Accurate Two-Dimensional Simulation of Advective-Diffusive-Reactive Transport
Publication: Journal of Hydraulic Engineering
Volume 127, Issue 9
Abstract
The present paper presents an accurate numerical algorithm for the simulation of 2D solute/heat transport by unsteady advection-diffusion-reaction. The model was specifically developed for the study of convective exchange processes in a cross section of lakes and ponds, when the currents are predominantly driven by density (temperature) gradients. The numerical scheme is based on the split-operator approach, in which advection and diffusion with chemical/biological kinetic processes are calculated separately at each time step. Special attention is given to the advection operator in order to avoid excessive numerical damping or oscillations, as well as to the source/sink term, which may cause numerical instability and inaccuracy if improperly treated. The model has been verified on standard test problems for a wide range of Courant, Fourier, Péclet, and Thiele numbers, and found to produce stable results of high accuracy.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Branski, J. M., and Holley, E. R. ( 1986). “Advection calculation using spline schemes.” Proc., Water Forum '86, M. Karamouz et al., eds., Vol. 2, ASCE, New York, 1807–1814.
2.
Brezonik, P. L. ( 1994). Chemical kinetics and process dynamics in aquatic systems, Lewis, Boca Raton, Fla.
3.
Cahyono, M. ( 1993). “Three-dimensional numerical modelling of sediment transport processes in non-stratified estuarine and coastal waters.” PhD thesis, Dept. of Civ. Engrg., University of Bradford, Bradford, England.
4.
Crockett, S. R. ( 1993). “A semi-Lagrangian scheme for solving the advection-diffusion equation in two-dimensional simply connected regions.” MASc thesis, University of Toronto, Toronto.
5.
DeBoor, C. ( 1978). A practical guide to splines, Springer, New York.
6.
Egan, B., and Mahoney, J. R. ( 1972). “Numerical modelling of advection and diffusion of urban area source pollutants.” J. Appl. Meteorology, 11(2), 312–321.
7.
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J., and Brooks, N. H. ( 1979). Mixing in inland and coastal waters, Academic, San Diego.
8.
Gebhart, B. ( 1988). Buoyancy-induced flows and transport, Hemisphere Publishing Corp., Bristol, Pa.
9.
Glass, J., Rodi, W. ( 1982). “A higher order numerical scheme for scalar transport.” Comput. Methods Appl. Mech. Engrg., 31, 337–358.
10.
Hirsch, C. ( 1988). Numerical computation of internal and external flows, Wiley, New York.
11.
Holly, F. M., Jr. ( 1975). “Two-dimensional mass dispersion in rivers.” Hydro. Paper No. 78, Colorado State University, Fort Collins, Colo.
12.
Holly, F. M., Jr. and Preissmann, A. (1977). “Accurate calculation of transport in two dimensions.”J. Hydr. Div., ASCE, 103(11), 1259–1277.
13.
Holly, F. M., Jr. and Rahuel, J. L. (1990). “New numerical/physical framework for mobile-bed modelling, part I—Numerical and physical principles.”J. Hydr. Res., Delft, The Netherlands, 28(4), 401–416.
14.
Holly, F. M., Jr. and Usseglio-Polatera, J. (1984). “Dispersion simulation in two-dimensional tidal flows.”J. Hydr. Engrg., ASCE, 110(7), 905–926.
15.
Karpik, S. R., and Crockett, S. R. (1997). “Semi-Lagrangian algorithm for two-dimensional advection-diffusion equation on curvilinear coordinate meshes.”J. Hydr. Engrg., ASCE, 123(5), 389–401.
16.
Komatsu, T., Holly, F. M., Jr. Nakashiki, N., and Ohgushi, K. ( 1985). “Numerical calculation of pollutant transport in one and two dimensions.” J. Hydrosci. and Hydr. Engrg., 3(2), 15–30.
17.
Lax, P. D., and Wendroff, B. ( 1964). “Difference schemes with high order of accuracy for solving hyperbolic equations.” Communication in Pure and Appl. Math., 17, 381–398.
18.
Leonard, B. P. ( 1979). “A stable and accurate convection modeling procedure based on quadratic upstream interpolation.” Comput. Methods Appl. Mech. Engrg., 19, 59–98.
19.
Leonard, B. P. ( 1991). “The Ultimate conservative difference scheme applied to unsteady one-dimensional advection.” Comput. Methods Appl. Mech. Engrg., 88, 17–74.
20.
Lin, B., and Falconer, R. A. (1997). “Tidal flow and transport modeling using Ultimate Quickest scheme.”J. Hydr. Engrg., ASCE, 123(4), 303–314.
21.
Malm, J., Bengtsson, L., and Terzhevik, A. ( 1998). “Field study on currents in a shallow, ice-covered lake.” Limnol. Oceanogr., 43(7), 1669–1679.
22.
Nassiri, M., and Babarutsi, S. (1997). “Computation of dye concentration in shallow recirculating flow.”J. Hydr. Engrg., ASCE, 123(9), 793–805.
23.
Neuman, S. P. ( 1981). “An Eulerian-Lagrangian scheme for the dispersion-convection equation using conjugate space-time grids.” J. Comp. Phys., 41, 270–294.
24.
Noye, J., ed. ( 1987). “Finite difference methods for solving the one-dimensional transport equation.” Numerical modelling: Applications to marine systems, North-Holland, Amsterdam, 231–256.
25.
Patankar, S. V. ( 1980). Numerical heat transfer and fluid flow, Hemisphere Publishing Corp., Bristol, Pa.
26.
Patankar, S. V., and Baliga, B. R. ( 1978). “A new finite-difference scheme for parabolic differential equations.” Numer. Heat Transfer, 1, 27.
27.
Peaceman, D. W., and Rachford, H. H. ( 1955). “The numerical solution of parabolic and elliptic differential equations.” J. Soc. Ind. Appl. Math., 3, 28.
28.
Prenter, P. M. ( 1975). Splines and variational methods, Wiley, New York.
29.
Riley, M. J., and Stefan, H. G. ( 1988). “MINLAKE: A dynamic lake water quality simulation model.” Ecological Modeling, 43.
30.
Schohl, G. A., and Holly, F. M., Jr. (1991). “Cubic-spline interpolation in Lagrangian advection computation.”J. Hydr. Engrg., ASCE, 117(2), 248–253.
31.
Sobey, R. J. ( 1983). “Fractional step algorithm for estuarine mass transport.” Int. J. Numer. Methods in Fluids, 3, 567–581.
32.
Sobey, R. J. (1984). “Numerical alternatives in transient stream response.”J. Hydr. Engrg., ASCE, 110(6), 749–772.
33.
Stefan, H. G., Horsch, G. M., and Barko, J. W. ( 1989). “A model for estimation of convective exchange in the littoral region of a shallow lake during cooling.” Hydrobiologia, Dordrecht, the Netherlands, 174, 225–234.
34.
Szalai, L., Krebs, P., and Rodi, W. (1994). “Simulation of a flow in circular clarifiers with and without swirl.”J. Hydr. Engrg., ASCE, 120(1), 4–21.
35.
Tannehill, J. C., Anderson, D. A., Pletcher, R. H. ( 1997). Computational fluid mechanics and heat transfer, Taylor and Francis, Bristol, Pa.
36.
Thomann, R. V., and Mueller, J. A. ( 1987). Principles of surface water quality modeling and control, HarperCollins, New York.
37.
Toda, K., and Holly, F. M., Jr. ( 1987). “Hybrid numerical method for linear advection-diffusion.” Microsoftware for Engrs., 3(4), 199–205.
38.
Tolstov, G. P. ( 1962). Fourier series, Prentice-Hall, Englewood Cliffs, N.J.
39.
Yanenko, N. N. ( 1971). The method of fractional steps, Springer, New York.
40.
Zhu, J. ( 1991). “A low-diffusive and oscillation-free convection scheme.” Communications in Appl. Numer. Methods, 7(3), 225–232.
41.
Zhu, J., and Rodi, W. ( 1991). “Zonal finite-volume computations of incompressible flows.” Comp. Fluids, 20(4), 411–420.
Information & Authors
Information
Published In
Journal of Hydraulic Engineering
Volume 127 • Issue 9 • September 2001
Pages: 728 - 737
History
Received: Mar 11, 1998
Published online: Sep 1, 2001
Published in print: Sep 2001
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.