Journal of Applied and Computational Mechanics
Random Walk Particle Tracking for Convection-diffusion Dominated Problems in Shallow Water Flows
Document Type : Research Paper
Authors
1 LME, Department of Physics, Faculty of Science, Mohammed First University, 60000 Oujda, Morocco
2 LPTPME, Department of Physics, Faculty of Science, Mohammed First University, 60000 Oujda, Morocco
Abstract
This paper deals with a Lagrangian stochastic approach to solve the advection-diffusion equation of a scalar tracer inflow based on random walk particle tracking (RWPT) of a fine number of particles that describe the tracer. The water flow is governed by the shallow water equations that are solved using a finite volume upwinding scheme on a non-structured triangular mesh. Results are presented for two problems, pure advection in a square cavity and pollutant advection in the strait of Gibraltar, that demonstrate the performance, accuracy, and the flexibility of the RWPT method to examine the process of pollutant convection by comparing predictions with those from the Eulerian approach. The Lagrangian approach is shown to have advantages in terms of the high level of simplicity and stability relative to the Eulerian approach.
Keywords
- Lagrangian random walk tracking
- Shallow water equations
- Convection-diffusion equation
- Strait of Gibraltar
Main Subjects
[1] Paster, A., Bolster, D., Benson, D.A., Connecting the dots: Semi-analytical and random walk numerical solutions of the diffusion-reaction equation with stochastic initial conditions, Journal of Computational Physics, 263, 2014, 91–112.
[2] Delay, F., Ackerer, P., Danquigny, C., Simulating solute transport in porous or fractured formations using random walk particle tracking A review, Vadose Zone Journal, 4(2), 2005, 360-379.
[3] Majumder, P., Eldho, T.I., Vectorized simulation of groundwater flow and contaminant transport using analytic element method and random walk particle tracking, Hydrological Processes, 31(5), 2017, 1144–1160.
[4] Scheidegger, A.E., Statistical hydrodynamics in porous media, Journal of Geophysical Research, 66, 1954, 3273–3278.
[5] De Josselin de Jong, G., Longitudinal and transverse diffusion in granular deposits, Transactions American Geophysical Union, 39, 1958, 67–74.
[6] Zhang, Z.A., Comparison of the Eulerian and Lagrangian methods for predicting particle transport in enclosed spaces, Atmospheric Environment, 41(25), 2007, 5236-5248.
[7] Wu, X.F., Liang, D., Study of pollutant transport in depth averaged flows using random walk method, Journal of Hydrodynamics, 31(2), 2019, 303-316.
[8] Park, C.H., Beyer, C., Bauer, S., Kolditz, O., Using global node-based velocity in random walk particle tracking in variably saturated porous media: Application to contaminant leaching from road constructions, Environmental Geology, 55(8), 2008, 1755–1766.
[9] Selle, B., Rink, K., Kolditz, O., Recharge and discharge controls on groundwater travel times and flow paths to production wells for the Ammer catchment in southwestern Germany, Environmental Earth Sciences, 69(2), 2013, 443–452.
[10] Bechtold, M., Vanderborght, J., Ippisch, O., Vereecken, H., Efficient random walk particle tracking algorithm for advective-dispersive transport in media with discontinuous dispersion coefficients and water, Water Resources Research, 47(10), 2011, W10526.
[11] Ahmed, M.I., Abd-Elmegeed, M.A., Hassan, A.E., Modelling transport in fractured media using the fracture continuum approach, Arabian Journal of Geosciences, 12, 2019, 172.
[12] LaBolle, E.M., Fogg, G.E., Tompson, A.F.B., Random-walk simulation of transport in heterogeneous porous media: local mass-conservation problem and implementation methods, Water Resources Research, 32(3), 1996, 583–593.
[13] Masciopinto, C., Particles transport in a single fracture under variable flow regimes, Advances in Engineering Software, 35(5), 1999, 327–337.
[14] Zhang, Z., Chen, Q., Comparison of the Eulerian and Lagrangian methods for predicting particle transport in enclosed spaces, Atmospheric Environment, 41(25), 2007, 5236-5248.
[15] Cheng, K., Acevedo-Bolton, V., Jiang, R. et al. Stochastic modeling of short-term exposure close to an air pollution source in a naturally ventilated room: An autocorrelated random walk method, Journal of Exposure Science and Environmental Epidemiology, 24, 2014, 311–318.
[16] Wu, X., Liang, D., Study of pollutant transport in depth-averaged flows using random walk method, Journal of Hydrodynamics, 31(2), 2019, 303-316.
[17] Yang, F., Liang, D., Wu, X., Xiao, Y., On the application of the depth-averaged random walk method to solute transport simulations, Journal of Hydroinformatics, 22(1), 2020, 33-45.
[18] Jalali, M.M., Borthwick, A.G.L., Tracer advection in a pair of adjacent side-wall cavities, and in a rectangular channel containing two groynes in series, Journal of Hydrodynamics, 30, 2018, 564–572.
[19] Chaabelasri, E.M., Borthwick, A.G.L., Salhi, N., Elmahi, I., Balanced adaptive simulation of pollutant transport in bay of tangier, World Journal of Modelling & Simulation, 10(1), 2014, 3-19.
[20] Chaabelasri, E., Jeyar, M., Salhi, N., Elmahi, I., A simple unstructured finite volume scheme for solving shallow water equations with wet/dry interface, International Journal of Mechanical Engineering and Technology, 10(1), 2019, 1849-1861.
[21] Chaabelasri, E.M., Amahmouj, A., Jeyar, M., Borthwick, A.G.L., Salhi, N., Elmahi, I., Numerical survey of contaminant transport and self-cleansing of water in nador lagoon, Morocco, Modelling and Simulation in Engineering, 2014, 2014, 179504.
[22] Liang, Q., Flood simulation using a well-balanced shallow flow model, Journal of Hydraulic Engineering, 136, 2010, 669–75
[23] Li, G., Song, L., Gao, J., High order well-balanced discontinuous Galerkin methods based on hydrostatic reconstruction for shallow water equations, Journal of Computational and Applied Mathematics, 340, 2018, 546-560.
[24] Xia, C., Cao, Z., Pender, G., Borthwick, A., Numerical algorithms for solving shallow water hydro sediment-morphodynamic equations, International Journal for Computer Aided Engineering and Software, 34(8), 2017, 2836-2861.
[25] Hou, I., Wang, R., Liang, Q., Li, Z., Huang, M.S., Hinkelmann, R., Efficient surface water flow simulation on static Cartesian grid with local refinement according to key topographic features, Computers and Fluids, 176, 2018, 117-134.
[26] Robinson, B.A., Dash, Z.V., Srinivasan, G., A Particle tracking transport method for the simulation of resident and flux-averaged concentration of solute plumes in groundwater models, Computational Geosciences, 14(4), 2014, 779–792.
[27] Srinivasan, G., Keating, E., Moulton, J.D., Dash, Z.V., Robinson, B.A., Convolution based particle tracking method for transient flow, Computational Geosciences, 16(3), 2012, 551-563.
[28] Salamon, P., Fernandez-Garcia, D., Gomez-Hernandez, J.J., A review and numerical assessment of the random walk particle tracking method, Journal of Contaminant Hydrology, 87(3-4), 2006, 277-305.
[29] Lattanzi, A.M., Yin, X., Hrenya, C.M., A fully-developed boundary condition for the random walk particle tracking Method, International Journal of Heat and Mass Transfer, 131, 2019, 604-610.
[30] Benkhaldoun, F., Elmahi, I., Seaid, M., Well-balanced finite volume schemes for pollutant transport by shallow water equations on unstructured meshes, Journal of Computational Physics, 226(1), 2017, 180–203.
)
