# A Beach Profile Evolution Model Driven by the Hybrid Shock-Capturing Boussinesq Wave Solver

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## Abstract

**:**

## 1. Introduction

## 2. Model Development

#### 2.1. Wave Module

_{d}in the above equations are shown below:

_{x}is the outward unit vector normal to Ω. The computation domain is discretized into finite rectangular cells, and integrating the control Equation (10) in the conservation form within the finite volume $[{x}_{i-1/2},{x}_{i+1/2}]\times [{t}_{n},{t}_{n+1}]$ and applying Green Theorem, the discrete equations can be obtained as follows:

#### 2.2. Wave Boundary Layer Module

_{t}is obtained by using the mixing-length theory:

#### 2.3. Morphology Module

_{d}is the topographic height; ρ

_{s}is the density of sediment and ρ is the density of water; and ξ is the sediment porosity, which is usually taken to be 0.4 in size.

## 3. Hydrodynamic Validations

#### 3.1. Validations of Wave Boundary Layer

_{0}is the amplitude of the flow velocity, and tanh is used to control the u

_{b}so that it can change smoothly. In Test 10, the amplitude of the free stream velocity is U

_{0}= 2 m/s, the orbital excursion amplitude is a = 3.1 m, the viscosity is υ = 0.0114 cm

^{2}/s, the period of free stream oscillatory flow is T = 9.72 s and the Reynolds number is Re = 6 × 10

^{6}. The vertical grid size in the boundary layer is Δz = 0.0015 m, and the time step is Δt = 0.01 s. In Test 13, the parameters are unchanged and the bottom bed roughness is set to K

_{s}= 0.84 mm.

#### 3.2. Validations of Solitary Wave

## 4. Morphology Validations

#### 4.1. Sandbar Evolution under Bragg Resonance

_{0}= 0.5 m and period T = 8.0 s. A sponge layer is placed at x = 860 m to absorb the wave. The shape amplitude of the underwater sandbar is given by the following equation:

_{b}= 0.64 m. In the numerical calculations, the time step is ∆t = 0.05 s, the spatial step is ∆x = 0.2 m, the sediment density is ρ

_{s}= 2650 kg/m

^{3}, the sediment grain size is d = 0.4 mm, the porosity is ξ = 0.4 and the coefficients are A = 11.0 and b = 1.65.

#### 4.2. Sandbar Generation under Solitary Waves

^{3}, the sediment grain size is D

_{50}= 0.4 mm, the sediment density is ρ

_{s}= 2650 kg/m

^{3}, the sediment settling speed is ${\omega}_{fall}$ = 0.02 m/s, the sediment porosity is ξ = 0.4, the suspending mass efficiency coefficient is ε

_{s}= 0.01, the bedload mass efficiency coefficient is ε

_{b}= 0.135, the sediment internal friction angle is $\varphi $ = 32° and the friction coefficient is ${c}_{f}$ = 0.02. In the experiment, the locations of four wave height gauges in the numerical model correspond to x = 151 m, 153 m, 155 m and 157 m, respectively, and the locations of the velocity instruments correspond to x = 151 m, 156 m and 157 m, respectively.

#### 4.3. Sandbar Generation and Migration under Regular Waves

_{s}= 2650 kg/m

^{3}, the water density is ρ = 1000 kg/m

^{3}, the bedload mass efficiency coefficient is ε

_{b}= 0.135, the suspending mass efficiency coefficient is ε

_{s}= 0.01, the sediment settling speed is ${\omega}_{fall}$ = 0.03 m/s, the sediment internal friction angle is $\varphi $ = 32° and the friction coefficient is ${c}_{f}$ = 0.05.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Comparisons of wave evaluations between numerical results and experimental data at different moments of the non-breaking condition.

**Figure 6.**Comparisons of wave evaluations between numerical results and experimental data at different moments of the breaking condition.

**Figure 9.**Time series of water surface elevation at four locations [37].

**Figure 10.**Time series of velocities and volumetric sediment concentration at three locations [37].

**Figure 11.**Profile and topographic changes due to 3 groups wave action. (

**a**) Profile change, (

**b**) Topographic change.

**Figure 12.**Profile and topographic changes due to 6 groups wave action. (

**a**) Profile change, (

**b**) Topographic change.

**Figure 13.**Profile and topographic changes due to 9 groups wave action. (

**a**) Profile change, (

**b**) Topographic change.

Case | Slope | Depth/ $\mathbf{m}$ | Height/ $\mathbf{m}$ | Period/s |
---|---|---|---|---|

Test R3 | 1:20 | 0.45 | 0.11 | 1.2 |

Test R4 | 1:10 | 0.45 | 0.13 | 1.2 |

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**MDPI and ACS Style**

Wang, P.; Fang, K.; Liu, Z.; Sun, J.; Zhou, L.
A Beach Profile Evolution Model Driven by the Hybrid Shock-Capturing Boussinesq Wave Solver. *Water* **2023**, *15*, 3799.
https://doi.org/10.3390/w15213799

**AMA Style**

Wang P, Fang K, Liu Z, Sun J, Zhou L.
A Beach Profile Evolution Model Driven by the Hybrid Shock-Capturing Boussinesq Wave Solver. *Water*. 2023; 15(21):3799.
https://doi.org/10.3390/w15213799

**Chicago/Turabian Style**

Wang, Ping, Kezhao Fang, Zhongbo Liu, Jiawen Sun, and Long Zhou.
2023. "A Beach Profile Evolution Model Driven by the Hybrid Shock-Capturing Boussinesq Wave Solver" *Water* 15, no. 21: 3799.
https://doi.org/10.3390/w15213799