Modeling the Interaction of Water Waves with Porous Coastal Structures
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 142, Issue 6
Abstract
The mathematical modeling of the interaction of water waves with porous coastal structures has continuously been among the most relevant challenges in coastal engineering research and practice. Finding a tool to better predict essential processes, relevant to the functionality and stability of breakwaters and jetties, and how they are affected by permeability, has been hampered by computational limitations that are being overcome. Over the last 60 years, the Journal of Waterway, Port, Coast, and Ocean Engineering has witnessed gradual developments leading from linearized solutions based on wave theories and constant friction coefficients to very sophisticated Eulerian or Lagrangian solvers of the Navier-Stokes (NS) equations, including turbulence within porous media. Today, although not without difficulty, the first steps are being made toward addressing the simulation of a fully three-dimensional interaction of complete sea states with porous structures at prototype scale. In this paper, after posing the mathematical foundations of the problem, the solution techniques available in the literature are reviewed. Linear solutions based on potential theory; depth-integrated solutions, including Boussinesq approximations; and solutions based on the NS equations, in both the Eulerian and Lagrangian frameworks, are covered. Then, turbulence modeling on porous media is discussed. Conclusions and a discussion of future research directions close the paper. It is shown that, after so many years, some fundamental questions still remain unanswered, leaving challenges open for future research.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The authors express their gratitude to the editorial board of the Journal of Waterway, Port, Coastal, and Ocean Engineering for inviting us to participate in this special issue for the journal’s 60th anniversary.
References
Akbari, H. (2014). “Modified moving particle method for modeling wave interaction with multi layered porous structures.” Coastal Eng., 89, 1–19.
Akbari, H., and Namin, M. M. (2013). “Moving particle method for modeling wave interaction with porous structures.” Coastal Eng., 74, 59–73.
Altomare, C., Crespo, A. J., Domínguez, J. M., Gómez-Gesteira, M., Suzuki, T., and Verwaest, T. (2015). “Applicability of smoothed particle hydrodynamics for estimation of sea wave impact on coastal structures.” Coastal Eng., 96, 1–12.
Becker, M., and Teschner, M. (2007). “Weakly compressible SPH for free surface flows.” Proc., 2007 ACM SIGGRAPH/Eurographics Symp. on Computer Animation, Eurographics Association, Aire-la-Ville, Switzerland, 209–217.
Bogdanov, I. I., Guerton, F., Kpahou, J., and Kamp, A. M. (2011). “Direct pore-scale modeling of two-phase flow through natural media.” Proc., 2011 COMSOL Conf., COMSOL, Inc., Burlington, MA.
Burcharth, H., and Andersen, O. (1995). “On the one-dimensional steady and unsteady porous flow equations.” Coastal Eng., 24(3–4), 233–257.
Carman, P. C. (1937). “Fluid flow through granular beds.” Trans. Inst. Chem. Eng., 15, 150–166.
ComFlow [Computer software]. MARIN, Wageningen, the Netherlands.
Cruz, E. C., and Chen, Q. (2006). “Fundamental properties of Boussinesq-type equations for wave motion over a permeable bed.” Coastal Eng., 48(3), 225–256.
Cruz, E. C., Isobe, M., and Watanabe, A. (1997). “Boussinesq equations for wave transformation on porous beds.” Coastal Eng., 30(1–2), 125–154.
Dalrymple, R. A., Hérault, A., Bilotta, G., and Farahani, R. J. (2010) “GPU-accelerated SPH model for water waves and other free surface flows.” Proc., 31st Int. Conf. on Coastal Engineering, Shanghai, ASCE, Reston, VA.
Dalrymple, R. A., Losada, M. A., and Martin, P. (1991). “Reflection and transmission from porous structures under oblique wave attack.” J. Fluid Mech., 224, 625–644.
Dalrymple, R. A., and Rogers, B. D. (2006). “Numerical modeling of water waves with the SPH method.” Coastal Eng., 53(2–3), 141–147.
de Lemos, M. J. S., and Pedras, M. H. J. (2001). “Recent mathematical models for turbulent flow in saturated rigid porous media.” J. Fluids Eng., 123(4), 935.
del Jesus, M., Lara, J. L., and Losada, I. J. (2012). “Three-dimensional interaction of waves and porous coastal structures: Part I: Numerical model formulation.” Coastal Eng., 64, 57–72.
Dentale, F., Donnarumma, G., and Pugliese-Carratelli, E. (2014). “Simulation of flow within armour blocks in a breakwater.” J. Coastal Res., 30(3), 528–536.
Didier, E., Neves, D., Martins, R., and Neves, M. (2014). “Wave interaction with a vertical wall: SPH numerical and experimental modeling.” Ocean Eng., 88, 330–341.
FLOW3D [Computer software]. FlowScience, Inc., Santa Fe, NM.
Gao, R., Ren, B., Wang, G., and Wang, Y. (2012). “Numerical modelling of regular wave slamming on subface of open-piled structures with the corrected SPH method.” Appl. Ocean Res., 34, 173–186.
Garcia, N., Lara, J. L., and Losada, I. J. (2004). “2-D numerical analysis of near-field flow at low-crested permeable breakwaters.” Coastal Eng., 51(10), 991–1020.
Gotoh, H., Khayyer, A., Ikari, H., Arikawa, T., and Shimosako, K. (2014). “On enhancement of incompressible SPH method for simulation of violent sloshing flows.” Appl. Ocean Res., 46, 104–115.
GPUSPH [Computer software]. GPUSPH 〈http://www.gpusph.org/〉.
Guanche, R., Iturrioz, A., and Losada, I. J. (2015). “Hybrid modeling of pore pressure damping in rubble mound breakwaters.” Coastal Eng., 99, 82–95.
Guanche, R., Losada, I. J., and Lara, J. L. (2009). “Numerical analysis of wave loads for coastal structure stability.” Coastal Eng., 56(5–6), 543–558.
Gui, Q., Dong, P., and Shao, S. (2015). “Numerical study of PPE source term errors in the incompressible SPH models.” Int. J. Numer. Methods Fluids, 77(6), 358–379.
Gui, Q., Shao, S., and Dong, P. (2013). “Wave impact simulations by an improved ISPH model.” J. Waterway, Port, Coastal, Ocean Eng., 04014005.
Hall, K. R., Smith, G. M., and Turcke, D. J. (1995). “Comparison of oscillatory flow and stationary flow through porous media.” Coastal Eng., 24(3–4), 217–232.
Herrera, P. A., Massabó, M., and Beckie, R. D. (2009). “A meshless method to simulate solute transport in heterogeneous porous media.” Adv. Water Resour., 32(3), 413–429.
Higuera, P. (2015) “Application of computational fluid dynamics to wave action on structures.” Ph.D. thesis, Univ. de Cantabria, Santander, Spain.
Higuera, P., Lara, J. L., and Losada, I. J. (2014a). “Three-dimensional interaction of waves and porous coastal structures using OpenFOAM. Part I: Formulation and validation.” Coastal Eng., 83, 243–258.
Higuera, P., Lara, J. L., and Losada, I. J. (2014b). “Three-dimensional interaction of waves and porous coastal structures using OpenFOAM. Part II: Applications.” Coastal Eng., 83, 259–270.
Hirt, C., and Nichols, B. (1981). “Volume of fluid (VOF) method for the dynamics of free boundaries.” J. Comput. Phys., 39(1), 201–225.
Hsiao, S. C., Hu, K. C., and Hwung, H. H. (2010). “Extended Boussinesq equations for water wave propagation in porous media.” J. Eng. Mech., 625–640.
Hsiao, S. C., Liu, P. L. F., Hwung, H. H., and Woo, S. B. (2005). “Nonlinear water waves over a three-dimensional porous bottom using Boussinesq-type model.” Coastal Eng., 47(04), 231–253.
Hsu, T., Sakakiyama, T., and Liu, P. L. F. (2002). “A numerical model for wave motions and turbulence flows in front of a composite breakwater.” Coastal Eng., 46(1), 25–50.
Hu, K. C., Hsiao, S. C., Hwung, H. H., and Wu, T. R. (2012). “Three-dimensional numerical modeling of the interaction of dam-break waves and porous media.” Adv. Water Resour., 47, 14–30.
Hur, D. S., Lee, K. H., and Yeom, G. S. (2008). “The phase difference effects on 3-D structure of wave pressure acting on a composite breakwater.” Ocean Eng., 35(17–18), 1826–1841.
IHFOAM [Computer software]. IHCantabria, Cantabria, Spain.
Jensen, B., Jacobsen, N. G., and Christensen, E. D. (2014). “Investigations on the porous media equations and resistance coefficients for coastal structures.” Coastal Eng., 84, 56–72.
Khayyer, A., Gotoh, H., and Shao, S. (2008). “Corrected incompressible SPH method for accurate water-surface tracking in breaking waves.” Coastal Eng., 55(3), 236–250.
Kobayashi, N., and de los Santos, F. (2007). “Irregular wave seepage and overtopping of permeable slopes.” J. Waterway, Port, Coastal, Ocean Eng., 245–254.
Kobayashi, N., de los Santos, F., and Kearney, P. (2008). “Time-averaged probabilistic model for irregular wave runup on permeable slopes.” J. Waterway, Port, Coastal, Ocean Eng., 88–96.
Kobayashi, N., Meigs, L. E., Ota, T., and Melby, J. A. (2007). “Irregular breaking wave transmission over submerged porous breakwater.” J. Waterway, Port, Coastal, Ocean Eng., 104–116.
Kothe, D., Williams, M., Lam, K., Korzekwa, D., Tubesing, P., and Puckett, E. (1999). “A second-order accurate, linearity-preserving volume tracking algorithm for free surface flows on 3-D unstructured meshes.” Proc., 3rd ASME/JSME Joint Fluids Engineering Conf., ASME, New York, 18–22.
Lara, J. L., del Jesus, M., and Losada, I. J. (2012). “Three-dimensional interaction of waves and porous coastal structures. Part II: Experimental validation.” Coastal Eng., 64, 26–46.
Lara, J. L., Garcia, N., and Losada, I. J. (2006a). “RANS modeling applied to random wave interaction with submerged permeable structures.” Coastal Eng., 53(5–6), 395–417.
Lara, J. L., Losada, I. J., and Guanche, R. (2008). “Wave interaction with low mound breakwaters using a RANS model.” Ocean Eng., 35(13), 1388–1400.
Lara, J. L., Losada, I. J., and Liu, P. L. F. (2006b). “Breaking waves over a mild gravel slope: experimental and numerical analysis.” J. Geophys. Res., 111, C11019.
Lara, J. L., Losada, I. J., Maza, M., and Guanche, R. (2011). “Breaking solitary wave evolution over a porous underwater step.” Coastal Eng., 58(9), 837–850.
Lin, P., and Karunarathna, S. (2007). “Numerical study of solitary wave interaction with porous breakwaters.” J. Waterway, Port, Coastal, Ocean Eng., 352–363.
Lin, P., and Liu, P. L. F. (1998). “A numerical study of breaking waves in the surf zone.” J. Fluid Mech., 359, 239–264.
Lind, S. J., Xu, R., Stansby, P. K., and Rogers, B. D. (2012). “Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves.” J. Comput. Phys., 231(4), 1499–1523.
Liu, P. L. F., Hsu, T. J., Lin, P. Z., Losada, I. J., Vidal, C., and Sakakiyama, T. (2000). “The Cornell breaking wave and structure (COBRAS) model.” Proc., Coastal Structures 99, Vol. 1., ASCE, Reston, VA, 169–175.
Liu, P. L. F., Pengzhi, L., Chang, K., and Sakakiyama, T. (1999). “Numerical modeling of wave interaction with porous structures.” J. Waterway, Port, Coastal, Ocean Eng., 322–330.
Liu, P. L. F., and Wen, J. (1997). “Nonlinear diffusive surface waves in porous media.” J. Fluid Mech., 347, 119–139.
Losada, I. J., Dalrymple, R., and Losada, M. (1993a). “Water waves on crown breakwaters.” J. Waterway, Port, Coastal, Ocean Eng., 367–380.
Losada, I. J., Lara, J. L., Christensen, E., and Garcia, N. (2005). “Modelling of velocity and turbulence fields around and within low-crested rubble-mound breakwaters.” Coastal Eng., 52(10–11), 887–913.
Losada, I. J., Lara, J. L., Guanche, R., and Gonzalez-Ondina, J. M. (2008). “Numerical analysis of wave overtopping of rubble-mound breakwaters.” Coastal Eng., 55(1), 47–62.
Losada, I. J., Losada, M. A., and Baquerizo, A. (1993b). “An analytical method to evaluate the efficiency of porous screens as wave dampers.” Appl. Ocean Res., 15(4), 207–215.
Losada, I. J., Losada, M. A., and Martin, F. L. (1995). “Experimental study of wave-induced flow in a porous structure.” Coastal Eng., 26(1–2), 77–98.
Losada, I. J., Patterson, M. D., and Losada, M. A. (1997a). “Harmonic generation past a submerged porous step.” Coastal Eng., 31, 281–304.
Losada, I. J., Silva, R., and Losada, M. A. (1996a). “3-D non-breaking regular wave interaction with submerged breakwaters.” Coastal Eng., 28(1–4), 229–248.
Losada, I. J., Silva, R., and Losada, M. A. (1996b). “Interaction of non-breaking directional random waves with submerged breakwaters.” Coastal Eng., 28(1–4), 248–265.
Losada, I. J., Silva, R., and Losada, M. A. (1997b). “Effects of reflective vertical structures permeability on random wave kinematics.” J. Waterway, Port, Coastal, Ocean Eng., 347–353.
Lynett, P., Liu, P., Losada, I. J., and Vidal, C. (2000). “Solitary wave interaction with porous breakwaters.” J. Waterway, Port, Coastal, Ocean Eng., 314–322.
Madsen, O. S. (1974). “Wave transmission through porous structures.” J. Waterway, Port, Coastal, Ocean Eng., 100(3), 169–188.
Madsen, P. A., Murray, R., and Sorensen, O. R. (1991). “A new form of Boussinesq equations with improved linear dispersion characteristics.” Coastal Eng., 15(4), 371–388.
Mase, H., and Takeba, K. (1994). “Bragg scattering of waves over porous rippled bed.” Proc., 24th Int. Conf. on Coastal Engineering, ASCE, Reston, VA, 635–649.
Massel, S. R., and Butowski, P. (1980). “Wind waves transmission through porous breakwaters.” Proc., 17th Int. Conf. on Coastal Engineering, ASCE, Reston, VA, 333–346.
Massel, S. R., and Mei, C. C. (1977). “Transmission of random wind waves through perforated or porous breakwaters.” Coastal Eng., 1, 63–78.
Masuoka, T., and Takatsu, Y. (1996). “Turbulence model for flow through porous media.” Int. J. Heat Mass Transfer, 39(13), 2803–2809.
McIver, P. (1998). “The dispersion relation and eigenfunction expansions for water waves in a porous structure.” J. Eng. Math., 34(3), 319–334.
McIver, P. (1999). “Water wave diffraction by thin porous breakwater.” J. Waterway, Port, Coastal, Ocean Eng., 66–70.
Méndez, F., Losada, I. J., and Losada, M. (2001). “Wave-induced mean magnitudes in permeable submerged breakwaters.” J. Waterway, Port, Coastal, Ocean Eng., 7–15.
Mendez, F. J., and Losada, I. J. (2004). “A perturbation method to solve dispersion equations for water waves over dissipative media.” Coastal Eng., 51(1), 81–89.
Menter, F. (1994). “Two-equation eddy–viscosity turbulence models for engineering applications.” AIAA J., 32(8), 1598–1605.
Monaghan, J. J. (1994). “Simulating free surface flows with SPH.” J. Comput. Phys., 110(2), 399–406.
Nakayama, A., and Kuwahara, F. (1999). “A macroscopic turbulence model for flow in a porous medium.” J. Fluids Eng., 121(2), 427–433.
Neves, M., Reis, M., Losada, I. J., and Hu, K. (2008). “Wave overtopping of Póvoa de Varzim breakwater: physical and numerical simulations.” J. Waterway, Port, Coastal, Ocean Eng., 134(4), 226–236.
OpenFOAM [Computer software]. OpenCFD Ltd, Paris.
Ovaysi, S., and Piri, M. (2010). “Direct pore-level modeling of incompressible fluid flow in porous media.” J. Comput. Phys., 229(19), 7456–7476.
Ovaysi, S., and Piri, M. (2012). “Multi-GPU acceleration of direct pore-scale modeling of fluid flow in natural porous media.” Comput. Phys. Commun., 183(9), 1890–1898.
Paulsen, B. T., Bredmose, H., and Bingham, H. B. (2014). “An efficient domain decomposition strategy for wave loads on surface piercing circular cylinders.” Coastal Eng., 86, 57–76.
Polubarinova-Kochina, P. (1962). Theory of ground water movement, Princeton University Press, Princeton, NJ.
Requejo, S., Vidal, C., and Losada, I. J. (2002). “Modelling of wave loads and hydraulic performance of vertical permeable structures.” Coastal Eng., 46(4), 249–276.
Ren, B., Wen, H., Dong, P., and Wang, Y. (2014). “Numerical simulation of wave interaction with porous structures using an improved smoothed particle hydrodynamic method.” Coastal Eng., 88, 88–100.
Ren, B., Wen, H., Dong, P., and Wang, Y. (2016). “Improved SPH simulation of wave motions and turbulent flows through porous media.” Coastal Eng., 107, 14–27.
Rogers, B. D., Dalrymple, R. A., and Stansby, P. K. (2010). “Simulation of caisson breakwater movement using 2-D SPH.” J. Hydraul. Res., 48(Supplement 1), 135–141.
Sakakiyama, T., and Kajima, R., (1992). “Numerical simulation of nonlinear waves interacting with permeable breakwaters.” Proc., 23rd Int. Conf. on Coastal Engineering, ASCE, Reston, VA, 1517–1530.
Shao, S. (2010). “Incompressible SPH flow model for wave interactions with porous media.” Coastal Eng., 57(3), 304–316.
Shih, T. H., Zhu, J., and Lumley, J. L. (1996). “Calculation of wall-bounded complex flows and free shear flows.” Int. J. Numer. Methods Fluids, 23(11), 1133–1144.
Silva, R., Salles, P., and Palacio, A. (2002). “Linear waves propagating over a rapidly varying finite porous bed.” Coastal Eng., 44(3), 239–260.
Slattery, J. C. (1967). “Flow of viscoelastic fluids through porous media.” AIChE J., 13(6), 1066–1071.
Smith, G. (1991). “Comparison of stationary and oscillatory flow through porous media.” M.Sc. thesis, Queens Univ., Kingston, Ontario, Canada.
Sollitt, C. K., and Cross, R. H. (1972). “Wave transmission through permeabl breakwaters.” Proc., 13th Int. Conf. on Coastal Engineering, ASCE, Reston, VA, 1827–1846.
SPHYSICS [Computer software]. “SPHYSICS home page.” 〈https://wiki.manchester.ac.uk/sphysics/index.php/SPHYSICS_Home_Page〉.
Sulisz, W. (1985). “Wave reflection and transmission at permeable breakwaters of arbitrary cross-section.” Coastal Eng., 9(4), 371–386.
Sulisz, W. (1997). “Wave loads on caisson founded on multilayered rubble base.” J. Waterway, Port, Coastal, Ocean Eng., 91–101.
Tartakovsky, A. M., and Meakin, P. (2005). “A smoothed particle hydrodynamics model for miscible flow in three-dimensional fractures and the two-dimensional Rayleigh–Taylor instability.” J. Comput. Phys., 207(2), 610–624.
Troch, P., and De Rouck, J. (1999). “An active wave generating–absorbing boundary condition for VOF type numerical model.” Coastal Eng., 38(4), 223–247.
van Gent, M. R. A. (1991). “Formulae to describe porous flow.” Communications on Hydraulic and Geotechnical Engineering, Rep. 92-2, Delft Univ. of Technology, Delft, the Netherlands.
van Gent, M. R. A. (1993). “Stationary and oscillatory flow through coarse porous media.” Communications on Hydraulic and Geotechnical Engineering, Rep. 93-9, Delft Univ. of Technology, Delft, the Netherlands.
van Gent, M. R. A. (1994). “The modelling of wave action on and in coastal structures.” Coastal Eng., 22(3–4), 311–339.
van Gent, M. R. A. (1995a). “Porous flow through rubble-mound material.” J. Waterway, Port, Coastal, Ocean Eng., 176–181.
van Gent, M. R. A. (1995b). “Wave interaction with berm breakwaters.” J. Waterway, Port, Coastal, Ocean Eng., 229–238.
Vanneste, D., and Troch, P. (2015). “2D numerical simulation of large-scale physical model tests of wave interaction with a rubble-mound breakwater.” Coastal Eng., 103, 22–41.
Ward, J. (1964). “Turbulent flow in porous media.” J. Hydraul. Div., ASCE, 90, 1–12.
Wellens, P. R., Borsboom, M. J. A., and van Gent, M. R. A. (2010) “3D simulation of wave interaction with permeable structures.” Proc., 32nd Int. Conf. on Coastal Engineering, ASCE, Reston, VA.
Whitaker, S. (1967). “Diffusion and dispersion in porous media.” AIChE J., 13(3), 420–427.
Wu, T. Y. (1981). “Long waves in ocean and coastal waters.” J. Eng. Mech. Div., 107, 501–522.
Wurjanto, A., and Kobayashi, N. (1993). “Irregular wave reflection and runup on permeable slopes.” J. Waterway, Port, Coastal, Ocean Eng., 537–557.
Yu, X. (1995). “Diffraction of water waves by porous breakwaters.” J. Waterway, Port, Coastal, Ocean Eng., 275–282.
Yu, X., and Togoshi, H. (1996). “Combined diffraction and transmission of water waves around a porous breakwater gap.” Proc., 25th Int. Conf. on Coastal Engineering, ASCE, Reston, VA, 2063–2076.
Zhu, Y., Fox, P. J., and Morris, J. P. (1999). “A pore-scale numerical model for flow through porous media.” Int. J. Numer. Anal. Methods Geomech., 23(9), 881–904.
Information & Authors
Information
Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 142 • Issue 6 • November 2016
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Jan 21, 2016
Accepted: Jun 6, 2016
Published online: Aug 4, 2016
Published in print: Nov 1, 2016
Discussion open until: Jan 4, 2017
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.