Water Science and Technology; Hydrodynamic modelling; Pollutant transport; Diffusivity
Abstract :
This article reviews the mathematical formulations of the depth‐integrated advection‐diffusion equation used so far for modeling scalar transport and mixing in shallow environmental flows. It also summarizes the main approaches developed to evaluate the diffusivity tensor in such models. Remarkably, seven different formulations were found for the depth‐averaged advection‐diffusion equation, and eight distinct approaches have been used for determining the coefficients of the diffusivity tensor. Besides, only a minority of the reviewed studies report calibration of the diffusivity tensor against experimental or field observations. The fragmentation of existing methodologies exhibits a lack of scientific consensus. Promising approaches are highlighted and possible paths for improving the calibration, validation and transferability of the diffusivity coefficients are outlined.
Research Center/Unit :
UEE - Urban and Environmental Engineering - ULiège
Disciplines :
Civil engineering
Author, co-author :
Mignot, Emmanuel ; LMFA University of Lyon INSA Lyon CNRS Ecole Centrale Lyon Université Claude Bernard Lyon and UMR5509 Villeurbanne France
Riviere, Nicolas ; LMFA University of Lyon INSA Lyon CNRS Ecole Centrale Lyon Université Claude Bernard Lyon and UMR5509 Villeurbanne France
Dewals, Benjamin ; Université de Liège - ULiège > Département ArGEnCo > Hydraulics in Environmental and Civil Engineering
Language :
English
Title :
Formulations and Diffusivity Coefficients of the 2D Depth‐Averaged Advection‐Diffusion Models: A Literature Review
Publication date :
08 December 2023
Journal title :
Water Resources Research
ISSN :
0043-1397
eISSN :
1944-7973
Publisher :
American Geophysical Union (AGU)
Volume :
59
Issue :
12
Pages :
e2023WR035053
Peer reviewed :
Peer Reviewed verified by ORBi
Development Goals :
6. Clean water and sanitation 11. Sustainable cities and communities 14. Life below water
scite shows how a scientific paper has been cited by providing the context of the citation, a classification describing whether it supports, mentions, or contrasts the cited claim, and a label indicating in which section the citation was made.
Bibliography
Addison-Atkinson, W., Chen, A. S., Memon, F. A., & Chang, T. J. (2022). Modelling urban sewer flooding and quantitative microbial risk assessment: A critical review. Journal of Flood Risk Management, 15(4), e12844. https://doi.org/10.1111/jfr3.12844
Alavian, V. (1986). Dispersion tensor in rotating flows. Journal of Hydraulic Engineering, 112(8), 771–777. https://doi.org/10.1061/(asce)0733-9429(1986)112:8(771)
Baek, K. O., & Seo, I. W. (2010). Routing procedures for observed dispersion coefficients in two-dimensional river mixing. Advances in Water Resources, 33(12), 1551–1559. https://doi.org/10.1016/j.advwatres.2010.09.005
Baek, K. O., Seo, I. W., & Jeong, S. J. (2006). Evaluation of dispersion coefficient in meandering channels from transient tracer tests. Journal of Hydraulic Engineering, 132(10), 1021e1032.
Behzadi, F., Shamsaei, B., & Newman, J. C., III. (2018). Solution of fully-coupled shallow water equations and contaminant transport using a primitive-variable Riemann method. Environmental Fluid Mechanics, 18(2), 515–535. https://doi.org/10.1007/s10652-017-9571-7
Boxall, J. B., & Guymer, I. (2003). Analysis and prediction of transverse mixing coefficients in natural channels. Journal of Hydraulic Engineering, 129(2), 129–139. https://doi.org/10.1061/(asce)0733-9429(2003)129:2(129)
British Transport Docks Board. (1980). Humber estuary sediment flux, part I: field measurements. Rep. No. 283, Middlesex, England (p. 91).
Chang, Y. C. (1971). Lateral mixing in meandering channels. (PhD thesis). University of Iowa.
Cheng, R. T., Casulli, V., & Milford, S. N. (1984). Eulerian-Lagrangian solution of the convection-dispersion equation in natural coordinates. Water Resources Research, 20(7), 944–952. https://doi.org/10.1029/wr020i007p00944
De Barros, F. P. J., Mills, W. B., & Cotta, R. M. (2006). Integral transform solution of a two-dimensional model for contaminant dispersion in rivers and channels with spatially variable coefficients. Environmental Modelling & Software, 21(5), 699–709. ISSN 1364-8152. https://doi.org/10.1016/j.envsoft.2005.02.002
Duan, J. G. (2004). Simulation of flow and mass dispersion in meandering channels. Journal of Hydraulic Engineering, 130(10), 964–976. https://doi.org/10.1061/(asce)0733-9429(2004)130:10(964)
Elder, J. (1959). The dispersion of marked fluid in turbulent shear flow. Journal of Fluid Mechanics, 5(4), 544–560. https://doi.org/10.1017/s0022112059000374
Fagour, C., Gond, L., Le Coz, J., Riviere, N., Gostiaux, L., Mignot, E., & Kateb, L. (2022). Transverse mixing coefficients in non-uniform flows: Laboratory experiments. In Proceedings of the 39th IAHR World Congress, 19–24 June 2022.
Falconer, R. A. (1986). A two-dimensional mathematical model study of the nitrate levels in an inland natural basin. International Conference on Water Quality Modelling in the Inland Natural Environment (pp. 325–344).
Falconer, R. A. (1991). Review of modelling flow and pollutant transport processes in hydraulic basins. Water Pollution: Modelling, Measuring and Prediction, 3–23. https://doi.org/10.1007/978-94-011-3694-5_1
Fang, S., Ji, Y., & Zhang, M. (2022). Numerical modeling the flood and pollutant transport processes in residential areas with different land use types. Advances in Meteorology, 2022, 1–16. https://doi.org/10.1155/2022/9320089
Fernández-Pato, J., & García-Navarro, P. (2021). An efficient GPU implementation of a coupled overland-sewer hydraulic model with pollutant transport. Hydrology, 8(4), 146. https://doi.org/10.3390/hydrology8040146
Ferziger, J. H., & PeriC, M. (2002). Computational methods for fluid dynamics. Springer Berlin.
Fischer, H. B. (1973). Longitudinal dispersion and turbulent mixing in open-channel flow. Annual Review of Fluid Mechanics, 5(1), 59–78. https://doi.org/10.1146/annurev.fl.05.010173.000423
Fischer, H. B. (1978). On the tensor form of the bulk dispersion coefficient in a bounded skewed shear flow. Journal of Geophysical Research, 83(C5), 2373–2375. https://doi.org/10.1029/jc083ic05p02373
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J., & Brooks, N. H. (1979). Mixing in inland and coastal waters. Academic.
Gond, L., Mignot, E., Le Coz, J., & Kateb, L. (2021). Transverse mixing in rivers with longitudinally varied morphology. Water Resources Research, 57(6), e2020WR029478. https://doi.org/10.1029/2020wr029478
Gualtieri, C. (2010). RANS-based simulation of transverse turbulent mixing in a 2D geometry. Environmental Fluid Mechanics, 10(1), 137–156. https://doi.org/10.1007/s10652-009-9119-6
Gualtieri, C., Angeloudis, A., Bombardelli, F., Jha, S., & Stoesser, T. (2017). On the values for the turbulent Schmidt number in environmental flows. Fluids, 2(2), 17. https://doi.org/10.3390/fluids2020017
Gualtieri, C., & Mucherino, C. (2007). 5th International Symposium on Environmental Hydraulics (ISEH 2007), Tempe, USA.
Hinz, S., Katopodes, N. D., Freedman, P., Sullivan, M. P., & Freudberg, S. A. (1989). A FEM model for non-conservative plumes in the Potomac River.’ In Estuarine and coastal modeling (pp. 233–240). ASCE.
Hu, D., Zhu, Y., Zhong, D., & Qin, H. (2017). Two-dimensional finite-volume Eulerian-Lagrangian method on unstructured grid for solving advective transport of passive scalars in free-surface flows. Journal of Hydraulic Engineering, 143(12), 04017051. https://doi.org/10.1061/(asce)hy.1943-7900.0001371
Huai, W., Shi, H., Yang, Z., & Zeng, Y. (2018). Estimating the transverse mixing coefficient in laboratory flumes and natural rivers. Water, Air. & Soil Pollution, 229(8), 1–17.
Jackson, T. R., Haggerty, R., & Apte, S. V. (2013). A fluid-mechanics based classification scheme for surface transient storage in riverine environments: Quantitatively separating surface from hyporheic transient storage. Hydrology and Earth System Sciences, 17(7), 2747–2779. https://doi.org/10.5194/hess-17-2747-2013
Jiang, J., Liang, Q., Xia, X., & Hou, J. (2021). A coupled hydrodynamic and particle-tracking model for full-process simulation of nonpoint source pollutants. Environmental Modelling & Software, 136, 104951. https://doi.org/10.1016/j.envsoft.2020.104951
Jung, S. H., Seo, I. W., Kim, Y. D., & Park, I. (2019). Feasibility of velocity-based method for transverse mixing coefficients in river mixing analysis. Journal of Hydraulic Engineering, 145(11), 04019040. https://doi.org/10.1061/(asce)hy.1943-7900.0001638
Kashefipour, S. M., & Falconer, R. A. (2002). Longitudinal dispersion coefficients in natural channels. Water Research, 36(6), 1596–1608. https://doi.org/10.1016/s0043-1354(01)00351-7
Lau, Y. L., & Krishnappan, B. G. (1977). Transverse dispersion in rectangular channel. Journal of the Hydraulics Division, 103(HY10), 1173–1189. https://doi.org/10.1061/jyceaj.0004851
Lee, M. E., & Kim, G. (2012). Influence of secondary currents on solute dispersion in curved open channels. Journal of Applied Mathematics, 2012. https://doi.org/10.1155/2012/781695
Lee, M. E., & Seo, I. W. (2007). Analysis of pollutant transport in the Han River with tidal current using a 2D finite element model. Journal of Hydro-Environment Research, 1(1), 30–42. https://doi.org/10.1016/j.jher.2007.04.006
Lee, M. E., & Seo, I. W. (2010). 2D finite element pollutant transport model for accidental mass release in rivers. KSCE Journal of Civil Engineering, 14(1), 77–86. https://doi.org/10.1007/s12205-010-0077-9
Lee, M. E., & Seo, I. W. (2013). Spatially variable dispersion coefficients in meandering channels. Journal of Hydraulic Engineering, 139(2), 141–153. https://doi.org/10.1061/(asce)hy.1943-7900.0000669
Li, S., & Duffy, C. J. (2012). Fully-coupled modeling of shallow water flow and pollutant transport on unstructured grids. Procedia Environmental Sciences, 13, 2098–2121. https://doi.org/10.1016/j.proenv.2012.01.200
Liang, D., Wang, X., Falconer, R. A., & Bockelmann-Evans, B. N. (2010). Solving the depth-integrated solute transport equation with a TVD-MacCormack scheme. Environmental Modelling & Software, 25(12), 1619–1629. https://doi.org/10.1016/j.envsoft.2010.06.008
Lin, B., & Falconer, R. A. (1997). Tidal flow and transport modeling using ULTIMATE QUICKEST scheme. Journal of Hydraulic Engineering, 123(4), 303–314. https://doi.org/10.1061/(asce)0733-9429(1997)123:4(303)
Mark, O., Jørgensen, C., Hammond, M., Khan, D., Tjener, R., Erichsen, A., & Helwigh, B. (2018). A new methodology for modelling of health risk from urban flooding exemplified by cholera—Case Dhaka, Bangladesh (pp. 28–42). Blackwell Publishing Inc.
Morales-Hernandez, M., Murillo, J., & Garcia-Navarro, P. (2019). Diffusion-dispersion numerical discretization for solute transport in 2D transient shallow flows. Environmental Fluid Mechanics, 19(5), 1217–1234. https://doi.org/10.1007/s10652-018-9644-2
Murillo, J., Garcia-Navarro, P., Burguete, J., & Brufau, P. (2006). A conservative 2D model of inundation flow with solute transport over dry bed. International Journal for Numerical Methods in Fluids, 52(10), 1059–1092. https://doi.org/10.1002/fld.1216
Odgaard, A. J. (1986). Meander flow model. I: Development. Journal of Hydraulic Engineering, 112(12), 1117–1136. https://doi.org/10.1061/(asce)0733-9429(1986)112:12(1117)
Palman, L. E., Trento, A. E., & Alvarez, A. M. (2021). Scalar dispersion in the Salado River through tracers test and two-dimensional model. Water, Air, & Soil Pollution, 232(12), 1–16. https://doi.org/10.1007/s11270-021-05436-1
Park, I., & Seo, I. W. (2018). Modeling non-Fickian pollutant mixing in open channel flows using two-dimensional particle dispersion model. Advances in Water Resources, 111, 105–120. https://doi.org/10.1016/j.advwatres.2017.10.035
Park, I., Seo, I. W., Do Kim, Y., & Song, C. G. (2016). Flow and dispersion analysis of shallow water problems with Froude number variation. Environmental Earth Sciences, 75(2), 1–12. https://doi.org/10.1007/s12665-015-4928-z
Park, I., Seo, I. W., Shin, J., & Song, C. G. (2020). Experimental and numerical investigations of spatially-varying dispersion tensors based on vertical velocity profile and depth-averaged flow field. Advances in Water Resources, 142, 103606. https://doi.org/10.1016/j.advwatres.2020.103606
Park, I., & Song, C. G. (2018). Analysis of two-dimensional flow and pollutant transport induced by tidal currents in the Han River. Journal of Hydroinformatics, 20(3), 551–563. https://doi.org/10.2166/hydro.2017.118
Pathirana, A., Maheng Dikman, M., & Brdjanovic, D. (2011). A two dimensional pollutant transport model for sewer overflow impact simulation. In Proceedings: 12th International Conference on Urban Drainage, Porto Alegre/Brazil (pp. 10–15).
Piasecki, M., & Katopodes, N. D. (1999). Identification of stream dispersion coefficients by adjoint sensitivity method. Journal of Hydraulic Engineering, 125(7), 714–724. https://doi.org/10.1061/(asce)0733-9429(1999)125:7(714)
Sämann, R., Graf, T., & Neuweiler, I. (2019). Modeling of contaminant transport during an urban pluvial flood event—The importance of surface flow. Journal of Hydrology, 568, 301–310. https://doi.org/10.1016/j.jhydrol.2018.10.002
Seo, I. W., Choi, H. J., Kim, Y. D., & Han, E. J. (2016). Analysis of two-dimensional mixing in natural streams based on transient tracer tests. Journal of Hydraulic Engineering, 142(8), 04016020. https://doi.org/10.1061/(asce)hy.1943-7900.0001118
Seo, I. W., Lee, M. E., & Baek, K. O. (2008). 2D modeling of heterogeneous dispersion in meandering channels. Journal of Hydraulic Engineering, 134(2), 196–204. https://doi.org/10.1061/(asce)0733-9429(2008)134:2(196)
Seo, I. W., & Park, S. W. (2009). Effects of velocity structures on tracer mixing in a meandering channel. Journal of the Korean Society of Civil Engineering, 29(1B), 35–45. (In Korean).
Shen, X., Li, R., Huang, J., Feng, J., Hodges, B. R., Li, K., & Xu, W. (2016). Shelter construction for fish at the confluence of a river to avoid the effects of total dissolved gas supersaturation. Ecological Engineering, 97, 642–648. https://doi.org/10.1016/j.ecoleng.2016.10.055
Shin, J., Seo, I. W., & Baek, D. (2020). Longitudinal and transverse dispersion coefficients of 2D contaminant transport model for mixing analysis in open channels. Journal of Hydrology, 583, 124302. https://doi.org/10.1016/j.jhydrol.2019.124302
Taylor, G. I. (1954). The dispersion of matter in turbulent flow through a pipe. In Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences (Vol. 223(1155), pp. 446–468).
Wan, H., Li, J., Li, R., Feng, J., & Sun, Z. (2020). The optimal power generation operation of a hydropower station for improving fish shelter area of low TDG level. Ecological Engineering, 147, 105749. https://doi.org/10.1016/j.ecoleng.2020.105749
Wang, H., Li, Y., Li, J., An, R., Zhang, L., & Chen, M. (2018). Influences of hydrodynamic conditions on the biomass of benthic diatoms in a natural stream. Ecological Indicators, 92, 51–60. https://doi.org/10.1016/j.ecolind.2017.05.061
Wang, Y., & Huai, W. (2016). Estimating the longitudinal dispersion coefficient in straight natural rivers. Journal of Hydraulic Engineering, 142(11), 04016048. https://doi.org/10.1061/(asce)hy.1943-7900.0001196
Ye, J., & McCorquodale, J. A. (1997). Depth-averaged hydrodynamic model in curvilinear collocated grid. Journal of Hydraulic Engineering, 123(5), 380–388. https://doi.org/10.1061/(asce)0733-9429(1997)123:5(380)
Zhang, L., Liang, Q., Wang, Y., & Yin, J. (2015). A robust coupled model for solute transport driven by severe flow conditions. Journal of Hydro-Environment Research, 9(1), 49–60. https://doi.org/10.1016/j.jher.2014.04.005
Ouro, P., Fraga, B., Viti, N., Angeloudis, A., Stoesser, T., & Gualtieri, C. (2018). Instantaneous transport of a passive scalar in a turbulent separated flow. Environmental Fluid Mechanics, 18(2), 487–513. https://doi.org/10.1007/s10652-017-9567-3
Pope, S. B. (2000). Turbulent flows. Cambridge University Press.
Rossi, R., & Iaccarino, G. (2009). Numerical simulation of scalar dispersion downstream of a square obstacle using gradient-transport type models. Atmospheric Environment, 43(16), 2518–2531. https://doi.org/10.1016/j.atmosenv.2009.02.044
Shen, X., Li, R., Hodges, B. R., Feng, J., Cai, H., & Ma, X. (2019). Experiment and simulation of supersaturated total dissolved gas dissipation: Focus on the effect of confluence types. Water Research, 155, 320–332. https://doi.org/10.1016/j.watres.2019.02.056
Tavoularis, S., & Corrsin, S. (1985). Effects of shear on the turbulent diffusivity tensor. International Journal of Heat and Mass Transfer, 28(1), 265–276. https://doi.org/10.1016/0017-9310(85)90028-6
Wu, W. (2007). Computational river dynamics. CRC Press.
Similar publications
Sorry the service is unavailable at the moment. Please try again later.
This website uses cookies to improve user experience. Read more
Save & Close
Accept all
Decline all
Show detailsHide details
Cookie declaration
About cookies
Strictly necessary
Performance
Strictly necessary cookies allow core website functionality such as user login and account management. The website cannot be used properly without strictly necessary cookies.
This cookie is used by Cookie-Script.com service to remember visitor cookie consent preferences. It is necessary for Cookie-Script.com cookie banner to work properly.
Performance cookies are used to see how visitors use the website, eg. analytics cookies. Those cookies cannot be used to directly identify a certain visitor.
Used to store the attribution information, the referrer initially used to visit the website
Cookies are small text files that are placed on your computer by websites that you visit. Websites use cookies to help users navigate efficiently and perform certain functions. Cookies that are required for the website to operate properly are allowed to be set without your permission. All other cookies need to be approved before they can be set in the browser.
You can change your consent to cookie usage at any time on our Privacy Policy page.
