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Shock trains on a planar beach

Published online by Cambridge University Press:  30 September 2011

Matteo Antuono
Matteo Antuono*
Affiliation:
CNR-INSEAN (The Italian Ship Model Basin), Via di Vallerano 139, 00128 Rome, Italy
*
†Email address for correspondence: [email protected]
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Abstract

Following on from the author’s previous work, the propagation of trains of shock waves on a planar beach is studied in the framework of the nonlinear shallow water equations. The analysis is based on the use of a quasi-analytical solution valid for a shock wave which is fed by a constant Riemann invariant. The asymptotic behaviour of a train of such shock waves is inspected and novel approximate analytical solutions are provided. These are useful both for representing fundamental physical scenarios (e.g. propagation of saw-tooth spilling breakers in the surf zone) and for benchmarking wave-resolving and wave-averaged theoretical/numerical solutions. Finally, a study of the energy dissipation induced by the shock train is provided.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave breaking Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 686 , 10 November 2011 , pp. 583 - 606
Copyright
Copyright © Cambridge University Press 2011

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References

1. Antuono, M. 2010 A shock solution for the nonlinear shallow water equations. J. Fluid Mech. 658, 166–187.CrossRefGoogle Scholar
2. Antuono, M. & Hogg, A. J. 2009 Run-up and backwash bore formation from dam-break flow on an inclined plane. J. Fluid Mech. 640, 151–164.CrossRefGoogle Scholar
3. Antuono, M., Hogg, A. J. & Brocchini, M. 2009 The early stages of a shallow flow in an inclined flume. J. Fluid Mech. 633, 285–309.CrossRefGoogle Scholar
4. Antuono, M. & Brocchini, M. 2007 The boundary value problem for the nonlinear Shallow Water equations. Stud. Appl. Maths 119, 73–93.CrossRefGoogle Scholar
5. Abreu, T., Silva, P. A., Sancho, F. & Temperville, A. 2010 Analytical approximate wave form for asymmetric waves. Coast. Engng 57, 656–667.CrossRefGoogle Scholar
6. Booij, N., Ris, R. C. & Holthuijsen, L. H. 1999 A third-generation wave model for coastal regions. 1. Model description and validation. J. Geophys. Res. 104, 7649–7666.CrossRefGoogle Scholar
7. Briganti, R., Musumeci, R. E., Bellotti, G., Brocchini, M. & Foti, E. 2004 Boussinesq modelling of breaking waves: description of turbulence. J. Geophys. Res. 109 (C7), 7015 doi:10.1029/2003JC002065.CrossRefGoogle Scholar
8. Brocchini, M. & Dodd, N. 2008 Nonlinear shallow water equations modelling for coastal engineering. J. Waterway Port Coast. Ocean Engng 134, 104–120.CrossRefGoogle Scholar
9. Brocchini, M., Kennedy, A., Soldini, L. & Mancinelli, A. 2004 Topographically-controlled, breaking wave-induced macrovortices. Part 1. Widely separated breakwaters. J. Fluid Mech. 507, 289–307.CrossRefGoogle Scholar
10. Brocchini, M., Bernetti, R., Mancinelli, A. & Albertini, G. 2001 An efficient solver for nearshore flows based on the WAF method. Coast. Engng 43 (2), 105–129.CrossRefGoogle Scholar
11. Brocchini, M. & Peregrine, D. H. 1996 Integral flow properties of the swash zone and averaging. J. Fluid Mech. 317, 241–273.CrossRefGoogle Scholar
12. Drake, T. G. & Calantoni, J. 2001 Discrete particle model for sheet flow sediment transport in the nearshore. J. Geophys. Res. 106 (C9), 19859–19868.CrossRefGoogle Scholar
13. Elfrink, B. & Baldock, T. E. 2002 Hydrodynamics and sediment transport in the swash zone: a review and perspectives. Coast. Engng 45, 149–167.CrossRefGoogle Scholar
14. Elgar, S. & Guza, R. T. 1985 Observations of bispectra of shoaling surface gravity waves. J. Fluid Mech. 161, 425–448.CrossRefGoogle Scholar
15. Guard, P. A. & Baldock, T. E. 2007 The influence of seaward boundary conditions on swash zone hydrodynamics. Coast. Engng 54, 321–331.CrossRefGoogle Scholar
16. Hogg, A. J., Baldock, T. E. & Pritchard, D. 2011 Overtopping a truncated planar beach. J. Fluid Mech. 666, 521–553.CrossRefGoogle Scholar
17. Hubbard, M. E. & Dodd, N. 2002 A 2D numerical model of wave run-up, overtopping and regeneration. Coast. Engng 47 (1), 1–126.CrossRefGoogle Scholar
18. Jeffrey, A. 1976 Quasilinear Hyperbolic Systems and Waves. Pitman.Google Scholar
19. Kaihatu, J. M., Veeramony, J., Edwards, K. L. & Kirby, J. T. 2007 Asymptotic behaviour of frequency and wavenumber spectra of nearshore shoaling and breaking waves. J. Geophys. Res. 112 (C6), C06016.CrossRefGoogle Scholar
20. Kennedy, A. B., Chen, Q., Kirby, J. T. & Darlymple, R. A. 2000 Boussinesq modelling of wave transformation, breaking and run-up. I: one dimension. J. Waterway Port Coast. Ocean Engng 126, 39–47.CrossRefGoogle Scholar
21. Kennedy, A., Brocchini, M., Soldini, L. & Gutierrez, E. 2006 Topographically-controlled, breaking wave-induced macrovortices. Part 2. Changing geometries. J. Fluid Mech. 559, 57–80.CrossRefGoogle Scholar
22. Lax, P. D. 1957 Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10, 537–566.CrossRefGoogle Scholar
23. Leveque, R. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.CrossRefGoogle Scholar
24. Park, K. Y. & Borthwich, A. G. L. 2001 Quadtree grid numerical model of nearshore wave-current interaction. Coast. Engng 42 (3), 219–239.CrossRefGoogle Scholar
25. Pritchard, D. & Hogg, A. J. 2005 On the transport of suspended sediment by a swash event on a plane beach. Coast. Engng 52, 1–23.CrossRefGoogle Scholar
26. Ryrie, S. C. 1983 Longshore motion generated on beaches by obliquely incident bores. J. Fluid Mech. 129, 193–212.CrossRefGoogle Scholar
27. Shen, M. C. & Meyer, R. E. 1963 Climb of a bore on a beach. Part 3. Run-up. J. Fluid Mech. 16, 113–125.CrossRefGoogle Scholar
28. Stoker, J. J. 1957 Water Waves. Interscience Publishers, Inc.Google Scholar
29. Suntoyo, , Tanaka, H. & Sana, A. 2008 Characteristics of turbulent boundary layers over a rough bed under saw-tooth waves and its application to sediment transport. Coast. Engng 55 (12), 1102–1102.CrossRefGoogle Scholar
30. Svendsen, I. A. 1994 Wave heights and set-up in a surf zone. Coast. Engng 8 (4), 303–329.CrossRefGoogle Scholar
31. Ting, F. C. K. & Kirby, J. T. 1994 Observation of undertow and turbulence in a laboratory surf zone. Coast. Engng 24 (1–2), 51–80.CrossRefGoogle Scholar
32. Toro, E. F. 2001 Shock Capturing Methods for Free-Surface Shallow Flows. John Wiley and Sons.Google Scholar
33. Van Dongeren, A. R., Sancho, F. E., Svendsen, I. A. & Petrevu, U. 1994 SHORECIRC: a quasi 3D nearshore model. In Proceedings of the 24th International Conference on Coastal Engineering, pp. 2741–2754. ASCE.Google Scholar
34. Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
35. Zhang, Q. & Liu, P. L.-F. 2008 A numerical study of swash flows generated by bores. Coast. Engng 55, 1113–1134.CrossRefGoogle Scholar