Advection Problems with Spatially Varying Velocity Fields: 1D and 2D Analytical and Numerical Solutions
Publication: Journal of Hydraulic Engineering
Volume 146, Issue 8
Abstract
The advection equation is one of the most commonly used models in environmental fluid dynamics. Despite this, there exist very few analytical solutions to this equation for the case where the velocity field is spatially varying. In this work a class of solutions to the conservation form of the advection equation is defined in one and two dimensions using a particular change of variables. Example solutions are developed for constant, varying, and discontinuous initial density profiles, as well as for continuous and discontinuous velocity fields. The accuracy of four common finite-volume numerical methods is evaluated for eight test cases and compared to the exact solutions. The first- and second-order upwind methods and the upwind method with the van Leer and Superbee slope limiters are used. It is found that the upwind method with either of the slope limiters is well suited to numerically solving the advection equation with spatially varying velocity fields for most test cases, while, as is the case for nondivergent velocity fields, the first- and second-order upwind methods have some serious drawbacks.
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Data Availability Statement
All code generated or used during the study are available from the corresponding author by request.
References
Al-Mosallam, M., and K. Yokoi. 2016. “Efficient implementation of volume/surface integrated average-based multi-moment method.” Int. J. Comput. Methods 14 (2): 1750010. https://doi.org/10.1142/S0219876217500104.
Aubinet, M., B. Heinesch, and M. Yernaux. 2003. “Horizontal and vertical advection in a sloping forest.” Boundary Layer Meteorol. 108 (3): 397–417. https://doi.org/10.1023/A:1024168428135.
Ball, M., H. Pinkerton, and A. Harris. 2008. “Surface cooling, advection and the development of different surface textures on active lavas on Kilauea, Hawai’i.” J. Volcanol. Geotherm. Res. 173 (1): 148–156. https://doi.org/10.1016/j.jvolgeores.2008.01.004.
Barkhudarov, M. R. 2004. Lagrangian VOF advection method for FLOW-3D, 1–11. Santa Fe, NM: Flow Science.
Bernard, B., et al. 2006. “Impact of partial steps and momentum advection schemes in a global ocean circulation model at eddy-permitting resolution.” Ocean Dyn. 56 (5–6): 543–567. https://doi.org/10.1007/s10236-006-0082-1.
Bouche, D., J.-M. Ghidaglia, and F. Pascal. 2005. “Error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation.” SIAM J. Numer. Anal. 43 (2): 578–603. https://doi.org/10.1137/040605941.
Connors, J. M., J. W. Banks, J. A. Hittinger, and C. S. Woodward. 2014. “Quantification of errors for operator-split advection–diffusion calculations.” Comput. Methods Appl. Mech. Eng. 272 (Apr): 181–197. https://doi.org/10.1016/j.cma.2014.01.005.
Darracq, A., G. Destouni, K. Persson, C. Prieto, and J. Jarsjö. 2010. “Quantification of advective solute travel times and mass transport through hydrological catchments.” Environ. Fluid Mech. 10 (1): 103–120. https://doi.org/10.1007/s10652-009-9147-2.
Dendy, E., N. Padial-Collins, and W. VanderHeyden. 2002. “A general-purpose finite-volume advection scheme for continuous and discontinuous fields on unstructured grids.” J. Comput. Phys. 180 (2): 559–583. https://doi.org/10.1006/jcph.2002.7105.
Deng, X., Y. Shimizu, and F. Xiao. 2019. “A fifth-order shock capturing scheme with two-stage boundary variation diminishing algorithm.” J. Comput. Phys. 386 (Jun): 323–349. https://doi.org/10.1016/j.jcp.2019.02.024.
Deng, X., B. Xie, R. Loubère, Y. Shimizu, and F. Xiao. 2018. “Limiter-free discontinuity-capturing scheme for compressible gas dynamics with reactive fronts.” Comput. Fluids 171 (Jul): 1–14. https://doi.org/10.1016/j.compfluid.2018.05.015.
Deng, Z.-Q., V. P. Singh, and L. Bengtsson. 2004. “Numerical solution of fractional advection-dispersion equation.” J. Hydraul. Eng. 130 (5): 422–431. https://doi.org/10.1061/(ASCE)0733-9429(2004)130:5(422).
Di Lorenzo, E., D. Mountain, H. Batchelder, N. Bond, and E. E. Hofmann. 2013. “Advances in marine ecosystem dynamics from US Globec: The horizontal-advection bottom-up forcing paradigm.” Oceanography 26 (4): 22–33. https://doi.org/10.5670/oceanog.2013.73.
Fan, Y., C. P. Schlick, P. B. Umbanhowar, J. M. Ottino, and R. M. Lueptow. 2014. “Modelling size segregation of granular materials: The roles of segregation, advection and diffusion.” J. Fluid Mech. 741: 252–279. https://doi.org/10.1017/jfm.2013.680.
Fernando, H., S. Lee, J. Anderson, M. Princevac, E. Pardyjak, and S. Grossman-Clarke. 2001. “Urban fluid mechanics: Air circulation and contaminant dispersion in cities.” Environ. Fluid Mech. 1 (1): 107–164. https://doi.org/10.1023/A:1011504001479.
Fuster, D., G. Agbaglah, C. Josserand, S. Popinet, and S. Zaleski. 2009. “Numerical simulation of droplets, bubbles and waves: State of the art.” Fluid Dyn. Res. 41 (6): 065001. https://doi.org/10.1088/0169-5983/41/6/065001.
Guerrero, J. P., L. C. G. Pimentel, T. Skaggs, and M. T. Van Genuchten. 2009. “Analytical solution of the advection–diffusion transport equation using a change-of-variable and integral transform technique.” Int. J. Heat Mass Transfer 52 (13): 3297–3304. https://doi.org/10.1016/j.ijheatmasstransfer.2009.02.002.
Horvat, Z., M. Isic, and M. Spasojevic. 2015. “Two dimensional river flow and sediment transport model.” Environ. Fluid Mech. 15 (3): 595–625. https://doi.org/10.1007/s10652-014-9375-y.
Kumar, A., D. K. Jaiswal, and N. Kumar. 2010. “Analytical solutions to one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media.” J. Hydrol. 380 (3): 330–337. https://doi.org/10.1016/j.jhydrol.2009.11.008.
Künzli, P., K. Tsunematsu, P. Albuquerque, J.-L. Falcone, B. Chopard, and C. Bonadonna. 2016. “Parallel simulation of particle transport in an advection field applied to volcanic explosive eruptions.” Comput. Geosci. 89 (Apr): 174–185. https://doi.org/10.1016/j.cageo.2016.02.005.
Leonard, B. 1991. “The ultimate conservative difference scheme applied to unsteady one-dimensional advection.” Comput. Methods Appl. Mech. Eng. 88 (1): 17–74. https://doi.org/10.1016/0045-7825(91)90232-U.
Lube, G., E. Breard, S. Cronin, and J. Jones. 2015. “Synthesizing large-scale pyroclastic flows: Experimental design, scaling, and first results from pele.” J. Geophys. Res. Solid Earth 120 (3): 1487–1502. https://doi.org/10.1002/2014JB011666.
Onodera, N., T. Aoki, and K. Yokoi. 2016. “A fully conservative high-order upwind multi-moment method using moments in both upwind and downwind cells.” Int. J. Numer. Methods Fluids 82 (8): 493–511. https://doi.org/10.1002/fld.4228.
Patankar, S. 1980. Numerical heat transfer and fluid flow. Boca Raton, FL: CRC Press.
Pedersen, O., E. Nilssen, L. Jørgensen, and D. Slagstad. 2006. “Advection of the red king crab larvae on the coast of North Norway—A Lagrangian model study.” Fish. Res. 79 (3): 325–336. https://doi.org/10.1016/j.fishres.2006.03.005.
Pelanti, M., and R. J. LeVeque. 2006. “High-resolution finite volume methods for dusty gas jets and plumes.” SIAM J. Sci. Comput. 28 (4): 1335–1360. https://doi.org/10.1137/050635018.
Petrova, S., H. Kirova, D. Syrakov, and M. Prodanova. 2008. “Some fast variants of trap scheme for solving advection equation—Comparison with other schemes.” Comput. Math. Appl. 55 (10): 2363–2380. https://doi.org/10.1016/j.camwa.2007.11.001.
Popinet, S. 2003. “Gerris: A tree-based adaptive solver for the incompressible Euler equations in complex geometries.” J. Comput. Phys. 190 (2): 572–600. https://doi.org/10.1016/S0021-9991(03)00298-5.
Popinet, S. 2009. “An accurate adaptive solver for surface-tension-driven interfacial flows.” J. Comput. Phys. 228 (16): 5838–5866. https://doi.org/10.1016/j.jcp.2009.04.042.
Popinet, S. 2011. “Introduction to numerical methods for interfacial flows.” Accessed November 1, 2019. http://gfs.sf.net/papers/course.pdf.
Qiu, J.-M., and C.-W. Shu. 2011. “Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow.” J. Comput. Phys. 230 (4): 863–889. https://doi.org/10.1016/j.jcp.2010.04.037.
Roe, P. L. 1986. “Characteristic-based schemes for the Euler equations.” Annu. Rev. Fluid Mech. 18 (1): 337–365. https://doi.org/10.1146/annurev.fl.18.010186.002005.
Rypina, I. I., L. J. Pratt, J. Pullen, J. Levin, and A. L. Gordon. 2010. “Chaotic advection in an archipelago.” J. Phys. Oceanogr. 40 (9): 1988–2006. https://doi.org/10.1175/2010JPO4336.1.
Saaltink, M. W., J. Carrera, and S. Olivella. 2004. “Mass balance errors when solving the convective form of the transport equation in transient flow problems.” Water Resour. Res. 40 (5): W05107. https://doi.org/10.1029/2003WR002866.
Seed, A. 2003. “A dynamic and spatial scaling approach to advection forecasting.” J. Appl. Meteorol. 42 (3): 381–388. https://doi.org/10.1175/1520-0450(2003)042%3C0381:ADASSA%3E2.0.CO;2.
Smolarkiewicz, P. K., and J. Szmelter. 2009. “Iterated upwind schemes for gas dynamics.” J. Comput. Phys. 228 (1): 33–54. https://doi.org/10.1016/j.jcp.2008.08.008.
Sofiev, M., J. Vira, R. Kouznetsov, M. Prank, J. Soares, and E. Genikhovich. 2015. “Construction of the SILAM Eulerian atmospheric dispersion model based on the advection algorithm of Michael Galperin.” Geosci. Model Dev. 8 (11): 3497–3522. https://doi.org/10.5194/gmd-8-3497-2015.
Stevens, D. E., and S. Bretherton. 1996. “A forward-in-time advection scheme and adaptive multilevel flow solver for nearly incompressible atmospheric flow.” J. Comput. Phys. 129 (2): 284–295. https://doi.org/10.1006/jcph.1996.0250.
Van Leer, B. 1974. “Towards the ultimate conservative difference scheme. II: Monotonicity and conservation combined in a second-order scheme.” J. Comput. Phys. 14 (4): 361–370. https://doi.org/10.1016/0021-9991(74)90019-9.
Xiao, F., and T. Yabe. 2001. “Completely conservative and oscillationless semi-Lagrangian schemes for advection transportation.” J. Comput. Phys. 170 (2): 498–522. https://doi.org/10.1006/jcph.2001.6746.
Zamani, K., and F. A. Bombardelli. 2014. “Analytical solutions of nonlinear and variable-parameter transport equations for verification of numerical solvers.” Environ. Fluid Mech. 14 (4): 711–742. https://doi.org/10.1007/s10652-013-9325-0.
Zhang, D., C. Jiang, D. Liang, and L. Cheng. 2015. “A review on TVD schemes and a refined flux-limiter for steady-state calculations.” J. Comput. Phys. 302 (Dec): 114–154. https://doi.org/10.1016/j.jcp.2015.08.042.
Zoppou, C., and J. Knight. 1997. “Analytical solutions for advection and advection-diffusion equations with spatially variable coefficients.” J. Hydraul. Eng. 123 (2): 144–148. https://doi.org/10.1061/(ASCE)0733-9429(1997)123:2(144).
Zoppou, C., and J. Knight. 1999. “Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions.” Appl. Math. Modell. 23 (9): 667–685. https://doi.org/10.1016/S0307-904X(99)00005-0.
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Published In
Journal of Hydraulic Engineering
Volume 146 • Issue 8 • August 2020
Copyright
©2020 American Society of Civil Engineers.
History
Received: Oct 8, 2019
Accepted: Feb 18, 2020
Published online: May 20, 2020
Published in print: Aug 1, 2020
Discussion open until: Oct 20, 2020
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