Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T20:00:10.479Z Has data issue: false hasContentIssue false
  • English
  • Français

A shock solution for the nonlinear shallow water equations

Published online by Cambridge University Press:  17 June 2010

MATTEO ANTUONO
MATTEO ANTUONO*
Affiliation:
US3, INSEAN, via di Vallerano 139, Rome 00128, Italy
*
†Email address for correspondence: matteoantuono@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

A global shock solution for the nonlinear shallow water equations (NSWEs) is found by assigning proper seaward boundary data that preserve a constant incoming Riemann invariant during the shock wave evolution. The correct shock relations, entropy conditions and asymptotic behaviour near the shoreline are provided along with an in-depth analysis of the main quantities along and behind the bore. The theoretical analysis is then applied to the specific case in which the water at the front of the shock wave is still. A comparison with the Shen & Meyer (J. Fluid Mech., vol. 16, 1963, p. 113) solution reveals that such a solution can be regarded as a specific case of the more general solution proposed here. The results obtained can be regarded as a useful benchmark for numerical solvers based on the NSWEs.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave breaking
Type
Papers
Information
Journal of Fluid Mechanics , Volume 658 , 10 September 2010 , pp. 166 - 187
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Antuono, M. & Brocchini, M. 2007 The boundary value problem for the nonlinear shallow water equations. Stud. Appl. Math. 119, 73–93.CrossRefGoogle Scholar
Antuono, M. & Brocchini, M. 2008 Maximum run-up, breaking conditions and dynamical forces in the swash zone: a boundary value approach. Coast. Engng 55 (9), 732–740.Google Scholar
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
Bokhove, O. 2005 Flooding and drying in discontinuous Galerkin finite element discretizations of shallow-water equations. Part 1. One dimension J. Sci. Comput. 22–23 (1–3), 47–82.CrossRefGoogle Scholar
Briganti, R. & Dodd, N. 2009 Shoreline motion in nonlinear shallow water coastal models. Coast. Engng 56, 495–505.CrossRefGoogle Scholar
Brocchini, M. & Dodd, N. 2008 Nonlinear shallow water equations modeling for coastal engineering. J. Waterway Port Coast. Ocean Engng 134, 104–120.CrossRefGoogle Scholar
Brocchini, M. & Peregrine, D. H. 1996 Integral flow properties of the swash zone and averaging. J. Fluid Mech. 317, 241–273.CrossRefGoogle Scholar
Chang, Y.-H., Hwang, K.-S. & Hwung, H.-H. 2009 Large-scale laboratory measurements of solitary wave inundation on a 1:20 slope. Coast. Engng. 56 (10), 1022–1034.CrossRefGoogle Scholar
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
Guard, P. A. & Baldock, T. E. 2007 The influence of seaward boundary conditions on swash zone hydrodynamics. Coast. Engng 54, 321–331.CrossRefGoogle Scholar
Hibberd, S. & Peregrine, D. H. 1979 Surf and run-up on a beach: a uniform bore. J. Fluid Mech. 95, 323–345.CrossRefGoogle Scholar
Kubatko, E. J., Bunya, S., Dawson, C., Westerink, J. J. & Mirabito, C. 2009 A performance comparison of continuous and discontinuous finite element shallow water models. J. Sci. Comput. 40 (1–3), 315–339.CrossRefGoogle Scholar
Peregrine, D. H. & Williams, S. H. 2001 Swash overtopping a truncated plane beach. J. Fluid Mech. 440, 391–399.CrossRefGoogle Scholar
Pritchard, D., Guard, P. A. & Baldock, T. E. 2008 An analytical model for bore-driven run-up. J. Fluid Mech. 610, 183–193.CrossRefGoogle Scholar
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
Ryrie, S. C. 1983 Longshore motion generated on beaches by obliquely incident bores. J. Fluid Mech. 129, 193–212.CrossRefGoogle Scholar
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
Stoker, J. J. 1957 Water Waves. Interscience.Google Scholar
Synolakis, C. E. 1987 The run-up of solitary waves. J. Fluid Mech. 185, 523–545.CrossRefGoogle Scholar
Toro, E. F. 1999 Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd edn.Springer.CrossRefGoogle Scholar
Toro, E. F. 2001 Shock Capturing Methods for Free-Surface Shallow Flows. Wiley.Google Scholar
Whitham, G. B. 1958 On the propagation of shock waves through regions of non-uniform area or flow. J. Fluid Mech. 4 (4), 337–360.CrossRefGoogle Scholar
Wu, Y. & Cheung, K. F. 2007 Explicit solution to the exact Riemann problem and application in nonlinear shallow-water equations. Intl J. Numer. Methods Fluids 57 (11), 1649–1668.CrossRefGoogle Scholar
Zhang, Q. & Liu, P. L.-F. 2008 A numerical study of swash flows generated by bores. Coast. Engng 55, 1113–1134.CrossRefGoogle Scholar