (

**b**) Neumann boundary condition.

**Figure 6.**Comparison between the numerical results and the analytical solution for individual velocity components and velocity magnitude for the Ekman profile.

**Figure 7.**Computational domain geometry for tidal flow in the west region of Bohai Bay. (

**a**) Computational domain geometry; (

**b**) Gauge points.

**Figure 8.**Flow field at different levels at the flood tide (left column) and the ebb tide (right column). (

**a**) surface ($\mathsf{\sigma}=0$); (

**b**) middle ($\mathsf{\sigma}=-0.5$); (

**c**) bottom ($\mathsf{\sigma}=-1.0$).

**Figure 9.**Comparison between the numerical results and the measured surface elevation at Gauges P1 and P2 for the Bohai Bay case. (

**a**) P1; (

**b**) P2.

**Figure 10.**Comparison between the simulated results and the measured data for velocity (left column) and flow direction (right column) at Gauges P3, P4, and P5 for the Bohai Bay case. (

**a**) P3; (

**b**) P4; (

**c**) P5.

Variables | Explanation |
---|---|

$u\left(x,y,z,t\right)$ | velocity of water in x direction |

$v\left(x,y,z,t\right)$ | velocity of water in y direction |

$w\left(x,y,z,t\right)$ | velocity of water in z direction |

g | acceleration due to gravity |

$\theta $ | angle of geographical latitude |

$\varpi $ | magnitude of the angular velocity of the Earth |

$f=2\varpi \mathrm{sin}\theta $ | Coriolis parameter |

${K}_{m}$ | vertical eddy viscosity coefficient |

${K}_{h}$ | horizontal eddy viscosity coefficient |

${\rho}_{0}$ | density |

${\tau}_{sx}$ | wind stress in x direction |

${\tau}_{sy}$ | wind stress in y direction |

$\eta \left(x,y,t\right)$ | the surface elevation |

${u}_{\eta}\left(x,y,t\right)$ | surface velocity of water in x direction |

${v}_{\eta}\left(x,y,t\right)$ | surface velocity of water in y direction |

$b\left(x,y\right)$ | bottom elevation |

${u}_{b}\left(x,y,t\right)$ | bottom velocity of water in x direction |

${v}_{b}\left(x,y,t\right)$ | bottom velocity of water in y direction |

${z}_{0}\left(x,y,t\right)$ | half height of the bottommost element |

${u}_{c}\left(x,y,t\right)$ | $\mathrm{velocity}\mathrm{of}\mathrm{water}\mathrm{at}{z}_{0}$ in x direction |

${v}_{c}\left(x,y,t\right)$ | $\mathrm{velocity}\mathrm{of}\mathrm{water}\mathrm{at}{z}_{0}$ in y direction |

${C}_{f}\left(x,y,t\right)$ | drag coefficient |

$\kappa $ | von Karman constant |

${z}_{0}^{b}\left(x,y\right)$ | bottom roughness parameter |

$\omega \left(x,y,\sigma ,t\right)$ | vertical velocity in the computational domain |

$\overline{U}\left(x,y,t\right)$ | depth-averaged velocity in x direction |

$\overline{V}\left(x,y,t\right)$ | depth-averaged velocity in y direction |

$D\left(x,y,t\right)=\eta \left(x,y,t\right)-b\left(x,y\right)$ | water depth |

Ne | N_{L} | Er(D) | CR(D) | Er(Du) | CR(Du) | Er(Dv) | CR(Dv) | Er(ω) | CR(ω) |
---|---|---|---|---|---|---|---|---|---|

1 | 1 | 0.00033 | 0.00169 | 0.00169 | $5.224\times 1{0}^{-5}$ | ||||

4 | 2 | $6.125\times 1{0}^{-5}$ | 2.42 | 0.00031 | 2.47 | 0.00031 | 2.47 | $1.277\times 1{0}^{-5}$ | 2.03 |

16 | 4 | $1.340\times 1{0}^{-5}$ | 2.19 | $6.086\times 1{0}^{-5}$ | 2.33 | $6.086\times 1{0}^{-5}$ | 2.33 | $3.748\times 1{0}^{-6}$ | 1.77 |

64 | 8 | $3.170\times 1{0}^{-6}$ | 2.08 | $1.350\times 1{0}^{-5}$ | 2.17 | $1.350\times 1{0}^{-5}$ | 2.17 | $1.239\times 1{0}^{-6}$ | 1.60 |

256 | 16 | $7.617\times 1{0}^{-7}$ | 2.06 | $3.083\times 1{0}^{-6}$ | 2.13 | $3.083\times 1{0}^{-6}$ | 2.13 | $4.527\times 1{0}^{-7}$ | 1.45 |

Ne | N_{L} | Er(D) | CR(D) | Er(Du) | CR(Du) | Er(Dv) | CR(Dv) | Er(ω) | CR(ω) |
---|---|---|---|---|---|---|---|---|---|

1 | 1 | $9.997\times 1{0}^{-6}$ | 0.00015 | 0.00015 | $4.95\times 1{0}^{-6}$ | ||||

4 | 2 | $1.353\times 1{0}^{-6}$ | 3.10 | $1.804\times 1{0}^{-5}$ | 3.10 | $1.804\times 1{0}^{-5}$ | 3.10 | $5.71\times 1{0}^{-7}$ | 3.11 |

16 | 4 | $1.703\times 1{0}^{-7}$ | 3.04 | $2.187\times 1{0}^{-6}$ | 3.04 | $2.187\times 1{0}^{-6}$ | 3.04 | $1.13\times 1{0}^{-7}$ | 2.33 |

64 | 8 | $2.196\times 1{0}^{-8}$ | 3.05 | $2.649\times 1{0}^{-7}$ | 3.05 | $2.649\times 1{0}^{-7}$ | 3.05 | $3.00\times 1{0}^{-8}$ | 1.91 |

256 | 16 | $3.276\times 1{0}^{-9}$ | 3.02 | $3.264\times 1{0}^{-8}$ | 3.02 | $3.264\times 1{0}^{-8}$ | 3.02 | $6.99\times 1{0}^{-9}$ | 2.10 |

N_{L} | 5 | 10 | 15 | 20 |
---|---|---|---|---|

Homogeneous Dirichlet boundary condition | ||||

RMSE | 0.0107 | 0.0051 | 0.0041 | 0.0037 |

Neumann boundary condition | ||||

RMSE | 0.0219 | 0.0138 | 0.0114 | 0.0103 |

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**MDPI and ACS Style**

Ran, G.; Zhang, Q.; Chen, Z. Development of a Three-Dimensional Hydrodynamic Model Based on the Discontinuous Galerkin Method. *Water* **2023**, *15*, 156.
https://doi.org/10.3390/w15010156

**AMA Style**

Ran G, Zhang Q, Chen Z. Development of a Three-Dimensional Hydrodynamic Model Based on the Discontinuous Galerkin Method. *Water*. 2023; 15(1):156.
https://doi.org/10.3390/w15010156

**Chicago/Turabian Style**

Ran, Guoquan, Qinghe Zhang, and Zereng Chen. 2023. "Development of a Three-Dimensional Hydrodynamic Model Based on the Discontinuous Galerkin Method" *Water* 15, no. 1: 156.
https://doi.org/10.3390/w15010156