Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T04:25:52.648Z Has data issue: false hasContentIssue false
  • English
  • Français

Direct numerical simulation of wind turbulence over breaking waves

Published online by Cambridge University Press:  04 July 2018

Zixuan Yang Open the ORCID record for Zixuan Yang [Opens in a new window] ,
Bing-Qing Deng  and
Lian Shen Open the ORCID record for Lian Shen [Opens in a new window]
Zixuan Yang
Affiliation:
Department of Mechanical Engineering and St. Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN 55455, USA
Bing-Qing Deng
Affiliation:
Department of Mechanical Engineering and St. Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN 55455, USA
Lian Shen*
Affiliation:
Department of Mechanical Engineering and St. Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN 55455, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study wind turbulence over breaking waves based on direct numerical simulation (DNS) of two-fluid flows. In the DNS, the air and water are simulated as a coherent system, with the interface captured using the coupled level-set and volume-of-fluid method. Because the wave breaking is an unsteady process, we use ensemble averaging over 100 runs to define turbulence statistics. We focus on analysing the turbulence statistics of the airflow over breaking waves. The effects of wave age and wave steepness are investigated. It is found that before wave breaking, the turbulence statistics are largely influenced by the wave age. The vertical gradient of mean streamwise velocity is positive at small and intermediate wave ages, but it becomes negative near the wave surface at large wave age as the pressure force changes from drag to thrust. Furthermore, wave-coherent motions make increasingly important contributions to the momentum flux and kinetic energy of velocity fluctuations (KE-F) as the wave age increases. During the wave breaking process, spilling breakers do not influence the wind field significantly; in contrast, plunging breakers alter the structures of wind turbulence near the wave surface drastically. It is observed from the DNS results that during wave plunging, a high pressure region occurs ahead of the wave front, which further accelerates the wind in the downstream direction. Meanwhile, a large spanwise vortex is generated, which greatly disturbs the airflow around it, resulting in large magnitudes of Reynolds stress and turbulence kinetic energy (TKE) below the wave crest. Above the crest, the magnitude of KE-F is enhanced during wave plunging at small and large wave ages, but at intermediate wave age, the transient enhancement of KE-F is absent. The effect of wave breaking on the magnitude of KE-F is further investigated through the analysis of the KE-F production. It is discovered that at small wave age, the transient enhancement of KE-F is caused by the appearance of a local maximum in the profile of total momentum flux; but at large wave age, it results from the change in the sign of the KE-F production from negative to positive, due to the sign change in the wave-coherent momentum flux. At intermediate wave age, neither of these two processes is present, and the transient growth of KE-F does not take place.

JFM classification

Geophysical and Geological Flows: Air/sea interactions Turbulent Flows: Turbulence simulation Waves/Free-surface Flows: Wave breaking
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 850 , 10 September 2018 , pp. 120 - 155
Check for updates
Copyright
© 2018 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

Babanin, A. 2011 Breaking and Dissipation of Ocean Surface Waves. Cambridge University Press.Google Scholar
Babanin, A., Chalikov, D., Young, I. & Savelyev, I. 2007 Predicting the breaking onset of surface water waves. Geophys. Res. Lett. 34, L07605.Google Scholar
Babanin, A. V., Chalikov, D., Young, I. R. & Savelyev, I. 2010 Numerical and laboratory investigation of breaking of steep two-dimensional waves in deep water. J. Fluid Mech. 644, 433–463.Google Scholar
Balay, S., Gropp, W. D., McInnes, L. C. & Smith, B. F. 1997 Efficient management of parallelism in object oriented numerical software libraries. In Modern Software Tools in Scientific Computing (ed. Arge, E., Bruaset, A. M. & Langtangen, H. P.), pp. 163–202. Birkhäuser.Google Scholar
Banner, M. L. 1990 The influence of wave breaking on the surface pressure distribution in wind–wave interactions. J. Fluid Mech. 211, 463–495.Google Scholar
Banner, M. L. & Melville, W. K. 1976 On the separation of air flow over water waves. J. Fluid Mech. 77, 825–842.Google Scholar
Banner, M. L. & Peirson, W. L. 1998 Tangential stress beneath wind-driven air–water interfaces. J. Fluid Mech. 364, 115–145.Google Scholar
Banner, M. L. & Peirson, W. L. 2007 Wave breaking onset and strength for two-dimensional deep-water wave groups. J. Fluid Mech. 585, 93–115.Google Scholar
Banner, M. L. & Peregrine, D. H. 1993 Wave breaking in deep water. Annu. Rev. Fluid Mech. 25, 373–397.Google Scholar
Banner, M. L. & Tian, X. 1998 On the determination of the onset of breaking for modulating surface gravity water waves. J. Fluid Mech. 367, 107–137.Google Scholar
Belcher, S. E. & Hunt, J. C. R. 1998 Turbulent flow over hills and waves. Annu. Rev. Fluid Mech. 30, 507–538.Google Scholar
Bell, M. M., Montgomery, M. T. & Emanuel, K. A. 2012 Air–sea enthalpy and momentum exchange at major hurricane wind speeds observed during CBLAST. J. Atmos. Sci. 69, 3197–3222.Google Scholar
Blenkinsopp, C. E. & Chaplin, J. R. 2007 Void fraction measurements in breaking waves. Proc. R. Soc. Lond. A 463, 3151–3170.Google Scholar
Bonmarin, P. 1989 Geometric properties of deep-water breaking waves. J. Fluid Mech. 209, 405–433.Google Scholar
Bonmarin, P. & Ramamonjiarisoa, A. 1985 Deformation to breaking of deep water gravity waves. Exp. Fluids 3, 11–16.Google Scholar
Brocchini, M. & Peregrine, D. H. 2001 The dynamics of strong turbulence at free surfaces. Part 2. Free-surface boundary conditions. J. Fluid Mech. 449, 255–290.Google Scholar
Buckley, M. P. & Veron, F. 2016 Structure of the airflow above surface waves. J. Phys. Oceanogr. 46, 1377–1397.Google Scholar
Caulliez, G. 2013 Dissipation regimes for short wind waves. J. Geophys. Res. 118 (2), 672–684.Google Scholar
Chang, K.-A. & Liu, P. L.-F. 1998 Velocity, acceleration and vorticity under a breaking wave. Phys. Fluids 10 (1), 327–329.Google Scholar
Chassaing, P., Antonia, R. A., Anselmet, F., Joly, L. & Sarkar, S. 2002 Variable Density Fluid Turbulence. Kluwer.Google Scholar
Chen, G., Kharif, C., Zaleski, S. & Li, J. 1999 Two-dimensional Navier–Stokes simulation of breaking waves. Phys. Fluids 11 (1), 121–133.Google Scholar
Chiang, W.-S. & Hwung, H.-H. 2007 Steepness effect on modulation instability of the nonlinear wave train. Phys. Fluids 19 (1), 014105.Google Scholar
Deane, G. B. & Stokes, M. D. 2002 Scale dependence of bubble creation mechanisms in breaking waves. Nature 418, 839–844.Google Scholar
Deike, L., Melville, W. K. & Popinet, S. 2016 Air entrainment and bubble statistics in breaking waves. J. Fluid Mech. 801, 91–129.Google Scholar
Deike, L., Popinet, S. & Melville, W. K. 2015 Capillary effects on wave breaking. J. Fluid Mech. 769, 541–569.Google Scholar
Derakhti, M. & Kirby, J. T. 2014 Bubble entrainment and liquid–bubble interaction under unsteady breaking waves. J. Fluid Mech. 761, 464–506.Google Scholar
Derakhti, M. & Kirby, J. T. 2016 Breaking-onset, energy and momentum flux in unsteady focused wave packets. J. Fluid Mech. 790, 553–581.Google Scholar
Diorio, J. D., Liu, X. & Duncan, J. H. 2009 An experimental investigation of incipient spilling breakers. J. Fluid Mech. 633, 271–283.Google Scholar
Donelan, M. A., Haus, B. K., Reul, N., Plant, W. J., Stiassnie, M., Graber, H. C., Brown, O. B. & Saltzman, E. S. 2004 On the limiting aerodynamic roughness of the ocean in very strong winds. Geophys. Res. Lett. 31, L18306.Google Scholar
Drazen, D. A., Melville, W. K. & Lenain, L. 2008 Inertial scaling of dissipation in unsteady breaking waves. J. Fluid Mech. 611, 307–332.Google Scholar
Druzhinin, O. A., Troitskaya, Y. I. & Zilitinkevich, S. S. 2012 Direct numerical simulation of a turbulent wind over a wavy water surface. J. Geophys. Res. 117, C00J05.Google Scholar
Duncan, J. H. 1981 An experimental investigation of breaking waves produced by a towed hydrofoil. Proc. R. Soc. Lond. A 377, 331–348.Google Scholar
Duncan, J. H. 1983 The breaking and non-breaking wave resistance of a two-dimensional hydrofoil. J. Fluid Mech. 126, 507–520.Google Scholar
Duncan, J. H. 2001 Spilling breakers. Annu. Rev. Fluid Mech. 33, 519–547.Google Scholar
Duncan, J. H., Qiao, H., Philomin, V. & Wenz, A. 1999 Gentle spilling breakers: crest profile evolution. J. Fluid Mech. 379, 191–222.Google Scholar
Eswaran, V. & Pope, S. B. 1988 An examination of forcing in direct numerical simulations of turbulence. Comput. Fluids 16 (3), 257–278.Google Scholar
Fedorov, A. V., Melville, W. K. & Rozenberg, A. 1998 An experimental and numerical study of parasitic capillary waves. Phys. Fluids 10 (6), 1315–1323.Google Scholar
Hara, T. & Sullivan, P. P. 2015 Wave boundary layer turbulence over surface waves in a strongly forced condition. J. Phys. Oceanogr. 45, 868–883.Google Scholar
Hendrickson, K., Weymouth, G., Banerjee, S. & Yue, D. K.-P. 2013 Air entraiment and multiphase turbulence in the bubbly wake of a transom stern. Int. Shipbuilding Progress 60 (1–4), 375–401.Google Scholar
Holthuijsen, L. H., Powell, M. D. & Pietrzak, J. D. 2012 Wind and waves in extreme hurricanes. J. Geophys. Res. 117, C09003.Google Scholar
Hong, W.-L. & Walker, D. T. 2000 Reynolds-averaged equations for free-surface flows with application to high-Froude-number jet spreading. J. Fluid Mech. 417, 183–209.Google Scholar
Hsu, C.-T., Hsu, E. Y. & Street, R. L. 1981 On the structure of turbulent flow over a progressive water wave: theory and experiment in a transformed, wave-following co-ordinate system. J. Fluid Mech. 105, 87–117.Google Scholar
Hu, Y., Guo, X., Lu, X., Liu, Y., Dalrymple, R. A. & Shen, L. 2012 Idealized numerical simulation of breaking water wave propagating over a viscous mud layer. Phys. Fluids 24 (11), 112104.Google Scholar
Iafrati, A. 2009 Numerical study of the effects of the breaking intensity on wave breaking flows. J. Fluid Mech. 622, 371–411.Google Scholar
Iafrati, A. 2011 Energy dissipation mechanisms in wave breaking processes: spilling and highly aerated plunging breaking events. J. Geophys. Res. 116, C07024.Google 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, 184504.Google Scholar
Jarosz, E., Mitchell, D. A., Wang, D. W. & Teague, W. J. 2007 Bottom-up determination of air–sea momentum exchange under a major tropical cyclone. Science 315 (5819), 1707–1709.Google Scholar
Jiang, L., Lin, H.-J., Schultz, W. W. & Perlin, M. 1999 Unsteady ripple generation on steep gravity–capillary waves. J. Fluid Mech. 386, 281–304.Google Scholar
Jiang, Q., Sullivan, P., Wang, S., Doyle, J. & Vincent, L. 2016 Impact of swell on air–sea momentum flux and marine boundary layer under low-wind conditions. J. Atmos. Sci. 73, 2683–2697.Google Scholar
Johannessen, T. B. & Swan, C. 2001 A laboratory study of the focusing of transient and directionally spread surface water waves. Proc. R. Soc. Lond. A 457, 971–1006.Google Scholar
Kawai, S. 1982 Structure of air flow separation over wind wave crests. Boundary-Layer Meteorol. 23, 503–521.Google Scholar
Kawamura, H. & Toba, Y. 1988 Ordered motion in the turbulent boundary layer over wind waves. J. Fluid Mech. 197, 105–138.Google Scholar
Kharif, C., Giovanangeli, J.-P., Touboul, J., Grare, L. & Pelinovsky, E. 2008 Influence of wind on extreme wave events: experimental and numerical approaches. J. Fluid Mech. 594, 209–247.Google Scholar
Kiger, K. T. & Duncan, J. H. 2012 Air-entrainment mechanisms in plunging jets and breaking waves. Annu. Rev. Fluid Mech. 44, 563–596.Google Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323.Google Scholar
Kudryavtsev, V. N. & Makin, V. K. 2001 The impact of air-flow separation on the drag of the sea surface. Boundary-Layer Meteorol. 98, 155–171.Google Scholar
Kudryavtsev, V. N. & Makin, V. K. 2007 Aerodynamic roughness of the sea surface at high winds. Boundary-Layer Meteorol. 125, 289–303.Google Scholar
Lamarre, E. & Melville, W. K. 1991 Air entrainment and dissipation in breaking waves. Nature 351, 469–472.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Liu, B., Guan, C. & Xie, L. 2012 The wave state and sea spray related parameterization of wind stress applicable from low to extreme winds. J. Geophys. Res. 117, C00J22.Google Scholar
Liu, X. & Duncan, J. H. 2003 The effects of surfactants on spilling breaking waves. Nature 421, 520–523.Google Scholar
Liu, X. & Duncan, J. H. 2006 An experimental study of surfactant effects on spilling breakers. J. Fluid Mech. 567, 433–455.Google Scholar
Liu, Y.2012 Numerical study of strong free surface flow and breaking waves. PhD thesis, The Johns Hopkins University.Google Scholar
Longuet-Higgins, M. & Tanaka, M. 1997a Crest instabilities of gravity waves. Part 3. Nonlinear development and breaking. J. Fluid Mech. 336, 33–50.Google Scholar
Longuet-Higgins, M. & Tanaka, M. 1997b On the crest instabilities of steep surface waves. J. Fluid Mech. 336, 51–68.Google Scholar
Longuet-Higgins, M. S. 1982 Parametric solutions for breaking waves. J. Fluid Mech. 121, 403–424.Google Scholar
Longuet-Higgins, M. S. 1992 Capillary rollers and bores. J. Fluid Mech. 240, 659–679.Google Scholar
López, J., Hernández, J., Gómez, P. & Faura, F. 2005 An improved PLIC-VOF method for tracking thin fluid structures in incompressible two-phase flows. J. Comput. Phys. 208, 51–74.Google Scholar
Lubin, P. & Glockner, S. 2015 Numerical simulations of three-dimensional plunging breaking waves: generation and evolution of aerated vortex filaments. J. Fluid Mech. 767, 364–393.Google Scholar
Mansour, N. N., Kim, J. & Moin, P. 1988 Reynolds-stress and dissipatio-rate budgets in a turbulent channel flow. J. Fluid Mech. 194, 15–44.Google Scholar
McLean, J. W., Ma, Y. C., Martin, D. U., Saffman, P. G. & Yuen, H. C. 1982 Three-dimensional instability of finite-amplitude water waves. Phys. Rev. Lett. 46 (13), 817–821.Google Scholar
Mei, C. C. 1983 The Applied Dynamics of Ocean Surface Waves. Wiley.Google Scholar
Melville, W. K. 1982 The instability and breaking of deep-water waves. J. Fluid Mech. 115, 165–185.Google Scholar
Melville, W. K. 1996 The role of surface-wave breaking in air–sea interaction. Annu. Rev. Fluid Mech. 28, 279–321.Google Scholar
Melville, W. K., Veron, F. & White, C. J. 2002 The velocity field under breaking waves: coherent structures and turbulence. J. Fluid Mech. 454, 203–233.Google Scholar
Meza, E., Zhang, J. & Seymour, R. J. 2000 Free-wave energy dissipation in experimental breaking waves. J. Phys. Oceanogr. 30, 2404–2418.Google Scholar
Moon, I.-J., Ginis, I. & Hara, T. 2004 Effect of surface waves on air–sea momentum exchange. Part II: behavior of drag coefficient under tropical cyclones. J. Atmos. Sci. 61, 2334–2348.Google Scholar
Mueller, J. A. & Veron, F. 2009 Nonlinear formulation of the bulk surface stress over breaking waves: feedback mechanisms from air-flow separation. Boundary-Layer Meteorol. 130, 117–134.Google Scholar
Nepf, H. M., Wu, C. H. & Chan, E. S. 1998 A comparison of two- and three-dimensional wave breaking. J. Phys. Oceanogr. 28, 1496–1510.Google Scholar
New, A. L. 1983 A class of elliptical free-surface flows. J. Fluid Mech. 130, 219–239.Google Scholar
Osher, S. & Sethian, J. A. 1988 Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49.Google Scholar
Perlin, M., Choi, W. & Tian, Z. 2013 Breaking waves in deep and intermediate waters. Annu. Rev. Fluid Mech. 45, 115–145.Google Scholar
Perlin, M., He, J. & Bernal, L. P. 1996 An experimental study of deep water plunging breakers. Phys. Fluids 8 (9), 2365–2374.Google Scholar
Perlin, M., Lin, H. & Ting, C.-L. 1993 On parasitic capillary waves generated by steep gravity waves: an experimental investigation with spatial and temporal measurements. J. Fluid Mech. 255, 597–620.Google Scholar
Perlin, M. & Schultz, W. W. 2000 Capillary effects on surface waves. Annu. Rev. Fluid Mech. 32, 241–274.Google Scholar
Pizzo, N. E., Deike, L. & Melville, W. K. 2016 Current generation by deep-water breaking waves. J. Fluid Mech. 803, 275–291.Google Scholar
Powell, M. D., Vickery, P. J. & Reinhold, T. A. 2003 Reduced drag coefficient for high wind speeds in tropical cyclones. Nature 422, 279–283.Google Scholar
Qiao, H. & Duncan, J. H. 2001 Gentle spilling breakers: crest flow-field evolution. J. Fluid Mech. 439, 57–85.Google Scholar
Rapp, R. J. & Melville, W. K. 1990 Laboratory measurements of deep-water breaking waves. Proc. R. Soc. Lond. A 331, 735–800.Google Scholar
Reul, N., Branger, H. & Giovanangeli, J.-P. 1999 Air flow separation over unsteady breaking waves. Phys. Fluids 11 (7), 1959–1961.Google Scholar
Reul, N., Branger, H. & Giovanangeli, J.-P. 2008 Air flow structure over short-gravity breaking water waves. Boundary-Layer Meteorol. 126, 477–505.Google Scholar
Scardovelli, R. & Zaleski, S. 1999 Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31, 567–603.Google Scholar
Schultz, W. W., Huh, J. & Griffin, O. M. 1994 Potential energy in steep and breaking waves. J. Fluid Mech. 278, 201–228.Google Scholar
Shu, C.-W. & Osher, S. 1989 Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys. 83, 32–78.Google Scholar
Skyner, D. 1996 A comparison of numerical predictions and experimental measurements of the internal kinematics of a deep-water plunging wave. J. Fluid Mech. 315, 51–64.Google Scholar
Smits, A. J., McKeon, B. J. & Marusic, I. 2011 High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43, 353–375.Google Scholar
Song, C. & Sirviente, A. I. 2004 A numerical study of breaking waves. Phys. Fluids 16 (7), 2649–2667.Google Scholar
Song, J.-B. & Banner, M. L. 2002 On determining the onset and strength of breaking for deep water waves. Part I: unforced irrotational wave groups. J. Phys. Oceanogr. 32, 2541–2558.Google Scholar
Stansell, P. & MacFarlane, C. 2002 Experimental investigation of wave breaking criteria based on wave phase speeds. J. Phys. Oceanogr. 32, 1269–1283.Google Scholar
Su, M.-Y., Bergin, M., Marler, P. & Myrick, R. 1982 Experiments on nonlinear instabilities and evolution of steep gravity-wave trains. J. Fluid Mech. 124, 45–72.Google Scholar
Sullivan, P. P., Banner, M., Morison, R. & Peirson, W. 2018 Turbulent flow over steep steady and unsteady waves under strong wind forcing. J. Phys. Oceanogr. 48, 3–27.Google Scholar
Sullivan, P. P., McWilliams, J. C. & Moeng, C.-H. 2000 Simulation of turbulent flow over idealized water waves. J. Fluid Mech. 404, 47–85.Google Scholar
Sullivan, P. P., McWilliams, J. C. & Patton, E. G. 2014 Large-eddy simulation of marine atmospheric boundary layers above a specturm of moving waves. J. Atmos. Sci. 71, 4001–4027.Google Scholar
Sussman, M., Fatemi, E., Smereka, P. & Osher, S. 1998 An improved level set method for incompressible two-phase flows. Comput. Fluids 27, 663–680.Google Scholar
Sussman, M. & Puckett, E. G. 2000 A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows. J. Comput. Phys. 162, 301–337.Google Scholar
Sussman, M., Smereka, P. & Osher, S. 1994 A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159.Google Scholar
Suzuki, N., Hara, T. & Sullivan, P. P. 2015 Impact of breaking wave form drag on near-surface turbulence and drag coefficient over young seas at high winds. J. Phys. Oceanogr. 43, 324–343.Google Scholar
Takagaki, N., Komori, S., Suzuki, N., Iwano, K., Kuramoto, T., Shimada, S., Kurose, R. & Takahashi, K. 2012 Strong correlation between the drag coefficient and the shape of the wind sea spectrum over a broad range of wind speeds. Geophys. Res. Lett. 39, L23604.Google Scholar
Tian, Z., Perlin, M. & Choi, W. 2008 Evaluation of a deep-water wave breaking criterion. Phys. Fluids 20 (6), 066604.Google Scholar
Tian, Z., Perlin, M. & Choi, W. 2010 Energy dissipation in two-dimensional unsteady plunging breakers and an eddy viscosity model. J. Fluid Mech. 655, 217–257.Google Scholar
Tian, Z., Perlin, M. & Choi, W. 2012 An eddy viscosity model for two-dimensional breaking waves and its validation with laboratory experiments. Phys. Fluids 24, 036601.Google Scholar
Toffoli, A., Babanin, A., Onorato, M. & Waseda, T. 2010 Maximum steepness of oceanic waves: field and laboratory experiments. Geophys. Res. Lett. 37, L05603.Google Scholar
Townsend, A. A. 1980 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Troitskaya, Y. I., Sergeev, D. A., Kandaurov, A. A., Baidakov, G. A., Vdovin, M. A. & Kazakov, V. I. 2012 Laboratory and theoretical modeling of air–sea momentum transfer under severe wind conditions. J. Geophys. Res. 117, C00J21.Google Scholar
Tulin, M. P. & Waseda, T. 1999 Laboratory observations of wave group evolution, including breaking effects. J. Fluid Mech. 378, 197–232.Google Scholar
Veron, F., Saxena, G. & Misra, S. K. 2007 Measurements of the viscous tangential stress in the airflow above wind waves. Geophys. Res. Lett. 34, L19603.Google Scholar
Wang, Z., Yang, J. & Stern, F. 2016 High-fidelity simulations of bubble, droplet and spray formation in breaking waves. J. Fluid Mech. 792, 307–327.Google Scholar
Weissman, M. A. 1986 Observations and measurements of air flow over water waves. In Wave Dynamics and Radio Probing of the Ocean Surface (ed. Phillips, O. M. & Hasselmann, K.), chap. 23. Springer.Google Scholar
Weymouth, G. D. & Yue, D. K.-P. 2010 Conservative volume-of-fluid method for free-surface simulations on Cartesian-grids. J. Comput. Phys. 229, 2853–2865.Google Scholar
Wu, C. H. & Nepf, H. M. 2002 Breaking criteria and energy losses for three-dimensional wave breaking. J. Geophys. Res. 107 (C10), 3177.Google Scholar
Wu, C. H. & Yao, A. 2004 Laboratory measurements of limiting freak waves on currents. J. Geophys. Res. 109, C12002.Google Scholar
Yang, D. & Shen, L. 2010 Direct-simulation-based study of turbulent flow over various waving boundaries. J. Fluid Mech. 650, 131–180.Google Scholar
Yang, Z., Lu, X.-H., Guo, X., Liu, Y. & Shen, L. 2017 Numerical simulation of sediment suspension and transport under plunging breaking waves. Comput. Fluids 158, 57–71.Google Scholar
Yao, A. & Wu, C. H. 2004 Energy dissipation of unsteady wave breaking on currents. J. Phys. Oceanogr. 34, 2288–2304.Google Scholar
Yao, A. & Wu, C. H. 2005 Incipient breaking of unsteady waves on sheared currents. Phys. Fluids 17, 082104.Google Scholar
Zhang, X. 1995 Capillary–gravity and capillary waves generated in a wind wave tank: observations and theories. J. Fluid Mech. 289, 51–82.Google Scholar
Zhao, Z.-K., Liu, C.-X., Li, Q., Dai, G.-F., Song, Q.-T. & Lv, W.-H. 2015 Typhoon air–sea drag coefficient in coastal regions. J. Geophys. Res. 120, 716–727.Google Scholar