Remarks on the Boundary Conditions for a Serre-Type Model Extended to Intermediate-Waters
Abstract
:1. Introduction
2. Mathematical Formulation
3. Serre Model Extension to Intermediate-Depths
3.1. Model Parameters
3.2. Boundary Conditions
Complete Theory for Plane Wavemakers—Intermediate-Waters
3.3. Wave Breaking Strategy
4. Results and Discussion
4.1. Solitary Wave
4.2. Periodic Wave Propagating over a Sybmerged Bar
4.3. Periodic Wave Propagating in Quasi-Deep-Water Conditions (h0/λ = 0.50)
5. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Berkhoff, J.; Booij, N.; Radder, A. Verification of numerical wave propagation models for simple harmonic linear water waves. Coast. Eng. 1982, 6, 255–279. [Google Scholar] [CrossRef]
- Kirby, J.T.; Dalrymple, R.A. A parabolic equation for the combined refraction–diffraction of Stokes waves by mildly varying topography. J. Fluid Mech. 1983, 136, 435–466. [Google Scholar] [CrossRef] [Green Version]
- Booij, N. A note on the accuracy of the mild-slope equation. Coast. Eng. 1983, 7, 191–203. [Google Scholar] [CrossRef]
- Kirby, J.T. A note on linear surface wave-current interaction over slowly varying topography. J. Geophys. Res. Space Phys. 1984, 89, 745. [Google Scholar] [CrossRef]
- Dalrymple, R.A. Model for Refraction of Water Waves. J. Waterw. Port Coastal Ocean Eng. 1988, 114, 423–435. [Google Scholar] [CrossRef]
- Carmo, J.S.A.D. Wave–current interactions over bottom with appreciable variations in both space and time. Adv. Eng. Softw. 2010, 41, 295–305. [Google Scholar] [CrossRef]
- Saint-Venant, B. Theory of unsteady water flow, with application to river floods and to propagation of tides in river channels. Computes Rendus. Acad. Sci. 1871, 73, 237–240. [Google Scholar]
- Seabra-Santos, F.J. Contribution a L’ètude des Ondes de gravité Bidimensionnelles en eau Peu Profonde. Ph.D. Thesis, Université Scientifique et Médicale et Institut National Polutechnique de Grenoble, Grenoble, France, 1985, unpublished (In French). [Google Scholar]
- Boussinesq, J. Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal. J. Mathématiques Pures Appliquées 1872, 17, 55–108. [Google Scholar]
- Serre, F. Contribution à l’étude des écoulements permanents et variables dans les canaux. Houille Blanche 1953, 39, 374–388. [Google Scholar] [CrossRef] [Green Version]
- Green, A.E.; Naghdi, P.M. A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 1976, 78, 237–246. [Google Scholar] [CrossRef]
- Antunes do Carmo, J.S.; Seabra-Santos, F.J. On breaking waves and wave-current interaction on shallow water: A 2DH finite element model. Int. J. Num. Meth. Fluids 1996, 22, 429–444. [Google Scholar] [CrossRef]
- Bonneton, P.; Barthelemy, E.; Chazel, F.; Cienfuegos, R.; Lannes, D.; Marche, F.; Tissier, T. Recent advances in Serre–Green Naghdi modelling for wave transformation, breaking and runup processes. Eur. J. Mech. B Fluids 2011, 30, 589–597. [Google Scholar] [CrossRef]
- Filippini, M.; Kazolea, A.G.; Ricchiuto, M. A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up. J. Comput. Phys. 2016, 310, 381–417. [Google Scholar] [CrossRef]
- Carmo, A.d.J.S.; Seabra-Santos, F.J.; Amado-Mendes, P. Sudden bed changes and wave–current interactions in coastal regions. Adv. Eng. Softw. 2002, 33, 97–107. [Google Scholar] [CrossRef]
- Pitt, J.P.A.; Zoppou, C.; Roberts, S.G. Behaviour of the Serre equations in the presence of steep gradients revisited. Wave Motion 2018, 76, 61–77. [Google Scholar] [CrossRef] [Green Version]
- Zoppou, C.; Pitt, J.; Roberts, S.G. Numerical solution of the fully non-linear weakly dispersive serre equations for steep gradient flows. Appl. Math. Model. 2017, 48, 70–95. [Google Scholar] [CrossRef]
- Tissier, M.; Bonneton, P.; Marche, F.; Chazel, F.; Lannes, D. A new approach to handle wave breaking in fully non-linear Boussinesq models. Coast. Eng. 2012, 67, 54–66. [Google Scholar] [CrossRef]
- Kazolea, M.; Ricchiuto, M. On wave breaking for Boussinesq-type models. Ocean Model. 2018, 123, 16–39. [Google Scholar] [CrossRef] [Green Version]
- Carmo, J.S.A.D.; Ferreira, J.A.; Pinto, L. On the accurate simulation of nearshore and dam break problems involving dispersive breaking waves. Wave Motion 2019, 85, 125–143. [Google Scholar] [CrossRef]
- Nwogu, O. Alternative Form of Boussinesq Equations for Nearshore Wave Propagation. J. Waterw. Port. Coastal. Ocean Eng. 1993, 119, 618–638. [Google Scholar] [CrossRef] [Green Version]
- Gobbi, M.; Kirby, J.; 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] [Green Version]
- Carmo, J.S.A.D. Boussinesq and Serre type models with improved linear dispersion characteristics: Applications. J. Hydraul. Res. 2013, 51, 719–727. [Google Scholar] [CrossRef]
- Lannes, D.; Marche, F. A new class of fully nonlinear and weakly dispersive Green–Naghdi models for efficient 2D simulations. J. Comput. Phys. 2015, 282, 238–268. [Google Scholar] [CrossRef] [Green Version]
- Clamond, D.; Dutykh, D.; Mitsotakis, D. Conservative modified Serre–Green–Naghdi equations with improved dispersion characteristics. Commun. Nonlinear Sci. Numer. Simul. 2017, 45, 245–257. [Google Scholar] [CrossRef] [Green Version]
- Antunes do Carmo, J.S.; Ferreira, J.A.; Pinto, L.; Romanazzi, G. An improved Serre model: Efficient simulation and comparative evaluation. Appl. Math. Model. 2018, 56, 404–423. [Google Scholar] [CrossRef]
- Antunes do Carmo, J.S.; Seabra-Santos, F.J.; Barthélemy, E. Surface waves propagation in shallow water: A finite element model. Int. J. Numer. Methods Fluids 1993, 16, 447–459. [Google Scholar] [CrossRef]
- Lynett, L.; Liu, P.L.-F. Modeling Wave Generation, Evolution, and Interaction with Depth Integrated, Dispersive Wave Equations COULWAVE Code Manual; Cornell University Long and Intermediate Wave Modeling Package; Cornell University: Ithaca, NY, USA, 2002. [Google Scholar]
- Tonelli, M.; Petti, M. Hybrid finite volume–finite difference scheme for 2DH improved Boussinesq equations. Coast. Eng. 2009, 56, 609–620. [Google Scholar] [CrossRef]
- Chazel, F.; Lannes, D.; Marche, F. Numerical Simulation of Strongly Nonlinear and Dispersive Waves Using a Green–Naghdi Model. J. Sci. Comput. 2011, 48, 105–116. [Google Scholar] [CrossRef] [Green Version]
- Mitsotakis, D.; Synolakis, C.; McGuinness, M. A modified Galerkin/finite element method for the numerical solution of the Serre-Green-Naghdi system. Int. J. Numer. Methods Fluids 2017, 83, 755–778. [Google Scholar] [CrossRef]
- Seabra-Santos, F.J. As aproximações de Wu e de Green & Naghdi no quadro geral da teoria da água pouco profunda. In Proceedings of the Simpósio Luso-Brasileiro de Hidráulica e Recursos Hídricos (4° SILUSBA) 209–219, Lisbon, Portugal, 14–16 June 1989, unpublished (In Portuguese). [Google Scholar]
- Antunes do Carmo, J.S. Nonlinear and dispersive wave effects in coastal processes. Management 2016, 16, 343–355. [Google Scholar] [CrossRef] [Green Version]
- Antunes do Carmo, J.S. Modeling of wave propagation from arbitrary depths to shallow waters-A review. In New Perspectives in Fluid Dynamics; Chaoqun, L., Ed.; InTech-Open Access Publisher: London, UK, 2015; pp. 23–66. [Google Scholar] [CrossRef] [Green Version]
- Madsen, P.A.; Sørensen, O.R. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part A slowly-varying bathymetry. Coast. Eng. 1992, 18, 183–204. [Google Scholar] [CrossRef]
- Beji, S.; Nadaoka, K. A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth. Ocean Eng. 1996, 23, 691–704. [Google Scholar] [CrossRef]
- 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]
- Dean, R.G.; Dalrymple, R.A. Water Wave Mechanics for Engineers and Scientists 1984; Prentice-Hall, Inc.: Englewood Cliffs, NJ, USA, 1984; ISBN 0-13-946038-1. [Google Scholar]
- Beji, S.; Battjes, J.A. Experimental investigation of wave propagation over a bar. Coast. Eng. 1993, 19, 151–162. [Google Scholar] [CrossRef]
- Carmo, J.S.A.D. Processos Físicos e Modelos Computacionais em Engenharia Costeira; Imprensa da Universidade de Coimbra/Coimbra University Press: Coimbra, Portugal, 2016; ISBN 978-989-26-1152-5. [Google Scholar] [CrossRef]
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Antunes Do Carmo, J.S. Remarks on the Boundary Conditions for a Serre-Type Model Extended to Intermediate-Waters. Modelling 2021, 2, 626-640. https://doi.org/10.3390/modelling2040033
Antunes Do Carmo JS. Remarks on the Boundary Conditions for a Serre-Type Model Extended to Intermediate-Waters. Modelling. 2021; 2(4):626-640. https://doi.org/10.3390/modelling2040033
Chicago/Turabian StyleAntunes Do Carmo, José Simão. 2021. "Remarks on the Boundary Conditions for a Serre-Type Model Extended to Intermediate-Waters" Modelling 2, no. 4: 626-640. https://doi.org/10.3390/modelling2040033