Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-29T06:07:12.217Z Has data issue: false hasContentIssue false
  • English
  • Français

Reynolds stress turbulence modelling of surf zone breaking waves

Published online by Cambridge University Press:  22 February 2022

Yuzhu Li Open the ORCID record for Yuzhu Li [Opens in a new window] ,
Bjarke Eltard Larsen Open the ORCID record for Bjarke Eltard Larsen [Opens in a new window]  and
David R. Fuhrman Open the ORCID record for David R. Fuhrman [Opens in a new window]
Yuzhu Li*
Affiliation:
Technical University of Denmark, Department of Mechanical Engineering, Section for Fluid Mechanics, Coastal and Maritime Engineering, 2800 Kgs. Lyngby, Denmark National University of Singapore, Department of Civil and Environmental Engineering, Singapore 117576, Republic of Singapore
Bjarke Eltard Larsen
Affiliation:
Technical University of Denmark, Department of Mechanical Engineering, Section for Fluid Mechanics, Coastal and Maritime Engineering, 2800 Kgs. Lyngby, Denmark
David R. Fuhrman
Affiliation:
Technical University of Denmark, Department of Mechanical Engineering, Section for Fluid Mechanics, Coastal and Maritime Engineering, 2800 Kgs. Lyngby, Denmark
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Computational fluid dynamics is increasingly used to investigate the inherently complicated phenomenon of wave breaking. To date, however, no single model has proved capable of accurately simulating the breaking process across the entirety of the surf zone for both spilling and plunging breakers. The present study newly considers the Reynolds stress–$\omega$ turbulence closure model for this purpose, where $\omega$ is the specific dissipation rate. Novel stability analysis proves that, unlike two-equation closures (at least in their standard forms), the stress–$\omega$ model is neutrally stable in the idealized potential flow region beneath surface waves. It thus naturally avoids unphysical exponential growth of turbulence prior to breaking, which has plagued numerous prior studies. The analysis is confirmed through simulation of a progressive surface wave train. The stress–$\omega$ model is then applied to simulate a turbulent wave boundary layer, demonstrating superior accuracy relative to a two-equation model, especially during flow deceleration. Finally, the stress–$\omega$ model is employed to simulate spilling and plunging breaking waves, with seemingly unprecedented accuracy. Specifically, the present work marks the first time that a single turbulence closure model collectively: (1) avoids turbulence over-production prior to breaking, (2) accurately predicts the breaking point, (3) provides reasonable evolution of turbulent normal stresses, while also (4) yielding accurate evolution of undertow velocity structure and magnitude across the surf zone, for both spilling and plunging cases. Differences in the predicted Reynolds shear stresses (hence flow resistance) are identified as key to the improved inner surf zone performance, relative to a state-of-the-art two-equation model.

JFM classification

Turbulent Flows: Turbulence modelling Waves/Free-surface Flows: Wave breaking Geophysical and Geological Flows: Coastal engineering
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 937 , 25 April 2022 , A7
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Berberović, E., van Hinsberg, N.P., Jakirlić, S., Roisman, I.V. & Tropea, C. 2009 Drop impact onto a liquid layer of finite thickness: dynamics of the cavity evolution. Phys. Rev. E 79 (3), 036306.CrossRefGoogle ScholarPubMed
Bradford, S.F. 2000 Numerical simulation of surf zone dynamics. ASCE J. Waterway Port Coastal Ocean Engng 126 (1), 1–13.CrossRefGoogle Scholar
Brocchini, M. 2002 Free surface boundary conditions at a bubbly/weakly splashing air–water interface. Phys. Fluids 14 (6), 1834–1840.CrossRefGoogle Scholar
Brocchini, M. & Peregrine, D.H. 2001 The dynamics of strong turbulence at free surfaces. Part 2. Free-surface boundary conditions. J. Fluid Mech. 449, 255–290.CrossRefGoogle Scholar
Brown, S.A., Greaves, D.M., Magar, V. & Conley, D.C. 2016 Evaluation of turbulence closure models under spilling and plunging breakers in the surf zone. Coast. Engng 114, 177–193.CrossRefGoogle Scholar
Burchard, H. 2002 Applied Turbulence Modelling in Marine Waters, Lecture Notes in Earth Sciences, vol. 100. Springer.CrossRefGoogle Scholar
Chan, W.H.R., Johnson, P.L., Moin, P. & Urzay, J. 2021 The turbulent bubble break-up cascade. Part 2. Numerical simulations of breaking waves. J. Fluid Mech. 912, A43.CrossRefGoogle Scholar
Chang, K.-A. & Liu, P.L.-F. 1999 Experimental investigation of turbulence generated by breaking waves in water of intermediate depth. Phys. Fluids 11 (11), 3390–3400.CrossRefGoogle Scholar
Chella, M.A., Bihs, H., Myrhaug, D. & Muskulus, M. 2015 Breaking characteristics and geometric properties of spilling breakers over slopes. Coast. Engng 95, 4–19.CrossRefGoogle Scholar
Christensen, E.D. 2006 Large eddy simulation of spilling and plunging breakers. Coast. Engng 53 (5–6), 463–485.CrossRefGoogle Scholar
Christensen, E.D. & Deigaard, R. 2001 Large eddy simulation of breaking waves. Coast. Engng 42, 53–86.CrossRefGoogle Scholar
De Serio, F. & Mossa, M. 2006 Experimental study on the hydrodynamics of regular breaking waves. Coast. Engng 53 (1), 99–113.CrossRefGoogle Scholar
Deike, L., Melville, W.K. & Popinet, S. 2016 Air entrainment and bubble statistics in breaking waves. J. Fluid Mech. 801, 91–129.CrossRefGoogle Scholar
Derakhti, M., Kirby, J.T., Shi, F. & Ma, G. 2016 a Wave breaking in the surf zone and deep-water in a non-hydrostatic RANS model. Part 1: organized wave motions. Ocean Model. 107, 125–138.CrossRefGoogle Scholar
Derakhti, M., Kirby, J.T., Shi, F. & Ma, G. 2016 b Wave breaking in the surf zone and deep-water in a non-hydrostatic RANS model. Part 2: turbulence and mean circulation. Ocean Model. 107, 139–150.CrossRefGoogle Scholar
Devolder, B., Troch, P. & Rauwoens, P. 2018 Performance of a buoyancy-modified $k$-$\omega$ and $k$-$\omega$ SST turbulence model for simulating wave breaking under regular waves using OpenFOAM®. Coast. Engng 138, 49–65.CrossRefGoogle Scholar
Fenton, J.D. 1988 The numerical solution of steady water wave problems. Comput. Geosci. 14 (3), 357–368.CrossRefGoogle Scholar
Fuhrman, D.R. & Li, Y. 2020 Instability of the realizable $k$-$\varepsilon$ turbulence model beneath surface waves. Phys. Fluids 32, 115108.CrossRefGoogle Scholar
Fuhrman, D.R. & Larsen, B.E. 2020 A discussion on “Numerical computations of resonant sloshing using the modified isoadvector method and the buoyancy-modified turbulence closure model” [Appl. Ocean Res. (2019), 93, article no. 101829, doi:10.1016.j.apor.2019.05.014]. Appl. Ocean Res. 99, 102159.CrossRefGoogle Scholar
Gibson, M.M. & Launder, B.E. 1978 Ground effects on pressure fluctuations in the atmospheric boundary layer. J. Fluid Mech. 86 (3), 491–511.CrossRefGoogle Scholar
Hsu, T.J., Sakakiyama, T. & Liu, P.L.-F. 2002 A numerical model for wave motions and turbulence flows in front of a composite breakwater. Coast. Engng 46 (1), 25–50.CrossRefGoogle Scholar
Iafrati, A. 2009 Numerical study of the effects of the breaking intensity on wave breaking flows. J. Fluid Mech. 622, 371–411.CrossRefGoogle Scholar
Iafrati, A. 2011 Energy dissipation mechanisms in wave breaking processes: spilling and highly aerated plunging breaking events. J. Geophys. Res.: Oceans 116, C07024.CrossRefGoogle Scholar
Jacobsen, N.G, Fredsoe, J. & Jensen, J.H. 2014 Formation and development of a breaker bar under regular waves. Part 1: model description and hydrodynamics. Coast. Engng 88, 182–193.CrossRefGoogle Scholar
Jacobsen, N.G., Fuhrman, D.R. & Fredsøe, J. 2012 A wave generation toolbox for the open-source CFD library: OpenFOAM®. Intl J. Numer. Meth. Fluids 70 (9), 1073–1088.CrossRefGoogle Scholar
Jensen, B.L, Sumer, B.M. & Fredsøe, J. 1989 Turbulent oscillatory boundary layers at high Reynolds numbers. J. Fluid Mech. 206, 265–297.CrossRefGoogle Scholar
Justesen, P. 1991 A note on turbulence calculations in the wave boundary layer. J. Hydraul. Res. 29 (5), 699–711.CrossRefGoogle Scholar
Lara, J.L., Losada, I.J. & Liu, P.L.-F. 2006 Breaking waves over a mild gravel slope: experimental and numerical analysis. J. Geophys. Res.: Oceans 111, C11019.CrossRefGoogle Scholar
Larsen, B.E. & Fuhrman, D.R. 2018 On the over-production of turbulence beneath surface waves in Reynolds-averaged Navier–Stokes models. J. Fluid Mech. 853, 419–460.CrossRefGoogle Scholar
Larsen, B.E., Fuhrman, D.R. & Roenby, J. 2019 Performance of interFoam on the simulation of progressive waves. Coast. Engng J. 61 (3), 380–400.CrossRefGoogle Scholar
Larsen, B.E., van der A, D.A., van der Zanden, J., Ruessink, G. & Fuhrman, D.R. 2020 Stabilized RANS simulation of surf zone kinematics and boundary layer processes beneath large-scale plunging waves over a breaker bar. Ocean Model. 155, 101705.CrossRefGoogle Scholar
Launder, B.E., Reece, G.J. & Rodi, W. 1975 Progress in the development of a Reynolds-stress turbulence closure. J. Fluid Mech. 68 (3), 537–566.CrossRefGoogle Scholar
Lin, P. & Liu, P.L.-F. 1998 A numerical study of breaking waves in the surf zone. J. Fluid Mech. 359, 239–264.CrossRefGoogle Scholar
Liu, S., Ong, M.C., Obhrai, C., Gatin, I. & Vukčević, V. 2020 Influences of free surface jump conditions and different $k$-$\omega$ SST turbulence models on breaking wave modelling. Ocean Engng 217, 107746.CrossRefGoogle Scholar
Lowe, R.J., Buckley, M.L., Altomare, C., Rijnsdorp, D.P., Yao, Y., Suzuki, T. & Bricker, J.D. 2019 Numerical simulations of surf zone wave dynamics using smoothed particle hydrodynamics. Ocean Model. 144, 101481.CrossRefGoogle Scholar
Lupieri, G. & Contento, G. 2015 Numerical simulations of 2-D steady and unsteady breaking waves. Ocean Engng 106, 298–316.CrossRefGoogle Scholar
Makris, C.V., Memos, C.D. & Krestenitis, Y.N. 2016 Numerical modeling of surf zone dynamics under weakly plunging breakers with SPH method. Ocean Model. 98, 12–35.CrossRefGoogle Scholar
Mayer, S. & Madsen, P.A. 2000 Simulation of breaking waves in the surf zone using a Navier–Stokes solver. In Proceedings of the 27th International Conference of Coastal Engineering, pp. 928–941. ASCE.Google Scholar
Menter, F. & Esch, T. 2001 Elements of industrial heat transfer predictions. In Proceedings of the 16th Brazilian Congress of Mechanical Engineering (COBEM), pp. 117–127.Google Scholar
Nadaoka, K., Hino, M. & Koyano, Y. 1989 Structure of the turbulent flow field under breaking waves in the surf zone. J. Fluid Mech. 204, 359–387.CrossRefGoogle Scholar
Parneix, S., Laurence, D. & Durbin, P.A. 1998 A procedure for using DNS databases. Trans. ASME J. Fluids Engng 120, 40–46.CrossRefGoogle Scholar
Popovac, M. & Hanjalic, K. 2007 Compound wall treatment for RANS computation of complex turbulent flows and heat transfer. Flow Turbul. Combust. 78 (2), 177–202.CrossRefGoogle Scholar
Schumann, U. & Gerz, T. 1995 Turbulent mixing in stably stratified shear flows. J. Appl. Meteorol. 34 (1), 33–48.CrossRefGoogle Scholar
Scott, C.P., Cox, D.T., Maddux, T.B. & Long, J.W. 2005 Large-scale laboratory observations of turbulence on a fixed barred beach. Meas. Sci. Technol. 16 (10), 1903–1912.CrossRefGoogle Scholar
Shadloo, M.S., Weiss, R., Yildiz, M. & Dalrymple, R.A. 2015 Numerical simulation of long wave runup for breaking and nonbreaking waves. Intl J. Offshore Polar Engng 25 (01), 1–7.Google Scholar
Shao, S. 2006 Simulation of breaking wave by SPH method coupled with $k$-$\varepsilon$ model. J. Hydraul. Res. 44 (3), 338–349.CrossRefGoogle Scholar
Shih, T.-H., Liou, W.W., Shabbir, A., Yang, Z. & Zhu, J. 1995 A new $k$-$\varepsilon$ eddy viscosity model for high Reynolds number turbulent flows. Comput. Fluids 24 (3), 227–238.CrossRefGoogle Scholar
Slotnick, J., Khodadoust, A., Alonso, J., Darmofal, D., Gropp, W., Lurie, E. & Mavriplis, D. 2014 CFD vision 2030 study: a path to revolutionary computational aerosciences. NASA Tech. Rep. NASA/CR-2014-218178.Google Scholar
Stansby, P.K. & Feng, T. 2005 Kinematics and depth-integrated terms in surf zone waves from laboratory measurement. J. Fluid Mech. 529, 279–310.CrossRefGoogle Scholar
Strogatz, S.H. 2018 Nonlinear Dynamics and Chaos with Student Solutions Manual: With Applications to Physics, Biology, Chemistry, and Engineering. CRC.CrossRefGoogle Scholar
Sumer, B.M. & Fuhrman, D.R. 2020 Turbulence in Coastal and Civil Engineering. World Scientific.CrossRefGoogle Scholar
Svendsen, Ib.A. 1984 Mass flux and undertow in a surf zone. Coast. Engng 8 (4), 347–365.CrossRefGoogle Scholar
Ting, F.C.K. & Kirby, J.T. 1994 Observation of undertow and turbulence in a laboratory surf zone. Coast. Engng 24 (1–2), 51–80.CrossRefGoogle Scholar
Ting, F.C.K. & Kirby, J.T. 1995 Dynamics of surf-zone turbulence in a strong plunging breaker. Coast. Engng 24 (3–4), 177–204.CrossRefGoogle Scholar
Ting, F.C.K. & Kirby, J.T. 1996 Dynamics of surf-zone turbulence in a spilling breaker. Coast. Engng 27 (3–4), 131–160.CrossRefGoogle Scholar
Van Rijn, L.C. 1993 Principles of Sediment Transport in Rivers, Estuaries and Coastal Seas, vol. 1006. Aqua.Google Scholar
Wang, Z., Yang, J. & Stern, F. 2016 High-fidelity simulations of bubble, droplet and spray formation in breaking waves. J. Fluid Mech. 792, 307–327.CrossRefGoogle Scholar
Watanabe, Y. & Saeki, H. 1999 Three-dimensional large eddy simulation of breaking waves. Coast. Engng J. 41 (3–4), 281–301.CrossRefGoogle Scholar
Wei, Z., Li, C., Dalrymple, R.A., Derakhti, M. & Katz, J. 2018 Chaos in breaking waves. Coast. Engng 140, 272–291.CrossRefGoogle Scholar
Wilcox, D.C. 2006 Turbulence Modeling for CFD, 3rd edn. DCW Industries.Google Scholar
Xie, Z. 2013 Two-phase flow modelling of spilling and plunging breaking waves. Appl. Math. Model. 37 (6), 3698–3713.CrossRefGoogle Scholar
van der A, D.A., van der Zanden, J., O'Donoghue, T., Hurther, D., Cáceres, I., McLelland, S.J. & Ribberink, J.S. 2017 Large-scale laboratory study of breaking wave hydrodynamics over a fixed bar. J. Geophys. Res.: Oceans 122 (4), 3287–3310.CrossRefGoogle Scholar
van der Zanden, J., van der A, D.A., Cáceres, I., Hurther, D., McLelland, S.J., Ribberink, J. & O'Donoghue, T. 2018 Near-bed turbulent kinetic energy budget under a large-scale plunging breaking wave over a fixed bar. J. Geophys. Res.: Oceans 123 (2), 1429–1456.CrossRefGoogle Scholar
van der Zanden, J., van der A, D.A., Cáceres, I., Larsen, B.E., Fromant, G., Petrotta, C., Scandura, P. & Li, M. 2019 Spatial and temporal distributions of turbulence under bichromatic breaking waves. Coast. Engng 146, 65–80.CrossRefGoogle Scholar
Zhou, Z., Hsu, T.J., Cox, D. & Liu, X. 2017 Large-eddy simulation of wave-breaking induced turbulent coherent structures and suspended sediment transport on a barred beach. J. Geophys. Res.: Oceans 122 (1), 207–235.CrossRefGoogle Scholar