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Turbulent windprint on a liquid surface

Published online by Cambridge University Press:  28 June 2019

Stéphane Perrard Open the ORCID record for Stéphane Perrard [Opens in a new window] ,
Adrián Lozano-Durán Open the ORCID record for Adrián Lozano-Durán [Opens in a new window] ,
Marc Rabaud Open the ORCID record for Marc Rabaud [Opens in a new window] ,
Michael Benzaquen Open the ORCID record for Michael Benzaquen [Opens in a new window]  and
Frédéric Moisy Open the ORCID record for Frédéric Moisy [Opens in a new window]
Stéphane Perrard
Affiliation:
FAST, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France LadHyX, UMR CNRS 7646, Ecole polytechnique, 91128 Palaiseau, France
Adrián Lozano-Durán
Affiliation:
Center for Turbulence Research Stanford University, Stanford, CA 94305, USA
Marc Rabaud
Affiliation:
FAST, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
Michael Benzaquen
Affiliation:
LadHyX, UMR CNRS 7646, Ecole polytechnique, 91128 Palaiseau, France
Frédéric Moisy
Affiliation:
FAST, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
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Abstract

We investigate the effect of a light turbulent wind on a liquid surface, below the onset of wave generation. In that regime, the liquid surface is populated by small disorganised deformations elongated in the streamwise direction. Formally identified recently by Paquier et al. (Phys. Fluids, vol. 27, 2015, art. 122103), the deformations that occur below the wave onset were named wrinkles. We provide here a theoretical framework for this regime, using the viscous response of a free liquid surface submitted to arbitrary normal and tangential interfacial stresses at its upper boundary. We relate the spatio-temporal spectrum of the surface deformations to that of the applied interfacial pressure and shear stress fluctuations. For that, we evaluate the spatio-temporal statistics of the turbulent forcing using direct numerical simulation of a turbulent channel flow, assuming no coupling between the air and the liquid flows. Combining theory and numerical simulation, we obtain synthetic wrinkles fields that reproduce the experimental observations. We show that the wrinkles are a multi-scale superposition of random wakes generated by the turbulent fluctuations. They result mainly from the nearly isotropic pressure fluctuations generated in the boundary layer, rather than from the elongated shear stress fluctuations. The wrinkle regime described in this paper naturally arises as the viscous-saturated asymptotic of the inviscid growth theory of Phillips (J. Fluid Mech., vol. 2 (05), 1957, pp. 417–445). We finally discuss the possible relation between wrinkles and the onset of regular quasi-monochromatic waves at larger wind velocity. Experiments indicate that the onset of regular waves increases with liquid viscosity. Our theory suggests that regular waves are triggered when the wrinkle amplitude reaches a fraction of the viscous sublayer thickness. This implies that the turbulent fluctuations near the onset may play a key role in the triggering of exponential wave growth.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 873 , 25 August 2019 , pp. 1020 - 1054
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Copyright
© 2019 Cambridge University Press 

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Footnotes

†

Present address: Département de Physique, Ecole Normale Supérieure/PSL Research University, CNRS, 24 rue Lhomond, 75005 Paris, France. Email address for correspondence: [email protected]

References

Banner, M. L. & Peirson, W. L. 1998 Tangential stress beneath wind-driven air–water interfaces. J. Fluid Mech. 364, 115–145.Google Scholar
Belcher, S. E. & Hunt, J. C. R. 1998 Turbulent flow over hills and waves. Annu. Rev. Fluid Mech. 30 (1), 507–538.Google Scholar
Benschop, H. O. G., Greidanus, A. J., Delfos, R., Westerweel, J. & Breugem, W. P. 2019 Deformation of a linear viscoelastic compliant coating in a turbulent flow. J. Fluid Mech. 859, 613–658.Google Scholar
Caulliez, G., Makin, V. & Kudryavtsev, V. 2008 Drag of the water surface at very short fetches: observations and modeling. J. Phys. Oceanogr. 38 (9), 2038–2055.Google Scholar
Choi, H. & Moin, P. 1990 On the space–time characteristics of wall-pressure fluctuations. Phys. Fluids A 2 (8), 1450–1460.Google Scholar
Corcos, G. M. 1963 The structure of the turbulent pressure field in boundary-layer flows. J. Fluid Mech. 18, 353–378.Google Scholar
Darmon, A., Benzaquen, M. & Raphaël, E. 2014 Kelvin wake pattern at large Froude numbers. J. Fluid Mech. 738, R3.Google Scholar
Druzhinin, O., Troitskaya, A. & Zilitinkevich, Y. I. 2012 Direct numerical simulation of a turbulent wind over a wavy water surface. J. Geophys. Res. 117 (C11), C00J05.Google Scholar
Eckart, C. 1953 The generation of wind waves on a water surface. J. Appl. Phys. 24 (12), 1485–1494.Google Scholar
Ellingsen, S. A. & Li, Y. 2017 Approximate dispersion relations for waves on arbitrary shear flows. J. Geophys. Res. 122, 9889–9905.Google Scholar
Francis, J. R. D. 1956 LXIX. Correspondence. Wave motions on a free oil surface. Phil. Mag. 1 (7), 685–688.Google Scholar
Funada, T. & Joseph, D. D. 2001 Viscous potential flow analysis of Kelvin–Helmholtz instability in a channel. J. Fluid Mech. 445, 263–283.Google Scholar
Gottifredi, J. & Jameson, G. 1970 The growth of short waves on liquid surfaces under the action of a wind. Proc. R. Soc. Lond. A 319 (1538), 373–397.Google Scholar
Havelock, T. H. 1919 Wave resistance: some cases of three-dimensional fluid motion. Proc. R. Soc. Lond. A 95, 354–365.Google Scholar
Janssen, P. 2004 The Interaction of Ocean Waves and Wind. Cambridge University Press.Google Scholar
Jimenez, J. 2013 Near wall turbulence. Phys. Fluids 25, 101302.Google Scholar
Jimenez, J., Del Alamo, J. C. & Flores, O. 2004 The large-scale dynamics of near-wall turbulence. J. Fluid Mech. 505, 179–199.Google Scholar
Jimenez, J. & Hoyas, S. 2008 Turbulent fluctuations above the buffer layer of wall-bounded flows. J. Fluid Mech. 611, 215–236.Google Scholar
Jimenez, J., Hoyas, S., Simens, M. P. & Mizuno, Y. 2010 Turbulent boundary layers and channels at moderate Reynolds numbers. J. Fluid Mech. 657, 335–360.Google Scholar
Kahma, K. & Donelan, M. A. 1988 A laboratory study of the minimum wind speed for wind wave generation. J. Fluid Mech. 192, 339–364.Google Scholar
Kawai, S. 1979 Generation of initial wavelets by instability of a coupled shear flow and their evolution to wind waves. J. Fluid Mech. 93 (4), 661–703.Google Scholar
Keulegan, G. H. 1951 Wind tides in small closed channels. J. Res. Natl Bur. Stand. 46, 358–381.Google Scholar
Kim, H., Padrino, J. C. & Joseph, D. D. 2011 Viscous effects on Kelvin–Helmholtz instability in a channel. J. Fluid Mech. 680, 398–416.Google Scholar
Kim, J. 1989 On the structure of pressure fluctuations in simulated turbulent channel flow. J. Fluid Mech. 205, 421–451.Google Scholar
Kim, J., Moin, P. & Moser, R. D. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.Google Scholar
Kirby, J. T. & Chen, T. M. 1989 Surface waves on vertically sheared flows: Approximate dispersion relations. J. Geophys. Res. 94 (C1), 1013–1027.Google Scholar
Kudryavtsev, V., Chapron, B. & Makin, V. 2014 Impact of wind waves on the air–sea fluxes: A coupled model. J. Geophys. Res. 119 (2), 1217–1236.Google Scholar
Kudryavtsev, V. N. & Makin, V. K. 2002 Coupled dynamics of short waves and the airflow over long surface waves. J. Geophys. Res. 107 (C12), 3209.Google Scholar
Lamb, H. 1995 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Langevin, P. 1908 Sur la théorie du mouvement brownien. C. R. Acad. Sci. 146, 530–533.Google Scholar
LeBlond, P. H. & Mainardi, F. 1987 The viscous damping of capillary-gravity waves. Acta Mechanica 68, 203–222.Google Scholar
Lee, M. & Moser, R. D. 2015 Direct numerical simulation of a turbulent channel flow up to Re 𝜏 ≈ 5200. J. Fluid Mech. 774, 395–415.Google Scholar
Liberzon, D. & Shemer, L. 2011 Experimental study of the initial stages of wind waves’ spatial evolution. J. Fluid Mech. 681, 462–498.Google 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.Google Scholar
Lindsay, K. A. 1984 The Kelvin–Helmholtz instability for a viscous interface. Acta Mechanica 52 (1), 51–61.Google Scholar
Lozano-Durán, A. & Jiménez, J. 2014 Time-resolved evolution of coherent structures in turbulent channels: characterization of eddies and cascades. J. Fluid Mech. 759, 432–471.Google Scholar
Manneville, P. 2010 Instabilities, Chaos and Turbulence. World Scientific.Google Scholar
Melville, W. K., Shear, R. & Veron, F. 1998 Laboratory measurements of the generation and evolution of langmuir circulations. J. Fluid Mech. 364, 31–58.Google Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185–204.Google Scholar
Miles, J. W. 1968 The Cauchy–Poisson problem for a viscous liquid. J. Fluid Mech. 34 (02), 359–370.Google Scholar
Miles, J. W. 1993 Surface-wave generation revisited. J. Fluid Mech. 256, 427–441.Google Scholar
Moisy, F. & Rabaud, M. 2014 Mach-like capillary-gravity wakes. Phys. Rev. E 90, 023009.Google Scholar
Moisy, F. & Rabaud, M. 2014 Scaling of far-field wake angle of non-axisymmetric pressure disturbance. Phys. Rev. E 89, 063004.Google Scholar
Moisy, F., Rabaud, M. & Salsac, K. 2009 A synthetic schlieren method for the measurement of the topography of a liquid interface. Exp. Fluids 46, 1021–1036.Google Scholar
Moser, R. D., Kim, J. & Mansour, N. N. 1999 Direct numerical simulation of turbulent channel flow up to Re = 590. Phys. Fluids 11 (4), 943–945.Google Scholar
Munk, W. 2009 An inconvenient sea truth: Spread, steepness, and skewness of surface slopes. Annu. Rev. Mar. Sci. 1, 377–415.Google Scholar
Paquier, A., Moisy, F. & Rabaud, M. 2015 Surface deformations and wave generation by wind blowing over a viscous liquid. Phys. Fluids 27, 122103.Google Scholar
Paquier, A., Moisy, F. & Rabaud, M. 2016 Viscosity effects in wind wave generation. Phys. Rev. Fluids 1, 083901.Google Scholar
Peregrine, D. H. 1976 Interaction of water waves and currents. Adv. Appl. Mech. 16, 9–117.Google Scholar
Phillips, O. M. 1957 On the generation of waves by turbulent wind. J. Fluid Mech. 2 (05), 417–445.Google Scholar
Plant, W. J. 1982 A relationship between wind stress and wave slope. J. Geophys. Res. 87 (C3), 1961–1967.Google Scholar
Pottier, N. 2014 Non Equilibrium Statistical Physics. Oxford Graduate texts.Google Scholar
Rabaud, M. & Moisy, F. 2013 Ship wakes: Kelvin or Mach angle?. Phys. Rev. Lett. 110, 214503.Google Scholar
Raphaël, E. & de Gennes, P-G. 1996 Capillary gravity waves caused by a moving disturbance: wave resistance. Phys. Rev. E 53 (4), 3448–3455.Google Scholar
Richard, D. & Raphaël, E. 1999 Capillary-gravity waves: The effect of viscosity on the wave resistance. Eur. Phys. Lett. 48 (1), 49–52.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639.Google Scholar
Russell, J. S. 1844 On waves. In Report of fourteenth meeting of the British Association for the Advancement of Science, York, pp. 311–390. John Murray.Google Scholar
Sajjadi, S. G., Robertson, S., Harvey, R. & Brown, M. 2017 Wave motion induced by turbulent shear flows over growing Stokes waves. J. Ocean Engng Marine Energy 3 (2), 97–112.Google Scholar
Schlichting, H. 2000 Boundary Layer Theory, 8th edn. Springer.Google Scholar
Sullivan, P. P. & McWilliams, J. C. 2010 Dynamics of winds and currents coupled to surface waves. Annu. Rev. Fluid Mech. 42, 19–42.Google Scholar
Veron, F. & Melville, W. K. 2001 Experiments on the stability and transition of wind-driven water surfaces. J. Fluid Mech. 446, 25–65.Google Scholar
Willmarth, W. W. & Wooldridge, C. E. 1962 Measurements of the fluctuating pressure at the wall beneath a thick turbulent boundary layer. J. Fluid Mech. 14, 187–210.Google Scholar
Yamamoto, Y. & Tsuji, Y. 2018 Numerical evidence of logarithmic regions in channel flow at Re 𝜏 = 8000. Phys. Rev. Fluids 3 (012602(R)).Google Scholar
Zavadasky, A. & Shemer, L. 2017 Water waves excited by near-impulsive wind forcing. J. Fluid Mech. 828, 459–495.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
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.Google Scholar