Gerasimos Kolokythas, Aggelos Dimakopoulos, Athanassios Dimas


In the present study, the three-dimensional, incompressible, turbulent, free-surface flow, developing by the propagation of nonlinear breaking waves over a rigid bed of constant slope, is numerically simulated. The main objective is to investigate the process of spilling wave breaking and the characteristics of the developing undertow current employing the large-wave simulation (LWS) method. According to LWS methodology, large velocity and free-surface scales are fully resolved, and subgrid scales are treated by an eddy viscosity model, similar to large-eddy simulation (LES) methodology. The simulations are based on the numerical solution of the unsteady, three-dimensional, Navier-Stokes equations subject to the fully-nonlinear free-surface boundary conditions and the appropriate bottom, inflow and outflow boundary conditions. The case of incoming second-order Stokes waves, normal to the shore, with wavelength to inflow depth ratio λ/dΙ = 6.6, wave steepness H/λ = 0.025, bed slope tanβ = 1/35 and Reynolds number (based on inflow water depth) Red = 250,000 is investigated. The predictions of the LWS model for the incipient wave breaking parameters - breaking depth and height - are in very good agreement with published experimental measurements. Profiles of the time-averaged horizontal velocity in the surf zone are also in good agreement with the corresponding measured ones, verifying the ability of the model to capture adequately the undertow current.


numerical simulation; Navier-Stokes equations; turbulent flow; spilling breaking; surf zone; undertow current

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Absi, R. 2008. Analytical solutions for the modeled k equation, Journal of Applied Mechanics, ASME, 75(4), 044501.

Absi, R. 2010. Concentration profiles for fine and coarse sediments suspended by waves over ripples: An analytical study with the 1-DV gradient diffusion model, Advances in Water Resources, 33, 411-418.

Absi, R. 2012. Analytical modeling of turbulent flows in a plan channel with smooth walls, Submitted.

Absi, R., S. Marchandon, and M. Lavarde. 2011. Turbulent diffusion of suspended particles: analysis of the turbulent Schmidt number, Defect and Diffusion Forum, Trans Tech Publications, 312-315, 794-799.

Beach, R.A., and R.W. Sternberg. 1988. Suspended sediment transport in the surf zone: response to cross-shore infragravity motion. Marine Geology, 80, 61–79.

Brevik, I. 1981. Oscillatory rough turbulent boundary layers, Journal of Waterways, Port, Coastal and Ocean Engineering. ASCE, 103, 175-188.

Fredsoe, J., and R. Deigaard. 1992. Mechanics of coastal sediment transport, World Scientific Publishing, 369 pp.

Gelfenbaum, G., and J.D. Smith. 1986. Experimental evaluation of a generalized suspended-sediment transport theory. In Shelf Sands and Sandstones (Knight, R. J. and McLean, J. R., eds.), Canadian Society of Petroleum Geologists, Memoir, pp. 133–144.

Grant, W.D., and O.S. Madsen. 1979. Combined wave and current interaction with a rough bottom. Journal of Geophysical Research. 84(C4), 1797-1808.

Hoyas, S., and J. Jiménez. 2006. Scaling of velocity fluctuations in turbulent channels up to Reτ = 2003, Phys. Fluids, 18, 011702.

Hsu, T.-W., and C.-D. Jan. 1998. Calibration of Businger-Arya type of eddy viscosity models parameters. Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, Vol. 124(5), 281-284.

Iwamoto, K., Y. Suzuki, and N. Kasagi. 2002. Reynolds number effect on wall turbulence: toward effective feedback control, International Journal of Heat Fluid Flow, 23, 678.

Jones, W.P., and B.E. Launder. 1972. The prediction of laminarization with a two-equation model of turbulence. International Journal of Heat Mass Transfer, 15, 301-314.

Kajiura, K. 1968. A model of the bottom boundary layer in water waves. Bulletin of the Earthquake Research Institute. 46, 75-123.

Liu, H., and S. Sato. 2006. A two-phase flow model for asymmetric sheet-flow conditions. Coastal Engineering, 53, 825-843.

Lundgren, H. 1972. Turbulent currents in the presence of waves, Proceedings of 13th International Conference on Coastal Engineering, ASCE, 623-634.

Madsen, O.S., and P. Salles. 1998. Eddy viscosity models for wave boundary layers, Proceedings of 26th International Conference on Coastal Engineering, ASCE, 2615-2627.

Menter, F.R. 1994. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 32 (8), 1598-1605.

Myrhaug, D. 1982. On a theoretical model of rough turbulent wave boundary layers, Ocean Engineering. 9(6), 547-565.

Nezu, I., and H. Nakagawa, 1993. Turbulence in Open-Channel Flows, A. A. Balkema, ed.

Nielsen, P. 1992. Coastal bottom boundary layers and sediment transport, World Scientific, 324 p.

Sleath, J.F.A. 1990. Seabed boundary layers, The Sea, Vol. 9: Ocean Engineering Science, Bernard Le Méhauté, Daniel M. Hanes (Eds.), 693-728.

Smith, J.D. 1977. Modeling of sediment transport on continental shelves, In the Sea, 6, Intersience, N.Y., 538-577.


Suntoyo, and H. Tanaka. 2009. Effect of bed roughness on turbulent boundary layer and net sediment transport under asymmetric waves, Coastal Engineering, 56(9), 960-969.

Tanaka, H. and A. Thu. 1994. Full-range equation of friction coefficient and phase difference in a wave-current boundary layer, Coastal Engineering, 22, 237-254.

van Driest, E.R. 1956. On turbulent flow near a wall, J. Aero. Sci., 23, 1007-1011.

van Rijn, L.C. 1993. Principles of sediment transport in River, Estuaries and Coastal Seas. Aqua Publishing, Amsterdam.


van Rijn, L.C. 2007. Unified view of sediment transport by currents and waves II: Suspended transport, Journal of Hydraulic Engineering, ASCE, 133(6), 668-689.

Wilcox, D.C. 1988. Reassessment of the scale-determining equation for advanced turbulent models. AIAA Journal, 26 (11), 1299-1310.

You, Z.J., D.L. Wilkinson, and P. Nielsen. 1992. Velocity distribution in turbulent oscillatory boundary layer, Coastal Engineering, 18, 21-38.