Difference between revisions of "Hydrodynamic numerical models of wave-structure interaction"

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This article reviews various numerical hydrodynamic model approaches for simulating the interaction between water waves and breakwaters in the surf zone.  
 
This article reviews various numerical hydrodynamic model approaches for simulating the interaction between water waves and breakwaters in the surf zone.  
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==Introduction==
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The most relevant hydraulic processes involved in wave-structure interaction in the surf zone encompass wave reflection, wave dissipation, wave transmission resulting from [[wave overtopping]] and wave penetration through the porous structures, wave diffraction, run-up and wave breaking. Focusing on overtopping, additional processes such as trapped air in broken waves and turbulence, induced by local effects at the armour stones and breakwater cover layers, play an important role in order to determine wave induced dynamics. Formulations derived from experimentations, are, in most of the cases, semi-empirical in nature with their form based on physical considerations and empirical constants determined by fitting to experimental data.
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The applicability of physical scale models is limited due to the various scaling factors for dissipation mechanisms due to wave breaking, turbulence and generation of eddies in the fluid region as well as turbulence and friction within the porous material. Besides the problem of the scaling technique and other features related with the duration of the experiment programs, wave flume dimensions or economical cost have to be considered. Due to poor repeatability, a large number of experiments have to be carried out in order to define confidence intervals. Moreover, experimental investigation on large-scale models is expensive and measurements within breaking waves can be very complex, due to the aerated and transient nature of the water surface. As a consequence, formulations extracted from the experimental tests present several restrictions. They can only be applied to a structure with a geometry similar or almost identical to the one tested and under identical wave characteristics. An analytical approach is not possible because of the complexity of the problem. A great effort has therefore been made over the last decades in the numerical modelling of wave interaction with coastal structures to overcome these limitations. Nonlinear Shallow Water (NSW), Boussinesq-type and Navier-Stokes equations models have traditionally been used. SPH models have also appeared more recently as an alternative.
  
  
 
===NonLinear Shallow Water (NLSW) models===
 
===NonLinear Shallow Water (NLSW) models===
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[https://en.wikipedia.org/wiki/Shallow_water_equations Nonlinear Shallow Water models] have been used to simulate surf zone hydrodynamics and wave structure interaction following two approaches, wave-averaged models and the more commonly used time domain models. The validity of these models for simulating surf zone processes is restricted to situations where the wave height is small in comparison to water depth and the change in depth is small over a characteristic horizontal length for motion (Svendsen, 2005<ref name="S05">Svendsen, I. A., 2005. Introduction to Nearshore Hydrodynamics, Advanced Series on Ocean Engineering, vol. 24, World Scientific, Singapore.</ref>). This problem has been partially solved by assuming a hydrostatic pressure distribution, a depth-uniform velocity profile, negligible vertical accelerations, and simple parameterization of the turbulent terms in the equations of motion and the flow within porous media. These assumptions allow the use of depth-integrated nonlinear shallow water equations models near coastal structures. Despite assumptions and limitations, models based on NLSW equations are very efficient and cheap. Simulations covering many wave periods can be calculated rapidly and can be a valid method under certain conditions. The capability of the one-dimensional depth-averaged nonlinear shallow water equations model with bore-like dissipation and quadratic bottom friction model for the study of wave overtopping was first tested by Kobayashi and Raichle (1994<ref name="K94">Kobayashi, N. and Raichle, A.W., 1994. Irregular Wave Overtopping of Revetments in Surf Zones. ''Journal of Waterway, Port, Coastal and Ocean Engineering''. '''120(1)''', 56-73.</ref>). The model, called RBREAK, was essentially based on the previous work of Kobayashi and Wurjanto (1989<ref name="K89">Kobayashi, N., Wurjanto, A., 1989. Wave transmission over submerged breakwaters. ''Journal of Waterways, Port, Coastal and Ocean Engineering'', '''115''': 662-680.</ref>), in which a system of conservation equations of mass, momentum and energy was derived from the two dimensional Reynolds equations. Artificially triggered wave breaking was simulated numerically in the flow field. The model underpredicted overtopping thickness by a factor of 5 and the mean overtopping rate was largely underpredicted by the model. Kobayashi and Wurjanto (1989<ref name="K89"/>) and subsequently Dodd (1998<ref>Dodd, N., 1998. Numerical model of wave run-up, overtopping, and regeneration. ''Journal of Waterway, Port, Coastal and Ocean Engineering''. '''124(2)''', 73–81.</ref>) validated their NLSW models against a data set of monochromatic wave overtopping of a sloping sea wall by Saville (1955<ref>Saville, T. J., 1955. Laboratory data on wave run-up and overtopping on 21 shore structures. Tech. Memo. 64, U.S. Army, Beach Erosion Board, Document Service Center, Dayton, Ohio.</ref>). The seawall was situated on a shallow foreshore, which accommodated the conditions for wave breaking. Again, the average discharge was considerably underpredicted by both models, particularly for tests of high wave steepness. The failure in reproducing the mean water level by the model was attributed to the inability of the NLSW model in describing the complex wave interactions in the surf zone, where wave setup occurs. More recently, Kobayashi et al. (2010<ref>Kobayashi, N., Farhadzadeh, A. and Melby, J.A., 2010. Wave Overtopping and Damage Progression of Stone Armor Layer. ''Journal of Waterway, Port, Coastal and Ocean Engineering''. '''136(5)''', 257-265.</ref>) have improved the model by introducing a probabilistic module to the NLSW equations. Another example of a well-known model based on the NLSW equations is AMAZON presented by Hu et al. (2000<ref>Hu, C.G. Mingham and D.M. Causon, 2000. Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations. ''Coast. Eng.'', '''41''', pp. 433–465.</ref>). The AMAZON model solutions were compared with analytical solutions and laboratory data for wave overtopping at sloping and vertical seawalls and fair agreement was found.
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[https://en.wikipedia.org/wiki/Shallow_water_equations Nonlinear Shallow Water models] have been used to simulate surf zone hydrodynamics and wave structure interaction following two approaches, wave-averaged models and the more commonly used time domain models. The validity of these models for simulating surf zone processes is restricted to situations where the wave height is small in comparison to water depth and the change in depth is small over a characteristic horizontal length for motion (Svendsen, 2005<ref name="S05">Svendsen, I. A., 2005. Introduction to Nearshore Hydrodynamics, Advanced Series on Ocean Engineering, vol. 24, World Scientific, Singapore.</ref>). This problem has been partially solved by assuming a hydrostatic pressure distribution, a depth-uniform velocity profile, negligible vertical accelerations, and simple parameterization of the turbulent terms in the equations of motion and the flow within porous media. These assumptions allow the use of depth-integrated nonlinear shallow water equations models near coastal structures. Despite assumptions and limitations, models based on NLSW equations are very efficient and cheap. Simulations covering many wave periods can be calculated rapidly and can be a valid method under certain conditions. The capability of the one-dimensional depth-averaged nonlinear shallow water equations model with bore-like dissipation and quadratic bottom friction model for the study of [[wave overtopping]] was first tested by Kobayashi and Raichle (1994<ref name="K94">Kobayashi, N. and Raichle, A.W., 1994. Irregular Wave Overtopping of Revetments in Surf Zones. ''Journal of Waterway, Port, Coastal and Ocean Engineering''. '''120(1)''', 56-73.</ref>). The model, called RBREAK, was essentially based on the previous work of Kobayashi and Wurjanto (1989<ref name="K89">Kobayashi, N., Wurjanto, A., 1989. Wave transmission over submerged breakwaters. ''Journal of Waterways, Port, Coastal and Ocean Engineering'', '''115''': 662-680.</ref>), in which a system of conservation equations of mass, momentum and energy was derived from the two dimensional Reynolds equations. Artificially triggered wave breaking was simulated numerically in the flow field. The model underpredicted overtopping thickness by a factor of 5 and the mean overtopping rate was largely underpredicted by the model. Kobayashi and Wurjanto (1989<ref name="K89"/>) and subsequently Dodd (1998<ref>Dodd, N., 1998. Numerical model of wave run-up, overtopping, and regeneration. ''Journal of Waterway, Port, Coastal and Ocean Engineering''. '''124(2)''', 73–81.</ref>) validated their NLSW models against a data set of monochromatic wave overtopping of a sloping sea wall by Saville (1955<ref>Saville, T. J., 1955. Laboratory data on wave run-up and overtopping on 21 shore structures. Tech. Memo. 64, U.S. Army, Beach Erosion Board, Document Service Center, Dayton, Ohio.</ref>). The seawall was situated on a shallow foreshore, which accommodated the conditions for wave breaking. Again, the average discharge was considerably underpredicted by both models, particularly for tests of high wave steepness. The failure in reproducing the mean water level by the model was attributed to the inability of the NLSW model in describing the complex wave interactions in the surf zone, where wave setup occurs. More recently, Kobayashi et al. (2010<ref>Kobayashi, N., Farhadzadeh, A. and Melby, J.A., 2010. Wave Overtopping and Damage Progression of Stone Armor Layer. ''Journal of Waterway, Port, Coastal and Ocean Engineering''. '''136(5)''', 257-265.</ref>) have improved the model by introducing a probabilistic module to the NLSW equations. Another example of a well-known model based on the NLSW equations is AMAZON presented by Hu et al. (2000<ref>Hu, C.G. Mingham and D.M. Causon, 2000. Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations. ''Coast. Eng.'', '''41''', pp. 433–465.</ref>). The AMAZON model solutions were compared with analytical solutions and laboratory data for [[wave overtopping]] at sloping and vertical seawalls and fair agreement was found.
  
 
Modeling of wave transformation over reefs involves energetic wave breaking and transition between sub- and super-critical flows. The conservative form of the nonlinear shallow-water equations and the associated numerical schemes are suitable to describe these processes (e.g., Toro, 2001<ref>Toro, E.F., 2001. Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley, New York.</ref>; LeVeque, 2002<ref>LeVeque, R.J., 2002. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge.</ref>). [https://en.wikipedia.org/wiki/Godunov%27s_scheme Godunov-type schemes] based on a Riemann solver have shock-capturing capabilities to describe breaking waves as bores and conserve flow volume and momentum across discontinuities. Researchers have applied shock-capturing finite volume schemes in coastal and riverine flood hazard modeling (e.g., Dodd, 1998<ref>Dodd, N., 1998. Numerical model of wave run-up, overtopping, and regeneration. ''Journal of Waterway, Port, Coastal and Ocean Engineering''. '''124(2)''', 73–81.</ref>; Pan et al., 2007<ref>Pan, C.H., Lin, B.Y., Mao, X.Z., (2007). Case study: numerical modeling of the tidal bore on the Qiantang River, China. ''Journal of Hydraulic Engineering''. '''133(2)''', 130–138.</ref>; Begnudelli et al., 2008<ref>Begnudelli, L., Sanders, B.F., Bradford, S.F., 2008. Adaptive Godunov-based model for flood simulation. ''Journal of Hydraulic Engineering''. '''134(6)''', 714–725.</ref>). Although these applications are relevant in some aspects to surf-zone processes, the lack of dispersion in the nonlinear shallow-water equations hampers their application to near shore wave modeling.
 
Modeling of wave transformation over reefs involves energetic wave breaking and transition between sub- and super-critical flows. The conservative form of the nonlinear shallow-water equations and the associated numerical schemes are suitable to describe these processes (e.g., Toro, 2001<ref>Toro, E.F., 2001. Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley, New York.</ref>; LeVeque, 2002<ref>LeVeque, R.J., 2002. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge.</ref>). [https://en.wikipedia.org/wiki/Godunov%27s_scheme Godunov-type schemes] based on a Riemann solver have shock-capturing capabilities to describe breaking waves as bores and conserve flow volume and momentum across discontinuities. Researchers have applied shock-capturing finite volume schemes in coastal and riverine flood hazard modeling (e.g., Dodd, 1998<ref>Dodd, N., 1998. Numerical model of wave run-up, overtopping, and regeneration. ''Journal of Waterway, Port, Coastal and Ocean Engineering''. '''124(2)''', 73–81.</ref>; Pan et al., 2007<ref>Pan, C.H., Lin, B.Y., Mao, X.Z., (2007). Case study: numerical modeling of the tidal bore on the Qiantang River, China. ''Journal of Hydraulic Engineering''. '''133(2)''', 130–138.</ref>; Begnudelli et al., 2008<ref>Begnudelli, L., Sanders, B.F., Bradford, S.F., 2008. Adaptive Godunov-based model for flood simulation. ''Journal of Hydraulic Engineering''. '''134(6)''', 714–725.</ref>). Although these applications are relevant in some aspects to surf-zone processes, the lack of dispersion in the nonlinear shallow-water equations hampers their application to near shore wave modeling.
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===Smoothed Particle Hydrodynamics (SPH) models===
 
===Smoothed Particle Hydrodynamics (SPH) models===
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SPH ([https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed Particle Hydrodynamics]) models have been applied to coastal engineering since around 2000. This approach solves the flow from the Lagrangian point of view, calculating the kinematics of each particle and its interaction with neighbour particles. The Lagrangian nature of SPH makes it well suited to simulate free surface flows with rapid changes of the flow field. However, SPH models are very expensive from the computational point of view and they cannot be applied to solve the large domains requested by wave-structure interaction. They cannot be used to simulate many waves to provide a statistical representation of wave overtopping. A detailed explanation of the SPH theories and formulations can be found in Monaghan (1992<ref>Monaghan, J.J., 1992. Smoothed particle hydrodynamics. ''Annual Review of Astronomy and Astrophysics''. '''30''', pp. 543–574</ref>), but the method was first applied to coastal engineering by Dalrymple et al. (2001<ref>Dalrymple, R.A., Knio, O., Cox, D.T., Gomez-Gesteira, M., Zou, S., 2001. Using a Lagrangian particle method for deck overtopping. ''Proc. Waves, American Society Of Civil Engineers'', pp. 1082–1091.</ref>). Several works have been published in the last decades such as Gotoh et al. (2004<ref>Gotoh, H., Shao, S.D., Memita, T., 2004. SPH–LES model for numerical investigation of wave interaction with partially immersed breakwater. ''Coastal Engineering of Japan''. '''46(1)''', pp. 39–63.</ref>), Shao et al. (2006<ref>Shao, S., Ji, C., Graham, D.I., Reeve, D.E., James, P.W., Chadwick, A.J., 2006. Simulation of wave overtopping by an incompressible SPH model. ''Coastal Engineering'', '''53''', pp. 723-735.</ref>) or Dalrymple et al. (2006<ref>Dalrymple, R.A., Knio, O., Cox, D.T., Gomez-Gesteira, M., Zou, S., 2001. Using a Lagrangian particle method for deck overtopping. ''Proc. Waves, American Society Of Civil Engineers''. pp. 1082–1091.</ref>) among others, showing an ongoing progress on the definition of the boundary conditions and the improvement of the numerical prediction. More recently, Shao (2010<ref>Shao, S., 2010. Incompressible SPH flow model for wave interactions with porous media. ''Coastal Engineering'', '''57 (3)''', pp. 304-316</ref>) has published the first work describing wave interaction with a porous coastal structure. Model comparisons have been proved to describe the wave induced hydrodynamics. However, due to the high computational cost, the model cannot be used as predictive tool for overtopping.
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SPH ([https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed Particle Hydrodynamics]) models have been applied to coastal engineering since around 2000. This approach solves the flow from the Lagrangian point of view, calculating the kinematics of each particle and its interaction with neighbour particles. The Lagrangian nature of SPH makes it well suited to simulate free surface flows with rapid changes of the flow field. However, SPH models are very expensive from the computational point of view and they cannot be applied to solve the large domains requested by wave-structure interaction. They cannot be used to simulate many waves to provide a statistical representation of [[wave overtopping]]. A detailed explanation of the SPH theories and formulations can be found in Monaghan (1992<ref>Monaghan, J.J., 1992. Smoothed particle hydrodynamics. ''Annual Review of Astronomy and Astrophysics''. '''30''', pp. 543–574</ref>), but the method was first applied to coastal engineering by Dalrymple et al. (2001<ref>Dalrymple, R.A., Knio, O., Cox, D.T., Gomez-Gesteira, M., Zou, S., 2001. Using a Lagrangian particle method for deck overtopping. ''Proc. Waves, American Society Of Civil Engineers'', pp. 1082–1091.</ref>). Several works have been published in the last decades such as Gotoh et al. (2004<ref>Gotoh, H., Shao, S.D., Memita, T., 2004. SPH–LES model for numerical investigation of wave interaction with partially immersed breakwater. ''Coastal Engineering of Japan''. '''46(1)''', pp. 39–63.</ref>), Shao et al. (2006<ref>Shao, S., Ji, C., Graham, D.I., Reeve, D.E., James, P.W., Chadwick, A.J., 2006. Simulation of wave overtopping by an incompressible SPH model. ''Coastal Engineering'', '''53''', pp. 723-735.</ref>) or Dalrymple et al. (2006<ref>Dalrymple, R.A., Knio, O., Cox, D.T., Gomez-Gesteira, M., Zou, S., 2001. Using a Lagrangian particle method for deck overtopping. ''Proc. Waves, American Society Of Civil Engineers''. pp. 1082–1091.</ref>) among others, showing an ongoing progress on the definition of the boundary conditions and the improvement of the numerical prediction. More recently, Shao (2010<ref>Shao, S., 2010. Incompressible SPH flow model for wave interactions with porous media. ''Coastal Engineering'', '''57 (3)''', pp. 304-316</ref>) has published the first work describing wave interaction with a porous coastal structure. Model comparisons have been proved to describe the wave induced hydrodynamics. However, due to the high computational cost, the model cannot be used as predictive tool for overtopping.
  
  
 
===Navier-Stokes models===
 
===Navier-Stokes models===
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Having less simplifying assumptions, [https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations Navier-Stokes equations] (NSE) are free of some of the limitations mentioned above, allowing the calculation of the velocity field in the whole computational domain for either rotational or irrotational flows. These models are able to calculate flows in complex geometries and provide very refined information on the velocity, pressure and turbulence field. Moreover, multi-layered structures and wave flow within the porous media can be simulated. One of the first numerical models of wave interaction with a permeable breakwater based on the bidimensional NSE is the one developed by Van Gent et al. (1994<ref name="Gent">van Gent, M.R.A., Tönjes, P., Petit, H.A.H., van den Bosh, P., 1994. Wave action on and in permeable structures. Proc. of 24th Int. Conf. on Coastal Engineering, ASCE, pp. 1739-1753.</ref>). The model, named SKYLLA, was able to simulate wave motions inside and outside porous structures, including breaking conditions. For free surface updating, the SKYLLA model uses the Volume Of Fluid (VOF) method introduced by Hirt and Nichols (1981<ref name="Hirt">Hirt, C.W., Nichols, B.D., 1981. Volume of Fluid (VOF) method for dynamics of free boundaries. ''Journal of Computational Physic'', '''39''': 201-225.</ref>) which consists of tracking the density change within each computational cell instead of the free surface location. Other models based on Navier-Stokes equations using the VOF method have been presented more recently. Iwata et al. (1996<ref name="Iwata">Iwata, K., Kawasaki, K., Kim D.S., 1996. Breaking limit, breaking and post-breaking wave deformation due to submerged structures, Proc. 25th Int. Coastal Eng. Conf., pp. 2338-2351.</ref>) investigated experimentally and numerically the breaking limit, breaking and post-breaking wave deformation due to three different types of submerged structures (i.e. bottom-seated, non-bottom-seated fixed and tautly moored) using a VOF-type model. This numerical model could not simulate the porous flow in rubble-type breakwaters. In order to improve the knowledge of the wave breaking mechanisms, Lin and Liu (1998<ref name="Lin">Lin, P., Liu, P.L.-F., 1998. A numerical study of breaking waves in the surf zone. ''Journal of Fluid Mechanics''. '''359''': 239-264.</ref>) developed a 2DV RANS-type numerical model named COBRAS (Cornell BReaking Waves And Structures). The model was a modified version of RIPPLE with a nonlinear [https://en.wikipedia.org/wiki/Turbulence_modeling#:~:text=K%2Depsilon%20(k%2D%CE%B5,two%20transport%20equations%20(PDEs). k-&epsilon turbulence module] for the closure of Reynolds stresses and a new set of boundary conditions to generate waves and VOF to track the free surface movements. Propagation, shoaling and breaking of a wave train in the surf zone was investigated, focusing on the turbulence transport mechanisms. Liu et al. (1999<ref>Liu, P.L.-F., Lin, P., Chang, K.A., Sakakiyama, T., 1999. Numerical modeling of wave interaction with porous structures. ''J. Waterway, Port, Coastal and Ocean Engineering''. '''125''': 322-330.</ref>) improved the capability of the model including wave interaction with porous structures by spatially averaged Navier-Stokes equations. The internal drag forces were modelled by the empirical linear and nonlinear frictional forms. The authors assumed that turbulence inside the porous media is negligible, which is the case when the permeability of the medium is very small, i.e. for fine sands, but not in the breakwaters armour layer under pre-breaking waves as experimentally checked by Losada (2001<ref name="Los">Losada, I.J., 2001. Recent advances in the modeling of wave and permeable structure interaction. ''Recent Advances in Coastal Engineering''. World Scientific. New Jersey. Vol. VII. Ed. P.L.F. Liu., 163-202.</ref>) and Sakakiyama and Liu (2001<ref name="SL">Sakakiyama, T., Liu, P.L.-F., 2001. Laboratory experiments for wave motions and turbulence flows in front of a break water. ''Coastal Engineering''. '''44''', 117-139.</ref>). The numerical model was calibrated by simple experiments of flow passing through a porous dam. Hsu et al. (2002<ref name="Hsu">Hsu, T-J, Sakakiyama, T. , Liu, P.L.-F., 2002. A Numerical model for wave motions and turbulence flows in front of a composite breakwater. ''Coastal Engineering''. '''46''', 25-50.</ref>) extended the previous work introducing the effects of the small-scale turbulence in the porous media by a volume-averaging process of the governing equations. In their model based on the Volume-Averaged/Reynolds Averaged Navier-Stokes (VARANS) equations, the volume-averaged Reynolds stresses term was modelled by adopting the nonlinear eddy viscosity assumption. The model equations for the volume-averaged turbulent kinetic energy and its dissipation rate were derived by taking the volume-average of the standard k-&epsilon; equations. The performance of the numerical model was checked by comparing numerical solutions with the experimental data related to a composite breakwater reported by Sakakiyama and Liu (2001<ref name="SL"/>), although detailed comparisons for turbulent magnitudes were not performed. The model was validated only for regular waves and for a limited number of experiments. In later years, a great effort has been made in improving numerical models to study wave-structure interaction problems, especially for wave overtopping. Simulations with a two-dimensional Reynolds Averaged Navier-Stokes (RANS) model (Losada et al., 2008<ref>Losada, I.J., J.L. Lara, R. Guanche and J.M. Gonzalez-Ondina, 2008. Numerical analysis of wave overtopping of rubble mound breakwaters. ''Coastal Engineering''. '''55(1)''', 47-62</ref>; Lara et al., 2008<ref>Lara, J.L., I.J. Losada and R. Guanche., 2008. Wave interaction with low mound breakwaters using a RANS model. ''Ocean engineering''. '''35''', 1388-1400.</ref>; Guanche et al., 2009<ref>Guanche, R., I.J. Losada and J.L. Lara., 2009. Numerical analysis of wave loads for coastal structure stability. ''Ocean engineering''. '''56''', 543-558</ref>), called IH-2VOF, have revealed that structural functionality and stability can be studied with a high degree of accuracy, even in the presence of granular material layers. VARANS models have been used to characterize wave induced flow within porous structures. VARANS models in a two dimensional form have proved to overcome the inherent limitations of NSLW and [https://en.wikipedia.org/wiki/Boussinesq_approximation_(water_waves) Boussinesq models] related with wave dispersion and breaking, vertical flow characterization, non-hydrostatic pressure field and flow inside porous coastal structures.  
+
Having less simplifying assumptions, [https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations Navier-Stokes equations] (NSE) are free of some of the limitations mentioned above, allowing the calculation of the velocity field in the whole computational domain for either rotational or irrotational flows. These models are able to calculate flows in complex geometries and provide very refined information on the velocity, pressure and turbulence field. Moreover, multi-layered structures and wave flow within the porous media can be simulated. One of the first numerical models of wave interaction with a permeable breakwater based on the bidimensional NSE is the one developed by Van Gent et al. (1994<ref name="Gent">van Gent, M.R.A., Tönjes, P., Petit, H.A.H., van den Bosh, P., 1994. Wave action on and in permeable structures. Proc. of 24th Int. Conf. on Coastal Engineering, ASCE, pp. 1739-1753.</ref>). The model, named SKYLLA, was able to simulate wave motions inside and outside porous structures, including breaking conditions. For free surface updating, the SKYLLA model uses the Volume Of Fluid (VOF) method introduced by Hirt and Nichols (1981<ref name="Hirt">Hirt, C.W., Nichols, B.D., 1981. Volume of Fluid (VOF) method for dynamics of free boundaries. ''Journal of Computational Physic'', '''39''': 201-225.</ref>) which consists of tracking the density change within each computational cell instead of the free surface location. Other models based on Navier-Stokes equations using the VOF method have been presented more recently. Iwata et al. (1996<ref name="Iwata">Iwata, K., Kawasaki, K., Kim D.S., 1996. Breaking limit, breaking and post-breaking wave deformation due to submerged structures, Proc. 25th Int. Coastal Eng. Conf., pp. 2338-2351.</ref>) investigated experimentally and numerically the breaking limit, breaking and post-breaking wave deformation due to three different types of submerged structures (i.e. bottom-seated, non-bottom-seated fixed and tautly moored) using a VOF-type model. This numerical model could not simulate the porous flow in rubble-type breakwaters. In order to improve the knowledge of the wave breaking mechanisms, Lin and Liu (1998<ref name="Lin">Lin, P., Liu, P.L.-F., 1998. A numerical study of breaking waves in the surf zone. ''Journal of Fluid Mechanics''. '''359''': 239-264.</ref>) developed a 2DV RANS-type numerical model named COBRAS (Cornell BReaking Waves And Structures). The model was a modified version of RIPPLE with a nonlinear [https://en.wikipedia.org/wiki/Turbulence_modeling#:~:text=K%2Depsilon%20(k%2D%CE%B5,two%20transport%20equations%20(PDEs). k-&epsilon turbulence module] for the closure of Reynolds stresses and a new set of boundary conditions to generate waves and VOF to track the free surface movements. Propagation, shoaling and breaking of a wave train in the surf zone was investigated, focusing on the turbulence transport mechanisms. Liu et al. (1999<ref>Liu, P.L.-F., Lin, P., Chang, K.A., Sakakiyama, T., 1999. Numerical modeling of wave interaction with porous structures. ''J. Waterway, Port, Coastal and Ocean Engineering''. '''125''': 322-330.</ref>) improved the capability of the model including wave interaction with porous structures by spatially averaged Navier-Stokes equations. The internal drag forces were modelled by the empirical linear and nonlinear frictional forms. The authors assumed that turbulence inside the porous media is negligible, which is the case when the permeability of the medium is very small, i.e. for fine sands, but not in the breakwaters armour layer under pre-breaking waves as experimentally checked by Losada (2001<ref name="Los">Losada, I.J., 2001. Recent advances in the modeling of wave and permeable structure interaction. ''Recent Advances in Coastal Engineering''. World Scientific. New Jersey. Vol. VII. Ed. P.L.F. Liu., 163-202.</ref>) and Sakakiyama and Liu (2001<ref name="SL">Sakakiyama, T., Liu, P.L.-F., 2001. Laboratory experiments for wave motions and turbulence flows in front of a break water. ''Coastal Engineering''. '''44''', 117-139.</ref>). The numerical model was calibrated by simple experiments of flow passing through a porous dam. Hsu et al. (2002<ref name="Hsu">Hsu, T-J, Sakakiyama, T. , Liu, P.L.-F., 2002. A Numerical model for wave motions and turbulence flows in front of a composite breakwater. ''Coastal Engineering''. '''46''', 25-50.</ref>) extended the previous work introducing the effects of the small-scale turbulence in the porous media by a volume-averaging process of the governing equations. In their model based on the Volume-Averaged/Reynolds Averaged Navier-Stokes (VARANS) equations, the volume-averaged Reynolds stresses term was modelled by adopting the nonlinear eddy viscosity assumption. The model equations for the volume-averaged turbulent kinetic energy and its dissipation rate were derived by taking the volume-average of the standard k-&epsilon; equations. The performance of the numerical model was checked by comparing numerical solutions with the experimental data related to a composite breakwater reported by Sakakiyama and Liu (2001<ref name="SL"/>), although detailed comparisons for turbulent magnitudes were not performed. The model was validated only for regular waves and for a limited number of experiments. In later years, a great effort has been made in improving numerical models to study wave-structure interaction problems, especially for [[wave overtopping]]. Simulations with a two-dimensional Reynolds Averaged Navier-Stokes (RANS) model (Losada et al., 2008<ref>Losada, I.J., J.L. Lara, R. Guanche and J.M. Gonzalez-Ondina, 2008. Numerical analysis of wave overtopping of rubble mound breakwaters. ''Coastal Engineering''. '''55(1)''', 47-62</ref>; Lara et al., 2008<ref>Lara, J.L., I.J. Losada and R. Guanche., 2008. Wave interaction with low mound breakwaters using a RANS model. ''Ocean engineering''. '''35''', 1388-1400.</ref>; Guanche et al., 2009<ref>Guanche, R., I.J. Losada and J.L. Lara., 2009. Numerical analysis of wave loads for coastal structure stability. ''Ocean engineering''. '''56''', 543-558</ref>), called IH-2VOF, have revealed that structural functionality and stability can be studied with a high degree of accuracy, even in the presence of granular material layers. VARANS models have been used to characterize wave induced flow within porous structures. VARANS models in a two dimensional form have proved to overcome the inherent limitations of NSLW and [https://en.wikipedia.org/wiki/Boussinesq_approximation_(water_waves) Boussinesq models] related with wave dispersion and breaking, vertical flow characterization, non-hydrostatic pressure field and flow inside porous coastal structures.  
  
 
Several approaches based on the use of Eulerian three dimensional Navier-Stokes sets of equations can be found in the literature, for example Li et al. (2004<ref name="Li">Li, T.,Troch, P., De Rouck, J., 2004. Wave overtopping over a sea dike. ''Journal of Computational Physics''. '''198(2)''', 686-726.</ref>), Liu et al., (2009<ref>Liu, D. and P. Lin. 2009. Three-dimensional liquid sloshing in a tank with baffles. ''Ocean engineering''. '''36(2)''', 202-212.</ref>), Wang et al., (2009<ref>Wang, Z., Q. Zou and D. Reeve. 2009. Simulation of spilling breaking waves using a two phase flow CFD model. ''Computers & Fluids''. '''38(10)''', 1995-2005.</ref>) or Christensen (2006 <ref>Christensen, E.D., 2006. Large eddy simulation of spilling and plunging breakers. ''Coastal Engineering''. '''53(5-6)''', 463-485.</ref>), among others. However, none of them solved porous media flow and coastal structures are considered as impermeable. Hur et al. (2003<ref>Hur, D.S. and N. Mizutani, 2003. Numerical estimation of the wave forces acting on a threedimensional body on submerged breakwater. ''Coastal Engineering''. '''47''', pp. 329–345.</ref>) and later Hur et al. (2008<ref>Hur, D.S., Kim, C.H., Kim, D.S., Yoon, J.S., 2008. Simulation of the nonlinear dynamic interactions between waves, a submerged breakwater and the seabed. ''Ocean Engineering''. '''35(5-6)''', 511-522.</ref>) presented wave interaction with permeable structures using a three dimensional approach. Porous flow equations used in the simulations follow a different approach than presented by the VARANS equations. Although frictional forces due to the flow through porous material are represented in the same form by drag and inertia terms, Navier-Stokes (NS) equations are not volume averaged. Aerial and volume porosities are defined and applied to the different terms of the NS equations, as aperture coefficients. More recently, Lara et al. (2010<ref>Lara, J. L., Losada I. J., del Jesus, M., Barajas, G., Guanche, R., 2010. IH-3VOF: a three-dimensional Navier-Stokes model for wave and structure interaction. Proceedings of the 32nd International Conference on Coastal Engineering. ASCE. (in press).</ref>) has presented a new set of volume averaged set of equations for porous media flow. A k-&epsilon; model was also integrated in the model. Validations were shown, obtaining a high degree of accuracy in the numerical predictions.
 
Several approaches based on the use of Eulerian three dimensional Navier-Stokes sets of equations can be found in the literature, for example Li et al. (2004<ref name="Li">Li, T.,Troch, P., De Rouck, J., 2004. Wave overtopping over a sea dike. ''Journal of Computational Physics''. '''198(2)''', 686-726.</ref>), Liu et al., (2009<ref>Liu, D. and P. Lin. 2009. Three-dimensional liquid sloshing in a tank with baffles. ''Ocean engineering''. '''36(2)''', 202-212.</ref>), Wang et al., (2009<ref>Wang, Z., Q. Zou and D. Reeve. 2009. Simulation of spilling breaking waves using a two phase flow CFD model. ''Computers & Fluids''. '''38(10)''', 1995-2005.</ref>) or Christensen (2006 <ref>Christensen, E.D., 2006. Large eddy simulation of spilling and plunging breakers. ''Coastal Engineering''. '''53(5-6)''', 463-485.</ref>), among others. However, none of them solved porous media flow and coastal structures are considered as impermeable. Hur et al. (2003<ref>Hur, D.S. and N. Mizutani, 2003. Numerical estimation of the wave forces acting on a threedimensional body on submerged breakwater. ''Coastal Engineering''. '''47''', pp. 329–345.</ref>) and later Hur et al. (2008<ref>Hur, D.S., Kim, C.H., Kim, D.S., Yoon, J.S., 2008. Simulation of the nonlinear dynamic interactions between waves, a submerged breakwater and the seabed. ''Ocean Engineering''. '''35(5-6)''', 511-522.</ref>) presented wave interaction with permeable structures using a three dimensional approach. Porous flow equations used in the simulations follow a different approach than presented by the VARANS equations. Although frictional forces due to the flow through porous material are represented in the same form by drag and inertia terms, Navier-Stokes (NS) equations are not volume averaged. Aerial and volume porosities are defined and applied to the different terms of the NS equations, as aperture coefficients. More recently, Lara et al. (2010<ref>Lara, J. L., Losada I. J., del Jesus, M., Barajas, G., Guanche, R., 2010. IH-3VOF: a three-dimensional Navier-Stokes model for wave and structure interaction. Proceedings of the 32nd International Conference on Coastal Engineering. ASCE. (in press).</ref>) has presented a new set of volume averaged set of equations for porous media flow. A k-&epsilon; model was also integrated in the model. Validations were shown, obtaining a high degree of accuracy in the numerical predictions.

Latest revision as of 15:03, 25 February 2023


This article reviews various numerical hydrodynamic model approaches for simulating the interaction between water waves and breakwaters in the surf zone.


Introduction

The most relevant hydraulic processes involved in wave-structure interaction in the surf zone encompass wave reflection, wave dissipation, wave transmission resulting from wave overtopping and wave penetration through the porous structures, wave diffraction, run-up and wave breaking. Focusing on overtopping, additional processes such as trapped air in broken waves and turbulence, induced by local effects at the armour stones and breakwater cover layers, play an important role in order to determine wave induced dynamics. Formulations derived from experimentations, are, in most of the cases, semi-empirical in nature with their form based on physical considerations and empirical constants determined by fitting to experimental data.

The applicability of physical scale models is limited due to the various scaling factors for dissipation mechanisms due to wave breaking, turbulence and generation of eddies in the fluid region as well as turbulence and friction within the porous material. Besides the problem of the scaling technique and other features related with the duration of the experiment programs, wave flume dimensions or economical cost have to be considered. Due to poor repeatability, a large number of experiments have to be carried out in order to define confidence intervals. Moreover, experimental investigation on large-scale models is expensive and measurements within breaking waves can be very complex, due to the aerated and transient nature of the water surface. As a consequence, formulations extracted from the experimental tests present several restrictions. They can only be applied to a structure with a geometry similar or almost identical to the one tested and under identical wave characteristics. An analytical approach is not possible because of the complexity of the problem. A great effort has therefore been made over the last decades in the numerical modelling of wave interaction with coastal structures to overcome these limitations. Nonlinear Shallow Water (NSW), Boussinesq-type and Navier-Stokes equations models have traditionally been used. SPH models have also appeared more recently as an alternative.


NonLinear Shallow Water (NLSW) models

Nonlinear Shallow Water models have been used to simulate surf zone hydrodynamics and wave structure interaction following two approaches, wave-averaged models and the more commonly used time domain models. The validity of these models for simulating surf zone processes is restricted to situations where the wave height is small in comparison to water depth and the change in depth is small over a characteristic horizontal length for motion (Svendsen, 2005[1]). This problem has been partially solved by assuming a hydrostatic pressure distribution, a depth-uniform velocity profile, negligible vertical accelerations, and simple parameterization of the turbulent terms in the equations of motion and the flow within porous media. These assumptions allow the use of depth-integrated nonlinear shallow water equations models near coastal structures. Despite assumptions and limitations, models based on NLSW equations are very efficient and cheap. Simulations covering many wave periods can be calculated rapidly and can be a valid method under certain conditions. The capability of the one-dimensional depth-averaged nonlinear shallow water equations model with bore-like dissipation and quadratic bottom friction model for the study of wave overtopping was first tested by Kobayashi and Raichle (1994[2]). The model, called RBREAK, was essentially based on the previous work of Kobayashi and Wurjanto (1989[3]), in which a system of conservation equations of mass, momentum and energy was derived from the two dimensional Reynolds equations. Artificially triggered wave breaking was simulated numerically in the flow field. The model underpredicted overtopping thickness by a factor of 5 and the mean overtopping rate was largely underpredicted by the model. Kobayashi and Wurjanto (1989[3]) and subsequently Dodd (1998[4]) validated their NLSW models against a data set of monochromatic wave overtopping of a sloping sea wall by Saville (1955[5]). The seawall was situated on a shallow foreshore, which accommodated the conditions for wave breaking. Again, the average discharge was considerably underpredicted by both models, particularly for tests of high wave steepness. The failure in reproducing the mean water level by the model was attributed to the inability of the NLSW model in describing the complex wave interactions in the surf zone, where wave setup occurs. More recently, Kobayashi et al. (2010[6]) have improved the model by introducing a probabilistic module to the NLSW equations. Another example of a well-known model based on the NLSW equations is AMAZON presented by Hu et al. (2000[7]). The AMAZON model solutions were compared with analytical solutions and laboratory data for wave overtopping at sloping and vertical seawalls and fair agreement was found.

Modeling of wave transformation over reefs involves energetic wave breaking and transition between sub- and super-critical flows. The conservative form of the nonlinear shallow-water equations and the associated numerical schemes are suitable to describe these processes (e.g., Toro, 2001[8]; LeVeque, 2002[9]). Godunov-type schemes based on a Riemann solver have shock-capturing capabilities to describe breaking waves as bores and conserve flow volume and momentum across discontinuities. Researchers have applied shock-capturing finite volume schemes in coastal and riverine flood hazard modeling (e.g., Dodd, 1998[10]; Pan et al., 2007[11]; Begnudelli et al., 2008[12]). Although these applications are relevant in some aspects to surf-zone processes, the lack of dispersion in the nonlinear shallow-water equations hampers their application to near shore wave modeling.


Smoothed Particle Hydrodynamics (SPH) models

SPH (Smoothed Particle Hydrodynamics) models have been applied to coastal engineering since around 2000. This approach solves the flow from the Lagrangian point of view, calculating the kinematics of each particle and its interaction with neighbour particles. The Lagrangian nature of SPH makes it well suited to simulate free surface flows with rapid changes of the flow field. However, SPH models are very expensive from the computational point of view and they cannot be applied to solve the large domains requested by wave-structure interaction. They cannot be used to simulate many waves to provide a statistical representation of wave overtopping. A detailed explanation of the SPH theories and formulations can be found in Monaghan (1992[13]), but the method was first applied to coastal engineering by Dalrymple et al. (2001[14]). Several works have been published in the last decades such as Gotoh et al. (2004[15]), Shao et al. (2006[16]) or Dalrymple et al. (2006[17]) among others, showing an ongoing progress on the definition of the boundary conditions and the improvement of the numerical prediction. More recently, Shao (2010[18]) has published the first work describing wave interaction with a porous coastal structure. Model comparisons have been proved to describe the wave induced hydrodynamics. However, due to the high computational cost, the model cannot be used as predictive tool for overtopping.


Navier-Stokes models

Having less simplifying assumptions, Navier-Stokes equations (NSE) are free of some of the limitations mentioned above, allowing the calculation of the velocity field in the whole computational domain for either rotational or irrotational flows. These models are able to calculate flows in complex geometries and provide very refined information on the velocity, pressure and turbulence field. Moreover, multi-layered structures and wave flow within the porous media can be simulated. One of the first numerical models of wave interaction with a permeable breakwater based on the bidimensional NSE is the one developed by Van Gent et al. (1994[19]). The model, named SKYLLA, was able to simulate wave motions inside and outside porous structures, including breaking conditions. For free surface updating, the SKYLLA model uses the Volume Of Fluid (VOF) method introduced by Hirt and Nichols (1981[20]) which consists of tracking the density change within each computational cell instead of the free surface location. Other models based on Navier-Stokes equations using the VOF method have been presented more recently. Iwata et al. (1996[21]) investigated experimentally and numerically the breaking limit, breaking and post-breaking wave deformation due to three different types of submerged structures (i.e. bottom-seated, non-bottom-seated fixed and tautly moored) using a VOF-type model. This numerical model could not simulate the porous flow in rubble-type breakwaters. In order to improve the knowledge of the wave breaking mechanisms, Lin and Liu (1998[22]) developed a 2DV RANS-type numerical model named COBRAS (Cornell BReaking Waves And Structures). The model was a modified version of RIPPLE with a nonlinear k-&epsilon turbulence module for the closure of Reynolds stresses and a new set of boundary conditions to generate waves and VOF to track the free surface movements. Propagation, shoaling and breaking of a wave train in the surf zone was investigated, focusing on the turbulence transport mechanisms. Liu et al. (1999[23]) improved the capability of the model including wave interaction with porous structures by spatially averaged Navier-Stokes equations. The internal drag forces were modelled by the empirical linear and nonlinear frictional forms. The authors assumed that turbulence inside the porous media is negligible, which is the case when the permeability of the medium is very small, i.e. for fine sands, but not in the breakwaters armour layer under pre-breaking waves as experimentally checked by Losada (2001[24]) and Sakakiyama and Liu (2001[25]). The numerical model was calibrated by simple experiments of flow passing through a porous dam. Hsu et al. (2002[26]) extended the previous work introducing the effects of the small-scale turbulence in the porous media by a volume-averaging process of the governing equations. In their model based on the Volume-Averaged/Reynolds Averaged Navier-Stokes (VARANS) equations, the volume-averaged Reynolds stresses term was modelled by adopting the nonlinear eddy viscosity assumption. The model equations for the volume-averaged turbulent kinetic energy and its dissipation rate were derived by taking the volume-average of the standard k-ε equations. The performance of the numerical model was checked by comparing numerical solutions with the experimental data related to a composite breakwater reported by Sakakiyama and Liu (2001[25]), although detailed comparisons for turbulent magnitudes were not performed. The model was validated only for regular waves and for a limited number of experiments. In later years, a great effort has been made in improving numerical models to study wave-structure interaction problems, especially for wave overtopping. Simulations with a two-dimensional Reynolds Averaged Navier-Stokes (RANS) model (Losada et al., 2008[27]; Lara et al., 2008[28]; Guanche et al., 2009[29]), called IH-2VOF, have revealed that structural functionality and stability can be studied with a high degree of accuracy, even in the presence of granular material layers. VARANS models have been used to characterize wave induced flow within porous structures. VARANS models in a two dimensional form have proved to overcome the inherent limitations of NSLW and Boussinesq models related with wave dispersion and breaking, vertical flow characterization, non-hydrostatic pressure field and flow inside porous coastal structures.

Several approaches based on the use of Eulerian three dimensional Navier-Stokes sets of equations can be found in the literature, for example Li et al. (2004[30]), Liu et al., (2009[31]), Wang et al., (2009[32]) or Christensen (2006 [33]), among others. However, none of them solved porous media flow and coastal structures are considered as impermeable. Hur et al. (2003[34]) and later Hur et al. (2008[35]) presented wave interaction with permeable structures using a three dimensional approach. Porous flow equations used in the simulations follow a different approach than presented by the VARANS equations. Although frictional forces due to the flow through porous material are represented in the same form by drag and inertia terms, Navier-Stokes (NS) equations are not volume averaged. Aerial and volume porosities are defined and applied to the different terms of the NS equations, as aperture coefficients. More recently, Lara et al. (2010[36]) has presented a new set of volume averaged set of equations for porous media flow. A k-ε model was also integrated in the model. Validations were shown, obtaining a high degree of accuracy in the numerical predictions.

Theseus.jpg


Related articles

Modelling coastal hydrodynamics
Overtopping resistant dikes
Stability of rubble mound breakwaters and shore revetments

Eurotop manual

A more recent review of numerical and physical models for determining wave characteristics at structures in the surf zone can be found in the EurOtop manual 2018, chapter 4.6 [37].


References

  1. ↑ Svendsen, I. A., 2005. Introduction to Nearshore Hydrodynamics, Advanced Series on Ocean Engineering, vol. 24, World Scientific, Singapore.
  2. ↑ Kobayashi, N. and Raichle, A.W., 1994. Irregular Wave Overtopping of Revetments in Surf Zones. Journal of Waterway, Port, Coastal and Ocean Engineering. 120(1), 56-73.
  3. ↑ 3.0 3.1 Kobayashi, N., Wurjanto, A., 1989. Wave transmission over submerged breakwaters. Journal of Waterways, Port, Coastal and Ocean Engineering, 115: 662-680.
  4. ↑ Dodd, N., 1998. Numerical model of wave run-up, overtopping, and regeneration. Journal of Waterway, Port, Coastal and Ocean Engineering. 124(2), 73–81.
  5. ↑ Saville, T. J., 1955. Laboratory data on wave run-up and overtopping on 21 shore structures. Tech. Memo. 64, U.S. Army, Beach Erosion Board, Document Service Center, Dayton, Ohio.
  6. ↑ Kobayashi, N., Farhadzadeh, A. and Melby, J.A., 2010. Wave Overtopping and Damage Progression of Stone Armor Layer. Journal of Waterway, Port, Coastal and Ocean Engineering. 136(5), 257-265.
  7. ↑ Hu, C.G. Mingham and D.M. Causon, 2000. Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations. Coast. Eng., 41, pp. 433–465.
  8. ↑ Toro, E.F., 2001. Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley, New York.
  9. ↑ LeVeque, R.J., 2002. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge.
  10. ↑ Dodd, N., 1998. Numerical model of wave run-up, overtopping, and regeneration. Journal of Waterway, Port, Coastal and Ocean Engineering. 124(2), 73–81.
  11. ↑ Pan, C.H., Lin, B.Y., Mao, X.Z., (2007). Case study: numerical modeling of the tidal bore on the Qiantang River, China. Journal of Hydraulic Engineering. 133(2), 130–138.
  12. ↑ Begnudelli, L., Sanders, B.F., Bradford, S.F., 2008. Adaptive Godunov-based model for flood simulation. Journal of Hydraulic Engineering. 134(6), 714–725.
  13. ↑ Monaghan, J.J., 1992. Smoothed particle hydrodynamics. Annual Review of Astronomy and Astrophysics. 30, pp. 543–574
  14. ↑ Dalrymple, R.A., Knio, O., Cox, D.T., Gomez-Gesteira, M., Zou, S., 2001. Using a Lagrangian particle method for deck overtopping. Proc. Waves, American Society Of Civil Engineers, pp. 1082–1091.
  15. ↑ Gotoh, H., Shao, S.D., Memita, T., 2004. SPH–LES model for numerical investigation of wave interaction with partially immersed breakwater. Coastal Engineering of Japan. 46(1), pp. 39–63.
  16. ↑ Shao, S., Ji, C., Graham, D.I., Reeve, D.E., James, P.W., Chadwick, A.J., 2006. Simulation of wave overtopping by an incompressible SPH model. Coastal Engineering, 53, pp. 723-735.
  17. ↑ Dalrymple, R.A., Knio, O., Cox, D.T., Gomez-Gesteira, M., Zou, S., 2001. Using a Lagrangian particle method for deck overtopping. Proc. Waves, American Society Of Civil Engineers. pp. 1082–1091.
  18. ↑ Shao, S., 2010. Incompressible SPH flow model for wave interactions with porous media. Coastal Engineering, 57 (3), pp. 304-316
  19. ↑ van Gent, M.R.A., Tönjes, P., Petit, H.A.H., van den Bosh, P., 1994. Wave action on and in permeable structures. Proc. of 24th Int. Conf. on Coastal Engineering, ASCE, pp. 1739-1753.
  20. ↑ Hirt, C.W., Nichols, B.D., 1981. Volume of Fluid (VOF) method for dynamics of free boundaries. Journal of Computational Physic, 39: 201-225.
  21. ↑ Iwata, K., Kawasaki, K., Kim D.S., 1996. Breaking limit, breaking and post-breaking wave deformation due to submerged structures, Proc. 25th Int. Coastal Eng. Conf., pp. 2338-2351.
  22. ↑ Lin, P., Liu, P.L.-F., 1998. A numerical study of breaking waves in the surf zone. Journal of Fluid Mechanics. 359: 239-264.
  23. ↑ Liu, P.L.-F., Lin, P., Chang, K.A., Sakakiyama, T., 1999. Numerical modeling of wave interaction with porous structures. J. Waterway, Port, Coastal and Ocean Engineering. 125: 322-330.
  24. ↑ Losada, I.J., 2001. Recent advances in the modeling of wave and permeable structure interaction. Recent Advances in Coastal Engineering. World Scientific. New Jersey. Vol. VII. Ed. P.L.F. Liu., 163-202.
  25. ↑ 25.0 25.1 Sakakiyama, T., Liu, P.L.-F., 2001. Laboratory experiments for wave motions and turbulence flows in front of a break water. Coastal Engineering. 44, 117-139.
  26. ↑ Hsu, T-J, Sakakiyama, T. , Liu, P.L.-F., 2002. A Numerical model for wave motions and turbulence flows in front of a composite breakwater. Coastal Engineering. 46, 25-50.
  27. ↑ Losada, I.J., J.L. Lara, R. Guanche and J.M. Gonzalez-Ondina, 2008. Numerical analysis of wave overtopping of rubble mound breakwaters. Coastal Engineering. 55(1), 47-62
  28. ↑ Lara, J.L., I.J. Losada and R. Guanche., 2008. Wave interaction with low mound breakwaters using a RANS model. Ocean engineering. 35, 1388-1400.
  29. ↑ Guanche, R., I.J. Losada and J.L. Lara., 2009. Numerical analysis of wave loads for coastal structure stability. Ocean engineering. 56, 543-558
  30. ↑ Li, T.,Troch, P., De Rouck, J., 2004. Wave overtopping over a sea dike. Journal of Computational Physics. 198(2), 686-726.
  31. ↑ Liu, D. and P. Lin. 2009. Three-dimensional liquid sloshing in a tank with baffles. Ocean engineering. 36(2), 202-212.
  32. ↑ Wang, Z., Q. Zou and D. Reeve. 2009. Simulation of spilling breaking waves using a two phase flow CFD model. Computers & Fluids. 38(10), 1995-2005.
  33. ↑ Christensen, E.D., 2006. Large eddy simulation of spilling and plunging breakers. Coastal Engineering. 53(5-6), 463-485.
  34. ↑ Hur, D.S. and N. Mizutani, 2003. Numerical estimation of the wave forces acting on a threedimensional body on submerged breakwater. Coastal Engineering. 47, pp. 329–345.
  35. ↑ Hur, D.S., Kim, C.H., Kim, D.S., Yoon, J.S., 2008. Simulation of the nonlinear dynamic interactions between waves, a submerged breakwater and the seabed. Ocean Engineering. 35(5-6), 511-522.
  36. ↑ Lara, J. L., Losada I. J., del Jesus, M., Barajas, G., Guanche, R., 2010. IH-3VOF: a three-dimensional Navier-Stokes model for wave and structure interaction. Proceedings of the 32nd International Conference on Coastal Engineering. ASCE. (in press).
  37. ↑ Van der Meer, J.W., Allsop, N.W.H., Bruce, T., De Rouck, J., Kortenhaus, A., Pullen, T., Schüttrumpf, H., Troch, P. and Zanuttigh, B. 2018. EurOtop manual


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