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A simple and practical model for combined wave–current canopy flows

  • Robert B. Zeller (a1), Francisco J. Zarama (a1), Joel S. Weitzman (a1) and Jeffrey R. Koseff (a1)

Laboratory experiments were used to evaluate and improve modelling of combined wave–current flow through submerged aquatic canopies. Horizontal in-canopy particle image velocimetry (PIV) and wavemaker-measurement synchronization allowed direct volume averaging and ensemble averaging by wave phase, which were used to fully resolve the volume-averaged unsteady momentum budget. Parameterizations for the drag, Reynolds stress, vertical advection, wake production and shear production were tested against the laboratory measurements. The drag was found to have small errors due to unsteadiness and the finite aspect ratio of the canopy elements. The Smagorinsky model for the Reynolds stress showed much better agreement with the measurements than the quadratic friction parameterization used in the literature. A proposed parameterization for the vertical advection based on linear wave theory was also found to be effective and is much more computationally efficient than solving the pressure Poisson equation. A simple 1D 0-equation Reynolds-averaged Navier–Stokes (RANS) model was developed to use these parameterizations. The basic framework of the model is an extrapolation from previous 2- and 3-box models to  $N$ boxes. While the resolution of the model is flexible, the filter length for the Smagorinsky parameterization has to be chosen appropriately. With the proper filter length, the $N$ -box model demonstrated good agreement with the measurements at both low and high resolution. Scaling analysis was used to establish a region of parameter space where the $N$ -box model is expected to be effective. The following conditions define this region: the wave-induced velocity is of similar or greater magnitude than the background current, the drag to shear length ratio is small enough to produce canopy behaviour, the wave orbital excursion is not much larger than the drag length, the Froude number is small and the canopy is under shallow submergence, yet far from emergent. Under these assumptions, the dominant balance is between pressure and unsteadiness, the drag is secondary, and the other terms are small. The simple Reynolds stress parameterization in the $N$ -box model is appropriate under these conditions because the Reynolds stress is unlikely to be the dominant source of error. This finding is important because the Reynolds stress is typically one of the dominant drivers of computational cost and model complexity. Based on these findings, the $N$ -box model is expected to be a practical tool for a wide range of combined wave–current canopy flows because of its simplicity and computational efficiency.

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Belcher, S. E., Harman, I. N. & Finnigan, J. J. 2012 The wind in the willows: flows in forest canopies in complex terrain. Annu. Rev. Fluid Mech. 44, 479504.
Bradley, K. & Houser, C. 2009 Relative velocity of seagrass blades: implications for wave attenuation in low-energy environments. J. Geophys. Res. 114, F1.
Calaf, M., Meneveau, C. & Meyers, J. 2010 Large eddy simulation study of fully developed wind-turbine array boundary layers. Phys. Fluids 22 (1), 015110.
Dalrymple, R. A. & Dean, R. G. 1991 Water Wave Mechanics for Engineers and Scientists. Prentice-Hall.
Falter, J. L., Atkinson, M. J. & Merrifield, M. A. 2004 Mass-transfer limitation of nutrient uptake by a wave-dominated reef flat community. Limnol. Oceanogr. 49 (5), 18201831.
Finnigan, J. 2000 Turbulence in plant canopies. Annu. Rev. Fluid Mech. 32 (1), 519571.
Ghisalberti, M. 2009 Obstructed shear flows: similarities across systems and scales. J. Fluid Mech. 641, 5161.
Ghisalberti, M. & Schlosser, T. 2013 Vortex generation in oscillatory canopy flow. J. Geophys. Res. 118 (3), 15341542.
Grimmond, C. S. B. & Oke, T. R. 1999 Aerodynamic properties of urban areas derived from analysis of surface form. J. Appl. Meteorol. 38 (9), 12621292.
Hansen, J. C. R. & Reidenbach, M. A. 2011 Wave and tidally driven flows in eelgrass beds and their effect on sediment suspension. Mar. Ecol. Prog. Ser. 448, 271287.
Hu, Z., Suzuki, T., Zitman, T., Uittewaal, W. & Stive, M. 2014 Laboratory study on wave dissipation by vegetation in combined current–wave flow. Coast. Engng 88, 131142.
King, A. T., Tinoco, R. O. & Cowen, E. A. 2012 A $k{-}{\it\varepsilon}$ turbulence model based on the scales of vertical shear and stem wakes valid for emergent and submerged vegetated flows. J. Fluid Mech. 701, 139.
Li, C. W. & Yan, K. 2007 Numerical investigation of wave–current–vegetation interaction. ASCE J. Hydraul. Engng 133 (7), 794803.
López, F. & García, M. 1998 Open-channel flow through simulated vegetation: suspended sediment transport modeling. Water Resour. Res. 34 (9), 23412352.
Lowe, R. J., Falter, J. L., Koseff, J. R., Monismith, S. G. & Atkinson, M. J. 2007 Spectral wave flow attenuation within submerged canopies: implications for wave energy dissipation. J. Geophys. Res. 112, C5.
Lowe, R. J., Koseff, J. R. & Monismith, S. G. 2005 Oscillatory flow through submerged canopies, part 1: velocity structure. J. Geophys. Res. 110, C10.
Lowe, R. J., Shavit, U., Falter, J. L., Koseff, J. R. & Monismith, S. G. 2008 Modeling flow in coral communities with and without waves: a synthesis of porous media and canopy flow approaches. Limnol. Oceanogr. 53 (6), 26682680.
Luhar, M., Coutu, S., Infantes, E., Fox, S. & Nepf, H. 2010 Wave-induced velocities inside a model seagrass bed. J. Geophys. Res. 115, C12.
Luhar, M., Infantes, E., Orfila, A., Terrados, J. & Nepf, H. M. 2013 Field observations of wave-induced streaming through a submerged seagrass (Posidonia oceanica) meadow. J. Geophys. Res. 118 (4), 19551968.
Marshall, P. A. 2000 Skeletal damage in reef corals: relating resistance to colony morphology. Mar. Ecol. Prog. Ser. 200, 177189.
Maza, M., Lara, J. L. & Losada, I. J. 2013 A coupled model of submerged vegetation under oscillatory flow using Navier–Stokes equations. Coast. Engng 80, 1634.
Moltchanov, S., Bohbot-Raviv, Y. & Shavit, U. 2011 Dispersive stresses at the canopy upstream edge. Boundary–Layer Meteorol. 139 (2), 333351.
Monismith, S. G. 2007 Hydrodynamics of coral reefs. Annu. Rev. Fluid Mech. 39, 3755.
Narasimhamurthy, V. D. & Andersson, H. I. 2009 Numerical simulation of the turbulent wake behind a normal flat plate. Intl J. Heat Fluid Flow 30 (6), 10371043.
Nepf, H. M. 2012 Flow and transport in regions with aquatic vegetation. Annu. Rev. Fluid Mech. 44, 123142.
Neumeier, U. & Ciavola, P. 2004 Flow resistance and associated sedimentary processes in a Spartina maritima salt-marsh. J. Coast. Res. 20 (2), 435447.
Orth, R. J., Carruthers, T. J. B., Dennison, W. C., Duarte, C. M., Fourqurean, J. W., Heck, K. L., Hughes, A. R., Kendrick, G. A., Kenworthy, W. J., Olyarnik, S., Short, F. T., Waycott, M. & Williams, S. L. 2006 A global crisis for seagrass ecosystems. Bioscience 56 (12), 987996.
Philips, D. A., Rossi, R. & Iaccarino, G. 2013 Large-eddy simulation of passive scalar dispersion in an urban-like canopy. J. Fluid Mech. 723, 404428.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Pujol, D. & Nepf, H. 2012 Breaker-generated turbulence in and above a seagrass meadow. Cont. Shelf Res. 49, 19.
Pujol, D., Serra, T., Colomer, J. & Casamitjana, X. 2013 Flow structure in canopy models dominated by progressive waves. J. Hydrol. 486, 281292.
Reidenbach, M. A., Koseff, J. R., Monismith, S. G., Steinbuck, J. V. & Genin, A. 2006 The effects of waves and morphology on mass transfer within branched reef corals. Limnol. Oceanogr. 51 (2), 11341141.
Ringuette, M. J.2004 Vortex formation and drag on low aspect ratio, normal flat plates. PhD thesis, California Institute of Technology.
Ringuette, M. J., Milano, M. & Gharib, M. 2007 Role of the tip vortex in the force generation of low-aspect-ratio normal flat plates. J. Fluid Mech. 581, 453468.
Stevens, A. W. & Lacy, J. R. 2012 The influence of wave energy and sediment transport on seagrass distribution. Estuar. Coast. 35 (1), 92108.
Storlazzi, C. D., Brown, E. K., Field, M. E., Rodgers, K. & Jokiel, P. L. 2005 A model for wave control on coral breakage and species distribution in the Hawaiian Islands. Coral Reefs 24 (1), 4355.
Taebi, S., Lowe, R. J., Pattiaratchi, C. B., Ivey, G. N., Symonds, G. & Brinkman, R. 2011 Nearshore circulation in a tropical fringing reef system. J. Geophys. Res. 116, C2.
Tanino, Y. & Nepf, H. M. 2008 Laboratory investigation of mean drag in a random array of rigid, emergent cylinders. J. Hydraul. Engng 134 (1), 3441.
Vreman, B., Geurts, B. & Kuerten, H. 1997 Large-eddy simulation of the turbulent mixing layer. J. Fluid Mech. 339, 357390.
Ward, L. G., Kemp, W. M. & Boynton, W. R. 1984 The influence of waves and seagrass communities on suspended particulates in an estuarine embayment. Mar. Geol. 59 (1), 85103.
Waycott, M., Duarte, C. M., Carruthers, T. J. B., Orth, R. J., Dennison, W. C., Olyarnik, S., Calladine, A., Fourqurean, J. W., Heck, K. L., Hughes, A. R., Kendrick, G. A., Kenworthy, W. J., Short, F. T. & Williams, S. L. 2009 Accelerating loss of seagrasses across the globe threatens coastal ecosystems. Proc. Natl Acad. Sci. USA 106 (30), 1237712381.
Weitzman, J. S.2013 Current-and wave-driven flow and nutrient exchange in natural and model submerged vegetated canopies. PhD thesis, Stanford University.
Weitzman, J. S., Aveni-Deforge, K., Koseff, J. R. & Thomas, F. I. M. 2013 Uptake of dissolved inorganic nitrogen by shallow seagrass communities exposed to wave-driven unsteady flow. Mar. Ecol. Prog. Ser. 475, 6583.
Weitzman, J. S., Zeller, R. B., Thomas, F. I. M. & Koseff, J. R. 2015 The attenuation of current- and wave-driven flow within submerged multispecific vegetative canopies. Limnol. Oceanogr. (submitted).
Whittlesey, R. W., Liska, S. & Dabiri, J. O. 2010 Fish schooling as a basis for vertical axis wind turbine farm design. Bioinspir. Biomim. 5 (3), 035005.
Zeller, R. B., Weitzman, J. S., Abbett, M. E., Zarama, F. J., Fringer, O. B. & Koseff, J. R. 2014 Improved parameterization of seagrass blade dynamics and wave attenuation based on numerical and laboratory experiments. Limnol. Oceanogr. 59, 251266.
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Journal of Fluid Mechanics
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