Rafik Absi, Hitoshi Tanaka, Loreline Kerlidou, Alicia André


Eddy viscosity in wave boundary layers is a key parameter in coastal engineering. Two analytical eddy viscosity profiles present a particular interest for practical applications: the parabolic-uniform profile (Myrhaug 1982, van Rijn 1993, Liu and Sato 2006) and the exponential-linear profile (Gelfenbaum and Smith 1986, Beach and Sternberg 1988, Hsu and Jan 1998, Absi 2010). The aim of our study is to assess and validate these two profiles by: (1) investigation of eddy viscosity in steady fully developed plane channel flow; (2) comparisons with numerical results of the two equation baseline (BSL) k-ω model (Menter 1994, Suntoyo and Tanaka 2009). Our study shows that these two profiles are able to describe the eddy viscosity distribution in the wave bottom boundary layer but for different wave conditions given by the parameter am/ks, where am is the wave orbital amplitude and ks the equivalent roughness. The exponential-linear profile is adequate for am/ks <500, while the parabolic-uniform profile is more appropriate for am/ks ≥500. We suggest empirical formulations for the different coefficients which appear in these two profiles based on numerical results of the BSL k-ω model.


wave boundary layers; eddy viscosity profiles; two-equation k-ω model, validation, calibration

Full Text:



Abadie S., Caltagirone J. P. & Watremez P. 1998. Splash-up generation in a plunging breaker. Comptes Rendus de l'Académie des Sciences - Series IIB - Mechanics-Physics-Astronomy, 326, 9, 553-559.

Abadie, S., Morichon, D., Grilli, S., Glockner, S. 2010. Numerical simulation of waves generated by landslides using a multiple-fluid Navier-Stokes model. Coastal engineering, 57, 779-794.http://dx.doi.org/10.1016/j.coastaleng.2010.03.003

Arai M., Cheng L., Kunamo A., Miyamoto T. 2002. A technique for stable numerical computation of hydrodynamic impact pressure in sloshing simulation. Journal of the society of naval architects of Japan, 191, pp 299–307.

Brosset, L, Lafeber, W., Bogaert, H., Marhem, M., Carden, P., Maguire, J. 2011. A Mark III panel subjected to a flip-through wave impact: results from the Sloshel project. Proc. 21st International Offshore and Polar Engineering Conference. 84-96.

Chen Y., Djidjeli, K., Price W. 2009. Numerical simulation of liquid sloshing phenomena in partially filled containers. Computers & fluids, 38, 830–842.http://dx.doi.org/10.1016/j.compfluid.2008.09.003

Hu, C. & Kashiwagi, M. 2004. C.I.P. method for violent free-surface flows, Journal of Marine Science and Technology, 9, 143–157.http://dx.doi.org/10.1007/s00773-004-0180-z

Lubin, P., Vincent, S., Abadie, S. & Caltagirone, J.P., 2006. Three-dimensional Large Eddy Simulation of air entrainment under plunging breaking waves, Coastal Engineering, 53, 8, 631-655.http://dx.doi.org/10.1016/j.coastaleng.2006.01.001

Lugni C., Miozzi M., Brocchini M., Faltinsen O.M. 2010. Evolution of the air cavity during a depressurized wave impact. I. The Kinematic Flow Field. Physics of fluids, 22, 56-101.

Kleefman K., Fekken, G., Veldman A., Iwanoski B., Buchner B. 2005. A volume-of-fluid based simulation method for wave impact problems. Journal of computational physics, 206, 363–393.http://dx.doi.org/10.1016/j.jcp.2004.12.007

Mokrani C. 2012. Impacts de vagues déferlantes sur un obstacle vertical. Modèle théorique et estimation numérique des pics de pression. PhD Thesis, Université de Pau et des Pays de l'Adour. (in french)

Mory, M., Abadie, S, Mauriet, S. and Lubin, P. 2011. Run-up flows of collapsing bores over a beach, European Journal of Mechanics – B/Fluids, 30, 6, 565-576.http://dx.doi.org/10.1016/j.euromechflu.2010.11.005

Plumerault L-R., Astruc, D., Maron, P., 2012. The influence of air on the impact of a plunging breaking wave on a vertical wall using a multi-fluid model. Coastal Engineering, 62, 62-74http://dx.doi.org/10.1016/j.coastaleng.2011.12.002

Richardson L.F. 1911. The approximate arithmetical solution by finite differences of physical problems including differential equations with an application to the stresses in a masonry dam. Philo Trans of Royal Soc of London, Series A, 210, pp 307-357.http://dx.doi.org/10.1098/rsta.1911.0009

Vinje, T. and Brevig P., 1981. Numerical Calculation of Breaking Waves", J. Adv. Water Res., 4, 77-82.http://dx.doi.org/10.1016/0309-1708(81)90027-0

Witte H.H. 1988. Wave induced impact loading in deterministic and stochastic reflection. Mitt Leichtweiss Inst. Wasserbau, vol. 102, Tech. Univ. Braunschweig, Braunschweig, Germany.

Wu G. 2007. Fluid impact on a solid boundary. Journal of fluids and structures, Vol. 23, pp 755–765.http://dx.doi.org/10.1016/j.jfluidstructs.2006.11.002

Youngs D.L. 1982. Time-dependent multimaterial flow with large fluid distorsion. K.W. Morton and M.J. Baines, Numerical methods for fluids dynamics, Academic Press, New York.


Yettou El-M., Desrochers A., Champoux Y. 2007. A new analytical model for pressure estimation of symmetrical water impact of a rigid wedge at variable velocities. Journal of fluids and structures, 23, pp 501-522.http://dx.doi.org/10.1016/j.jfluidstructs.2006.10.001

DOI: https://doi.org/10.9753/icce.v33.waves.63