Focus is on the turbulence for a plunging breaker. Laser Doppler anemometer point measurements were used to determine the velocity matrix of a breaking wave on a sloping bottom. Using the Reynolds stress anisotropy for incompressible fluid, it was found that the ensemble averaged measured velocity predicted eddy viscosity is associated with peaks, which are absent in the broadly accepted empirical predictions. The instantaneous eddy viscosity coefficient was determined according to the Reynolds stresses, modified mean velocity and its gradient components and turbulent kinetic energy. The modified mean velocity and its derivatives improve eddy viscosity predictions during the wave period, which gives evidence that the velocity used corresponds well to a rotational part. In addition to the measurement predictions, empirical formulae were used to estimate the eddy viscosity values during the wave period. Furthermore, a meshless numerical model is proposed to determine artificial viscosity and demonstrate its dependence on eddy viscosity in the case of weakly compressible fluid.
Batchelor, G. K. 1967. An Introduction to fluid dynamics. Cambridge University Press, Cambridge, United Kingdom.
Battjes, J. A. 1988. Surf-zone dynamics. Annu. Rev. Fluid Mech., 20, 257–291.
https://doi.org/10.1146/annurev.fl.20.010188.001353
Bertin, J. J., Periaux, J., and Ballmann, J. 1992. Advances in Hypersonics. Modeling Hypersonic Flows. – Volume 2. Birkhäuser, Boston, USA.
https://doi.org/10.1007/978-1-4612-0371-1
Briganti, R., Musumeci, R. E., Bellotti, G., Brocchini, M., and Foti, E. 2004. Boussinesq modeling of breaking waves: description of turbulence. J. Geophys. Res., 109(C0701).
https://doi.org/10.1029/2003JC002065
Canuto, V. M. and Cheng, Y. 1997. Determination of the Smagorinsky-Lilly constant CS. Phys. Fluids, 9, 1368.
https://doi.org/10.1063/1.869251
Capone, T., Panizzo, A., and Monaghan, J. J. 2010. SPH modelling of water waves generated by submarine landslides. J. Hydraul. Res., 48, 80–84.
https://doi.org/10.1080/00221686.2010.9641248
Chang, K.-A. and Liu, P. L.-F. 1999. Experimental investigation of turbulence generated by breaking waves in water of intermediate depth. Phys. Fluids, 11, 3339–3400.
https://doi.org/10.1063/1.870198
Dalrymple, R. A. and Rogers, B. D. 2006. Numerical modelling of water waves with the SPH method. Coastal Eng., 53, 141–147.
https://doi.org/10.1016/j.coastaleng.2005.10.004
Davies, A. G. and Villaret, C. 1999. Eulerian drift induced by progressive waves above rippled and very rough beds. J. Geophys. Res., 104(C1), 1465–1488.
https://doi.org/10.1029/1998JC900016
De Padova, D., Dalrymple, R. A., Mossa, M., and Petrillo, A. F. 2009. SPH simulations of regular and irregular waves and their comparison with experimental data. arXiv:0911.1872v1
Fredsøe, J. and Deigaard, R. 1992. Mechanics of coastal sediment transport. Advanced Series on Ocean Engineering – Volume 3. World Scientific Publishing, Singapore.
https://doi.org/10.1142/9789812385314
Gomez-Gesteira, M., Rogers, B. D., Dalrymple, R. A., and Crespo, A. J. C. 2010. State-of-the-art of classical SPH for free-surface flows. J. Hydraul. Res., 48, 6–27.
https://doi.org/10.1080/00221686.2010.9641242
Ihmsen, M., Akinci, N., Becker, M., and Teschner, M. 2011. A parallel SPH implementation on multi-core CPUs. Comput. Graphics Forum, 30(1), 99–112.
https://doi.org/10.1111/j.1467-8659.2010.01832.x
Laanearu, J., Koppel, T., Soomere, T., and Davies, P. A. 2007. Joint influence of river stream, water level and wind waves on the height of sand bar in a river mouth. Nord. Hydrol., 38(3), 287–302.
https://doi.org/10.2166/nh.2007.012
Laanearu, J., Vassiljev, A., and Davies, P. A. 2011. Hydraulic modelling of stratified bi-directional flow in a river mouth. In Proceedings of the Institution of Civil Engineers: Engineering and Computational Mechanics, 164(4), 207–216.
https://doi.org/10.1680/eacm.2011.164.4.207
Launder, B. E. and Sharma, B. I. 1974. Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transfer, 1(2), 131–138.
https://doi.org/10.1016/0094-4548(74)90150-7
Lee, E. S., Violeau, D., Issa, R., and Ploix, S. 2010. Application of weakly compressible and truly incompressible SPH to 3-D water collapse in waterworks. J. Hydraul. Res., 48, 50–60.
https://doi.org/10.1080/00221686.2010.9641245
Liiv, T. 2007. An experimental investigation of the oscillatory boundary layer around the breaking point. Proc. Est. Acad. Sci., 13(3), 215–233.
Liiv, T. and Lagemaa, P. 2008. The variation of the velocity and turbulent kinetic energy field in the wave in the vicinity of the breaking point. Est. J. Eng., 14(1), 42–64.
https://doi.org/10.3176/eng.2008.1.04
Lundgren, H. and Sorensen, T. 1958. A pulsating water tunnel. In Proceedings 6th Coastal Engineering Conference, ASCE, 356–358.
Madsen, O. S. and Grant, W. D. 1976. Quantitative description of sediment transport by waves. Coastal Eng. Proc., 1(15), 1093–1112.
https://doi.org/10.9753/icce.v15.64
Malarkey, J. and Davies, A. G. 2004. An eddy viscosity formulation for oscillatory flow over vortex ripples. J. Geophys. Res., 109, C12016.
https://doi.org/10.1029/2003JC002086
Monaghan, J. J. 1992. Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys., 30, 543–574.
https://doi.org/10.1146/annurev.aa.30.090192.002551
Monaghan, J. J. 1994. Simulating free surface flows with SPH. J. Comput. Phys., 110, 399–406.
https://doi.org/10.1006/jcph.1994.1034
Monaghan, J. J. 2000. SPH without a Tensile Instability. J. Comput. Phys., 159, 290–311.
https://doi.org/10.1006/jcph.2000.6439
Morris, J. P. 2000. Simulating surface tension with smoothed particle hydrodynamics. Int. J. Numer. Methods Fluids, 33(3), 333–353.
https://doi.org/10.1002/1097-0363(20000615)33:3<333::AID-FLD11>3.0.CO;2-7
Oldekop, N. and Liiv, T. 2013. Measurement of the variation of shear velocity on bed during a wave cycle. J. Earth Sci. Eng., 3(5), 322–330.
Oldekop, N., Liiv, T., and Lagemaa, P. 2015. The variation of turbulent eddy viscosity during a wave cycle. E-Proceedings of the 36th IAHR World Congress, June 28–July 3, 2015, Hague, Netherlands, 1−5.
Peregrine, D. H. 1983. Breaking waves on beaches. Annu. Rev. Fluid Mech., 15, 149–178.
https://doi.org/10.1146/annurev.fl.15.010183.001053
Perrier, G., Villaret, C., Davies, A. G., and Hansen, E. A. 1995. Numerical modelling of the oscillatory boundary layer over ripples. MAST G8-M Coastal Morphodynamics Project, Final Overall Meeting, Delft Hydraul., Gdansk, Poland, 4.26–4.29.
Rodi, W. 1980. Turbulence models and their application in hydraulics – A state of the art review. International Association for Hydraulic Research, Delft.
Rogers, B. D., Dalrymple, R. A., and Stansby, P. K. 2010. Simulation of caisson breakwater movement using SPH. J. Hydraul. Res., 48, 135–141.
https://doi.org/10.1080/00221686.2010.9641254
Rosswog, S. 2015. SPH methods in the modelling of compact objects. Living Rev. Comput. Astrophys., 1(1).
https://doi.org/10.1007/lrca-2015-1
Shao, S. 2006. Simulation of breaking wave by SPH method coupled with k – ε model. J. Hydraul. Res., 44(3), 338–349.
https://doi.org/10.1080/00221686.2006.9521686
Shih, T.-H., Zhu, J., and Lumley, J. L. 1996. Calculation of wall-bounded complex flows and free shear flows. Int. J. Numer. Methods Fluids, 23, 1133–1144.
https://doi.org/10.1002/(SICI)1097-0363(19961215)23:11<1133::AID-FLD456>3.0.CO;2-A
Svendsen, I. A. and Putrevu, V. 1996. Surf-zone hydrodynamics. Adv. Coastal Ocean Eng., 2, 1–78.
https://doi.org/10.1142/9789812797575_0001
Ting, F. C. K. and Kirby, J. T. 1994. Observation of undertow and turbulence in a laboratory surf zone. Coastal Eng., 24, 51–80.
https://doi.org/10.1016/0378-3839(94)90026-4
Ting, F. C. K. and Kirby, J. T. 1995. Dynamics of surf-zone turbulence in a strong plunging breaker. Coastal Eng., 24, 177–204.
https://doi.org/10.1016/0378-3839(94)00036-W
Ting, F. C. K. and Kirby, J. T. 1996. Dynamics of surf-zone turbulence in a spilling breaker. Coastal Eng., 27, 131–160.
https://doi.org/10.1016/0378-3839(95)00037-2
Violeau, D. and Rogers, B. D. 2016. Smoothed particle hydrodynamics (SPH) for free-surface flows: past, present and future. J. Hydraul. Res., 54(1), 1–26.
https://doi.org/10.1080/00221686.2015.1119209