On Higher-Order Boussinesq-Type Wave Models
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 142, Issue 1
Abstract
Three enhanced versions of two existing nonlinear Boussinesq-type models are herein derived. These models, along with two other similar solvers, were investigated with respect to their nonlinear and dispersive characteristics. In particular, this study comprises Fourier analysis at first-order, second-order, and third-order harmonics; an investigation of linear and nonlinear dispersion; linear shoaling analysis; and an estimation of transfer functions for subharmonics and superharmonics. The models are also validated against demanding experimental tests of wave propagation over a submerged bar. Conclusions of both a special and a general nature are drawn and discussed concerning the scope of nonlinear upgrade of Boussinesq-type equations, their numerical implementation, and the limitations of such models.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The authors acknowledge the support provided by the director and staff of the laboratory of Harbour Works, National Technical University of Athens. The second author (G.K.) gratefully acknowledges the scholarship provided by the Onassis Foundation. He also acknowledges the support by the Danish Council for Strategic Research under the project Danish Coasts and Climate Adaptation—Flooding Risk and Coastal Protection, project No. 09-066869. The third author (M.Ch.) gratefully acknowledges the support provided by the State Scholarships Foundation of Greece. The authors also thank S. Beji and M. Gobbi for kindly providing the Delft University’s relevant full set of experimental data and the model’s results, respectively. Also, many thanks to Th. Karambas for fruitful discussions at an earlier stage of this study.
References
Agnon, Y., Madsen, P. A., and Schäffer, H. A. (1999). “A new approach to high-order Boussinesq models.” J. Fluid Mech., 399, 319–333.
Antuono, M., and Brocchini, M. (2013). “Beyond Boussinesq-type equations: Semi-integrated models for coastal dynamics.” Phys. Fluids, 25(1), 016603.
Battjes, J. A., and Beji, S. (1991). “Spectral evolution in waves traveling over a shoal.” Proc., Nonlinear Water Waves Workshop, D. H. Peregrine, ed., Univ. of Bristol, Bristol, U.K., 11–19.
Beji, S., and Battjes, J. A. (1993). “Experimental investigation of wave propagation over a bar.” Coast. Eng., 19(1–2), 151–162.
Beji, S., and Battjes, J. A. (1994). “Numerical simulation of nonlinear wave propagation over a bar.” Coast. Eng., 23(1–2), 1–16.
Beji, S., and Nadaoka, K. (1996). “A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth.” Ocean Eng., 23(8), 691–704.
Beji, S., and Nadaoka, K. (1999). “A spectral model for unidirectional nonlinear wave propagation over arbitrary depths.” Coast. Eng., 36(1), 1–16.
Benjamin, T. B., and Feir, J. E. (1967). “The disintegration of wave trains on deep water: Part 1.” J. Fluid Mech., 27(3), 417–430.
Bingham, H. B., and Agnon, Y. (2005). “A Fourier–Boussinesq method for nonlinear water waves.” Eur. J. Mech. B, 24(2), 255–274.
Boussinesq, J. (1871). “Théorie de l’ intumescence liquede appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire.” C. R. Acad. Sci., 72, 755–759 (in French).
Brocchini, M. (2013). “A reasoned overview on Boussinesq-type models: The interplay between physics, mathematics and numerics.” Proc. R. Soc. A, 469(2160), 20130496.
Castro, A., and Lannes, D. (2014). “Fully nonlinear long-wave models in the presence of vorticity.” J. Fluid Mech., 759, 642–675.
Chen, Y., and Liu, P. L.-F. (1995). “Modified Boussinesq equations and associated parabolic models for water wave propagation.” J. Fluid Mech., 288, 351–381.
Chondros, M. K., Koutsourelakis, I. G., and Memos, C. D. (2011). “A Boussinesq-type model incorporating random wave breaking.” J. Hydraul. Res., 49(4), 529–538.
Chondros, M. K., and Memos, C. D. (2014). “A 2DH nonlinear Boussinesq-type wave model of improved dispersion, shoaling, and wave generation characteristics.” Coast. Eng., 91, 99–122.
Chondros, M. K., and Memos, C. D. (2015). “Discussion of ‘A 2DH nonlinear Boussinesq-type wave model of improved dispersion, shoaling, and wave generation characteristics’.” Coast. Eng., 95, 181–182.
Dingemans, M. W. (1994a). “Comparison of computations with Boussinesq-like models and laboratory measurements.” MAST-G8M Note H1684, Delft Hydraulics, Delft, Netherlands.
Dingemans, M. W. (1994b). “Water wave propagation over uneven bottoms.” Ph.D. thesis, Delft Univ. of Technology, Delft, Netherlands.
Galan, A., Simarro, G., Orfila, A., Simarro, J., and Liu, P. L.-F. (2012). “Fully nonlinear model for water wave propagation from deep to shallow waters.” J. Waterway, Port, Coastal, Ocean Eng., 362–371.
Gobbi, M. F., and Kirby, J. T. (1999). “Wave evolution over submerged sills: Tests of a high-order Boussinesq model.” Coast. Eng., 37(1), 57–96.
Gobbi, M. F., Kirby, J. T., and Wei, G. (2000). “A fully nonlinear Boussinesq model for surface waves: Part 2.” J. Fluid Mech., 405, 181–210.
Infeld, E., and Rowlands, G. (1990). Nonlinear waves, solitons and chaos, Cambridge Univ. Press, Cambridge, U.K.
Karambas, Th. V., and Koutitas, C. (2002). “Surf and swash zone morphology evolution induced by nonlinear waves.” J. Waterway, Port, Coastal, Ocean Eng., 102–113.
Karambas, Th. V., and Memos, C. D. (2009). “Boussinesq model for weakly nonlinear fully dispersive water waves.” J. Waterway, Port, Coastal, Ocean Eng., 187–199.
Klonaris, G. Th., Memos, C. D., and Drønen, N. K. (2015). “A high order Boussinesq-type model integrating nearshore dynamics.” (to be submitted).
Klonaris, G. Th., Memos, C. D., and Karambas, Th. V. (2013). “A Boussinesq-type model including wave-breaking terms in both continuity and momentum equations.” Ocean Eng., 57, 128–140.
Li, B. (2008). “Wave equations for regular and irregular water wave propagation.” J. Waterway, Port, Coastal, Ocean Eng., 121–142.
Lighthill, J. (1978). Waves in fluids, Cambridge Univ. Press, Cambridge, U.K.
Lynett, P., and Liu, P. L.-F. (2004). “A two-layer approach to wave modelling.” Proc. R. Soc. Lond. A, 460(2049), 2637–2669.
Madsen, P. A., Bingham, H. B., and Liu, H. (2002). “A new Boussinesq method for fully nonlinear waves from shallow to deep water.” J. Fluid Mech., 462, 1–30.
Madsen, P. A., Fuhrman, D. R., and Wang, B. (2006). “A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry.” Coast. Eng., 53(5–6), 487–504.
Madsen, P. A., Murray, R., and Sørensen, O. R. (1991). “A new form of the Boussinesq equations with improved linear dispersion characteristics.” Coast. Eng., 15(4), 371–388.
Madsen, P. A., and Schäffer, H. A. (1998). “Higher-order Boussinesq-type equations for surface gravity waves: Derivation and analysis.” Phil. Trans. R. Soc. Lond. A, 356(1749), 3123–3184.
Madsen, P. A., and Sørensen, O. R. (1992). “A new form of the Boussinesq equations with improved linear dispersion characteristics: Part 2.” Coast. Eng., 18(3–4), 183–204.
Madsen, P. A., and Sørensen, O. R. (1993). “Bound waves and triad interactions in shallow water.” Ocean Eng., 20(4), 359–388.
Madsen, P. A., Sørensen, O. R., and Schäffer, H. A. (1997). “Surf zone dynamics simulated by a Boussinesq type model: Part I.” Coast. Eng., 32(4), 255–287.
Nwogu, O. (1993). “Alternative form of Boussinesq equations for nearshore wave propagation.” J. Waterway, Port, Coastal, Ocean Eng., 618–638.
Ohyama, T., Beji, S., Nadaoka, K., and Battjes, J. A. (1994). “Experimental verification of numerical model for nonlinear wave evolutions.” J. Waterway, Port, Coastal, Ocean Eng., 637–644.
Ottesen-Hansen, N. E. (1978). “Long period waves in natural wave trains.” Progress Rep. No. 46, Institute of Hydrodynamics and Hydraulic Engineering, Technical Univ. of Denmark, Kongens Lyngby, Denmark.
Peregrine, D. H. (1967). “Long waves on a beach.” J. Fluid Mech., 27(4), 815–827.
Sand, S. E., and Mansard, E. P. D. (1986). “Reproduction of higher harmonics in irregular waves.” Ocean Eng., 13(1), 57–83.
Schäffer, H. A., and Madsen, P. A. (1995). “Further enhancements of Boussinesq-type equations.” Coast. Eng., 26(1–2), 1–14.
Schröter, A., Mayerle, R., and Zielke, W. (1994). “Optimized dispersion characteristics of the Boussinesq wave equations.” Proc., Waves: Physical and Numerical Modelling, M. Isaacson and M. Quick, eds., Univ. of British Columbia, Vancouver, BC, Canada, 416–425.
Simarro, G., Galan, A., Minguez, R., and Orfila, A. (2013). “Narrow banded wave propagation from very deep waters to the shore.” Coast. Eng., 77, 140–150.
Skjelbreia, L., and Hendrickson, J. (1960). “Fifth order gravity wave theory.” Proc., 7th Int. Conf. on Coastal, J. W. Johnson, ed., Vol. 1, Council on Wave Research, The Engineering Foundation, The Hague, Netherlands, 184–196.
Tsutsui, S., Suzuyama, K., and Ohki, H. (1998). “Model equations of nonlinear dispersive waves in shallow water and an application of its simplified version to wave evolution on the step-type reef.” Coast. Eng. J., 40(1), 41–60.
Ursell, F. (1953). “The long-wave paradox in the theory of gravity waves.” Proc., Camb. Philos. Soc., 49(4), 685–694.
Veeramony, J., and Svendsen, I. A. (2000). “The flow in surf-zone waves.” Coast. Eng., 39(2–4), 93–122.
Wei, G., and Kirby, J. T. (1995). “Time-dependent numerical code for extended Boussinesq equations.” J. Waterway, Port, Coastal, Ocean Eng., 251–261.
Wei, G., Kirby, J. T., Grilli, S. T., and Subramanya, R. (1995). “A fully nonlinear Boussinesq model for surface waves: Part 1.” J. Fluid Mech., 294, 71–92.
Witting, J. M. (1984). “A unified model for the evolution nonlinear water waves.” J. Comput. Phys., 56(2), 203–236.
Zou, Z. L. (1999). “Higher order Boussinesq equations.” Ocean Eng., 26(8), 767–792.
Information & Authors
Information
Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 142 • Issue 1 • January 2016
Copyright
© 2015 American Society of Civil Engineers.
History
Received: Dec 30, 2014
Accepted: Apr 28, 2015
Published online: Jun 30, 2015
Discussion open until: Nov 30, 2015
Published in print: Jan 1, 2016
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.