Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-18T10:29:16.218Z Has data issue: false hasContentIssue false
  • English
  • Français

Solving the nonlinear shallow-water equations in physical space

Published online by Cambridge University Press:  23 December 2009

M. ANTUONO  and
M. BROCCHINI
M. ANTUONO*
Affiliation:
INSEAN (The Italian Ship Model Basin), via di Vallerano 139, 00128 Rome, Italy
M. BROCCHINI
Affiliation:
Dipartimento ISAC, Università Politecnica delle Marche, via Brecce Bianche 12, 60131 Ancona, Italy
*
†Email address for correspondence: matteoantuono@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The boundary value problem for the nonlinear shallow-water equations with a beach source term is solved by direct use of physical variables, so that solutions are more easily inspected than those obtained by means of hodograph transformations. Beyond an overall description of the near-shoreline flows in terms of the nonlinear shallow-water equations, significant results are provided by means of a perturbation approach which enables much of the information on the flow to be retained. For sample waves of interest (periodic and solitary), first-order solutions of the shoreline motion and of the near-shoreline flows are computed, illustrated and successfully compared with the equivalent ones obtained through a hodograph transformation method previously developed by the authors. Wave–wave interaction, both at the seaward boundary and within the domain, is also accurately described. Analytical conditions for wave breaking within the domain are provided. These, compared with the authors' hodograph model, show that the first-order condition of the present model is comparable to the second-order condition of that model.

JFM classification

Waves/Free-surface Flows: Solitary waves Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave breaking
Type
Papers
Information
Journal of Fluid Mechanics , Volume 643 , 25 January 2010 , pp. 207 - 232
Copyright
Copyright © Cambridge University Press 2009

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

Abramowitz, M. & Stegun, I. 1964 Handbook of Mathematical Functions. Dover.Google Scholar
Antuono, M. & Brocchini, M. 2007 The boundary value problem for the nonlinear shallow water equation. Stud. Appl. Maths. 119 71–91.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
Bellotti, G. & Brocchini, M. 2002 On using Boussinesq-type equations near the shoreline: a note of caution. Ocean Engng 29 1569–1575.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
Carrier, G. F. & Greenspan, H. P. 1958 Water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4, 97–109.CrossRefGoogle Scholar
Hubbard, M. E. & Dodd, N. 2002 A two-dimensional numerical model of wave run-up and overtopping. Coast. Engng 47, 1–26.CrossRefGoogle Scholar
Mei, C. C. 1983 The Applied Dynamics of Ocean Surface Waves. John Wiley.Google Scholar
Meyer, E. R. 1986 a On the shore singularity of water waves. Part I. The local model. Phys. Fluids 29 (10), 3152–3163.CrossRefGoogle Scholar
Meyer, E. R. 1986 b On the shore singularity of water waves. Part II. Small waves do not break on gentle beaches. Phys. Fluids 29 (10), 3164–3171.CrossRefGoogle Scholar
Pritchard, D. & Dickinson, L. 2007 The near-shore behaviour of shallow-water waves with localized initial conditions. J. Fluid Mech. 591, 413–436.CrossRefGoogle Scholar
Shen, M. C. & Meyer, E. R. 1963 Climb of a bore on a beach. Part 3. Run-up. J. Fluid Mech. 16, 113–125.CrossRefGoogle Scholar
Synolakis, C. E. 1987 The run-up of solitary waves. J. Fluid Mech. 185, 523–545.CrossRefGoogle Scholar
Tuck, E. O. & Hwang, L. S. 1972 Long wave generation on a sloping beach. J. Fluid Mech. 51, 449–461.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar