Boussinesq Modeling of Transcritical Flows over Steep Topography
Publication: Journal of Hydraulic Engineering
Volume 150, Issue 1
Abstract
Nonlinear shallow water equations effectively work for various fluvial hydraulic problems. However, the underlying assumption of hydrostatic pressure is violated by flows with significant vertical acceleration around steep landforms or manmade structures. Hence, unless the hydrostatic assumption is relaxed, refining the grid to resolve finer-scale topography does not guarantee enhancement in the accuracy of flow models. A potential solution to this issue is to employ the Boussinesq-type equations (BTE), which are suitable when the vertical flow scale is approaching the horizontal one. However, the classical BTE have been known to yield excessive undulations under certain conditions, which can lead to model robustness and stability issues. This study aims to develop a BTE model based on modified BTE free from erroneous undulations for fluvial applications. The model incorporates a free parameter responsible for controlling wave dispersion properties, which is optimized to ensure accurate results in flow simulations over steep beds. With the aid of artificial dissipation, the parameter-optimized BTE model successfully simulates transcritical flows, generating surface undulations within an appropriate range of flow conditions.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
The data set of the embankment overflow experiments was provided by a third party as indicated in the acknowledgments.
The model results that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
This work was supported by JSPS KAKENHI Grant No. 22H01597. The author would like to thank Dr. Kato (National Institute for Land and Infrastructure Management, Japan) for providing valuable data from their experiments. The author expresses gratitude to the editors and the two anonymous reviewers for their valuable comments and suggestions, which significantly contributed to the improvement of the manuscript.
References
Bouchut, F., A. Mangeney-Castelnau, B. Perthame, and J.-P. Vilotte. 2003. “A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows.” C. R. Math. 336 (6): 531–536. https://doi.org/10.1016/S1631-073X(03)00117-1.
Castro-Orgaz, O., F. N. Cantero-Chinchilla, and H. Chanson. 2022a. “Shallow fluid flow over an obstacle: Higher-order non-hydrostatic modeling and breaking waves.” Environ. Fluid Mech. 22 (4): 971–1003. https://doi.org/10.1007/s10652-022-09875-0.
Castro-Orgaz, O., W. H. Hager, and F. N. Cantero-Chinchilla. 2022b. “Shallow flows over curved beds: Application of the Serre–Green–Naghdi theory to weir flow.” J. Hydraul. Eng. 148 (1): 04021053. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001954.
Castro-Orgaz, O., W. H. Hager, and N. D. Katopodes. 2023. “Variational models for nonhydrostatic free-surface flow: A unified outlook to maritime and open-channel hydraulics developments.” J. Hydraul. Eng. 149 (7): 04023014. https://doi.org/10.1061/JHEND8.HYENG-13338.
Cienfuegos, R. 2023. “Surfing waves from the ocean to the river with the Serre-Green-Naghdi equations.” J. Hydraul. Eng. 149 (9): 04023032. https://doi.org/10.1061/JHEND8.HYENG-13487.
Clamond, D., D. Dutykh, and D. Mitsotakis. 2017. “Conservative modified Serre–Green–Naghdi equations with improved dispersion characteristics.” Commun. Nonlinear Sci. Numer. Simul. 45 (Apr): 245–257. https://doi.org/10.1016/j.cnsns.2016.10.009.
Dressler, R. F. 1978. “New nonlinear shallow-flow equations with curvature.” J. Hydraul. Res. 16 (3): 205–222. https://doi.org/10.1080/00221687809499617.
Favre, H. 1935. Etude théorique et expérimentale des ondes de translation dans les canaux découverts. Paris: Dunod.
Fenton, J. D. 1996. “Channel flow over curved boundaries and a new hydraulic theory.” In Proc., 10th Congress, Asia and Pacific Division of Int. Association for Hydraulic Research, 266–273. Langkawi, Malaysia: International Association for Hydraulic Research.
Fritz, H. M., and W. H. Hager. 1998. “Hydraulics of embankment weirs.” J. Hydraul. Eng. 124 (9): 963–971. https://doi.org/10.1061/(ASCE)0733-9429(1998)124:9(963).
Green, A. E., and P. M. Naghdi. 1976. “A derivation of equations for wave propagation in water of variable depth.” J. Fluid Mech. 78 (2): 237–246. https://doi.org/10.1017/S0022112076002425.
Harten, A., and J. M. Hyman. 1983. “Self adjusting grid methods for one-dimensional hyperbolic conservation laws.” J. Comput. Phys. 50 (2): 235–269. https://doi.org/10.1016/0021-9991(83)90066-9.
Kato, F., Y. Suwa, K. Watanabe, and S. Hatogai. 2012. “Mechanisms of coastal dike failure induced by the great east Japan earthquake tsunami.” Coastal Eng. Proc. 1 (33): 1–9. https://doi.org/10.9753/icce.v33.structures.40.
Keller, J. B. 2003. “Shallow-water theory for arbitrary slopes of the bottom.” J. Fluid Mech. 489 (Jun): 345–348. https://doi.org/10.1017/S0022112003005342.
Kennedy, A. B., Q. Chen, J. T. Kirby, and R. A. Dalrymple. 2000. “Boussinesq modeling of wave transformation, breaking, and runup. I: 1D.” J. Waterw. Port Coastal Ocean Eng. 126 (1): 39–47. https://doi.org/10.1061/(ASCE)0733-950X(2000)126:1(39).
Lenau, C. W. 1966. “The solitary wave of maximum amplitude.” J. Fluid Mech. 26 (2): 309–320. https://doi.org/10.1017/S0022112066001253.
Lin, C., M.-J. Kao, J.-M. Yuan, R. V. Raikar, S.-C. Hsieh, P.-Y. Chuang, J.-M. Syu, and W.-C. Pan. 2020. “Similarities in the free-surface elevations and horizontal velocities of undular bores propagating over a horizontal bed.” Phys. Fluids 32 (6): 063605. https://doi.org/10.1063/5.0010321.
Longuet-Higgins, M. S., and J. D. Fenton. 1974. “On the mass, momentum, energy and circulation of a solitary wave. II.” Proc. R. Soc. London, Ser. A 340 (1623): 471–493. https://doi.org/10.1098/rspa.1974.0166.
Madsen, P., R. Murray, and O. Sorensen. 1991. “A new form of the Boussinesq equations with improved linear dispersion characteristics.” Coastal Eng. 15 (4): 371–388. https://doi.org/10.1016/0378-3839(91)90017-B.
Madsen, P., and H. Schäffer. 1998. “Higher-order Boussinesq–type equations for surface gravity waves: Derivation and analysis.” Philos. Trans. R. Soc. London, Ser. A 356 (1749): 3123–3181. https://doi.org/10.1098/rsta.1998.0309.
Meselhe, E. A., F. Sotiropoulos, and F. M. Holly. 1997. “Numerical simulation of transcritical flow in open channels.” J. Hydraul. Eng. 123 (9): 774–783. https://doi.org/10.1061/(ASCE)0733-9429(1997)123:9(774).
Mignot, E., and R. Cienfuegos. 2009. “On the application of a Boussinesq model to river flows including shocks.” Coastal Eng. 56 (1): 23–31. https://doi.org/10.1016/j.coastaleng.2008.06.007.
Naghdi, P. M., and L. Vongsarnpigoon. 1986. “The downstream flow beyond an obstacle.” J. Fluid Mech. 162 (Jan): 223–236. https://doi.org/10.1017/S0022112086002021.
Ozmen-Cagatay, H., and S. Kocaman. 2010. “Dam-break flows during initial stage using SWE and RANS approaches.” J. Hydraul. Res. 48 (5): 603–611. https://doi.org/10.1080/00221686.2010.507342.
Peregrine, D. H. 1966. “Calculations of the development of an undular bore.” J. Fluid Mech. 25 (2): 321–330. https://doi.org/10.1017/S0022112066001678.
Roe, P. L. 1981. “Approximate Riemann solvers, parameter vectors, and difference schemes.” J. Comput. Phys. 43 (2): 357–372. https://doi.org/10.1016/0021-9991(81)90128-5.
Seabra-Santos, F. J., D. P. Renouard, and A. M. Temperville. 1987. “Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle.” J. Fluid Mech. 176 (Mar): 117–134. https://doi.org/10.1017/S0022112087000594.
Serre, F. 1953. “Contribution à l’étude des écoulements permanents et variables dans les canaux.” Houille Blanche 39 (6): 830–872. https://doi.org/10.1051/lhb/1953058.
Shimozono, T., H. Ikewaza, and S. Sato. 2017. “Non-hydrostatic modeling of coastal levee overflows.” In Proc., Coastal Dynamics 2017, 1606–1615. Tokyo: Univ. of Tokyo.
Stoker, J. J. 1957. Water waves. New York: Interscience.
Su, C. H., and C. S. Gardner. 1969. “Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation.” J. Math. Phys. 10 (3): 536–539. https://doi.org/10.1063/1.1664873.
Sweby, P. K. 1984. “High resolution schemes using flux limiters for hyperbolic conservation laws.” SIAM J. Numer. Anal. 21 (5): 995–1011. https://doi.org/10.1137/0721062.
Tissier, M., P. Bonneton, F. Marche, F. Chazel, and D. Lannes. 2011. “Nearshore dynamics of tsunami-like undular bores using a fully nonlinear Boussinesq model.” J. Coastal Res. 64: 603–607.
Wei, G., and J. Kirby. 1995. “Time-dependent numerical code for extended Boussinesq equations.” J. Waterw. Port Coastal Ocean Eng. 121 (5): 251–261. https://doi.org/10.1061/(ASCE)0733-950X(1995)121:5(251).
Yee, H. C. 1989. A class of high resolution explicit and implicit shock-capturing methods. Washington, DC: National Aeronautics and Space Administration.
Yee, H. C., N. D. Sandham, and M. J. Djomehri. 1999. “Low-dissipative high-order shock-capturing methods using characteristic-based filters.” J. Comput. Phys. 150 (1): 199–238. https://doi.org/10.1006/jcph.1998.6177.
Zerihun, Y. T., and J. D. Fenton. 2007. “A Boussinesq-type model for flow over trapezoidal profile weirs.” J. Hydraul. Res. 45 (4): 519–528. https://doi.org/10.1080/00221686.2007.9521787.
Information & Authors
Information
Published In
Journal of Hydraulic Engineering
Volume 150 • Issue 1 • January 2024
Copyright
© 2023 American Society of Civil Engineers.
History
Received: Dec 19, 2022
Accepted: Aug 2, 2023
Published online: Oct 18, 2023
Published in print: Jan 1, 2024
Discussion open until: Mar 18, 2024
ASCE Technical Topics:
- Boussinesq equations
- Critical flow
- Engineering fundamentals
- Flow (fluid dynamics)
- Flow simulation
- Fluid dynamics
- Fluid mechanics
- Geomatics
- Geomechanics
- Geotechnical engineering
- Hydraulic models
- Hydrologic engineering
- Hydrostatics
- Model accuracy
- Models (by type)
- Optimization models
- Soil mechanics
- Soil properties
- Soil stress
- Surveying methods
- Topography
- Water and water resources
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.