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Numerical study of the effect of surface waves on turbulence underneath. Part 1. Mean flow and turbulence vorticity

Published online by Cambridge University Press:  25 September 2013

Xin Guo  and
Lian Shen
Xin Guo
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: shen@umn.edu
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Abstract

Direct numerical simulation is performed to study the effect of progressive gravity waves on turbulence underneath. The Navier–Stokes equations subject to fully nonlinear kinematic and dynamic free-surface boundary conditions are simulated on a surface-following mapped grid using a fractional-step scheme with a pseudo-spectral method in the horizontal directions and a finite-difference method in the vertical direction. To facilitate a mechanistic study that focuses on the fundamental physics of wave–turbulence interaction, the wave and turbulence fields are set up precisely in the simulation: a pressure-forcing method is used to generate and maintain the progressive wave being investigated and to suppress other wave components, and a random forcing method is used to produce statistically steady, homogeneous turbulence in the bulk flow beneath the surface wave. Cases with various moderate-to-large turbulence-to-wave time ratios and wave steepnesses are considered. Study of the turbulence velocity spectrum shows that the turbulence is dynamically forced by the surface wave. Mean flow and turbulence vorticity are studied in both the Eulerian and Lagrangian frames of the wave. In the Eulerian frame, statistics of the underlying turbulence field indicates that the magnitude of turbulence vorticity and the inclination angle of vortices are dependent on the wave phase. In the Lagrangian frame, wave properties and the accumulative effect on turbulence vorticity are studied. It is shown that vertical vortices are tilted in the wave propagation direction due to the cumulative effects of both the Stokes drift velocity and the correlation between turbulence fluctuations and wave strain rate, whereas for streamwise vortices, these two factors offset each other and result in a negligible tilting effect.

JFM classification

Turbulent Flows: Turbulent Flows Waves/Free-surface Flows: Waves/Free-surface Flows

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Papers
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Journal of Fluid Mechanics , Volume 733 , 25 October 2013 , pp. 558 - 587
Copyright
©2013 Cambridge University Press 

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References

Andrews, D. G. & McIntyre, M. E. 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.Google Scholar
Borue, V., Orszag, S. A. & Staroselsky, I. 1995 Interaction of surface waves with turbulence: direct numerical simulations of turbulent open-channel flow. J. Fluid Mech. 286, 1–23.Google Scholar
Brumley, B. H. & Jirka, G. H. 1987 Near-surface turbulence in a grid-stirred tank. J. Fluid Mech. 183, 235–263.Google Scholar
Campagne, G., Cazalbou, J.-B., Joly, L. & Chassaing, P. 2009 The structure of a statistically steady turbulent boundary layer near a free-slip surface. Phys. Fluids 21, 065111.CrossRefGoogle Scholar
Cavaleri, L., Alves, J.-H. G. M., Ardhuin, F., Babanin, A., Banner, M., Belibassakis, K., Benoit, M., Donelan, M., Groeneweg, J., Herbers, T. H. C., Hwang, P., Janssen, P. A. E. M., Janssen, T., Lavrenov, I. V., Magne, R., Monbaliu, J., Onorato, M., Polnikov, V., Resio, D., Rogers, W. E., Sheremet, A., McKee Smith, J., Tolman, H. L., van Vledder, G., Wolf, J. & Young, I. 2007 Wave modelling: the state of the art. Prog. Oceanogr. 75 (4), 603–674.CrossRefGoogle Scholar
Chang, H.-K., Chen, Y.-Y & Liou, J.-C. 2009 Particle trajectories of nonlinear gravity waves in deep water. Ocean Engng 36, 324–329.CrossRefGoogle Scholar
Chen, J., Meneveau, C. & Katz, J. 2006 Scale interactions of turbulence subjected to a straining–relaxation–destraining cycle. J. Fluid Mech. 562, 123–150.Google Scholar
Craik, A. D. D. 1977 The generation of Langmuir circulations by an instability mechanism. J. Fluid Mech. 81, 209–223.Google Scholar
Craik, A. D. D. & Leibovich, S. 1976 A rational model for Langmuir circulations. J. Fluid Mech. 73, 401–426.Google Scholar
De Angelis, V., Lombardi, P. & Banerjee, S. 1997 Direct numerical simulation of turbulent flow over a wavy wall. Phys. Fluids 9 (8), 2429–2442.Google Scholar
Elliott, J. G. 1953 Interim report. Tech. Rep. contract NOy-12561, US Navy, Bureau Yards and Docks. Hydrodynamics Laboratory, California Institute of Technology.Google Scholar
Fenton, J. D. 1985 A fifth-order Stokes theory for steady waves. J. Waterways Port Coast. Ocean Engng 111 (2), 216–234.Google Scholar
Fulgosi, M., Lakehal, D., Banerjee, S. & De Angelis, V. 2003 Direct numerical simulation of turbulence in a sheared air–water flow with a deformable interface. J. Fluid Mech. 482, 319–345.CrossRefGoogle Scholar
Grant, A. L. M. & Belcher, S. E. 2009 Characteristics of Langmuir turbulence in the ocean mixed layer. J. Phys. Oceanogr. 39 (8), 1871–1887.Google Scholar
Guo, X. & Shen, L. 2009 On the generation and maintenance of waves and turbulence in simulations of free-surface turbulence. J. Comput. Phys. 228, 7313–7332.Google Scholar
Guo, X. & Shen, L. 2010 Interaction of a deformable free surface with statistically steady homogeneous turbulence. J. Fluid Mech. 658, 33–62.Google Scholar
Guo, X. & Shen, L. 2013 Numerical study of the effect of surface wave on turbulence underneath. Part 2. Eulerian and Lagrangian properties of turbulence kinetic energy. J. Fluid Mech. (submitted).CrossRefGoogle Scholar
Handler, R. A., Swean, T. F. Jr, Leighton, R. I. & Swearingen, J. D. 1993 Length scales and the energy balance for turbulence near a free surface. AIAA J. 31, 1998–2007.Google Scholar
Hodges, B. R. & Street, R. L. 1999 On simulation of turbulent nonlinear free-surface flows. J. Comput. Phys. 151, 425–457.Google Scholar
Hunt, J. C. R. 1984 Turbulence structure in thermal convection and shear-free boundary layers. J. Fluid Mech. 138, 161–184.Google Scholar
Hunt, J. C. R. & Graham, J. M. R. 1978 Free-stream turbulence near plane boundaries. J. Fluid Mech. 84, 209–235.Google Scholar
Iskandarani, M. & Liu, P. L.-F. 1991 Mass transport in two-dimensional water waves. J. Fluid Mech. 231, 395–415.Google Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.Google Scholar
Jiang, J.-Y. & Street, R. L. 1991 Modulated flows beneath wind-ruffled, mechanically generated water waves. J. Geophys. Res. 96, 2711–2721.Google Scholar
Jiang, J.-Y., Street, R. L. & Klotz, S. P. 1990 A study of wave–turbulence interaction by use of a nonlinear water wave decomposition technique. J. Geophys. Res. 95, 16037–16054.CrossRefGoogle Scholar
Kawamura, T. 2000 Numerical investigation of turbulence near a sheared air–water interface. Part 2. Interaction of turbulent shear flow with surface waves. J. Mar. Sci. Technol. 5, 161–175.CrossRefGoogle Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323.CrossRefGoogle Scholar
Kitaigorodskii, S. A., Donelan, M. A., Lumley, J. L. & Terray, E. A. 1983 Wave–turbulence interactions in the upper ocean. Part 2. Statistical characteristics of wave and turbulent components of the random velocity field in the marine surface layer. J. Phys. Oceanogr. 13, 1988–1999.Google Scholar
Kitaigorodskii, S. A. & Lumley, J. L. 1983 Wave–turbulence interactions in the upper ocean. Part 1. The energy balance of the interacting fields of surface wind waves and wind-induced three-dimensional turbulence. J. Phys. Oceanogr. 13, 1977–1987.Google Scholar
Komori, S., Kurose, R., Iwano, K., Ukai, T. & Suzuki, N. 2010 Direct numerical simulation of wind-driven turbulence and scalar transfer at sheared gas–liquid interfaces. J. Turbul. 11, 1–20.Google Scholar
Komori, S., Nagaosa, N., Murakami, Y., Chiba, S., Ishii, K. & Kuwahara, K. 1993 Direct numerical simulation of three-dimensional open-channel flow with zero-shear gas–liquid interface. Phys. Fluids A 5, 115–125.Google Scholar
Kumar, S., Gupta, R. & Banerjee, S. 1998 An experimental investigation of the characteristics of free-surface turbulence in channel flow. Phys. Fluids 10, 437–456.Google Scholar
Leibovich, S. 1977 Convective instability of stably stratified water in the ocean. J. Fluid Mech. 82, 561–581.Google Scholar
Leibovich, S. 1980 On wave–current interaction theories of Langmuir circulations. J. Fluid Mech. 99, 715–724.Google Scholar
Li, M., Garrett, C. & Skyllingstad, E. 2005 A regime diagram for classifying turbulent large eddies in the upper ocean. Deep-Sea Res. A 52, 259–278.Google Scholar
Lombardi, P., De Angelis, V. & Banerjee, S. 1996 Direct numerical simulation of near-interface turbulence in coupled gas–liquid flow. Phys. Fluids 8 (6), 1643–1665.Google Scholar
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. A 245, 535–581.Google Scholar
Longuet-Higgins, M. S. 1986 Eulerian and Lagrangian aspects of surface waves. J. Fluid Mech. 173, 683–707.Google Scholar
Longuet-Higgins, M. S. 1992 Capillary rollers and bores. J. Fluid Mech. 240, 659–679.CrossRefGoogle Scholar
Lumley, J. L. & Terray, E. A. 1983 Kinematics of turbulence converted by a random wave field. J. Phys. Oceanogr 13, 2000–2007.Google Scholar
Lundgren, T. 2003 Linearly forced isotropic turbulence. Tech. Rep. 461. Center for Turbulence Research, Stanford, CA.Google Scholar
Magnaudet, J. & Thais, L. 1995 Orbital rotational motion and turbulence below laboratory wind water waves. J. Geophys. Res. 100, 757–771.CrossRefGoogle Scholar
McWilliams, J. C., Sullivan, P. P. & Moeng, C.-H. 1997 Langmuir turbulence in the ocean. J. Fluid Mech. 334, 1–30.CrossRefGoogle Scholar
Moin, P. & Mahesh, K. 1998 Direct numerical simulation: a tool in turbulence research. Annu. Rev. Fluid. Mech. 30 (1), 539–578.CrossRefGoogle Scholar
Morinishi, Y., Lund, T. S., Vasilyev, O. V. & Moin, P. 1998 Fully conservative higher-order finite difference schemes for incompressible flow. J. Comput. Phys. 143 (1), 90–124.Google Scholar
Morison, J. R. & Crooke, R. C. 1953 The mechanics of deep water, shallow water, and breaking waves. Tech. Rep. 40. US Army, Corps of Engineers, Beach Erosion Board.Google Scholar
Nagaosa, R. 1999 Direct numerical simulation of vortex structures and turbulent scalar transfer across a free surface in a fully developed turbulence. Phys. Fluids 11, 1581–1595.Google Scholar
Ölmez, H. S. & Milgram, J. H. 1992 An experimental study of attenuation of short water waves by turbulence. J. Fluid Mech. 239, 133–156.Google Scholar
Pan, Y. & Banerjee, S. 1995 A numerical study of free-surface turbulence in channel flow. Phys. Fluids 7, 1649–1664.Google Scholar
Perot, B. & Moin, P. 1995 Shear-free turbulent boundary layers. Part 1. Physical insights into near-wall turbulence. J. Fluid Mech. 295, 199–227.CrossRefGoogle Scholar
Phillips, O. M. 1961 A note on the turbulence generated by gravity waves. J. Geophys. Res. 66, 2889–2893.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Rashidi, M., Hetsroni, G. & Banerjee, S. 1992 Wave–turbulence interaction in free-surface channel flows. Phys. Fluids A 4, 2727–2738.Google Scholar
Rosales, C. & Meneveau, C. 2005 Linear forcing in numerical simulations of isotropic turbulence: physical space implementations and convergence properties. Phys. Fluids 17, 095106.Google Scholar
Shen, L., Zhang, X., Yue, D. K. P. & Triantafyllou, G. S. 1999 The surface layer for free-surface turbulent flows. J. Fluid Mech. 386, 167–212.Google Scholar
Teixeira, M. A. C. & Belcher, S. E. 2000 Dissipation of shear-free turbulence near boundaries. J. Fluid Mech. 422, 167–191.Google Scholar
Teixeira, M. A. C. & Belcher, S. E. 2002 On the distortion of turbulence by a progressive surface wave. J. Fluid Mech. 458, 229–267.CrossRefGoogle Scholar
Teixeira, M. A. C. & Belcher, S. E. 2010 On the structure of Langmuir turbulence. Ocean Model. 31, 105–119.Google Scholar
Tejada-Martínez, A. E., Grosch, C. E., Gargett, A. E., Polton, J. A., Smith, J. A. & MacKinnon, J. A. 2009 A hybrid spectral/finite-difference large-eddy simulator of turbulent processes in the upper ocean. Ocean Model. 30, 115–142.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.Google Scholar
Thais, L. & Magnaudet, J. 1996 Turbulent structure beneath surface gravity waves sheared by the wind. J. Fluid Mech. 328, 313–344.Google Scholar
Variano, E. A. & Cowen, E. A. 2008 A random-jet-stirred turbulence tank. J. Fluid Mech. 604, 1–32.Google Scholar
Veron, F., Melville, W. K. & Lenain, L. 2009 Measurements of ocean surface turbulence and wave–turbulence interactions. J. Phys. Oceanogr. 39, 2310–2323.Google Scholar
Walker, D. T., Leighton, R. I. & Garza-Rios, L. O. 1996 Shear-free turbulence near a flat free surface. J. Fluid Mech. 320, 19–51.Google Scholar
Yang, D. & Shen, L. 2011 Simulation of viscous flows with undulatory boundaries. Part 1. Basic solver. J. Comput. Phys. 230, 5488–5509.Google Scholar
Yoshikawa, I., Kawamura, H., Okuda, K. & Toba, Y. 1988 Turbulence structure in water under laboratory wind waves. J. Phys. Soc. Japan 44, 143–156.Google Scholar
Zhou, H. 1999 Numerical simulation of Langmuir circulations in a wavy domain and its comparison with the Craik–Leibovich theory. PhD thesis, Stanford University.Google Scholar