# High-Order Boussinesq Equations for Water Wave Propagation in Porous Media

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Derivation of High Order of Boussinesq-Type Equations

#### 2.1. Governing Equations and Boundary Conditions

#### 2.2. Dimensionless Equations

_{0}, and characteristic wavelength l

_{0}as follows

_{r}= 1 and α = 0, Equation (18) recovers to the expressions presented in the original paper of Madsen and Schäffer [45].

#### 2.3. Power Series Solution to the Laplace Equations

#### 2.4. Boussinesq Equations in Terms of Velocity at the Free Surface ${\widehat{\mathit{u}}}_{s}$

#### 2.5. Boussinesq Equations in Terms of Depth-Averaged Velocity ${\overline{\mathit{u}}}_{s}$

#### 2.6. Boussinesq Equations in Terms of Velocity at an Arbitrary Water Column ${\mathit{u}}_{sa}$

#### 2.7. Boussinesq Models Extended to Deeper Water Depth

#### 2.7.1. The Improvement of Equations in Terms of Depth-Averaged Velocity

- (1)
- Keeping at order $O({\mu}^{2})$

- (2)
- Keeping at order $O({\mu}^{4})$

#### 2.7.2. The Improvement of Equations in Terms of ${\mathit{u}}_{s\alpha}$

## 3. Dispersive Analysis on a Horizontal Bottom

_{r}) of the different models is compared to the analytic solution for α

_{1}= 0.2, 2, and 4 s

^{−1}in Figure 2, Figure 3 and Figure 4 (where c is normalized by the analytic value), where Model 1 uses the two set group parameters in Table 1. It can be seen from the figures that Model 2 presents much more accurate results than Model 1, and the maximum error is only 2.5% when the applicable range of water depth is ${k}_{r}h=6.0$ The applicable range of water depth for Model 1 using the second set of parameter value is ${k}_{r}h\le 4.05,4.57,1.58$ with a 5% tolerance error for three considered cases, which is higher than that using the first set of parameter value ${k}_{r}h\le 3.2,3.75,1.67$.

^{−1}. Model 2 again presents better results than Model 1. The figures show that Model 2 is applicable for water depth ${k}_{r}h\le 4.19,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}4.41,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}4.98$ for three cases within 2% error, whilst Model 1 with the first set of parameter values is only applicable for ${k}_{r}h\le 2.0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2.16,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.55$ with 5% error, and with the second set of values only for ${k}_{r}h\le 2.62,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2.68,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.52$. According to the above analysis, Model 2 is better adapted to deep water conditions, and its accuracy in deep water varies for different linear force coefficients.

## 4. Numerical Implementation and Validation

#### 4.1. Numerical Implementation

#### 4.2. Model Validation

_{50}, and n are the porous structure width, mean diameter, and material porosity. Due to the transient nature of solitary waves, incident and reflected waves do not superimpose in the flume. Wave heights of reflected and transmitted waves on the left and right sides of the porous structure can be directly obtained. Therefore, the calculated reflection coefficient is defined as K

_{r}= H

_{L}/H

_{0}, and the transmission coefficient is defined as K

_{t}= H

_{R}/H

_{0}, where H

_{0}is the input wave height of the solitary wave in the wave generation region, and H

_{L}is the wave height on the left side of the solitary wave after passing through the porous structure, H

_{R}is the wave height on the right side of the solitary wave. The numerical results were also compared with the experimental data.

_{50}= 1.43 cm and 2.43 cm. In the numerical simulation, the space interval is 0.01 m, and the time interval is 0.005 s. By setting ${\alpha}_{1}$ = 18 s

^{−1}and ${\alpha}_{2}$ = 234.45 m for the d

_{50}= 1.43 cm case, and ${\alpha}_{1}$ = 6.23 s

^{−1}and ${\alpha}_{2}$ = 137.97 m for the d

_{50}= 2.43 cm case, the reflection coefficient and transmission coefficient are calculated numerically using Model 2, as shown in Figure 10. For the breakwater with d

_{50}= 2.43 cm, both the calculated reflection and transmission coefficients are in good agreement with the experimental results. However, for the breakwater with d

_{50}= 1.43 cm, the calculated reflection coefficients are slightly larger than the experimental data, and the calculated transmission coefficients are slightly smaller than the experimental data, but the trend with wave height (H/h) is consistent with the measured values. Considering the errors introduced by the modeling assumptions and parameters, the model developed in this paper can better simulate the wave interaction with porous structures which has surface-piercing boundaries.

_{50}= 1.6 cm, ${\alpha}_{1}$ = 9.5 s

^{−1}and ${\alpha}_{2}$ = 200 m are used, and for d

_{50}= 2.0 cm, ${\alpha}_{1}$ = 8 s

^{−1}and ${\alpha}_{2}$ = 160 m are used. Figure 11 shows the numerical reflection and transmission coefficients and the measurements from Lynett et al. [4]. For the 15 cm breakwater, both the calculated reflection and transmission coefficients are in good agreement with the experimental results. However, for the 30 cm breakwater, only the reflection coefficients are in good agreement with the experimental data. The numerical results for the thin breakwater are slightly better than those for the wide breakwater. In general, the agreements between the model results and the experiment data are acceptable after choosing the appropriate coefficients.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Losada, I.J.; Patterson, M.D.; Losada, M.A. Harmonic generation past a submerged porous step. Coast. Eng.
**1997**, 31, 281–304. [Google Scholar] [CrossRef] - Gu, Z.; Wang, H. Gravity waves over porous bottoms. Coast. Eng.
**1991**, 15, 497–524. [Google Scholar] [CrossRef] - Lara, J.L.; Losada, I.J.; Liu, P.L.F. Breaking waves over a mild gravel slope: Experimental and numerical analysis. J. Geophys. Res.
**2006**, 111, C11019. [Google Scholar] [CrossRef] - Lynett, P.J.; Liu, P.L.F.; Losada, I.J. Solitary wave interaction with porous breakwaters. J. Waterw. Port Coast. Ocean Eng.
**2000**, 126, 314–322. [Google Scholar] [CrossRef] - Losada, I.J.; Lara, J.L.; Guanche, R.; Gonzalez-Ondina, J.M. Numerical analysis of wave overtopping of rubble mound breakwaters. Coast. Eng.
**2008**, 55, 47–62. [Google Scholar] [CrossRef] - Cheng, Y.Z.; Jiang, C.B.; Wang, Y.Y. A coupled numerical model of wave interaction with porous medium. Ocean Eng.
**2009**, 36, 952–959. [Google Scholar] [CrossRef] - Higuera, P.; Lara, J.L.; Losada, I.J. Three-dimensional interaction of waves and porous coastal structures using OpenFOAM
^{®}. Part I: Formulation and validation. Coast. Eng.**2014**, 83, 243–258. [Google Scholar] [CrossRef] - Sasikumar, A.; Kamath, A.; Bihs, H. Modeling porous coastal structures using a level set method based VRANS-solver on staggered grids. Coast. Eng. J.
**2020**, 62, 198–216. [Google Scholar] [CrossRef] - Mohamed, K. A finite volume method for numerical simulation of shallow water models with porosity. Comput. Fluids
**2014**, 104, 9–19. [Google Scholar] [CrossRef] - Ren, B.; Wen, H.; Dong, P.; Wang, Y. Improved SPH simulation of wave motions and turbulent flows through porous media. Coast. Eng.
**2016**, 107, 14–27. [Google Scholar] [CrossRef] - Gao, J.L.; Zhou, X.J.; Zhou, L.; Zang, J.; Chen, H.Z. Numerical investigation on effects of fringing reefs on low-frequency oscillations within a harbor. Ocean Eng.
**2019**, 172, 86–95. [Google Scholar] [CrossRef] - Gao, J.L.; Ma, X.Z.; Zang, J.; Dong, G.H.; Ma, X.J.; Zhu, Y.Z.; Zhou, L. Numerical investigation of harbor oscillations induced by focused transient wave groups. Coast. Eng.
**2020**, 158, 103670. [Google Scholar] [CrossRef] - Gao, J.L.; Ma, X.Z.; Chen, H.Z.; Zang, J.; Dong, G.H. On hydrodynamic characteristics of transient harbor resonance excited by double solitary waves. Ocean Eng.
**2021**, 219, 108345. [Google Scholar] [CrossRef] - Buccino, M.; Tuozzo, S.; Ciccaglione, M.C.; Calabrese, M. Predicting Crenulate Bay Profiles from Wave Fronts: Numerical Experiments and Empirical Formulae. Geosciences
**2021**, 11, 208. [Google Scholar] [CrossRef] - Kirby, J.T. Boussinesq models and applications to nearshore wave propagation, surf zone processes and wave induced-current. In Advances in Coastal Modelling; Lakhan, V.C., Ed.; Elsevier Science: Amsterdam, The Netherlands, 2003; pp. 1–41. [Google Scholar]
- Madsen, P.A.; Fuhrman, D.R. High-order Boussinesq-type modeling of nonlinear wave phenomena in deep and shallow water. In Advances in Numerical Simulation of Nonlinear Water Waves; Ma, Q.W., Ed.; World Scientific Publishing Co. Pte. Ltd.: Singapore, 2010; pp. 245–285. [Google Scholar]
- Brocchini, M.A. reasoned overview on Boussinesq-type models: The interplay between physics, mathematics and numerics. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.
**2013**, 469, 20130496. [Google Scholar] [CrossRef] - Kirby, J.T. Boussinesq models and their application to coastal processes across a wide range of scales. J. Waterw. Port Coast. Ocean Eng.
**2016**, 142, 03116005. [Google Scholar] [CrossRef] - Sun, J.W.; Fang, K.Z.; Liu, Z.B.; Fan, H.X.; Sun, Z.C.; Wang, P. A review on the theory and application of Boussinesq-type equations for water waves. Haiyang Xuebao
**2020**, 42, 1–11. (In Chinese) [Google Scholar] - Gao, J.L.; Shi, H.B.; Zang, J.; Liu, Y.Y. Mechanism analysis on the mitigation of harbor resonance by periodic undulating topography. Ocean Eng.
**2023**, 281, 114923. [Google Scholar] [CrossRef] - Gao, J.L.; Ma, X.Z.; Dong, G.H.; Chen, H.Z.; Liu, Q.; Zang, J. Investigation on the effects of Bragg reflection on harbor oscillations. Coast. Eng.
**2021**, 170, 103977. [Google Scholar] [CrossRef] - Madsen, P.A.; Murray, R.; Sørensen, O.R. A new form of the Boussinesq equations with improved linear dispersion characteristics. Coast. Eng.
**1991**, 15, 371–388. [Google Scholar] [CrossRef] - Madsen, P.A.; Sørensen, O.R. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry. Coast. Eng.
**1992**, 18, 183–204. [Google Scholar] [CrossRef] - Nwogu, O. Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw. Port Coast. Ocean Eng.
**1993**, 119, 618–638. [Google Scholar] [CrossRef] - Schäffer, H.A.; Madsen, P.A. Further enhancements of Boussinesq-type equations. Coast. Eng.
**1995**, 26, 1–14. [Google Scholar] [CrossRef] - Zou, Z.L. Higher-order Boussinesq equations for rapidly varying topography. Haiyang Xuebao
**2001**, 23, 109–119. (In Chinese) [Google Scholar] - Liu, Z.B.; Sun, Z.C. Two sets of higher-order Boussinesq-type equations for water waves. Ocean Eng.
**2005**, 32, 1296–1310. [Google Scholar] [CrossRef] - Wei, G.; Kirby, J.T.; Grilli, S.T.; Subramanya, R. A fully nonlinear Boussinesq model for surface waves. Part I. Highly nonlinear unsteady waves. J. Fluid Mech.
**1995**, 294, 71–92. [Google Scholar] [CrossRef] - Gobbi, M.F.; Kirby, J.T.; Wei, G. A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)4. J. Fluid Mech.
**2000**, 405, 181–210. [Google Scholar] [CrossRef] - Lynett, P.J.; Liu, P.L.F. Linear analysis of the multi-layer model. Coast. Eng.
**2004**, 51, 439–454. [Google Scholar] [CrossRef] - Zou, Z.L.; Fang, K.Z. Alternative forms of the higher-order Boussinesq equations: Derivations and validations. Coast. Eng.
**2008**, 55, 506–521. [Google Scholar] [CrossRef] - Liu, Z.B.; Fang, K.Z. Two-layer Boussinesq models for coastal water waves. Wave Motion
**2015**, 57, 88–111. [Google Scholar] [CrossRef] - Madsen, P.A.; Bingham, H.B.; Liu, H. A new method for fully nonlinear waves from shallow water to deep water. J. Fluid Mech.
**2002**, 462, 1–30. [Google Scholar] [CrossRef] - Chazel, F.; Benoit, M.; Ern, A.; Piperno, S. A double-layer Boussinesq-type model for highly nonlinear and dispersive waves. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.
**2009**, 465, 2319–2346. [Google Scholar] [CrossRef] - Liu, Z.B.; Fang, K.Z. A new two-layer Boussinesq model for coastal waves from deep to shallow water: Derivation and analysis. Wave Motion
**2016**, 67, 1–14. [Google Scholar] [CrossRef] - Liu, Z.B.; Fang, K.Z.; Cheng, Y.Z. A new multi-layer irrotational Boussinesq-type model for highly nonlinear and dispersive surface waves over a mildly sloping seabed. J. Fluid Mech.
**2018**, 842, 323–853. [Google Scholar] [CrossRef] - Liu, Z.B.; Han, P.X.; Fang, K.Z.; Liu, Y. A high-order nonlinear Boussinesq-type model for internal waves over a mildly-sloping topography in a two-fluid system. Ocean Eng.
**2023**, 285, 115283. [Google Scholar] [CrossRef] - Cruz, E.C.; Isobe, M.; Watanabe, A. Boussinesq equations for wave transformation on porous beds. Coast. Eng.
**1997**, 30, 125–156. [Google Scholar] [CrossRef] - Hsiao, S.; Liu, P.L.F.; Chen, Y. Nonlinear water waves propagating over a permeable bed. Philos. Trans. R. Soc. Lond. Ser. A
**2002**, 458, 1291–1322. [Google Scholar] [CrossRef] - Chen, Q. Fully nonlinear Boussinesq-type equations for waves and currents over porous beds. J. Eng. Mech.
**2006**, 132, 220–230. [Google Scholar] [CrossRef] - Liu, Z.B.; Sun, Z.C. Wave propagating model over a porous seabed. China Sci. Pap.
**2011**, 6, 374–379. (In Chinese) [Google Scholar] - Klonaris, G.T.; Memos, C.D. Compound Boussinesq-type modelling over porous beds. Appl. Ocean Res.
**2020**, 105, 102422. [Google Scholar] [CrossRef] - Hsiao, S.; Hu, K.; Hwung, H. Extended Boussinesq Equations for Water-Wave Propagation in Porous Media. J. Eng. Mech.
**2010**, 136, 625–640. [Google Scholar] [CrossRef] - Fang, K.Z.; Huang, M.H.; Chen, G.L.; Wu, J.; Wu, H.; Jiang, T. Boussinesq Simulation of Coastal Wave Interaction with Bottom-Mounted Porous Structures. J. Mar. Sci. Eng.
**2022**, 10, 1367. [Google Scholar] [CrossRef] - Madsen, P.A.; Schäffer, H.A. Higher-order Boussinesq-type equations for surface gravity waves: Derivation and analysis. Philos. Trans. R. Soc. Lond. Ser. A
**1998**, 356, 3123–3184. [Google Scholar] [CrossRef] - Zou, Z.L.; Liu, Z.B.; Fang, K.Z. Further improvements to the higher-order Boussinesq equations: Bragg reflection. Coast. Eng.
**2009**, 56, 672–687. [Google Scholar] [CrossRef] - Peregrine, D.H. Long waves on a beach. J. Fluid Mech.
**1967**, 27, 815–827. [Google Scholar] [CrossRef] - Liu, Z.B.; Fang, K.Z.; Zou, Z.L. Boussinesq wave equations with full nonlinear characteristics at order O(µ2). J. Harbin Eng. Univ.
**2012**, 33, 556–561. (In Chinese) [Google Scholar] - Kirby, J.T.; Wei, G.; Chen, Q.; Kennedy, A.B.; Dalrymple, R.A. FUNWAVE 1.0 Fully Nonlinear BOUSSINESQ Wave Model Documentation and User’s Manual; Report, Center for Applied Coastal Research; University of Delawar: Newwark, NJ, USA, 1998. [Google Scholar]
- Vidal, C.; Losada, M.A.; Medina, R.; Rubio, J. Solitary wave transmission through porous breakwaters. In Proceedings of the 24th International Conference on Coastal Engineering, Costa del Sol-Malaga, Spain, 20–25 June 1988; pp. 1073–1083. [Google Scholar]
- Lin, P.; Karunarathna, S.A. Numerical study of Solitary wave interaction with porous breakwater. J. Waterw. Port Coast. Ocean Eng.
**2007**, 133, 352–363. [Google Scholar] [CrossRef]

**Figure 2.**Non-dimensional wave celerity versus water depth (${\alpha}_{1}=0.2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_{r}=1$).

**Figure 3.**Non-dimensional wave celerity versus water depth (${\alpha}_{1}=2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_{r}=1$).

**Figure 4.**Non-dimensional wave celerity versus water depth (${\alpha}_{1}=4,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_{r}=1$).

**Figure 5.**Non-dimensional imaginary wave number versus water depth (${\alpha}_{1}=0.2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_{r}=1$).

**Figure 6.**Non-dimensional imaginary wave number versus water depth (${\alpha}_{1}=2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_{r}=1$).

**Figure 7.**Non-dimensional imaginary wave number versus water depth (${\alpha}_{1}=4,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{c}_{r}=1$).

**Figure 10.**The comparison between computed (line) and measured (points) reflection coefficient (solid line and rhombus) and transmission coefficient (dashed line and circle) in Vidal et al.’s experiment [50].

**Figure 11.**The comparison between computed (line) and measured (points) reflection coefficient (solid line and rhombus) and transmission coefficient (dashed line and circle) in Lynett et al.’s experiment [4].

Model Sets | Parameter Values |
---|---|

1 | β_{1} = −0.0013, β_{2} = −0.0654 (I) or β_{1} = 0.0073, β_{2} = −0.064 (II) |

2 | γ_{1} = 1/9, γ_{2} = 0.146488, γ_{3} = 1/945, γ_{4} = 0.00798359 |

3 | B = −0.4 or B = −0.395 |

4 | δ_{1}= 0.101, δ_{3} = 0.039, B = −0.305, δ_{3} = 0.082 *, δ_{4} = 0.162 * |

*****The two parameters are determined by reanalysis of the shoaling property, which differs from Madsen and Schäffer [45].

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, P.; Liu, Z.; Fang, K.; Sun, J.; Gou, D.
High-Order Boussinesq Equations for Water Wave Propagation in Porous Media. *Water* **2023**, *15*, 3900.
https://doi.org/10.3390/w15223900

**AMA Style**

Wang P, Liu Z, Fang K, Sun J, Gou D.
High-Order Boussinesq Equations for Water Wave Propagation in Porous Media. *Water*. 2023; 15(22):3900.
https://doi.org/10.3390/w15223900

**Chicago/Turabian Style**

Wang, Ping, Zhongbo Liu, Kezhao Fang, Jiawen Sun, and Daxun Gou.
2023. "High-Order Boussinesq Equations for Water Wave Propagation in Porous Media" *Water* 15, no. 22: 3900.
https://doi.org/10.3390/w15223900