# Stabilized Formulation for Modeling the Erosion/Deposition Flux of Sediment in Circulation/CFD Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modified Erosion/Deposition Equations Based on Moving-Bottom Formulation

#### 2.1. Deposition

#### 2.2. Erosion

#### 2.3. Modified Erosion/Deposition Formula

## 3. Numerical Tests

#### 3.1. 1DV Model

^{−2}s

^{−1}, $\alpha =6.5$, $\beta =1.0$, and ${\tau}_{CE}=0.3$ N m

^{−2}, which represents erosion over the soft mud layer. Although the choice of these values is arbitrary, it does not affect the conclusions of the present study. Just beneath the water column is a partially-consolidated mud layer with ${C}_{dry}=530$ g L

^{−1}. In Equation (21), $\tau $ is the result of combined wave current forces. Assuming that the wave orbital velocity is aligned with the current velocity, $\tau $ can be written as [36]

^{−1}is used corresponding to the particle diameter ${d}_{0}\approx 46$ µm based on the Stokes law. A steady current is forced by a pressure gradient resulting in a depth-averaged steady current velocity = 0.2 m s

^{−1}. Using the log law with ${z}_{0}={d}_{0}/12$, the simulated ${\tau}_{c}$ is equal to $0.0014$ N m

^{−2}, which is less than ${\tau}_{CE}$ and would not induce any suspension. Three cases associated with different wave conditions are simulated using ${u}_{orb}=0.55$, 0.70, and 0.82 m s

^{−1}. Assuming that the wave period is 10 s and the wavelength is 30 m, thtese ${u}_{orb}$s correspond to wave heights of 1.17, 1.49, and 1.75 m, respectively. Using the formulation for ${f}_{w}$ given in Madsen et al. [37], these three cases correspond to ${\tau}_{w}=0.85$, 1.33, and 1.74 N m

^{−2}. For each wave condition, simulations are conducted according to various modeling approaches. The first is the moving-grid simulation using the original deposition/erosion formulation (Equation (2)). Because it strictly conserves volume, it gives the most realistic and accurate SSC results. Both the second and third modeling approaches are performed on fixed grids. In the second case, we apply the modified deposition/erosion formulation (Equations (17) and (18)). In the third case, the standard approach that directly employs the original deposition/erosion formulation is applied. Sediment erosion begins after the current reaches a steady state, during which eddy viscosity is of a parabolic shape. Simulations are then performed for a three-hour duration, which gives a significant change in depth due to erosion in the case of the strongest wave.

^{−1}(Figure 2a), the wave-induced shear stress is relatively weak and different modeling approaches give almost the same results. As ${\tau}_{w}$ increases, the difference among cases using different approaches becomes visible, as shown in Figure 2b. In the case when the wave-induced stress is the largest, as shown in Figure 2c, a significant overestimate of SSC can be found using the original erosion/deposition formula. Figure 2c also shows that the modified erosion/deposition formula greatly eliminates the overestimates, which demonstrates the importance of applying the modified erosion/deposition formula when suspended sediment modeling is conducted on the fixed-grid framework. Only little differences are found between cases of the original erosion/deposition formula on moving grid and of the modified erosion/deposition formula, which is due to the difference of the bed shear stresses induced by changing depth in the moving-grid case. Comparison of SSC profiles in different wave conditions in Figure 2 shows that SSC profiles dramatically increase as ${\tau}_{w}$ increases. This is due to the exponential formula (Equation (21)) employed for bed erosion, which is potentially ill posed when applied to the strong wave case on fixed grids without the present modified erosion/deposition formula.

^{−1}here) at approximately 10 h.

#### 3.2. Three-Dimensional Sediment Transport Modeling of San Francisco Bay, California, USA

^{−1}(coarse), ${R}_{1}$ = 5% (${R}_{2}$ = 95%) in regions where the depth < 5 m and ${R}_{1}$ = 30% (${R}_{2}$ = 70%) in regions where the depth ≥ 5 m give the best agreement with the field data. Here, following previous studies, we use these setup conditions with the original and modified resupension formulas for comparison.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic plot to show modeled deposition (

**a**); real deposition (

**b**); modeled erosion (

**c**); and real erosion (

**d**).

**Figure 2.**Vertical profiles of SSC for ${\tau}_{w}=0.85$ (

**a**), 1.32 (

**b**), and 1.74 N m

^{−2}(

**c**) in the cases of modified erosion/deposition formulation with fixed grids (black solid line), original formulation with moving grids (black dashed line), and original formulation with fixed grids (gray solid line) in different wave conditions.

**Figure 3.**Time series of SSC at the bottommost cell in the cases of modified erosion/deposition formulation with fixed grids (black solid line), original formulation with moving grids (black dashed line), and original formulation with fixed grids (gray solid line).

**Figure 4.**Model bathymetry (in m below the mean higher high water (MHHW)) of North San Francisco Bay (

**a**), following Chua and Fringer [42], in which the white contours indicate a depth 5 m to separate shallow-water shoals (depth < 5 m) from the deep channel (depth > 5 m), and modeled surface elevation (

**b**) and RMS wave height (${h}_{w,rms}$) (

**c**) at 4:30 p.m. on 13 September 2009. The cross in (

**b**) indicates the location of the point measurement of SSC and wave height. Distances are relative to the point measurement marked by the cross in (

**b**). The gray solid line ($\overline{AB}$) in (

**b**) indicates the transect of SSC profiles presented in Figure 7.

**Figure 5.**San Francisco Bay coastlines along with the locations of wind stations (black triangles) and the tidal gauge (black square).

**Figure 6.**Time series of the modeled root mean squared (RMS) wave height (${h}_{w,rms}$) measured at the point indicated by the cross in Figure 4b). The gray dashed line indicates the time point of 4:30 p.m. on 13 September 2009, which corresponds to the time point in Figure 4, Figure 7, and Figure 8a.

**Figure 7.**Modeled SSC along the transect shown in Figure 4b at 4:30 p.m. on 13 September 2009 in cases with the modified resuspension formula (

**a**) and with the original resuspension formula (

**b**).

**Figure 8.**Representative snapshots of the near-surface SSC at 04:30 p.m. (

**a**), 05:30 p.m. (

**b**), 06:30 p.m. (

**c**), and 07:30 p.m. (

**d**) on 13 September 2009 predicted using the original resuspension formula without the present modification.

**Figure 9.**Comparison of time series of the modeled near-surface SSC measured at the point indicated by the cross in Figure 4b) using the original resuspension formula with (gray solid) and without (black dashed) waves and the modified resuspension formula with waves (black solid).

**Table 1.**Parameters of the multilayer bed model. Here, ${\rho}_{bd}$ is the dry density, ${T}_{c}$ is the consolidation rate, ${h}_{L}$ is the layer height, and the other parameters are defined in Equation (21).

Layer No. | ${\mathit{\rho}}_{\mathit{bd}}$ (g m^{−3}) | ${\mathit{\tau}}_{\mathit{CE}}$ (N m^{−2}) | $\mathit{\alpha}$ | ${\mathit{E}}_{\mathit{b}\mathbf{,}\mathbf{0}}$ (g m^{−2} s^{−1}) | $\mathit{\beta}$ | ${\mathit{T}}_{\mathit{c}}$ (g m^{−2} s^{−1}) | ${\mathit{h}}_{\mathit{L}}$ (m) |
---|---|---|---|---|---|---|---|

1 | 75,000 | 0.1 | 4.5 | 0.01 | 1 | 0.0002 | 0.1 |

2 | 530,000 | 0.4 | 4.5 | 0.01 | 1 | 0.0002 | 0.5 |

3 | 1,200,000 | 1.2 | 4.5 | 0.01 | 1 | 0.0002 | 4.0 |

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**MDPI and ACS Style**

Chou, Y.-J.; Shao, Y.-C.; Sheng, Y.-H.; Cheng, C.-J.
Stabilized Formulation for Modeling the Erosion/Deposition Flux of Sediment in Circulation/CFD Models. *Water* **2019**, *11*, 197.
https://doi.org/10.3390/w11020197

**AMA Style**

Chou Y-J, Shao Y-C, Sheng Y-H, Cheng C-J.
Stabilized Formulation for Modeling the Erosion/Deposition Flux of Sediment in Circulation/CFD Models. *Water*. 2019; 11(2):197.
https://doi.org/10.3390/w11020197

**Chicago/Turabian Style**

Chou, Yi-Ju, Yun-Chuan Shao, Yi-Hao Sheng, and Che-Jung Cheng.
2019. "Stabilized Formulation for Modeling the Erosion/Deposition Flux of Sediment in Circulation/CFD Models" *Water* 11, no. 2: 197.
https://doi.org/10.3390/w11020197