# Study on Numerical Modeling for Pollutants Movement Based on the Wave Parabolic Mild Slope Equation in Curvilinear Coordinates

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Numerical Model of Wave Propagation

_{g}is the wave propagation group velocity; p = CC

_{g}; h is the local water depth; r is the density of seawater; g is the acceleration of gravity; E = ρgh

^{2}/8, is the wave energy; E

_{s}is the wave energy where the wave height is stable after wave breaking [20]; D is the factor considering the nonlinear effect of waves; D

_{b}is the energy loss rate of wave breaking; K ≈ 0.15 is the empirical coefficient. ξ and η are independent variables in the transformed image domain; and g

_{ξ}and g

_{η}are Lame coefficients.

_{b}can be determined by the following equation [21].

_{0}, wave number k

_{0}and wave incident angle in the cartesian coordinate system with respect to the main wave propagation direction a

_{0}at the boundary, the wave incident boundary can be set as,

_{r}is the reflection coefficient; and ε

_{r}is the phase difference. For a fully reflective boundary, the reflection coefficient is taken as 1, and Equation (4) is transformed into

_{1}is the left boundary tangent direction angle and δ

_{2}is the right boundary tangent direction angle.

#### 2.2. Numerical Model of Nearshore Currents

_{Sξ}and F

_{Sη}are the wave radiation stress terms in the ξ and η directions, respectively; τ

_{ξ}and τ

_{η}are the surface shear stress components in the ξ and η directions, respectively; τ

_{bξ}and τ

_{bη}are the bottom shear stress components in the wave-current fields in the ξ and η directions, respectively; A

_{mξ}and A

_{mη}are the lateral turbulent stress terms in the ξ and η directions, respectively.

_{ξξ}, S

_{ξη}, S

_{ηξ}and S

_{ηη}are the wave radiation stress components in the orthogonal curve coordinate system, respectively, and the expressions [23] are

_{0}= 2πa

_{0}/T derived from the linear wave theory is the horizontal velocity of bottom wave water quality point; T is the wave period; a

_{0}= H/(2sinhkh) is the horizontal trajectory amplitude of bottom wave water quality point; and c

_{f}for the bottom friction coefficient under the combined effect of current.

_{ξξ}, σ

_{ξη}, σ

_{ηξ}, σ

_{ηη}are the lateral turbulent stress components.

_{l}is the shore distance of the wave breaking point and N is the uncaused quantity. Outside the breaking zone, μ is usually taken as a constant, and its value is the value at the wave breaking point.

#### 2.3. Numerical Model of Nearshore Pollutant Transport under Co-Action of Waves and Currents

_{cwξ}and Γ

_{cwη}are the diffusion coefficients of pollutants in the ξ and η directions, respectively, under the co-action of waves and currents; S

_{m}is the pollutant source-sink term.

_{wc}is the diffusion coefficient of pollutants under the co-action of waves and currents. D

_{w}is the diffusion coefficient of pollutants under the action of waves alone [27].

_{wξ}and D

_{wη}are the diffusion coefficients of pollutants in the direction of ξ and η under the wave action, respectively; α is the empirical coefficient with different values in different zones, and the following expression is often used,

_{c}is the depth-averaged diffusion coefficient of pollutants under the action of pure currents, and the following empirical formula can be used [28].

_{cξ}and D

_{cη}are the diffusion coefficients of pollutants in ξ and η directions under the action of pure currents, respectively; C is the Chezy coefficient; k

_{m}and k

_{n}[29] can be approximated as 5.93 and 0.15, respectively.

## 3. Results and Discussion

#### 3.1. Validation of Physical Experiments on Nearshore Currents and Pollutant Transport

_{0}is the incident wave height; T is the incident wave period; γ is the ratio parameter for guiding the breaking waves; C

_{f}is the bottom friction coefficient under wave action; λ is the unfactored amount in the lateral turbulence mixing factor equation; L is the shore distance of the pollutant discharge point.

^{−3}. From Figure 6, Figure 7 and Figure 8, it can be seen that the pollutant transport trend is basically steady under the certain given wave incidence angle. In Case 1, the incident wave height and current velocity are relatively large, and the pollutant transport velocity is significantly faster than other cases. Compared with Case 2 and Case 3, the pollutant transport velocity is accelerated in the steep slope case under the same wave incidence condition. In general, the numerical results of pollutant distribution at each moment are in good agreement with the distribution trend of the measured data. Due to the difficulty of measurement in the experiment itself and the fact that the numerical results of pollutant distribution are averaged along the vertical direction of water depth, while the measured results are superimposed, these numerical results inevitably have some differences with the measured data.

#### 3.2. Numerical Simulation of Pollutant Transport in Gourlay Experiment

^{−2}. From the figure, it can be seen that the distribution range of pollutants with relative concentration values above 0.6 is relatively small, mainly concentrated in the area around 1.5 m from the discharge point, while pollutants with relative concentration values between 0.2 and 0.6 occupy a larger distribution range. The pollutants show an obvious rotational development in the fan-shaped mild sloping region depending on the boundary of the computational domain, which is consistent with the eddy motion of the nearshore current in this region. In the process of pollutant transport from the discharge point to the left end of the breakwater, the pollutants show more and more obvious diffusion trend while convective propagation. Especially at the intersection of the fan-shaped mild slope and the left end of the breakwater, the diffusion effect of pollutants distribution is very significant. As can be seen from Figure 14a, this phenomenon is not only related to the diffusion effect in the process of pollutant transport, but also may be related to the presence of small-scale eddy at the intersection leading to the intensification of pollutant diffusion. In the process of pollutants leaving the breakwater and propagating to the upper mild sloping area, the distribution of pollutants becomes significantly narrower, which is not only related to the reduction of pollutants concentration in the area away from the discharge point, but also consistent with the relatively narrow distribution of nearshore currents developing in the same direction in the area. In general, the numerical results of the wav-current fields in this case reasonably explain the pollutant transport phenomenon under this topographic condition.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Comparison of simulated and measured wave height, surface elevation, and current ve−locity (

**a**) Case1, (

**b**) Case2, (

**c**) Case3.

**Figure 6.**Comparison of the simulated and measured data of the pollutant concentration field in Case 1.

**Figure 7.**Comparison of the simulated and measured data of the pollutant concentration field in Case 2.

**Figure 8.**Comparison of the simulated and measured data of the pollutant concentration field in Case 3.

**Figure 9.**Layout of Gourlay experiment [16].

**Figure 10.**Orthogonal calculation grids. (The grid density used in the numerical calculations is three times the grid shown in this figure).

**Figure 14.**Comparison of velocity vectors (

**a**) Gourlay experimental results, (

**b**) Nielsen [31] numerical results, (

**c**) Numerical results in this paper.

Case | Plan Slope | h/m | θ/(°) | H_{0}/m | T/s | γ | C_{f} | λ | L/m |
---|---|---|---|---|---|---|---|---|---|

1 | 1:40 | 0.45 | 0 | 0.09 | 1.0 | 0.65 | 0.023 | 0.007 | 3.0 |

2 | 1:40 | 0.45 | 0 | 0.05 | 2.0 | 0.65 | 0.023 | 0.007 | 3.0 |

3 | 1:100 | 0.18 | 0 | 0.05 | 2.0 | 0.60 | 0.017 | 0.006 | 4.5 |

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**MDPI and ACS Style**

Cui, L.; Jiang, H.; Qu, L.; Dai, Z.; Wu, L.
Study on Numerical Modeling for Pollutants Movement Based on the Wave Parabolic Mild Slope Equation in Curvilinear Coordinates. *Water* **2023**, *15*, 3251.
https://doi.org/10.3390/w15183251

**AMA Style**

Cui L, Jiang H, Qu L, Dai Z, Wu L.
Study on Numerical Modeling for Pollutants Movement Based on the Wave Parabolic Mild Slope Equation in Curvilinear Coordinates. *Water*. 2023; 15(18):3251.
https://doi.org/10.3390/w15183251

**Chicago/Turabian Style**

Cui, Lei, Hengzhi Jiang, Limei Qu, Zheng Dai, and Liguo Wu.
2023. "Study on Numerical Modeling for Pollutants Movement Based on the Wave Parabolic Mild Slope Equation in Curvilinear Coordinates" *Water* 15, no. 18: 3251.
https://doi.org/10.3390/w15183251