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Inertial energy dissipation in shallow-water breaking waves

Published online by Cambridge University Press:  11 March 2020

W. Mostert Open the ORCID record for W. Mostert [Opens in a new window]  and
L. Deike Open the ORCID record for L. Deike [Opens in a new window]
W. Mostert*
Affiliation:
Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
L. Deike*
Affiliation:
Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Princeton Environmental Institute, Princeton University, Princeton, NJ 08544, USA
*
†Email addresses for correspondence: wmostert@princeton.edu, ldeike@princeton.edu
†Email addresses for correspondence: wmostert@princeton.edu, ldeike@princeton.edu
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Abstract

We present direct numerical simulations of breaking solitary waves in shallow water to quantify the energy dissipation during the active breaking time. We find that this dissipation can be predicted by an inertial model based on Taylor’s hypothesis as a function of the local wave height, depth and the beach slope. We obtain a relationship that gives the dissipation rate of a breaking wave on a shallow slope as a function of local breaking parameters. Next, we use empirical relations to relate the local wave parameters to the offshore conditions. This enables the energy dissipation to be predicted in terms of the initial conditions. We obtain good collapse of the numerical data with respect to the theoretical scaling.

JFM classification

Waves/Free-surface Flows: Wave breaking Geophysical and Geological Flows: Shallow water flows Geophysical and Geological Flows: Air/sea interactions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 890 , 10 May 2020 , A12
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Arthur, R. S. & Fringer, O. B. 2014 The dynamics of breaking internal solitary waves on slopes. J. Fluid Mech. 761, 360–398.CrossRefGoogle Scholar
Battjes, J. A. & Janssen, JPFM. 1978 Energy loss and set-up due to breaking of random waves. Coast. Engng Proc. 1 (16), 569–587.Google Scholar
Bell, J. B., Colella, P. & Glaz, H. M. 1989 A second-order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 85 (2), 257–283.CrossRefGoogle 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. Pure Appl. Series 2 17, 55–108.Google Scholar
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100 (2), 335–354.CrossRefGoogle Scholar
Camfield, F. E. & Street, R. L 1969a The effects of bottom configuration on the deformation, breaking and run-up of solitary waves. In Coastal Engineering 1968, pp. 173–189. ASCE Library.CrossRefGoogle Scholar
Camfield, F. E. & Street, R. L. 1969b Shoaling of solitary waves on small slopes. J. Waterways Harbors Div. 95 (1), 1–22.Google Scholar
Chanson, H. 2009 Current knowledge in hydraulic jumps and related phenomena. A survey of experimental results. Eur. J. Mech. (B/Fluids) 28 (2), 191–210.CrossRefGoogle Scholar
Deike, L. & Melville, W. K. 2018 Gas transfer by breaking waves. Geophys. Res. Lett. 45 (19), 10482–10492.CrossRefGoogle Scholar
Deike, L., Melville, W. K. & Popinet, S. 2016 Air entrainment and bubble statistics in breaking waves. J. Fluid Mech. 801, 91–129.CrossRefGoogle Scholar
Deike, L., Pizzo, N. & Melville, W. K. 2017 Lagrangian transport by breaking surface waves. J. Fluid Mech. 829, 364–391.CrossRefGoogle Scholar
Deike, L., Popinet, S. & Melville, W. K. 2015 Capillary effects on wave breaking. J. Fluid Mech. 769, 541–569.CrossRefGoogle Scholar
Derakhti, M., Banner, M. L. & Kirby, J. T. 2018 Predicting the breaking strength of gravity water waves in deep and intermediate depth. J. Fluid Mech. 848, R2.CrossRefGoogle Scholar
Derakhti, M., Kirby, J. T., Shi, F. & Ma, G. 2016 Wave breaking in the surf zone and deep-water in a non-hydrostatic RANS model. Part 1: organized wave motions. Ocean Model. 107, 125–138.CrossRefGoogle Scholar
Derakhti, M. & Kirby, J. T. 2014 Bubble entrainment and liquid–bubble interaction under unsteady breaking waves. J. Fluid Mech. 761, 464–506.CrossRefGoogle Scholar
Derakhti, M. & Kirby, J. T. 2016 Breaking-onset, energy and momentum flux in unsteady focused wave packets. J. Fluid Mech. 790, 553–581.CrossRefGoogle Scholar
Drazen, D. A., Melville, W. K. & Lenain, L. 2008 Inertial scaling of dissipation in unsteady breaking waves. J. Fluid Mech. 611, 307–332.CrossRefGoogle Scholar
Duncan, J. H. 1981 An experimental investigation of breaking waves produced by a towed hydrofoil. Proc. R. Soc. Lond. A 377 (1770), 331–348.Google Scholar
Elghobashi, S. 2019 Direct numerical simulation of turbulent flows laden with droplets or bubbles. Annu. Rev. Fluid Mech. 51, 217–244.CrossRefGoogle Scholar
Emarat, N., Forehand, D. I. M., Christensen, E. D. & Greated, C. A. 2012 Experimental and numerical investigation of the internal kinematics of a surf-zone plunging breaker. Eur. J. Mech. (B/Fluid) 32, 1–16.CrossRefGoogle Scholar
Fuster, D. & Popinet, S. 2018 An all-Mach method for the simulation of bubble dynamics problems in the presence of surface tension. J. Comput. Phys. 374, 752–768.CrossRefGoogle Scholar
Grilli, S. T., Svendsen, I. A. & Subramanya, R. 1997 Breaking criterion and characteristics for solitary waves on slopes. J. Waterways Port C-ASCE 123 (3), 102–112.CrossRefGoogle Scholar
Grimshaw, R. 1971 The solitary wave in water of variable depth. Part 2. J. Fluid Mech. 46 (3), 611–622.CrossRefGoogle Scholar
van Hooft, J. A., Popinet, S., van Heerwaarden, C. C., van der Linden, S. J. A., de Roode, S. R. & van de Wiel, B. J. H. 2018 Towards adaptive grids for atmospheric boundary-layer simulations. Boundary-Layer Meteorol. 167 (3), 421–443.CrossRefGoogle ScholarPubMed
Iafrati, A. 2009 Numerical study of the effects of the breaking intensity on wave breaking flows. J. Fluid Mech. 622, 371–411.CrossRefGoogle Scholar
Iafrati, A. 2011 Energy dissipation mechanisms in wave breaking processes: spilling and highly aerated plunging breaking events. J. Geophys. Res. 116 (C7), C07024.CrossRefGoogle Scholar
Iafrati, A., Babanin, A. & Onorato, M. 2013 Modulational instability, wave breaking, and formation of large-scale dipoles in the atmosphere. Phys. Rev. Lett. 110 (18), 184504.CrossRefGoogle ScholarPubMed
Kirby, J. 2016 Boussinesq models and their application to coastal processes across a wide range of scales. J. Waterways Port C-ASCE 142 (6), 03116005.Google Scholar
Kirby, J. T. 2017 Recent advances in nearshore wave, circulation, and sediment transport modeling. J. Mar. Res. 75 (3), 263–300.CrossRefGoogle Scholar
Knowles, J. & Yeh, H. 2018 On shoaling of solitary waves. J. Fluid Mech. 848, 1073–1097.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Le Métayer, O., Gavrilyuk, S. & Hank, S. 2010 A numerical scheme for the Green–Naghdi model. J. Comput. Phys. 229 (6), 2034–2045.CrossRefGoogle Scholar
Li, Y. & Raichlen, F. 2001 Solitary wave runup on plane slopes. J. Waterways Port Coast. ASCE 127 (1), 33–44.CrossRefGoogle Scholar
Lin, N. & Shullman, E. 2017 Dealing with hurricane surge flooding in a changing environment: part I. Risk assessment considering storm climatology change, sea level rise, and coastal development. Stochastic Environ. Res. Risk Assessment 31 (9), 2379–2400.CrossRefGoogle Scholar
Lopez-Herrera, J. M., Popinet, S. & Castrejón-Pita, A. A. 2019 An adaptive solver for viscoelastic incompressible two-phase problems applied to the study of the splashing of slightly viscoelastic droplets. J. Non-Newtonian Fluid Mech. 264, 144–158.CrossRefGoogle Scholar
Lubin, P. & Caltagirone, J.-P. 2010 Large eddy simulation of the hydrodynamics generated by breaking waves. In Advances in Numerical Simulation of Nonlinear Water Waves, pp. 575–604. World Scientific.CrossRefGoogle Scholar
Lubin, P. & Lemonnier, H. 2004 Test-case no 33: Propagation of solitary waves in constant depths over horizontal beds (PA, PN, PE). Multiphase Sci. Technol. 16 (1-3), 239–250.CrossRefGoogle Scholar
Martins, K., Blenkinsopp, C. E., Deigaard, R. & Power, H. E. 2018 Energy dissipation in the inner surf zone: New insights from Li DAR-based roller geometry measurements. J. Geophys. Res.-Oceans 123, 3386–3407.CrossRefGoogle Scholar
McCowan, J. 1894 XXXIX. On the highest wave of permanent type. London, Edinb. Dublin Philos. Mag. J. Sci. 38 (233), 351–358.CrossRefGoogle Scholar
Melville, W. K. 1996 The role of surface-wave breaking in air-sea interaction. Annu. Rev. Fluid Mech. 28 (1), 279–321.CrossRefGoogle Scholar
Memery, L. & Merlivat, L. 1985 Modelling of gas flux through bubbles at the air-water interface. Tellus B: Chem. Phys. Meteorol. 37 (4–5), 272–285.CrossRefGoogle Scholar
Monahan, E. C. & Dam, H. G. 2001 Bubbles: an estimate of their role in the global oceanic flux of carbon. J. Geophys. Res. 106 (C5), 9377–9383.CrossRefGoogle Scholar
Pedersen, G. K., Lindstrøm, E., Bertelsen, A. F., Jensen, A., Laskovski, D. & Sælevik, G. 2013 Runup and boundary layers on sloping beaches. Phys. Fluids 25 (1), 012102.CrossRefGoogle Scholar
Peregrine, D. H. 1998 Surf zone currents. Theor. Comput. Fluid Dyn. 10 (1–4), 295–309.CrossRefGoogle Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27 (4), 815–827.CrossRefGoogle Scholar
Perlin, M., Choi, W. & Tian, Z. 2013 Breaking waves in deep and intermediate waters. Annu. Rev. Fluid Mech. 45, 115–145.CrossRefGoogle Scholar
Pizzo, N. E. & Melville, W. K. 2013 Vortex generation by deep-water breaking waves. J. Fluid Mech. 734, 198–218.CrossRefGoogle Scholar
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190 (2), 572–600.CrossRefGoogle Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228 (16), 5838–5866.CrossRefGoogle Scholar
Popinet, S. 2015 A quadtree-adaptive multigrid solver for the Serre–Green–Naghdi equations. J. Comput. Phys. 302, 336–358.CrossRefGoogle Scholar
Popinet, S.2019 Basilisk flow solver and PDE library. Available at: http://basilisk.fr.Google Scholar
Rapp, R. J. & Melville, W. K. 1990 Laboratory measurements of deep-water breaking waves. Phil. Trans. R. Soc. Lond. A 331 (1622), 735–800.Google Scholar
Robertson, B., Hall, K., Zytner, R. & Nistor, I. 2013 Breaking waves: Review of characteristic relationships. Coast. Engng J. 55 (01), 1350002.Google Scholar
Romero, L., Melville, W. K. & Kleiss, J. M. 2012 Spectral energy dissipation due to surface wave breaking. J. Phys. Oceanogr. 42 (9), 1421–1444.CrossRefGoogle Scholar
Scardovelli, R. & Zaleski, S. 1999 Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31 (1), 567–603.CrossRefGoogle Scholar
Song, C. & Sirviente, A. I. 2004 A numerical study of breaking waves. Phys. Fluids 16 (7), 2649–2667.CrossRefGoogle Scholar
Synolakis, C. E. 1987 The runup of solitary waves. J. Fluid Mech. 185, 523–545.CrossRefGoogle Scholar
Synolakis, C. E. & Skjelbreia, J. E. 1993 Evolution of maximum amplitude of solitary waves on plane beaches. J. Waterways Port Coast. ASCE 119 (3), 323–342.CrossRefGoogle Scholar
Tanaka, M., Dold, J. W., Lewy, M. & Peregrine, D. H. 1987 Instability and breaking of a solitary wave. J. Fluid Mech. 185, 235–248.CrossRefGoogle Scholar
Taylor, G. I. 1935 Statistical theory of turbulence. Proc. R. Soc. Lond. A 151 (873), 421–444.CrossRefGoogle Scholar
Thornton, E. B. & Guza, R. T. 1983 Transformation of wave height distribution. J. Geophys. Res. 88 (C10), 5925–5938.CrossRefGoogle Scholar
Tian, Z., Perlin, M. & Choi, W. 2011 Frequency spectra evolution of two-dimensional focusing wave groups in finite depth water. J. Fluid Mech. 688, 169–194.CrossRefGoogle Scholar
Vassilicos, J. C. 2015 Dissipation in turbulent flows. Annu. Rev. Fluid Mech. 47, 95–114.CrossRefGoogle Scholar
Yamada, H. & Shiotani, T.1968 On the highest water waves of permanent type.Google Scholar
Yin, J., Lin, N. & Yu, D. 2016 Coupled modeling of storm surge and coastal inundation: a case study in New York City during Hurricane Sandy. Water Resour. Res. 52 (11), 8685–8699.CrossRefGoogle Scholar

Mostert and Dieke supplementary movie 1

Video of spilling breaker at α=2^∘,h_0/d_0 =0.15, coloured by vorticity content in the liquid phase, with effective resolution Δx=L_0/2^14. (See main document for definition of nomenclature.)
Download Mostert and Dieke supplementary movie 1(Video)
Video 2 MB

Mostert and Dieke supplementary movie 2

Video of plunging breaker at α=3^∘,h_0/d_0 =0.3, coloured by vorticity content in the liquid phase, with effective resolution Δx=L_0/2^14. (See main document for definition of nomenclature.)

Download Mostert and Dieke supplementary movie 2(Video)
Video 2.3 MB

Mostert and Dieke supplementary movie 3

Video of strong plunging breaker at α=4^∘,h_0/d_0 =0.4, coloured by vorticity content in the liquid phase, with effective resolution Δx=L_0/2^14. (See main document for definition of nomenclature.)

Download Mostert and Dieke supplementary movie 3(Video)
Video 2.7 MB

Mostert and Dieke supplementary movie 4

Video of collapsing breaker at α=6^∘,h_0/d_0 =0.5, coloured by vorticity content in the liquid phase, with effective resolution Δx=L_0/2^14. (See main document for definition of nomenclature.)

Download Mostert and Dieke supplementary movie 4(Video)
Video 1.8 MB

Mostert and Dieke supplementary movie 5

Video of surging breaker at α=7^∘,h_0/d_0 =0.3, coloured by vorticity content in the liquid phase, with effective resolution Δx=L_0/2^13. (See main document for definition of nomenclature.)

Download Mostert and Dieke supplementary movie 5(Video)
Video 1.5 MB