Two-Dimensional Finite-Volume Eulerian-Lagrangian Method on Unstructured Grid for Solving Advective Transport of Passive Scalars in Free-Surface Flows
Publication: Journal of Hydraulic Engineering
Volume 143, Issue 12
Abstract
A two-dimensional (2D) finite-volume Eulerian-Lagrangian method (FVELM) on unstructured grid is proposed for solving the advection equation in free-surface scalar transport models. A backtracking band is defined along the backward trajectory of a side center as the dependence domain of a cell face, over which scalar concentration distribution is integrated to evaluate the advective flux through the cell face. Using the cell-face advective fluxes, a finite-volume cell update is finally carried out to obtain new cell concentrations, when mass is conserved both locally and globally by the unique flux at a cell face. The FVELM is then tested by a solid-body rotation experiment, a laboratory bend-flume experiment, and a real-river test (in a 365-km reach of the Yangtze River). In solid-body rotation tests, the FVELM is revealed to at least achieve a performance of existing second-order accuracy advection schemes. Relative to explicit Eulerian methods, the FVELM extends the dependence domain of a cell face from the upwind cell to the backtracking band, and therefore allows large time steps for which the Courant–Friedrichs–Lewy number (CFL) can be greater than 1. Accurate and stable FVELM simulations can be achieved at a CFL as large as 2–5 in these three tests. Efficiency issues of the FVELM are clarified by using the bend-flume test with refined grids (193,536 cells) on a computer with 16 cores; a parallel run using the FVELM is 14.2 times faster than a sequential run. In solving a transport problem (using 16 kinds of scalars and 16 cores), a parallel run using the FVELM is 2.3 times faster than a parallel run using an existing subcycling finite-volume method.
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Acknowledgments
Financial support from China’s National Natural Science Foundation (51339001, 51109009, 91647114), the Fundamental Research Funds for the Central Universities (2017KFYXJJ197), and the Public Welfare Scientific Research Funding Project of the MWR of China (201401011) are acknowledged.
References
Adcroft, A., Hill, C., and Marshall, J. (1999). “A new treatment of the Coriolis terms in C-grid models at both high and low resolution.” Mon. Weather Rev., 127(8), 1928–1936.
Allen, M. B., and Khosravani, A. (1992). “Solute transport via alternating-direction collocation using the modified method of characteristics.” Adv. Water Resour., 15(2), 125–132.
Baptista, A. M. (1987). “Solution of advection-dominated transport by Eulerian–Lagrangian methods using the backwards method of characteristics.” Ph.D. thesis, MIT, Cambridge, MA.
Bermejo, R., and Staniforth, A. (1992). “The conversion of semi-Lagrangian advection schemes to quasi-monotone schemes.” Mon. Weather Rev., 120(11), 2622–2632.
Binning, P., and Celia, M. A. (1996). “A finite volume Eulerian-Lagrangian localized adjoint method for solution of the contaminant transport equations in two-dimensional multiphase flow systems.” Water Resour. Res., 32(1), 103–114.
Budgell, W. P., Oliveira, A., and Skogen, M. D. (2007). “Scalar advection schemes for ocean modeling on unstructured triangular grids.” Ocean Dyn., 57(4), 339–361.
Cao, Z. X., Hu, P. H., Hu, K. H., Pender, G., and Liu, Q. (2015). “Modelling roll waves with shallow water equations and turbulent closure.” J. Hydraul. Res., 53(2), 161–177.
Casulli, V., and Zanolli, P. (2002). “Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems.” Math. Comput. Modell., 36(9–10), 1131–1149.
Celia, M. A., Russell, T. F., Herrera, I., and Ewing, R. E. (1990). “An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation.” Adv. Water. Resour., 13(4), 187–206.
Chang, Y. C. (1971). Lateral mixing in meandering channels, Univ. of Iowa, Iowa City, IA.
Cheng, R. T., Casulli, V., and Milford, S. (1984). “Eulerian-Lagrangian solution of the convection-dispersion equation in natural coordinates.” Water Resour. Res., 20(7), 944–952.
Dimou, K. (1992). “3-D hybrid Eulerian–Lagrangian/particle tracking model for simulating mass transport in coastal water bodies.” Ph.D. thesis, Dept. of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA.
Dukowicz, J. K., and Baumgardner, J. R. (2000). “Incremental remapping as a transport/advection algorithm.” J. Comput. Phys., 160(1), 318–335.
ECOMSED. (2002). “A primer for ECOMSED (version 1.3).” HydroQual, Inc., Mahwah, NJ.
Falconer, R. A., Wolanski, E., and Mardapitta-Hadjipandeli, L. (1986). “Modeling tidal circulation in an Island’s Wake.” J. Waterw. Port Coastal Ocean Eng., 234–254.
Fischer, H. B. (1973). “Longitudinal dispersion and turbulent mixing in open channel flow.” Ann. Rev. Fluid Mech., 5(1), 59–78.
Frolkovič, P. (2002). “Flux-based method of characteristics for contaminant transport in flowing groundwater.” Comput. Visual. Sci., 5(2), 73–83.
Gross, E. S., Bonaventura, L., and Giorgio, R. (2002). “Consistency with continuity in conservative advection schemes for free-surface models.” Int. J. Numer. Methods Fluids, 38(4), 307–327.
Gross, E. S., Koseff, J. R., and Monismith, S. G. (1999). “Evaluation of advective schemes for estuarine salinity simulations.” J. Hydraul. Eng., 32–46.
Harris, L. M., Lauritzen, P. H., and Mittal, R. (2011). “A flux-form version of the conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed sphere grid.” J. Comput. Phys., 230(4–230), 1215–1237.
Healy, R. W., and Russell, T. F. (1993). “A finite-volume Eulerian-Lagrangian localized adjoint method for solution of the advection-dispersion equation.” Water Resour. Res., 29(7), 2399–2413.
Healy, R. W., and Russell, T. F. (1998). “Solution of the advection-dispersion equation in two dimensions by a finite-volume Eulerian-Lagrangian localized adjoint method.” Adv. Water Resour., 21(1), 11–26.
Hu, D. C., Zhang, H. W., and Zhong, D. Y. (2009). “Properties of the Eulerian-Lagrangian method using linear interpolators in a three dimensional shallow water model using z-level coordinates.” Int. J. Comput. Fluid Dyn., 23(3), 271–284.
Hu, D. C., Zhong, D. Y., Zhang, H. W., and Wang, G. Q. (2015a). “Prediction—Correction method for parallelizing implicit 2D hydrodynamic models. I: Scheme.” J. Hydraul. Eng., 1–12.
Hu, D. C., Zhong, D. Y., Zhu, Y. H., and Wang, G. Q. (2015b). “Prediction—Correction method for parallelizing implicit 2D hydrodynamic models. II: Application.” J. Hydraul. Eng., 1–10.
Kaas, E. (2008). “A simple and efficient locally mass conserving semi-Lagrangian transport scheme.” Tellus A, 60(2), 305–320.
Laprise, J. P. R., and Plante, A. (1995). “A class of semi-Lagrangian integrated-mass (SLM) numerical transport algorithms.” Mon. Weather Rev., 123(2), 553–565.
Lauritzen, P. H., Kaas, E., and Machenhauer, B. (2006). “A mass-conservative semi-implicit semi-Lagrangian limited-area shallow-water model on the sphere.” Mon. Weather Rev., 134(4), 1205–1221.
Lauritzen, P. H., Nair, R. D., and Ullrich, P. A. (2010). “A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid.” J. Comput. Phys., 229(5), 1401–1424.
Lentine, M., Grétarsson, J. T., and Fedkiw, R. (2011). “An unconditionally stable fully conservative semi-Lagrangian method.” J. Comput. Phys., 230(8), 2857–2879.
Leonard, B. P., Macvean, M. K., and Lock, A. P. (1995). “The flux integral method for multidimensional convection and diffusion.” Appl. Math. Modell., 19(6), 333–342.
Lin, S. J., and Rood, R. B. (1996). “Multidimensional flux-form semi-Lagrangian transport schemes.” Mon. Weather Rev., 124(9), 2046–2070.
Machenhauer, B., and Olk, M. (1996). “On the development of a cell-integrated semi-Lagrangian shallow water model on the sphere.” Proc., ECMWF Workshop on Semi-Lagrangian Methods, European Centre for Medium-Range Weather Forecasts (ECMWF), Reading, U.K., 213–228.
Manson, J. R., and Wallis, S. G. (1998). “Accurate simulation of transport processes in two-dimensional shear flow.” Commun. Numer. Methods Eng., 14(9), 863–869.
Manson, J. R., and Wallis, S. G. (2000). “A conservative, semi-Lagrangian fate and transport model for fluvial systems. I: Theoretical development.” Water Res., 34(15), 3769–3777.
Nair, R., Côté, J., and Staniforth, A. (1999). “Monotonic cascade interpolation for semi-Lagrangian advection.” Q. J. R. Meteorol. Soc., 125(553), 197–212.
Neubauer, T., and Bastian, P. (2005). “On a monotonicity preserving Eulerian-Lagrangian localized adjoint method for advection-diffusion equations.” Adv. Water Res., 28(12), 1292–1309.
Oliveira, A., and Baptista, A. M. (1998). “On the role of tracking on Eulerian-Lagrangian solutions of the transport equation.” Adv. Water Resour., 21(7), 539–554.
Oliveira, A., and Fortunato, A. B. (2002). “Toward an oscillation-free, mass conservative, Eulerian-Lagrangian transport model.” J. Comput. Phys., 183(1), 142–164.
Priestley, A. (1993). “A quasi-conservative version of the semi-Lagrangian advection scheme.” Mon. Weather Rev., 121(2), 621–629.
Purser, R. J., and Leslie, L. M. (1994). “An efficient semi-Lagrangian scheme using third-order semi-implicit time integration and forward trajectories.” Mon. Weather Rev., 122(4), 745–756.
Rancic, M. (1992). “Semi-Lagrangian piecewise biparabolic scheme for two-dimensional horizontal advection of a passive scalar.” Mon. Weather Rev., 120(7), 1394–1406.
Rancic, M. (1995). “An efficient, conservative, monotonic remapping for semi-Lagrangian transport algorithms.” Mon. Weather Rev., 123(4), 1213–1217.
Rasch, P. J. (1994). “Conservative shape-preserving two-dimensional transport on a spherical reduced grid.” Mon. Weather Rev., 122(6), 1337–1350.
Rastogi, A. K., and Rodi, W. (1978). “Predictions of heat and mass transfer in open channels.” J. Hydraul. Div., 104(3), 397–420.
Roache, P. J. (1992). “A flux-based modified method of characteristics.” Int. J. Numer. Methods Fluids, 15(11), 1259–1275.
Russell, T. F., and Celia, M. A. (2002). “An overview of research on Eulerian-Lagrangian localized adjoint methods (ELLAM).” Adv. Water Resour., 25(8), 1215–1231.
Smagorinsky, J. (1963). “General circulation experiments with the primitive equations. I: The basic experiment.” Mon. Weather Rev., 91(3), 99–164.
Tanaka, R., Nakamura, T., and Yabe, T. (2000). “Constructing exactly conservative scheme in non-conservative form.” Comput. Phys. Commun., 126(3), 232–243.
Tennekes, H., and Lumley, J. L. (1983). A first course in turbulence, MIT Press, Cambridge, MA.
Wang, C., Wang, H., and Kuo, A. (2008). “Mass conservative transport scheme for the application of the ELCIRC model to water quality computation.” J. Hydraul. Eng., 1166–1171.
Wang, H., Dahle, H. K., Ewing, R. E., Espedal, M. S., Sharpley, R. C., and Man, S. (1999). “An ELLAM scheme for advection—Diffusion in two dimensions.” SIAM J. Sci. Comput., 20(6), 2160–2194.
Warner, J. C., Sherwood, C. R., Arango, H. G., and Signell, R. P. (2005). “Performance of four turbulence closure models implemented using a generic length scale method.” Ocean Modell., 8(1), 81–113.
Wolfram, P. J., and Fringer, O. B. (2013). “Mitigating horizontal divergence ‘checker-board’ oscillations on unstructured triangular C-grids for nonlinear hydrostatic and nonhydrostatic flows.” Ocean Modell., 69, 64–78.
Wu, W. M., Rodi, W., and Wenka, T. (2000). “3D numerical model for suspended sediment transport in channels.” J. Hydraul. Eng., 4–15.
Wu, W. M., and Wang, S. Y. (2004). “Depth-averaged 2-D calculation of flow and sediment transport in curved channels.” Int. J. Sediment. Res., 19(4), 241–257.
Xiao, F., and Yabe, T. (2001). “Completely conservative and oscillationless semi-Lagrangian schemes for advection transportation.” J. Comput. Phys., 170(2), 498–522.
Xing, Y. L., Zhang, X. X., and Shu, C. W. (2010). “Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations.” Adv. Water Resour., 33(12), 1476–1493.
Yabe, T., Tanaka, R., Nakamura, T., and Xiao, F. (2001). “An exactly conservative semi-Lagrangian scheme (CIP CSL) in one dimension.” Mon. Weather Rev., 129(2), 332–344.
Younes, A., Ackerer, P., and Lehmann, F. (2006). “A new efficient Eulerian-Lagrangian localized adjoint method for solving the advection—Dispersion equation on unstructured meshes.” Adv. Water Resour., 29(7), 1056–1074.
Zerroukat, M., Wood, N., and Staniforth, A. (2002). “SLICE: A semi-Lagrangian inherently conserving and efficient scheme for transport problems.” Q. J. R. Meteorol. Soc., 128(586), 2801–2820.
Zerroukat, M., Wood, N., and Staniforth, A. (2005). “A monotonic and positive-definite filter for a semi-Lagrangian inherently conserving and efficient (SLICE) scheme.” Q. J. R. Meteorolog. Soc., 131(611), 2923–2936.
Zhang, Y. L., Baptista, A. M., and Myers, E. P. (2004). “A cross-scale model for 3D baroclinic circulation in estuary-plume-shelf systems. I: Formulation and skill assessment.” Cont. Shelf Res., 24(18), 2187–2214.
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Journal of Hydraulic Engineering
Volume 143 • Issue 12 • December 2017
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©2017 American Society of Civil Engineers.
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Received: Jul 12, 2016
Accepted: May 18, 2017
Published online: Oct 9, 2017
Published in print: Dec 1, 2017
Discussion open until: Mar 9, 2018
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