# A Wetting and Drying Approach for a Mode-Nonsplit Discontinuous Galerkin Hydrodynamic Model with Application to Laizhou Bay

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Three-Dimensional Model of the DG Method

#### 2.1. Governing Equations and Boundary Conditions

#### 2.2. Discontinuous Galerkin Discretization

## 3. Wetting and Drying Treatments

#### 3.1. Wet–Dry Nodes and Elements

_{p}as the total number of interpolating nodes in each element. The maximal free surface elevation and the maximal bottom elevation in each element are denoted as ${\eta}_{\mathrm{max}}$ and ${z}_{b,\mathrm{max}}$, respectively, where ${\eta}_{\mathrm{max}}=\mathrm{max}\left\{D+{z}_{b}\right\}$. Usually, a positive threshold ${D}_{\mathrm{crit}}$ of a minimum water depth, which is similar to the thickness of the viscous bottom sublayer [13,30], is defined, and then all the nodes can be divided into two categories:

- wet nodes, for $D\ge {D}_{\mathrm{crit}}$;
- dry nodes, for $D<{D}_{\mathrm{crit}}$.

- wet elements, for $n={N}_{p}$;
- dry elements, for $n=0$;
- semidry elements for $0<n<{N}_{p}$.

**Figure 2.**Judgement process of wet and dry elements. Wet elements can be first distinguished by Judge1. Two conditions of elements are defined as dry elements with Judge1 and Judge2. Semidry elements can be divided into flooding elements and dam-breaking elements by Judge3 after reconstructing the water depth and horizontal momentum.

#### 3.2. Calculation of Elements in the Model

#### 3.3. Combined 2D and 3D Limiters

## 4. Tests and Analysis of the Wet–Dry Approach

#### 4.1. Well-Balanced Test

^{1}errors and the L

^{∞}errors of water depth and momentum in the whole domain, which are calculated by the values of all the nodes at 30 s and the initial time, are listed in Table 1, whose orders of magnitude are 10

^{−12}and 10

^{−11}, respectively.

#### 4.2. Long Waves Climbing a Sloping Beach

#### 4.3. Basin with Variable Slope under a Tidal Cycle

#### 4.4. Discussion on the Use of 2D and 3D Limiters

^{1}errors and the L

^{∞}errors of water depth and momentum are fixed (or may be zero), while the employment of 2D and 3D limiters causes variable errors first from the reconstruction of semidry elements and gradually expands to the whole region.

## 5. Application of the Model for Tidal Flow in Laizhou Bay

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Basic elements in the model. Black nodes at the vertices are interpolating nodes for one order interpolating basis functions.

**Figure 3.**Calculation process in the model. The explicit solution contains the processes of solving convection, horizontal diffusion, bottom topography terms, and the calculation of surface elevation. The implicit solution contains the processes of solving vertical diffusion with GOTM calculating vertical diffusion coefficients and Coriolis acceleration terms.

**Figure 4.**Grids and the bottom topography of the well-balanced test in [45].

**Figure 6.**Simulated (blue dots) and analytical (red lines) water surface elevations of a long wave run-up at four different times shown in (

**a**–

**d**). Dashed lines are at the locations of the initial wet and dry interface, and dashed-dotted lines are at the locations of the current wet and dry front.

**Figure 7.**The initial water depth of the basin in [47].

**Figure 9.**Computational domain, WD regions, and locations of the tidal elevation station (T) and four tidal flow velocity stations (V1–V4) in Laizhou Bay.

**Figure 10.**The simulated and measured results of (

**a**) the tidal elevation and layered currents. (

**b1**–

**d4**) give the layered velocity of the V1–V4 stations, and (

**e1**–

**g4**) show the layered direction of the four stations in Laizhou Bay.

**Figure 11.**Flow fields and water depth at ebb tide (

**a**) and flood tide (

**b**) in the southwest area of Laizhou Bay in 2003. The local flow field at the jetty head is given. Arrows show the depth-averaged current velocity with directions.

**Figure 12.**Variation in surface water elevation at ebb tide (

**a**) and flood tide (

**b**) around a jetty in the southwest area of Laizhou Bay. The cyan surface represents the surface water elevation, and the yellow–blue gradient surface is the bottom terrain. The grey structure is a jetty with a harbor at its head.

**Table 1.**Error analysis of the case in [45].

L^{1} error | L^{∞} error | ||||
---|---|---|---|---|---|

D | Du | Dv | D | Du | Dv |

3.52 × 10^{−12} | 3.77 × 10^{−13} | 3.79 × 10^{−13} | 7.48 × 10^{−11} | 3.76 × 10^{−11} | 3.74 × 10^{−11} |

Limiters in Model | No Limiter | 2D Limiter | 3D Limiter | Combined 2D and 3D Limiters |
---|---|---|---|---|

Status | Breakdown | Breakdown | Breakdown | Normal |

Running time (s) | 3070.5 | 3104.0 | 3074.0 | 7200.0 (end) |

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**MDPI and ACS Style**

Chen, Z.; Zhang, Q.; Ran, G.; Nie, Y.
A Wetting and Drying Approach for a Mode-Nonsplit Discontinuous Galerkin Hydrodynamic Model with Application to Laizhou Bay. *J. Mar. Sci. Eng.* **2024**, *12*, 147.
https://doi.org/10.3390/jmse12010147

**AMA Style**

Chen Z, Zhang Q, Ran G, Nie Y.
A Wetting and Drying Approach for a Mode-Nonsplit Discontinuous Galerkin Hydrodynamic Model with Application to Laizhou Bay. *Journal of Marine Science and Engineering*. 2024; 12(1):147.
https://doi.org/10.3390/jmse12010147

**Chicago/Turabian Style**

Chen, Zereng, Qinghe Zhang, Guoquan Ran, and Yang Nie.
2024. "A Wetting and Drying Approach for a Mode-Nonsplit Discontinuous Galerkin Hydrodynamic Model with Application to Laizhou Bay" *Journal of Marine Science and Engineering* 12, no. 1: 147.
https://doi.org/10.3390/jmse12010147