Modeling Swash Zone Hydrodynamics Using Discontinuous Galerkin Finite-Element Method
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 147, Issue 2
Abstract
A two-dimensional numerical model for the solution of the nonlinear shallow water equations (NSWEs) using the discontinuous Galerkin finite element method (DGFEM) is presented. A new adaptation of the thin-film approach is developed for the wetting/drying treatment. The model is applied to a number of test cases that can be characterized as swash flows, or as cases that are particularly useful for swash flow modeling. The DGFEM model shows robustness and provides accurate predictions of water depth, velocities, and shoreline movement. For the case of bore collapse on a plane beach the model performs well against a state-of-the-art finite volume swash code. The new wetting/drying algorithm is tested against a previous algorithm within the same framework for simulating a solitary wave propagating on a beach with bottom friction, showing a noticeable improvement in the shoreline prediction. The model is also tested against a more subtle test case, including generation of subharmonic edge waves, in order to test the effectiveness of DGFEM in reproducing second-order effects. The model simulates the excitation and development of the subharmonic edge waves when compared with the analytical solutions in the literature. Overall, it is shown here for the first time that the DGFEM technique can be used to simulate accurately a wide range of swash zone flows and therefore swash zone processes.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
The following data, models, or code generated or used during the study are available from the corresponding author by request:
•
DGFEM model code.
•
The model results for the periodic solution of Carrier and Greenspan (1958).
•
The model results for the case of bore collapse at the shoreline.
•
The model results for the case of the shock solution of Antuono (2010).
•
The model results for the case of the solitary wave over a frictional beach.
•
The model results for the case of the subharmonic edge wave generation.
Acknowledgments
This research is sponsored by Newton-Mosharafa scholarship programme. The authors would like to thank the University of Nottingham for allowing use of its high-performance computer, which allowed the use of parallel processing over both CPU and GPU. The authors also would like to thank Dr. FangFang Zhu for providing the quasi-analytical solution data for the solitary wave case.
References
Aizinger, V., and C. Dawson. 2002. “A discontinuous Galerkin method for two-dimensional flow and transport in shallow water.” Adv. Water Resour. 25 (1): 67–84. https://doi.org/10.1016/S0309-1708(01(00019-7.
Antuono, M. 2010. “A shock solution for the nonlinear shallow water equations.” J. Fluid Mech. 658: 166–187. https://doi.org/10.1017/S002211201000159X.
Bakhtyar, R., D. A. Barry, L. Li, D. S. Jeng, and A. Yeganeh-Bakhtiary. 2009. “Modeling sediment transport in the swash zone: A review.” Ocean Eng. 36 (9–10): 767–783. https://doi.org/10.1016/j.oceaneng.2009.03.003.
Baldock, T. E., and P. Holmes. 1999. “Simulation and prediction of swash oscillations on a steep beach.” Coastal Eng. 36 (3): 219–242. https://doi.org/10.1016/S0378-3839(99(00011-3.
Borthwick, A. G. L., M. Ford, B. P. Weston, P. H. Taylor, and P. K. Stansby. 2006. “Solitary wave transformation, breaking and run-up at a beach.” Proc. Inst. Civ. Eng. Marit. Eng. 159 (3): 97–105.
Briganti, R., and N. Dodd. 2009. “Shoreline motion in nonlinear shallow water coastal models.” Coastal Eng. 56 (5): 495–505. https://doi.org/10.1016/j.coastaleng.2008.10.008.
Briganti, R., A. Torres-Freyermuth, T. Baldock, M. Brocchini, N. Dodd, T.-J. Hsu, Z. Jiang, Y. Kim, J. Pintado-Patiño, and M. Postacchini. 2016. “Advances in numerical modelling of swash zone dynamics.” Coastal Eng. 115: 26–41. https://doi.org/10.1016/j.coastaleng.2016.05.001.
Briggs, M. J., C. E. Synolakis, G. S. Harkins, and D. R. Green. 1995. “Laboratory experiments of tsunami run-up on circular island.” Pure Appl. Geophys. 144 (3/4): 569–593. https://doi.org/10.1007/BF00874384.
Brocchini, M., R. Bernetti, A. Mancinelli, and G. Albertini. 2001. “An efficient solver for nearshore flows based on the WAF method.” Coastal Eng. 43 (2): 105–129. https://doi.org/10.1016/S0378-3839(01(00009-6.
Brus, S. R., D. Wirasaet, E. J. Kubatko, J. J. Westerink, and C. Dawson. 2019. “High-order discontinuous Galerkin methods for coastal hydrodynamics applications.” Comput. Methods Appl. Mech. Eng. 355: 860–899. https://doi.org/10.1016/j.cma.2019.07.003.
Bunya, S., E. J. Kubatko, J. J. Westerink, and C. Dawson. 2009. “A wetting and drying treatment for the Runge–Kutta discontinuous Galerkin solution to the shallow water equations.” Comput. Methods Appl. Mech. Eng. 198 (17–20): 1548–1562. https://doi.org/10.1016/j.cma.2009.01.008.
Carrier, G. F., and H. P. Greenspan. 1958. “Water waves of finite amplitude on a sloping beach.” J. Fluid Mech. 4 (1): 97–109. https://doi.org/10.1017/S0022112058000331.
Chardón-Maldonado, P., J. C. Pintado-Patiño, and J. A. Puleo. 2016. “Advances in swash-zone research: Small-scale hydrodynamic and sediment transport processes.” Coastal Eng. 115: 8–25. https://doi.org/10.1016/j.coastaleng.2015.10.008.
Dawson, R. J., R. Peppe, and M. Wang. 2011. “An agent-based model for risk-based flood incident management.” Nat. Hazard. 59 (1): 167–189. https://doi.org/10.1007/s11069-011-9745-4.
de Brye, B., A. de Brauwere, O. Gourgue, T. Karna, J. Lambrechts, R. Comblen, and E. Deleersnijder. 2010. “A finite-element, multi-scale model of the Scheldt tributaries, river, estuary and ROFI.” Coastal Eng. 57 (9): 850–863. https://doi.org/10.1016/j.coastaleng.2010.04.001.
Duran, A., Q. Liang, and F. Marche. 2013. “On the well-balanced numerical discretization of shallow water equations on unstructured meshes.” J. Computat. Phys. 235: 565–586. https://doi.org/10.1016/j.jcp.2012.10.033.
Duran, A., and F. Marche. 2014. “Recent advances on the discontinuous Galerkin method for shallowwater equations with topography source term.” Comput. Fluids 101: 88–104. https://doi.org/10.1016/j.compfluid.2014.05.031.
Ern, A., S. Piperno, and K. Djadel. 2008. “A well-balanced Runge–Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying.” Int. J. Numer. Methods Fluids 58 (1): 1–25. https://doi.org/10.1002/fld.v58:1.
Giraldo, F. X., J. S. Hesthaven, and T. Warburton. 2002. “Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations.” J. Comput. Phys. 181 (2): 499–525. https://doi.org/10.1006/jcph.2002.7139.
Gourgue, O., R. Comblen, J. Lambrechts, T. Karna, V. Legat, and E. Deleersnijder. 2009. “A flux-limiting wetting–drying method for finite-element shallow-water models, with application to the Scheldt Estuary.” Adv. Water Resour. 32 (12): 1726–1739. https://doi.org/10.1016/j.advwatres.2009.09.005.
Guza, R. T., and R. E. Davis. 1974. “Excitation of edge waves by waves incident on a beach.” J. Geophys. Res. 79 (9): 1285–1291. https://doi.org/10.1029/JC079i009p01285.
Harten, A. 1983. “High resolution schemes for hyperbolic conservation laws.” J. Comput. Phys. 49 (3): 357–393. https://doi.org/10.1016/0021-9991(83(90136-5.
Hubbard, M. E., and N. Dodd. 2002. “A 2D numerical model of wave run-up and overtopping.” Coastal Eng. 47 (1): 1–26. https://doi.org/10.1016/S0378-3839(02(00094-7.
Huynh, V. L., N. Dodd, and F. Zhu. 2017. “Coastal morphodynamical modelling in nonlinear shallow water framework using a coordinate transformation method.” Adv. Water Resour. 107: 326–335. https://doi.org/10.1016/j.advwatres.2017.07.003.
Incelli, G., N. Dodd, C. E. Blenkinsopp, F. Zhu, and R. Briganti. 2016. “Morphodynamical modelling of field-scale swash events.” Coastal Eng. 115: 42–57. Swash-zone Processes. https://doi.org/10.1016/j.coastaleng.2015.09.006.
Karna, T., B. de Brye, O. Gourgue, J. Lambrechts, R. Comblen, V. Legat, and E. Deleersnijder. 2011. “A fully implicit wetting–drying method for DG-FEM shallow water models, with an application to the Scheldt Estuary.” Comput. Methods Appl. Mech. Eng. 200 (5–8): 509–524. https://doi.org/10.1016/j.cma.2010.07.001.
Kesserwani, G., J. L. Ayog, and D. Bau. 2018. “Discontinuous Galerkin formulation for 2D hydrodynamic modelling: Trade-offs between theoretical complexity and practical convenience.” Comput. Methods Appl. Mech. Eng. 342: 710–741. https://doi.org/10.1016/j.cma.2018.08.003.
Kesserwani, G., and Q. Liang. 2010. “Well-balanced RKDG2 solutions to the shallow water equations over irregular domains with wetting and drying.” Comput. Fluids 39 (10): 2040–2050. https://doi.org/10.1016/j.compfluid.2010.07.008.
Kobayashi, N., A. K. Otta, and I. Roy. 1987. “Wave reflection and runup on rough slopes.” J. Waterway, Port, Coastal, Ocean Eng. 113 (3): 282–298. https://doi.org/10.1061/(ASCE(0733-950X(1987(113:3(282(.
Kubatko, E. J., S. Bunya, C. Dawson, J. J. Westerink, and C. Mirabito. 2009. “A performance comparison of continuous and discontinuous finite element shallow water models.” J. Sci. Comput. 40 (1–3): 315–339. https://doi.org/10.1007/s10915-009-9268-2.
Kubatko, E. J., J. J. Westerink, and C. Dawson. 2006. “hp discontinuous Galerkin methods for advection dominated problems in shallow water flow.” Comput. Methods Appl. Mech. Eng. 196 (1–3): 437–451. https://doi.org/10.1016/j.cma.2006.05.002.
Lai, W., and A. A. Khan. 2012. “Discontinuous Galerkin method for 1D shallow water flows in natural rivers.” Eng. Appl. Comput. Fluid Mech. 6 (1): 74–86.
Le Bars, Y., V. Vallaeys, J. Deleersnijder, E. Hanert, L. Carrere, and C. Channeliere. 2016. “Unstructured-mesh modeling of the Congo river-to-sea continuum.” Ocean Dyn. 66 (4): 589–603. https://doi.org/10.1007/s10236-016-0939-x.
Lee, H., and N. Lee. 2016. “Wet-dry moving boundary treatment for Runge–Kutta discontinuous Galerkin shallow water equation model.” KSCE J. Civ. Eng. 20 (2): 978–989. https://doi.org/10.1007/s12205-015-0389-x.
LeVeque, R. J. 2002. Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge, UK: Cambridge University Press.
Li, B. Q. 2006. Discontinuous finite elements in fluid dynamics and heat transfer. Computational Fluid and Solid Mechanics. London: Springer.
Marras, S., M. A. Kopera, E. M. Constantinescu, J. Suckale, and F. X. Giraldo. 2018. “A residual-based shock capturing scheme for the continuous/discontinuous spectral element solution of the 2D shallow water equations.” Adv. Water Resour. 114: 45–63. https://doi.org/10.1016/j.advwatres.2018.02.003.
Mase, H. 1995. “Frequency down-shift of swash oscillations compared to incident waves.” J. Hydraul. Res. 33 (3): 397–411. https://doi.org/10.1080/00221689509498580.
Mulamba, T., P. Bacopoulos, E. J. Kubatko, and G. F. Pinto. 2019. “Sea-level rise impacts on longitudinal salinity for a low-gradient estuarine system.” Clim. Change 152 (3–4): 533–550. https://doi.org/10.1007/s10584-019-02369-x.
Özkan-Haller, H. T., and J. T. Kirby. 1997. “A Fourier-Chebyshev collocation method for the shallow water equations including shoreline runup.” Appl. Ocean Res. 19 (1): 21–34. https://doi.org/10.1016/S0141-1187(97(00011-4.
Peregrine, D. H., and S. M. Williams. 2001. “Swash overtopping a truncated plane beach.” J. Fluid Mech. 440: 391–399. https://doi.org/10.1017/S002211200100492X.
Pritchard, D., and A. J. Hogg. 2003. “Cross-shore sediment transport and the equilibrium morphology of mudflats under tidal currents.” J. Geophys. Res. Oceans 108 (C10): 3313. https://doi.org/10.1029/2002JC001570.
Rockliff, N. 1978. “Finite amplitude effects in free and forced edge waves.” Math. Proc. Cambridge Philos. Soc. 83 (3): 463–479. https://doi.org/10.1017/S030500410005475X.
Roe, P. 1981. “Approximate Riemann solvers, parameter vectors, and difference schemes.” J. Computat. Phys. 43 (2): 357–372. https://doi.org/10.1016/0021-9991(81(90128-5.
Rogers, B., A. Borthwick, and P. Taylor. 2003. “Mathematical balancing of flux gradient and source terms prior to using Roe’s approximate Riemann solver.” J. Comput. Phys. 192 (2): 422–451. https://doi.org/10.1016/j.jcp.2003.07.020.
Shen, M. C., and R. E. Meyer. 1963. “Climb of a bore on a beach Part 3. Run-up.” J. Fluid Mech. 16 (1): 113–125. https://doi.org/10.1017/S0022112063000628.
Thacker, W. C. 1981. “Some exact solutions to the nonlinear shallow-water wave equations.” J. Fluid Mech. 107 (-1): 499. https://doi.org/10.1017/S0022112081001882.
Wen, X., Z. Gao, W. S. Don, Y. Xing, and P. Li. 2016. “Application of positivity-preserving well-balanced discontinuous Galerkin method in computational hydrology.” Comput. Fluids 139: 112–119. https://doi.org/10.1016/j.compfluid.2016.04.020.
Zelt, J. A. 1986. “Tsunamis: The response of harbors with sloping boundaries to long wave excitation.” Rep. No. KH-R-47. Pasadena, CA: California Institute of Technology.
Zhu, F., and N. Dodd. 2013. “Net beach change in the swash zone: A numerical investigation.” Adv. Water Resour. 53: 12–22. https://doi.org/10.1016/j.advwatres.2012.10.002.
Zhu, F., and N. Dodd. 2015. “The morphodynamics of a swash event on an erodible beach.” J. Fluid Mech. 762: 110–140. https://doi.org/10.1017/jfm.2014.610.
Information & Authors
Information
Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 147 • Issue 2 • March 2021
Copyright
© 2020 American Society of Civil Engineers.
History
Received: Apr 17, 2020
Accepted: Sep 1, 2020
Published online: Nov 19, 2020
Published in print: Mar 1, 2021
Discussion open until: Apr 19, 2021
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.