TECHNICAL PAPERS
Mar 1, 2008

Nonlinear Shallow Water Equation Modeling for Coastal Engineering

Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 134, Issue 2

Abstract

Modeling of nearshore flows by means of the classic nonlinear shallow water equations is reviewed and analyzed with a specific focus on the interdependence between physical phenomena, model equations, and numerical schemes. Numerical benchmarking solutions for coastal applications are summarized. Strengths and limits of the nonlinear shallow water equations to reproduce the fundamental features of coastal flows are illustrated along with indications on needed improvements.

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Acknowledgments

The writers dedicate this review paper to Professor D.H. Peregrine, of the University of Bristol, whose contributions in the field of the nonlinear shallow water equations led directly to much of the work presented here. His research continues to be an inspiration for many. This paper is based on work in the HUMOR project, in the framework of the EU-sponsored Marine Science and Technology Programme (MAST-III), under Contract No. UNSPECIFIEDMAS3-CT97-0081. Mr. M. Antuono and Ms. G. Grosso are thanked for their help with the preparation of Fig. 7.

References

Alcrudo, F., and Garcia-Navarro, P. (1993). “A high-resolution Godunov-type scheme in finite volumes for the 2D shallow-water equations.” Int. J. Numer. Methods Fluids, 16, 489–505.
Bellotti, G., and Brocchini, M. (2002). “On using Boussinesq-type equations near the shoreline: A note of caution.” Ocean Eng., 29, 1569–1575.
Bermudez, A., and Vazquez, M. E. (1994). “Upwind methods for hyperbolic conservation laws with source terms.” Comput. Fluids, 23(8), 1049–1071.
Bernetti, R., Toro, E. F., and Brocchini, M. (2003). “An operator-splitting method for long waves.” Proc., Long Waves Symposium’03, Vol. 1, 49–56.
Bills, P., and Noye, J. (1987). “An investigation of open boundary conditions for tidal models of shallow seas.” Numerical modelling: Applications to marine systems, Vol. 1, Elsevier, New York, 159–194.
Blumberg, A. F., and Kantha, L. H. (1985). “Open boundary condition for circulation models.” J. Hydraul. Eng., 111(2), 237–255.
Briggs, M. J., Synolakis, C. E., Harkins, G. S., and Green, D. R. (1995). “Laboratory experiments of tsunami run-up on circular island.” Pure Appl. Geophys., 144, 569–593.
Brocchini, M., Bernetti, R., Mancinelli, A., and Albertini, G. (2001). “An efficient solver for nearshore flows based on the WAF method.” Coastal Eng., 43, 105–129.
Brocchini, M., and Colombini, M. (2004). “A note on the decay of vorticity in shallow flows.” Phys. Fluids, 16, 2469–2475.
Brocchini, M., Kennedy, A. B., Soldini, L., and Mancinelli, A. (2004). “Topographically-controlled, breaking wave-induced macrovortices. Part 1. Widely separated breakwaters.” J. Fluid Mech., 507, 289–307.
Brocchini, M., and Peregrine, D. H. (1996). “Integral flow properties of the swash zone and averaging.” J. Fluid Mech., 317, 241–273.
Carrier, G. F., and Greenspan, H. P. (1958). “Water waves of finite amplitude on a sloping beach.” J. Fluid Mech., 4, 97–109.
Carrier, G. F., Wu, T. T., and Yeh, H. (2003). “Tsunami run-up and draw-down on a plane beach.” J. Fluid Mech., 475, 79–97.
Chen, Q., Kirby, J. T., Dalrymple, R. A., Shi, F., and Thornton, E. B. (2003). “Boussinesq modeling of longshore currents.” J. Geophys. Res.-Oceans, 108(C11), 3062.
Chen, Q., and Svendsen, I. A. (2003). “Effects of cross-shore boundary condition errors in nearshore circulation models.” Coastal Eng., 48, 243–256.
Dodd, N. (1998). “A numerical model of wave run-up, overtopping and regeneration.” J. Waterway, Port, Coastal, Ocean Eng., 124(2), 73–81.
Erduran, K., Ilic, S., and Kutija, V. (2005). “Hybrid finite-volume finite-difference scheme for the solution of Boussinesq equations.” Int. J. Numer. Methods Fluids, 49, 1213–1232.
Glaister, P. (1988). “Approximate Riemann solutions of the shallow water equations.” J. Hydraul. Res., 26(3), 293–306.
Guinot, V. (2003). “Riemann solvers and boundary conditions for two-dimensional shallow water simulations.” Int. J. Numer. Methods Fluids, 41, 1191–1219.
Hibberd, S., and Peregrine, D. H. (1979). “Surf and run-up on a beach: A uniform bore.” J. Fluid Mech., 95, 323–345.
Higdon, R. L. (1987). “Numerical absorbing boundary conditions for the wave equation.” Math. Comput., 49, 65–90.
Ho, D. V., and Meyer, R. E. (1962). “Climb of a bore on a beach. Part 1: Uniform beach slope.” J. Fluid Mech., 14, 305–318.
Hogg, A. J., and Pritchard, D. (2004). “The effects of hydraulic resistance on dam-break and other shallow inertial flows.” J. Fluid Mech., 501, 179–212.
Hu, K., Mingham, C. G., and Causon, D. M. (2000). “Numerical simulation of wave over-topping of coastal structures using the nonlinear shallow water equations.” Coastal Eng., 41, 433–465.
Hubbard, M. E., and Dodd, N. (2002). “A 2-d numerical model of wave run-up and overtopping.” Coastal Eng., 47(1), 1–26.
Hubbard, M. E., and Garcia-Navarro, P. (2000). “Flux difference splitting and the balancing of source terms and flux gradients.” J. Comput. Phys., 165(1), 89–125.
Kanoglu, U. (2004). “Nonlinear evolution and runup-rundown of long waves on a sloping beach.” J. Fluid Mech., 512, 363–372.
Kobayashi, N., De Silva, G. S., and Watson, K. D. (1989). “Wave transformation and swash oscillations on gentle and steep slopes.” J. Geophys. Res.-Oceans, 94, 951–966.
Kobayashi, N., Otta, A. K., and Roy, I. (1987). “Wave reflection and run-up on rough slopes.” J. Waterway, Port, Coastal, Ocean Eng., 113(3), 282–298.
Lesser, G. R., Roelvink, J. A., van Kester, J. A. T. M., and Stelling, G. S. (2004). “Development and validation of a three-dimensional morphological model.” Coastal Eng., 51(8–9), 883–915.
LeVeque, R. J. (1992). Numerical methods for conservation laws, lectures in mathematics, 2nd Ed., Birkhäuser Verlag, Basel, Switzerland.
Lie, I. (2001). “Well-posed transparent boundary conditions for the shallow water equations.” Appl. Numer. Math., 38(4), 445–474.
Lin, G.-F., Lai, J.-S., and Guo, W.-D. (2003). “Finite-volume component-wise tvd schemes for 2d shallow water equations.” Adv. Water Resour., 26(8), 861–873.
Liu, P.L.-F., Lynett, P., and Synolakis, C. E. (2005). “Analytical solutions for forced long waves on a sloping beach.” J. Fluid Mech., 478, 101–109.
Lynett, P. J., Wu, T., and Liu, P. L. (2002). “Modeling wave runup with depth-integrated equations.” Coastal Eng., 46, 89–107.
Madsen, P. A., Murray, R., and Sörensen, O. R. (1991). “A new form of the Boussinesq equations with improved linear dispersion characteristics.” Coastal Eng., 15, 371–388.
Madsen, P. A., Sørensen, O. R., and Schäffer, H. A. (1997). “Surf zone dynamics simulated by a Boussinesq type model. Part II. Surf beat and swash oscillations for wave groups and irregular waves.” Coastal Eng., 32, 289–319.
Mei, C. C. (1990). The applied dynamics of ocean surface waves, Vol. 1 of Advanced series on ocean engineering, World Scientific, Singapore, 2nd Ed.
Oliger, R., and Sundstrom, A. (1978). “Theoretical and practical apsects of some initial boundary value problems in fluid dynamics.” SIAM J. Appl. Math., 35(3), 419–446.
Orlanski, I. (1976). “A simple boundary condition for unbounded hyperbolic flows.” J. Comput. Phys., 21, 251–269.
Otta, A. K., and Pedroso-Acuna, A. (2004). “Swash boundary and cross-shore variation of horizontal velocity on a slope.” Proc., 29th Int. Conf. of Coastal Engineering, Vol. 3, 1616–1628.
Özkan-Haller, H. T., and Kirby, J. T. (1997). “A Fourier-Chebyshev collocation method for the shallow water equations including shoreline runup.” Appl. Ocean. Res., 19, 21–34.
Peregrine, D. H. (1967). “Long waves on beaches.” J. Fluid Mech., 27, 815–827.
Peregrine, D. H. (1998). “Surf zone currents.” Theor. Comput. Fluid Dyn., 10, 295–309.
Peregrine, D. H., and Williams, S. M. (2001). “Swash overtopping a truncated beach.” J. Fluid Mech., 440, 391–399.
Pratt, L. J. (1983). “On inertial flow over topography, Part 1. Semigeostrophic adjustment to an obstacle.” J. Fluid Mech., 131, 195–218.
Roe, P. L. (1986). “Characteristic-based schemes for the Euler equations.” Annu. Rev. Fluid Mech., 18, 337–365.
Rogers, B. D., Borthwick, A. G. L., and Taylor, P. H. (2003). “Mathematical balancing of flux gradient and source terms prior to using Roe’s approximate Riemann solver.” J. Comput. Phys., 192, 422–451.
Schär, C., and Smith, R. B. (1993). “Shallow-water flow past isolated topography, Part I. Vorticity production and wake formation.” J. Atmos. Sci., 50, 1373–1400.
Shen, M. C., and Meyer, R. E. (1963). “Climb of a bore on a beach, Part 3: Run-up.” J. Fluid Mech., 16, 113–125.
Shiach, J. B., Mingham, C. G., Ingram, D. M., and Bruce, T. (2004). “The applicability of the shallow water equations for modelling violent wave overtopping.” Coastal Eng., 51, 1–15.
Sommerfeld, A. (1949). Partial differential equations, Vol. 6 of Lecture notes on theoretical physics, Academic Press, San Francisco.
Stansby, P. K. (2003). “A mixing-length model for shallow turbulent wakes.” J. Fluid Mech., 495, 369–384.
Stoker, J. J. (1957). Water waves, Interscience, New York.
Stoker, J. T. (1948). “Transformation of breaking waves and bores.” Commun. Pure Appl. Math., 1, 1–87.
Synolakis, C. E. (1987). “The run-up of solitary waves.” J. Fluid Mech., 185, 523–555.
Tao, J. H. (1984). “Numerical modelling of wave runup and breaking on the beach.” Acta Oceanoligica Sini’ca, 6(5), 692–700.
Thacker, W. C. (1981). “Some exact solutions of the nonlinear shallow-water wave equations.” J. Fluid Mech., 107, 499–508.
Titov, V. V., and Synolakis, C. E. (1995). “Modeling of breaking and nonbreaking long-wave evolution and runup using VTCS-2.” J. Waterway, Port, Coastal, Ocean Eng., 121(6), 308–316.
Toro, E. (2001). Shock-capturing methods for free-surface shallow/flows, Wiley, New York.
Toro, E. F. (1989). “A weighted average flux method for hyperbolic conservation laws.” Proc. R. Soc. London, Ser. A, 423, 401–418.
Toro, E. F. (1997). Riemann solvers and numerical methods for fluid dynamics, Springer, Berlin, Germany.
Tsynkov, S. V. (1998). “Numerical solution of problems on unbounded domains. A review.” Appl. Numer. Math., 27, 465–532.
Van Dongeren, A. R., and Svendsen, I. A. (1997). “Absorbing-generating boundary condition for shallow water models.” J. Waterway, Port, Coastal, Ocean Eng., 123(6), 303–313.
Vaźquez-Cendón, M. E. (1999). “Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry.” J. Comput. Phys., 148(2), 497–526.
Vincent, S., Caltagirone, J.-P., and Bonneton, P. (2001). “Numerical modelling of bore propagation and run-up on sloping beaches using a Maccormack TVD scheme.” J. Hydraul. Res., 39, 41–49.
Warming, R. F., and Hyett, B. J. (1974). “The modified equation approach to the stability and accuracy analysis of finite-difference method.” J. Comput. Phys., 14, 159–179.
Watson, G., Peregrine, D. H., and Toro, E. (1992). “Numerical solution of the shallow water equations on a beach using the weighted average flux method.” Computational fluid dynamics, Vol. 1, C. Hirsch, ed., Elsevier, New York.
Watts, P., Imamura, F., and Grilli, S. T. (2000). “Comparing model simulations of three benchmark tsunami generation cases.” Sci. Tsunami Hazards, 18(2), 107–124.
Wei, Y., Mao, X., and Cheung, K. F. (2006). “Well-balanced finite volume model for long-wave runup.” J. Waterway, Port, Coastal, Ocean Eng., 132(2), 114–124.
Whitham, G. B. (1974). Linear and nonlinear waves, Wiley-Interscience, New York.
Wurjanto, A., and Kobayashi, N. (1993). “Irregular wave reflection and runup on permeable slopes.” J. Waterway, Port, Coastal, Ocean Eng., 119(5), 537–557.
H. Yeh, P. L. Liu, and C. E. Synolakis, eds. (1996). Long wave run-up models, World Scientific, Singapore.
Zelt, J. A. (1986). “Tsunamis: The response of harbours with sloping boundaries to long wave excitation.” Technical Rep. No. KH-R-47, California Institute of Technology.
Zhao, D. H., Shen, H. W., Lai, J. S., and Tabios, G. Q. (1996). “Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling.” J. Hydraul. Eng., 122(12), 692–702.
Zhou, J. G., Causon, D. M., Mingham, C. G., and Ingram, D. M. (2001). “The surface gradient method for the treatment of source terms in the shallow water equations.” J. Comput. Phys., 168, 1–25.

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Go to Journal of Waterway, Port, Coastal, and Ocean Engineering
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 134Issue 2March 2008
Pages: 104 - 120

History

Received: May 8, 2006
Accepted: Dec 28, 2006
Published online: Mar 1, 2008
Published in print: Mar 2008

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Maurizio Brocchini
Istituto di Idraulica e Infrastrutture Viarie, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, Italy. E-mail: [email protected]
Nicholas Dodd
School of Civil Engineering, Univ. of Nottingham, University Park, Nottingham NG7 2RD, U.K. E-mail: [email protected]

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