Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-17T07:23:58.879Z Has data issue: false hasContentIssue false
  • English
  • Français

Wind–wave coupling study using LES of wind and phase-resolved simulation of nonlinear waves

Published online by Cambridge University Press:  09 July 2019

Xuanting Hao Open the ORCID record for Xuanting Hao [Opens in a new window]  and
Lian Shen Open the ORCID record for Lian Shen [Opens in a new window]
Xuanting Hao
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 present a study on the interaction between wind and water waves with a broad-band spectrum using wave-phase-resolved simulation with long-term wave field evolution. The wind turbulence is computed using large-eddy simulation and the wave field is simulated using a high-order spectral method. Numerical experiments are carried out for turbulent wind blowing over a wave field initialised using the Joint North Sea Wave Project spectrum, with various wind speeds considered. The results show that the waves, together with the mean wind flow and large turbulent eddies, have a significant impact on the wavenumber–frequency spectrum of the wind turbulence. It is found that the shear stress contributed by sweep events in turbulent wind is greatly enhanced as a result of the waves. The dependence of the wave growth rate on the wave age is consistent with the results in the literature. The probability density function and high-order statistics of the wave surface elevation deviate from the Gaussian distribution, manifesting the nonlinearity of the wave field. The shape of the change in the spectrum of wind-waves resembles that of the nonlinear wave–wave interactions, indicating the dominant role played by the nonlinear interactions in the evolution of the wave spectrum. The frequency downshift phenomenon is captured in our simulations wherein the wind-forced wave field evolves for $O(3000)$ peak wave periods. Using the numerical result, we compute the universal constant in a wave-growth law proposed in the literature, and substantiate the scaling of wind–wave growth based on intrinsic wave properties.

JFM classification

Geophysical and Geological Flows: Air/sea interactions Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 874 , 10 September 2019 , pp. 391 - 425
Check for updates
Copyright
© 2019 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

Agafontsev, D. S. & Zakharov, V. E. 2015 Intermittency in generalized NLS equation with focusing six-wave interactions. Phys. Lett. A 379 (40–41), 2586–2590.10.1016/j.physleta.2015.05.042Google Scholar
Annenkov, S. Y. & Shrira, V. I. 2013 Large-time evolution of statistical moments of wind-wave fields. J. Fluid Mech. 726, 517–546.10.1017/jfm.2013.243Google Scholar
Annenkov, S. Y. & Shrira, V. I. 2014 Evaluation of skewness and kurtosis of wind waves parameterized by JONSWAP spectra. J. Phys. Oceanogr. 44 (6), 1582–1594.10.1175/JPO-D-13-0218.1Google Scholar
Badulin, S. I., Babanin, A. V., Zakharov, V. E. & Resio, D. T. 2007 Weakly turbulent laws of wind-wave growth. J. Fluid Mech. 591, 339–378.10.1017/S0022112007008282Google Scholar
Badulin, S. I., Pushkarev, A. N., Resio, D. T. & Zakharov, V. E. 2005 Self-similarity of wind-driven seas. Nonlinear Process. Geophys. 12 (6), 891–945.10.5194/npg-12-891-2005Google Scholar
Banner, M. L. & Melville, W. K. 1976 On the separation of air flow over water waves. J. Fluid Mech. 77 (04), 825–842.10.1017/S0022112076002905Google Scholar
Belcher, S. E. & Hunt, J. C. R. 1993 Turbulent shear flow over slowly moving waves. J. Fluid Mech. 251, 109–148.10.1017/S0022112093003350Google Scholar
Buckley, M. P. & Veron, F. 2016 Structure of the airflow above surface waves. J. Phys. Oceanogr. 46 (5), 1377–1397.10.1175/JPO-D-15-0135.1Google Scholar
Buckley, M. P. & Veron, F. 2017 Airflow measurements at a wavy air–water interface using PIV and LIF. Exp. Fluids 58 (11), 161.10.1007/s00348-017-2439-2Google Scholar
Burling, R. W. 1959 The spectrum of waves at short fetches. Dtsch. Hydrogr. Z. 12 (3), 96–117.10.1007/BF02019818Google Scholar
Calaf, M., Meneveau, C. & Meyers, J. 2010 Large eddy simulation study of fully developed wind-turbine array boundary layers. Phys. Fluids 22 (1), 015110.10.1063/1.3291077Google Scholar
Campbell, B. K., Hendrickson, K. & Liu, Y. 2016 Nonlinear coupling of interfacial instabilities with resonant wave interactions in horizontal two-fluid plane Couette–Poiseuille flows: numerical and physical observations. J. Fluid Mech. 809, 438–479.10.1017/jfm.2016.636Google Scholar
Caulliez, G. 2013 Dissipation regimes for short wind waves. J. Geophys. Res. Ocean. 118 (2), 672–684.10.1029/2012JC008402Google Scholar
Chalikov, D., Babanin, A. V. & Sanina, E. 2014 Numerical modeling of 3D fully nonlinear potential periodic waves. Ocean Dyn. 64 (10), 1469–1486.Google Scholar
Chalikov, D. V. 2016 Numerical Modeling of Sea Waves. Springer International Publishing.10.1007/978-3-319-32916-1Google Scholar
DeLeonibus, P. S. & Simpson, L. S. 1972 Case study of duration-limited wave spectra observed at an open ocean tower. J. Geophys. Res. 77 (24), 4555–4569.10.1029/JC077i024p04555Google Scholar
Dommermuth, D. G. & Yue, D. K. P. 1987 A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267–288.10.1017/S002211208700288XGoogle Scholar
Donelan, M. 1979 On the fraction of wind momentum retained by waves. In Mar. Forecast. (ed. Nihoul, J. C. J.), pp. 141–159. Elsevier.Google Scholar
Donelan, M. A. 1999 Wind-induced growth and attenuation of laboratory waves. In Wind. Couplings Perspect. Prospect. (ed. Sajjadi, S. G., Thomas, N. H. & Hunt, J. C. R.), pp. 183–194. Clarendon.Google Scholar
Donelan, M. A. 2004 On the limiting aerodynamic roughness of the ocean in very strong winds. Geophys. Res. Lett. 31 (18), L18306.10.1029/2004GL019460Google Scholar
Donelan, M. A., Babanin, A. V., Young, I. R. & Banner, M. L. 2006 Wave-follower field measurements of the wind-input spectral function. Part II. Parameterization of the wind input. J. Phys. Oceanogr. 36 (8), 1672–1689.10.1175/JPO2933.1Google Scholar
Donelan, M. A. & Pierson, W. J. 1987 Radar scattering and equilibrium ranges in wind-generated waves with application to scatterometry. J. Geophys. Res. 92 (C5), 4971–5029.10.1029/JC092iC05p04971Google Scholar
Duncan, J. H. 2001 Spilling breakers. Annu. Rev. Fluid Mech. 33 (1), 519–547.10.1146/annurev.fluid.33.1.519Google Scholar
Edson, J., Crawford, T., Crescenti, J., Farrar, T., Frew, N., Gerbi, G., Helmis, C., Hristov, T., Khelif, D., Jessup, A. et al. 2007 The coupled boundary layers and air–sea transfer experiment in low winds. Bull. Am. Meteorol. Soc. 88 (3), 341–356.10.1175/BAMS-88-3-341Google Scholar
Engsig-Karup, A. P., Bingham, H. B. & Lindberg, O. 2009 An efficient flexible-order model for 3D nonlinear water waves. J. Comput. Phys. 228 (6), 2100–2118.10.1016/j.jcp.2008.11.028Google Scholar
Gagnaire-Renou, E., Benoit, M. & Badulin, S. I. 2011 On weakly turbulent scaling of wind sea in simulations of fetch-limited growth. J. Fluid Mech. 669, 178–213.10.1017/S0022112010004921Google Scholar
García-Nava, H., Ocampo-Torres, F. J., Osuna, P. & Donelan, M. A. 2009 Wind stress in the presence of swell under moderate to strong wind conditions. J. Geophys. Res. Ocean. 114 (C12), C12008.10.1029/2009JC005389Google Scholar
Germano, M., Piomelli, U., Moin, P. & Cabot, W. H. 1991 A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A 3 (7), 1760.10.1063/1.857955Google Scholar
Grare, L., Peirson, W. L., Branger, H., Walker, J. W., Giovanangeli, J.-P. & Makin, V. 2013 Growth and dissipation of wind-forced, deep-water waves. J. Fluid Mech. 722, 5–50.10.1017/jfm.2013.88Google Scholar
Hasselmann, K. 1962 On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory. J. Fluid Mech. 12 (04), 481–500.10.1017/S0022112062000373Google Scholar
Hasselmann, K. 1963a On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems; wave-particle analogy; irreversibility. J. Fluid Mech. 15 (02), 273–281.10.1017/S0022112063000239Google Scholar
Hasselmann, K. 1963b On the non-linear energy transfer in a gravity-wave spectrum. Part 3. Evaluation of the energy flux and swell-sea interaction for a Neumann spectrum. J. Fluid Mech. 15 (03), 385–398.10.1017/S002211206300032XGoogle Scholar
Hasselmann, K., Barnett, T. P., Bouws, E., Carlson, H., Cartwright, D. E., Enke, K., Ewing, J. A., Gienapp, H., Hasselmann, D. E., Kruseman, P. et al. 1973 Measurements of Wind-Wave Growth and Swell Decay During the Joint North Sea Wave Project (JONSWAP). Deutches Hydrographisches Institut.Google Scholar
He, G., Jin, G. & Yang, Y. 2017 Space–time correlations and dynamic coupling in turbulent flows. Annu. Rev. Fluid Mech. 49 (1), 51–70.10.1146/annurev-fluid-010816-060309Google Scholar
He, G. W. & Zhang, J. B. 2006 Elliptic model for space–time correlations in turbulent shear flows. Phys. Rev. E 73 (5), 2–5.10.1103/PhysRevE.73.055303Google Scholar
Holthuijsen, L. H. 2007 Waves in Oceanic and Coastal Waters. Cambridge University Press.10.1017/CBO9780511618536Google Scholar
Hwang, P. A. & Wang, D. W. 2004 Field measurements of duration-limited growth of wind-generated ocean surface waves at young stage of development. J. Phys. Oceanogr. 34, 2316–2326.10.1175/1520-0485(2004)034<2316:FMODGO>2.0.CO;22.0.CO;2>Google Scholar
Iafrati, A., De Vita, F. & Verzicco, R. 2019 Effects of the wind on the breaking of modulated wave trains. Eur. J. Mech. (B/Fluids) 73, 6–23.10.1016/j.euromechflu.2018.03.012Google Scholar
Janssen, P.A.E.M. 2009 On some consequences of the canonical transformation in the Hamiltonian theory of water waves. J. Fluid Mech. 637, 1–44.10.1017/S0022112009008131Google Scholar
Jeffreys, H. 1925 On the formation of water waves by wind. Proc. R. Soc. Lond. A 107 (742), 189–206.Google Scholar
Jeffreys, H. 1926 On the formation of water waves by wind (second paper). Proc. R. Soc. Lond. A 110 (754), 241–247.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 (7), 2683–2697.10.1175/JAS-D-15-0200.1Google Scholar
Kaihatu, J. M., Veeramony, J., Edwards, K. L. & Kirby, J. T. 2007 Asymptotic behavior of frequency and wave number spectra of nearshore shoaling and breaking waves. J. Geophys. Res. 112 (C6), C06016.10.1029/2006JC003817Google Scholar
Kats, A. V. & Kontorovich, V. M. 1973 Symmetry properties of the collision integral and non-isotropic stationary solutions in weak turbulence theory. J. Expl Theor. Phys. 37 (1), 80–85.Google Scholar
Kats, A. V. & Kontorovich, V. M. 1974 Anisotropic turbulent distributions for waves with a nondecay dispersion law. J. Expl Theor. Phys. 38 (1), 102–107.Google Scholar
Kawai, S. 1981 Visualization of airflow separation over wind-wave crests under moderate wind. Boundary-Layer Meteorol. 21 (1), 93–104.10.1007/BF00119370Google Scholar
Kihara, N., Hanazaki, H., Mizuya, T. & Ueda, H. 2007 Relationship between airflow at the critical height and momentum transfer to the traveling waves. Phys. Fluids 19 (1), 015102.10.1063/1.2409736Google Scholar
Kirby, J. T. & Kaihatu, J. M. 1996 Structure of frequency domain models for random wave breaking. Coast. Eng. Proc. 25, 1144–1155.Google Scholar
Kolassa, J. E. 2006 Series Approximation Methods in Statistics, Lecture Notes in Statistics, vol. 88. Springer.Google Scholar
Komen, G. J., Cavaleri, L., Donelan, M. A., Hasselmann, K., Hasselmann, S. & Janssen, P. A. E. M. 1994 Dynamics and Modelling of Ocean Waves. Cambridge University Press.10.1017/CBO9780511628955Google Scholar
Korotkevich, A. O., Pushkarev, A., Resio, D. & Zakharov, V. E. 2008 Numerical verification of the weak turbulent model for swell evolution. Eur. J. Mech. (B/Fluids) 27 (4), 361–387.10.1016/j.euromechflu.2007.08.004Google Scholar
Li, P. Y., Xu, D. & Taylor, P. A. 2000 Numerical modelling of turbulent airflow over water waves. Boundary-Layer Meteorol. 95 (3), 397–425.10.1023/A:1002677312259Google Scholar
Lilly, D. K. 1992 A proposed modification of the Germano-subgrid-scale closure method. Phys. Fluids A 4 (3), 633–635.10.1063/1.858280Google Scholar
Lin, M.-Y., Moeng, C.-H., Tsai, W.-T., Sullivan, P. P. & Belcher, S. E. 2008 Direct numerical simulation of wind-wave generation processes. J. Fluid Mech. 616, 1–30.10.1017/S0022112008004060Google Scholar
Liu, P. C. 1985 Testing parametric correlations for wind waves in the Great Lakes. J. Great Lakes Res. 11 (4), 478–491.10.1016/S0380-1330(85)71791-1Google Scholar
Liu, Y., Yang, D., Guo, X. & Shen, L. 2010 Numerical study of pressure forcing of wind on dynamically evolving water waves. Phys. Fluids 22 (4), 041704.10.1063/1.3414832Google Scholar
Longuet-Higgins, M. S. 1963 The effect of non-linearities on statistical distributions in the theory of sea waves. J. Fluid Mech. 17 (3), 459–480.10.1017/S0022112063001452Google Scholar
Lu, S. S. & Willmarth, W. W. 1973 Measurements of the structure of the Reynolds stress in a turbulent boundary layer. J. Fluid Mech. 60 (03), 481–511.10.1017/S0022112073000315Google Scholar
Mastenbroek, C., Makin, V. K., Garat, M. H. & Giovanangeli, J. P. 1996 Experimental evidence of the rapid distortion of turbulence in the air flow over water waves. J. Fluid Mech. 318, 273–302.10.1017/S0022112096007124Google Scholar
Melville, W. K. 1996 The role of surface-wave breaking in air–sea interaction. Annu. Rev. Fluid Mech. 28 (1), 279–321.10.1146/annurev.fl.28.010196.001431Google Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3 (2), 185–204.10.1017/S0022112057000567Google Scholar
Miles, J. W. 1993 Surface-wave generation revisited. J. Fluid Mech. 256, 427–441.10.1017/S0022112093002836Google Scholar
Nikolayeva, Y. I. & Tsimring, L. S. 1986 Kinetic model of the wind generation of waves by turbulent wind. Izv. Atmos. Ocean. Phys. 22, 102–107.Google Scholar
Nilsson, E. O., Rutgersson, A., Smedman, A.-S. & Sullivan, P. P. 2012 Convective boundary-layer structure in the presence of wind-following swell. Q. J. R. Meteorol. Soc. 138 (667), 1476–1489.10.1002/qj.1898Google Scholar
Ochi, M. K. & Wang, W.-C. 1984 Non-Gaussian characteristics of coastal waves. In Proceedings of the 19th Conference on Coast. Eng.Google Scholar
Onorato, M., Cavaleri, L., Fouques, S., Gramstad, O., Janssen, P.A.E.M., Monbaliu, J., Osborne, A. R., Pakozdi, C., Serio, M., Stansberg, C. T. et al. 2009 Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in a three-dimensional wave basin. J. Fluid Mech. 627, 235–257.10.1017/S002211200900603XGoogle Scholar
Perlin, M., Choi, W. & Tian, Z. 2013 Breaking waves in deep and intermediate waters. Annu. Rev. Fluid Mech. 45 (1), 115–145.10.1146/annurev-fluid-011212-140721Google Scholar
Phillips, O. M. 1957 On the generation of waves by turbulent wind. J. Fluid Mech. 2 (5), 417–445.10.1017/S0022112057000233Google Scholar
Phillips, O. M. 1958 The equilibrium range in the spectrum of wind-generated waves. J. Fluid Mech. 4, 426–434.10.1017/S0022112058000550Google Scholar
Piomelli, U. & Balaras, E. 2002 Wall-layer models for large-eddy simulations. Annu. Rev. Fluid Mech. 34 (1), 349–374.10.1146/annurev.fluid.34.082901.144919Google Scholar
Plant, W. J. 1982 A relationship between wind stress and wave slope. J. Geophys. Res. Ocean. 87 (C3), 1961–1967.10.1029/JC087iC03p01961Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.10.1017/CBO9780511840531Google Scholar
Raupach, M. R. 1981 Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers. J. Fluid Mech. 108, 363–382.10.1017/S0022112081002164Google Scholar
Reul, N., Branger, H. & Giovanangeli, J.-P. 1999 Air flow separation over unsteady breaking waves. Phys. Fluids 11 (7), 1959–1961.10.1063/1.870058Google Scholar
Reul, N., Branger, H. & Giovanangeli, J.-P. 2008 Air flow structure over short-gravity breaking water waves. Boundary-Layer Meteorol. 126 (3), 477–505.10.1007/s10546-007-9240-3Google Scholar
Romero, L. & Melville, W. K. 2010 Numerical modeling of fetch-limited waves in the Gulf of Tehuantepec. J. Phys. Oceanogr. 40 (3), 466–486.10.1175/2009JPO4128.1Google Scholar
Sergeev, D., Kandaurov, A., Troitskaya, Y., Caulliez, G., Bopp, M. & Jaehne, B. 2017 Laboratory modelling of the wind–wave interaction with modified PIV-method. EPJ Web Conf. 143, 02101.10.1051/epjconf/201714302101Google Scholar
Smedman, A., Högström, U., Bergström, H. & Rutgersson, A. 1999 A case study of air–sea interaction during swell conditions. J. Geophys. Res. Ocean. 104, 25833–25851.10.1029/1999JC900213Google Scholar
Socquet-Juglard, H., Dysthe, K. B., Trulsen, K., Krogstad, H. E. & Liu, J. 2005 Probability distributions of surface gravity waves during spectral changes. J. Fluid Mech. 542, 195–216.10.1017/S0022112005006312Google Scholar
Sullivan, P. P., Edson, J. B., Hristov, T. & McWilliams, J. C. 2008 Large-eddy simulations and observations of atmospheric marine boundary layers above nonequilibrium surface waves. J. Atmos. Sci. 65 (4), 1225–1245.10.1175/2007JAS2427.1Google Scholar
Sullivan, P. P. & McWilliams, J. C. 2010 Dynamics of winds and currents coupled to surface waves. Annu. Rev. Fluid Mech. 42, 19–42.10.1146/annurev-fluid-121108-145541Google Scholar
Sullivan, P. P., Mcwilliams, J. C. & Moeng, C. 2000 Simulation of turbulent flow over idealized water waves. J. Fluid Mech. 404, 47–85.10.1017/S0022112099006965Google Scholar
Sverdrup, H. U. & Munk, W. H. 1947 Wind, Sea, and Swell: Theory of Relations for Forecasting. U.S. Hydrographic Office.Google Scholar
Tanaka, M. 2001 Verification of Hasselmann’s energy transfer among surface gravity waves by direct numerical simulations of primitive equations. J. Fluid Mech. 444, 199–221.10.1017/S0022112001005389Google Scholar
Tayfun, M. A. 1980 Narrow-band nonlinear sea waves. J. Geophys. Res. Ocean. 85 (C3), 1548–1552.10.1029/JC085iC03p01548Google Scholar
Tayfun, M. A. & Fedele, F. 2007 Wave-height distributions and nonlinear effects. Ocean Engng 34 (11-12), 1631–1649.10.1016/j.oceaneng.2006.11.006Google Scholar
Taylor, G. I. 1938 The spectrum of turbulence. Proc. R. Soc. Lond. A 164 (919), 476–490.Google Scholar
The WAVEWATCH III® Development Group2016 User manual and system documentation of WAVEWATCH III® version 5.16. Tech. Note 329, NOAA/NWS/NCEP/MMAB, College Park, MD, USA, 326 pp. + Appendices.Google Scholar
Tian, Z., Perlin, M. & Choi, W. 2010 Observation of the occurrence of air flow separation over water waves. In 29th International Conference on Ocean. Offshore Arct. Eng., vol. 4, pp. 333–341. ASME.10.1115/OMAE2010-20576Google Scholar
Toba, Y. 1972 Local balance in the air–sea boundary processes. J. Oceanogr. 28 (3), 109–120.10.1007/BF02109772Google Scholar
Toffoli, A., Benoit, M., Onorato, M. & Bitner-Gregersen, E. M. 2009 The effect of third-order nonlinearity on statistical properties of random directional waves in finite depth. Nonlinear Process. Geophys. 16 (1), 131–139.10.5194/npg-16-131-2009Google Scholar
Tracy, B. A. & Resio, D. T.1982 Theory and calculation of the nonlinear energy transfer between sea waves in deep water. WES Rep. 11. Hydraulics Laboratory, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS.Google Scholar
Troitskaya, Y., Sergeev, D., Ermakova, O. & Balandina, G. 2011a Statistical parameters of the air turbulent boundary layer over steep water waves measured by the PIV technique. J. Phys. Oceanogr. 41 (8), 1421–1454.10.1175/2011JPO4392.1Google Scholar
Troitskaya, Y., Sergeev, D., Kandaurov, A. & Kazakov, V. 2011b Air–sea interaction under hurricane wind conditions. In Recent Hurric. Res. – Clim. Dyn. Soc. Impacts (ed. Lupo, A.), InTech.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. Ocean. 117 (C11), C00J21.10.1029/2011JC007778Google Scholar
Tulin, M. P. & Waseda, T. 1999 Laboratory observations of wave group evolution, including breaking effects. J. Fluid Mech. 378, 197–232.10.1017/S0022112098003255Google 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 (19), L19603.10.1029/2007GL031242Google Scholar
Wallace, J. M., Eckelmann, H. & Brodkey, R. S. 1972 The wall region in turbulent shear flow. J. Fluid Mech. 54 (1), 39–48.10.1017/S0022112072000515Google Scholar
Webb, D. J. 1978 Non-linear transfers between sea waves. Deep-Sea Res. 25, 279–298.10.1016/0146-6291(78)90593-3Google Scholar
Wilczek, M. & Narita, Y. 2012 Wave-number-frequency spectrum for turbulence from a random sweeping hypothesis with mean flow. Phys. Rev. E 86 (6), 1–8.10.1103/PhysRevE.86.066308Google Scholar
Wu, G.2004 Direct simulation and deterministic prediction of large-scale nonlinear ocean wave-field. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
Xiao, W., Liu, Y., Wu, G. & Yue, D. K. P. 2013 Rogue wave occurrence and dynamics by direct simulations of nonlinear wave-field evolution. J. Fluid Mech. 720, 357–392.10.1017/jfm.2013.37Google Scholar
Yang, D., Meneveau, C. & Shen, L. 2013 Dynamic modelling of sea-surface roughness for large-eddy simulation of wind over ocean wavefield. J. Fluid Mech. 726, 62–99.10.1017/jfm.2013.215Google Scholar
Yang, D., Meneveau, C. & Shen, L. 2014a Effect of downwind swells on offshore wind energy harvesting – a large-eddy simulation study. Renew. Energy 70, 11–23.10.1016/j.renene.2014.03.069Google Scholar
Yang, D., Meneveau, C. & Shen, L. 2014b Large-eddy simulation of offshore wind farm. Phys. Fluids 26 (2), 025101.10.1063/1.4863096Google Scholar
Yang, D. & Shen, L. 2010 Direct-simulation-based study of turbulent flow over various waving boundaries. J. Fluid Mech. 650, 131–180.10.1017/S0022112009993557Google Scholar
Yang, D. & Shen, L. 2011a Simulation of viscous flows with undulatory boundaries. Part I. Basic solver. J. Comput. Phys. 230 (14), 5488–5509.10.1016/j.jcp.2011.02.036Google Scholar
Yang, D. & Shen, L. 2011b Simulation of viscous flows with undulatory boundaries. Part II. Coupling with other solvers for two-fluid computations. J. Comput. Phys. 230 (14), 5510–5531.10.1016/j.jcp.2011.02.035Google Scholar
Yang, Z., Deng, B.-Q. & Shen, L. 2018 Direct numerical simulation of wind turbulence over breaking waves. J. Fluid Mech. 850, 120–155.10.1017/jfm.2018.466Google Scholar
Young, I. R. 1999 Wind Generated Ocean Waves, 1st edn. Elsevier.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190–194.10.1007/BF00913182Google Scholar
Zakharov, V. E., Badulin, S. I., Hwang, P. A. & Caulliez, G. 2015 Universality of sea wave growth and its physical roots. J. Fluid Mech. 780, 503–535.10.1017/jfm.2015.468Google Scholar
Zakharov, V. E. & Zaslavskii, M. M. 1982 Kinetic equation and Kolmogorov spectra in the weak turbulence theory of wind waves. Izv. Akad. Nauk SSSR Fiz. Atmos. i okeana 18 (9), 970–979.Google Scholar
Zonta, F., Soldati, A. & Onorato, M. 2015 Growth and spectra of gravity–capillary waves in countercurrent air/water turbulent flow. J. Fluid Mech. 777, 245–259.10.1017/jfm.2015.356Google Scholar