Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-28T21:50:46.767Z Has data issue: false hasContentIssue false

The swash of solitary waves on a plane beach: flow evolution, bed shear stress and run-up

Published online by Cambridge University Press:  18 August 2015

Nimish Pujara*
Affiliation:
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA
Philip L.-F. Liu
Affiliation:
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA Institute of Hydrological and Oceanic Sciences, National Central University, Jhongli, Taoyuan 320, Taiwan
Harry Yeh
Affiliation:
School of Civil and Construction Engineering, Oregon State University, Corvallis, OR 97331, USA
*
Email address for correspondence: np277@cornell.edu

Abstract

The swash of solitary waves on a plane beach is studied using large-scale experiments. Ten wave cases are examined which range from non-breaking waves to plunging breakers. The focus of this study is on the influence of breaker type on flow evolution, spatiotemporal variations of bed shear stresses and run-up. Measurements are made of the local water depths, flow velocities and bed shear stresses (using a shear plate sensor) at various locations in the swash zone. The bed shear stress is significant near the tip of the swash during uprush and in the shallow flow during the later stages of downrush. In between, the flow evolution is dominated by gravity and follows an explicit solution to the nonlinear shallow water equations, i.e. the flow due to a dam break on a slope. The controlling scale of the flow evolution is the initial velocity of the shoreline immediately following waveform collapse, which can be predicted by measurements of wave height prior to breaking, but also shows an additional dependence on breaker type. The maximum onshore-directed bed shear stress increases significantly onshore of the stillwater shoreline for non-breaking waves and onshore of the waveform collapse point for breaking waves. A new normalization for the bed shear stress which uses the initial shoreline velocity is presented. Under this normalization, the variation of the maximum magnitudes of the bed shear stress with distance along the beach, which is normalized using the run-up, follows the same trend for different breaker types. For the uprush, the maximum dimensionless bed shear stress is approximately 0.01, whereas for the downrush, it is approximately 0.002.

Type
Papers
Copyright
© 2015 Cambridge University Press 

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

Adityawan, M. B., Tanaka, H. & Lin, P. 2013 Boundary layer approach in the modeling of breaking solitary wave runup. Coast. Engng 73, 167177.Google Scholar
Bakhtyar, R., Barry, D. A., Li, L., Jeng, D. S. & Yeganeh-Bakhtiary, A. 2009 Modeling sediment transport in the swash zone: a review. Ocean Engng 36, 767783.CrossRefGoogle Scholar
Baldock, T. E., Cox, D., Maddux, T., Killian, J. & Fayler, L. 2009 Kinematics of breaking tsunami wavefronts: a data set from large scale laboratory experiments. Coast. Engng 56, 506516.CrossRefGoogle Scholar
Baldock, T. E., Grayson, R., Torr, B. & Power, H. E. 2014 Flow convergence at the tip and edges of a viscous swash front – experimental and analytical modeling. Coast. Engng 88, 123130.Google Scholar
Baldock, T. E. & Hughes, M. G. 2006 Field observations of instantaneous water slopes and horizontal pressure gradients in the swash-zone. Cont. Shelf Res. 26, 574588.Google Scholar
Baldock, T. E., Peiris, D. & Hogg, A. J. 2012 Overtopping of solitary waves and solitary bores on a plane beach. Proc. R. Soc. Lond. A 468, 34943516.Google Scholar
Barker, J. W. & Whitham, G. B. 1980 The similarity solution for a bore on a beach. Commun. Pure Appl. Maths 33, 447460.Google Scholar
Barnes, M. P., O’Donoghue, T., Alsina, J. M. & Baldock, T. E. 2009 Direct bed shear stress measurements in bore-driven swash. Coast. Engng 56, 853867.CrossRefGoogle Scholar
Battjes, J. A. 1974 Surf similarity. In Proceedings of 14th Coastal Engineering Conference, vol. 1, pp. 466480. American Society of Civil Engineers (ASCE).Google Scholar
Boussinesq, J. 1872 Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55102.Google Scholar
Briganti, R., Dodd, N., Pokrajac, D. & O’Donoghue, T. 2011 Non linear shallow water modelling of bore-driven swash: description of the bottom boundary layer. Coast. Engng 58, 463477.CrossRefGoogle Scholar
Briggs, M. J., Synolakis, C. E., Harkins, G. S. & Green, D. R. 1995 Laboratory experiments of tsunami runup on a circular island. In Tsunamis 1992–1994, pp. 569593. Springer.Google Scholar
Brocchini, M. & Baldock, T. E. 2008 Recent advances in modeling swash zone dynamics: influence of surf–swash interaction on nearshore hydrodynamics and morphodynamics. Rev. Geophy. 46, RG3003.CrossRefGoogle Scholar
Brocchini, M. & Dodd, N. 2008 Nonlinear shallow water equation modeling for coastal engng. J. Waterway Port Coastal Ocean Engng 134, 104120.Google Scholar
Butt, T. & Russell, P. 2000 Hydrodynamics and cross-shore sediment transport in the swash-zone of natural beaches: a review. J. Coast. Res. 16, 255268.Google Scholar
Camfield, F. E. & Street, R. L. 1969 Shoaling of solitary waves on small slopes. J. Waterway Port Coastal Ocean Engng 95, 122.Google Scholar
Carrier, G. F. & Greenspan, H. P. 1958 Water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4, 97109.Google 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, 10221034.CrossRefGoogle Scholar
Chen, Y. & Yeh, H. 2014 Laboratory experiments on counter-propagating collisions of solitary waves. Part 1. Wave interactions. J. Fluid Mech. 749, 577596.Google Scholar
Conley, D. C. & Griffin, J. G. 2004 Direct measurements of bed stress under swash in the field. J. Geophys. Res. 109, C03050.Google Scholar
Cowen, E. A., Sou, I. M., Liu, P. L.-F. & Raubenheimer, B. 2003 Particle image velocimetry measurements within a laboratory-generated swash zone. J. Engng Mech. 129, 11191129.Google Scholar
Cox, D., Kobayashi, N. & Okayasu, A. 1996 Bottom shear stress in the surf zone. J. Geophys. Res. 101, C00942.CrossRefGoogle Scholar
Dibble, T. L. & Sollitt, C. K.1989 New designs for acoustic and resistive wave profiles, Proceedings, Workshop on Instrumentation for Hydraulic Laboratories, IAHR.Google Scholar
Dingemans, M. W. 1997 Water wave propagation over uneven bottoms: non-linear wave propagation. In Advanced Series on Ocean Engineering, vol. 13. World Scientific.Google Scholar
Efron, B. & Tibshirani, R. 1993 An Introduction to the Bootstrap. Chapman & Hall.Google Scholar
Elfrink, B. & Baldock, T. E. 2002 Hydrodynamics and sediment transport in the swash zone: a review and perspectives. Coast. Engng 45, 149167.Google Scholar
Fenton, J. D. & Rienecker, M. M. 1982 A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions. J. Fluid Mech. 118, 411443.CrossRefGoogle Scholar
Fuhrman, D. R. & Madsen, P. A. 2008 Surf similarity and solitary wave runup. J. Waterway Port Coastal Ocean Engng 134, 195198.CrossRefGoogle Scholar
Goring, D. G.1978 Tsunamis – the propagation of long waves onto a shelf. PhD thesis, California Institute of Technology.Google Scholar
Grilli, S. T., Subramanya, R., Svendsen, I. A. & Veeramony, J. 1994 Shoaling of solitary waves on plane beaches. J. Waterway Port Coastal Ocean Engng 120, 609628.Google Scholar
Grilli, S. T. & Svendsen, I. A. 1991 The propagation and runup of solitary waves on steep slopes. In Center for Applied Coastal Research, University of Delaware, Research Report, pp. 135. University of Delaware.Google Scholar
Grilli, S. T., Svendsen, I. A. & Subramanya, R. 1997 Breaking criterion and characteristics for solitary waves on slopes. J. Waterway Port Coastal Ocean Engng 123, 102112.Google Scholar
Grimshaw, R. 1971 The solitary wave in water of variable depth. Part 2. J. Fluid Mech. 46, 611622.CrossRefGoogle Scholar
Guard, P. A. & Baldock, T. E. 2007 The influence of seaward boundary conditions on swash zone hydrodynamics. Coast. Engng 54, 321331.CrossRefGoogle Scholar
Hall, J. V. & Watts, G. M. 1953 Laboratory investigation of the vertical rise of solitary waves on impermeable slopes. Tech. Rep. DTIC Document.Google Scholar
Hammack, J. L. & Segur, H. 1978 Modelling criteria for long water waves. J. Fluid Mech. 84, 359373.CrossRefGoogle Scholar
Hanratty, T. J. & Campbell, J. A. 1996 Measurement of wall shear stress. In Fluid Mechanics Measurements, pp. 575648. Taylor & Francis.Google Scholar
Hibberd, S. & Peregrine, D. H. 1979 Surf and run-up on a beach: a uniform bore. J. Fluid Mech. 95, 323345.Google Scholar
Ho, D. V. & Meyer, R. E. 1962 Climb of a bore on a beach. Part 1. Uniform beach slope. J. Fluid Mech. 14, 305318.Google Scholar
Horn, D. P. 2006 Measurements and modelling of beach groundwater flow in the swash-zone: a review. Cont. Shelf Res. 26, 622652.Google Scholar
Hsiao, S. C., Hsu, T. W., Lin, T. C. & Chang, Y. H. 2008 On the evolution and run-up of breaking solitary waves on a mild sloping beach. Coast. Engng 55, 975988.Google Scholar
Ippen, A. T. & Kulin, G. 1954 The shoaling and breaking of the solitary wave. In Proceedings of the 5th Coastal Engineering Conference, vol. 1, pp. 2747. ASCE.Google Scholar
Jensen, A., Pedersen, G. K. & Wood, D. J. 2003 An experimental study of wave run-up at a steep beach. J. Fluid Mech. 468, 161188.Google Scholar
Keller, H. B., Levine, D. A. & Whitham, G. B. 1960 Motion of a bore over a sloping beach. J. Fluid Mech. 7, 302316.Google Scholar
Keller, J. B. & Keller, H. B. 1964 Water wave run-up on a beach. In ONR Research Rep. Contract NONR-3828(00), pp. 140. Department of the Navy, Washington, DC.Google Scholar
Keulegan, G. H. 1948 Gradual damping of solitary waves. Natl Bur. Sci. J. Res. 40, 487498.Google Scholar
Kikkert, G. A., O’Donoghue, T., Pokrajac, D. & Dodd, N. 2012 Experimental study of bore-driven swash hydrodynamics on impermeable rough slopes. Coast. Engng 60, 149166.CrossRefGoogle Scholar
Kikkert, G. A., Pokrajac, D. & O’Donoghue, T. 2009 Bed shear stress in bore-generated swash on steep beaches. In Proceedings of the 6th International Conference on Coastal Dynamics, pp. U56U57. World Scientific.Google Scholar
Kobayashi, N. & Karjadi, E. A. 1994 Surf-similarity parameter for breaking solitary-wave runup. J. Waterway Port Coastal Ocean Engng 120, 645650.Google Scholar
Kobayashi, N. & Lawrence, A. R. 2004 Cross-shore sediment transport under breaking solitary waves. J. Geophys. Res. 109, C03047.Google Scholar
Langsholt, M.1981 Experimental study of wave run-up. Cand. Real. PhD thesis, Department of Mathematics, University of Oslo.Google Scholar
Li, Y. & Raichlen, F. 1998 Discussion of ‘breaking criterion and characteristics for solitary waves on slopes’. J. Waterway Port Coastal Ocean Engng 124, 329335.Google Scholar
Li, Y. & Raichlen, F. 2001 Solitary wave runup on plane slopes. J. Waterway Port Coastal Ocean Engng 127, 3344.CrossRefGoogle Scholar
Li, Y. & Raichlen, F. 2002 Non-breaking and breaking solitary wave run-up. J. Fluid Mech. 456, 295318.CrossRefGoogle Scholar
Li, Y. & Raichlen, F. 2003 Energy balance model for breaking solitary wave runup. J. Waterway Port Coastal Ocean Engng 129, 4759.Google Scholar
Lin, P., Chang, K. A. & Liu, P. L.-F. 1999 Runup and rundown of solitary waves on sloping beaches. J. Waterway Port Coastal Ocean Engng 125, 247255.Google Scholar
Liu, P. L.-F., Cho, Y. S., Briggs, M. J., Kanoglu, U. & Synolakis, C. E. 1995 Runup of solitary waves on a circular island. J. Fluid Mech. 302, 259285.Google Scholar
Liu, P. L.-F., Park, Y. S. & Cowen, E. A. 2007 Boundary layer flow and bed shear stress under a solitary wave. J. Fluid Mech. 574, 449463.Google Scholar
Liu, P. L.-F., Simarro, G., Vandever, J. & Orfila, A. 2006 Experimental and numerical investigation of viscous effects on solitary wave propagation in a wave tank. Coast. Engng 53, 181190.Google Scholar
Liu, P. L.-F., Synolakis, C. E. & Yeh, H. 1991 Report on the international workshop on long-wave run-up. J. Fluid Mech. 229, 675688.CrossRefGoogle Scholar
Lo, H. Y., Park, Y. S. & Liu, P. L.-F. 2013 On the run-up and back-wash processes of single and double solitary waves – an experimental study. Coast. Engng 80, 114.Google Scholar
Longo, S., Petti, M. & Losada, I. J. 2002 Turbulence in the swash and surf zones: a review. Coast. Engng 45, 129147.Google Scholar
Longuet-Higgins, M. S. 1974 On the mass, momentum, energy and circulation of a solitary wave. Proc. R. Soc. Lond. A 337, 113.Google Scholar
Madsen, P. A., Fuhrman, D. R. & Schäffer, H. A. 2008 On the solitary wave paradigm for tsunamis. J. Geophys. Res. 113, C12012.Google Scholar
Madsen, P. A. & Schäffer, H. A. 2010 Analytical solutions for tsunami runup on a plane beach: single waves, N-waves and transient waves. J. Fluid Mech. 645, 2757.Google Scholar
Mahony, J. J. & Pritchard, W. G. 1980 Wave reflexion from beaches. J. Fluid Mech. 101, 809832.CrossRefGoogle Scholar
Masselink, G. & Puleo, J. A. 2006 Swash-zone morphodynamics. Cont. Shelf Res. 26, 661680.CrossRefGoogle Scholar
Mei, C. C. 1989 The applied dynamics of ocean surface waves. In Advanced Series on Ocean Engineering, vol. 1. World Scientific.Google Scholar
Mory, M., Abadie, S., Mauriet, S. & Lubin, P. 2011 Run-up flow of a collapsing bore over a beach. Eur. J. Mech. (B/Fluids) 30, 565576.Google Scholar
Nielsen, P. 1992 Coastal Bottom Boudary Layers and Sediment Transport. World Scientific.CrossRefGoogle Scholar
Nielsen, P. 2002 Shear stress and sediment transport calculations for swash zone modelling. Coast. Engng 45, 5360.Google Scholar
O’Donoghue, T., Pokrajac, D. & Hondebrink, L. J. 2010 Laboratory and numerical study of dambreak-generated swash on impermeable slopes. Coast. Engng 57, 513530.Google Scholar
Park, Y. S., Verschaeve, J., Pedersen, G. K. & Liu, P. L.-F. 2014 Boundary-layer flow and bed shear stress under a solitary wave: revision. J. Fluid Mech. 753, 554559.Google Scholar
Pedersen, G. K. & Gjevik, B. 1983 Run-up solitary waves. J. Fluid Mech. 135, 283299.Google Scholar
Pedersen, G. K., Lindstrom, E., Bertelsen, A. F., Jensen, A., Laskovski, D. & Sælevik, G. 2013 Runup and boundary layers on sloping beaches. Phys. Fluids 25, 012102.Google Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815827.Google Scholar
Peregrine, D. H. 1972 Equations for water waves and the approximations behind them. In Waves on Beaches and Resulting Sediment Transport, pp. 95121. Academic.CrossRefGoogle Scholar
Peregrine, D. H. 1983 Breaking waves on beaches. Annu. Rev. Fluid Mech. 15, 149178.CrossRefGoogle Scholar
Peregrine, D. H. & Williams, S. M. 2001 Swash overtopping a truncated plane beach. J. Fluid Mech. 440, 391399.Google Scholar
Pritchard, D., Guard, P. A. & Baldock, T. E. 2008 An analytical model for bore-driven run-up. J. Fluid Mech. 610, 183193.Google Scholar
Pujara, N. & Liu, P. L.-F. 2014 Direct measurements of local bed shear stress in the presence of pressure gradients. Exp. Fluids 55, 1767.Google Scholar
Puleo, J. A. & Butt, T. 2006 The first international workshop on swash-zone processes. Cont. Shelf Res. 26, 556560.Google Scholar
Raubenheimer, B. 2004 Observations of swash zone velocities: a note on friction coefficients. J. Geophys. Res. 109, C01027.CrossRefGoogle Scholar
Saeki, H. S., Hanayasu, A. O. & Takgi, K. 1971 The shoaling and run-up height of the solitary wave. Coast. Engng Japan 14, 2542.Google Scholar
Seelam, J. K., Guard, P. A. & Baldock, T. E. 2011 Measurement and modeling of bed shear stress under solitary waves. Coast. Engng 58, 937947.Google Scholar
Shen, M. C. & Meyer, R. E. 1963 Climb of a bore on a beach. Part 3. Run-up. J. Fluid Mech. 16, 113125.Google Scholar
Skjelbreia, J. E.1987 Observations of breaking waves on sloping bottoms by use of laser Doppler velocimetry. PhD thesis, California Institute of Technology.Google Scholar
Sou, I. M., Cowen, E. A. & Liu, P. L.-F. 2010 Evolution of the turbulence structure in the surf and swash zones. J. Fluid Mech. 644, 193216.Google Scholar
Sou, I. M. & Yeh, H. 2011 Laboratory study of the cross-shore flow structure in the surf and swash zones. J. Geophys. Res. 116, C03002.Google Scholar
Stoker, J. J. 1957 Water Waves, Interscience.Google Scholar
Sumer, B. M., Jensen, P. M., Sørensen, L. B., Fredsøe, J., Liu, P. L.-F. & Carstensen, S. 2010 Coherent structures in wave boundary layers. Part 2. Solitary motion. J. Fluid Mech. 646, 207231.CrossRefGoogle Scholar
Sumer, B. M., Sen, M. B., Karagali, I., Ceren, B., Fredsøe, J., Sottile, M., Zilioli, L. & Fuhrman, D. R. 2011 Flow and sediment transport induced by a plunging solitary wave. J. Geophys. Res. 116, C01008.Google Scholar
Synolakis, C. E.1986 The runup of long waves. PhD thesis, California Institute of Technology.Google Scholar
Synolakis, C. E. 1987 The runup of solitary waves. J. Fluid Mech. 185, 523545.Google Scholar
Synolakis, C. E. 1991 Green’s law and the evolution of solitary waves. Phys. Fluids A 3, 490491.Google Scholar
Synolakis, C. E. & Skjelbreia, J. E. 1993 Evolution of maximum amplitude of solitary waves on plane beaches. J. Waterway Port Coastal Ocean Engng 119, 323342.Google Scholar
Taylor, J. R. 1997 An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books.Google Scholar
Whitham, G. B. 1958 On the propagation of shock waves through regions of non-uniform area or flow. J. Fluid Mech. 4, 337360.Google Scholar
Yeh, H. & Ghazali, A. 1988 On bore collapse. J. Geophys. Res. 93, 69306936.Google Scholar
Yeh, H., Ghazali, A. & Marton, I. 1989 Experimental study of bore run-up. J. Fluid Mech. 206, 563578.CrossRefGoogle Scholar
Zelt, J. A. 1991 The run-up of nonbreaking and breaking solitary waves. Coast. Engng 15, 205246.Google Scholar
Zhang, Q. & Liu, P. L.-F. 2008 A numerical study of swash flows generated by bores. Coast. Engng 55, 11131134.Google Scholar