Environmental Modeling & Assessment

, Volume 22, Issue 1, pp 91–100 | Cite as

Development and Testing of 2D Finite Difference Model in Open Channels

  • Dang Truong An
  • Park Sang Deog


This study develops a depth-averaged two-dimensional (2D) numerical model using a finite difference method (FDM) on a staggered grid. The governing equations were solved using the Marker and Cell method that was developed at the Los Alamos laboratories by Harlow and Welch in 1965. In the paper, an explicit FDM was used to solve the governing equations. A first-order approximation was used for the temporal derivative. Second-order central difference approximations were used for space discretization. The time step is limited by the Courant–Friedrichs–Lewy (CFL) condition. The time step used in this study depends on the grid spacing and velocity components in the x- and y-directions. The study is divided into two steps: the first step is to develop a depth-averaged 2D numerical model to simulate the flow process. The second constructs a module to calculate the bed load transport and simulate the river morphology in the areas that have steep slopes, torrents, and mountain rivers. Developed model was applied to the artificial channel and a flood event in the Asungjun River section of the mountainous Yangyang Namdae River (South Korea). General simulation results showed that the developed model was in good agreement with the observed data.


Numerical model Finite difference Staggered grid Depth-averaged Open channels 



An important part of this work was realized during the author’s 3-year tenure at the Gangneung-Wonju National University, South Korea. The study was supported by the Institute for Disaster Prevention, Gangneung-Wonju National University, South Korea.


  1. 1.
    Gulkac, V. (2005). On the finite differences schemes for the numerical solution of two-dimensional moving boundary problem. Applied Mathematics and Computation, 168, 549–556.CrossRefGoogle Scholar
  2. 2.
    McKee, S., Tome, M. F., Ferreira, V. G., Cuminato, J. A., Castelo, A., Sousa, F. S., & Mangiavacchi, N. (2008). The marker and cell method. Journal of Computers & Fluids, 37, 907–930.CrossRefGoogle Scholar
  3. 3.
    MacDonald, I. (1996). Analysis and computation of steady open channel flow. U. K: Dissertation, University of Reading.Google Scholar
  4. 4.
    McGuirk, J. J., & Rodi, W. (1978). A depth averaged mathematical model for near field of side discharge into open channel flow. Journal Fluid Mechanic, 86(4), 761–781.CrossRefGoogle Scholar
  5. 5.
    Rao, P. (2003). Two-dimensional multiple grid algorithm for modeling transient open channel flows. Advances in Water Resources, 26, 685–690.CrossRefGoogle Scholar
  6. 6.
    Tomé, M. F., Mckee, S., Barratt, L., Jarvis, D. A., & Patrick, A. J. (1998). An experimental and numerical investigation of container filling with viscous liquids. International Journal Numerical Methods in Fluids, 31, 1333–1353.CrossRefGoogle Scholar
  7. 7.
    Jia, Y., & Wang, S. S. Y. (2001). CCHE2D: two-dimensional hydrodynamic and sediment transport model for unsteady open channel flows over loose bed. Technical Rep. No. NCCHE-TR-2001-1. Oxford: School of Engineering, Univ. of Mississippi.Google Scholar
  8. 8.
    Jin, Y. C., & Steffler, P. M. (1993). Predicting flow in curved open channel by depth-averaged method. Journal of Hydraulic Engineering, 119, 109–124.CrossRefGoogle Scholar
  9. 9.
    Duc, B. M. (2004). Numerical modeling of bed deformation in laboratory channels. Journal of Hydraulic Engineering ASCE, 130, 894–904.CrossRefGoogle Scholar
  10. 10.
    Horritt, M. (2004). Development and testing of a simple 2D finite volume model of sub critical shallow water flow. Journal of Numerical Mathematics, 44, 1231–1255.Google Scholar
  11. 11.
    Lai, Y. G. (2010). Two-dimensional depth-averaged flow modeling with an unstructured hybrid mesh. Journal of Hydraulic Engineering, 136(1), 12–23.CrossRefGoogle Scholar
  12. 12.
    Peric, M., Kessler, R., & Scheuerer, G. (1988). Comparison of finite volume numerical methods with staggered and collocated grids. Journal of Computational Fluids, 16(4), 389–403.CrossRefGoogle Scholar
  13. 13.
    Wu, W. (2004). Depth-averaged two-dimensional numerical modeling of unsteady flow and non-uniform sediment transport in open channels. Journal Hydraulic Engineering, 130(10), 1013–1024.CrossRefGoogle Scholar
  14. 14.
    Ye, J., & McCorquodale, J. A. (1997). Depth averaged hydrodynamic model in curvilinear collocated grid. Journal Hydraulic Engineering, 123(5), 380–388.CrossRefGoogle Scholar
  15. 15.
    Ahmadi, M. M., Ayyoubzadeh, S. A., Namin, M. M., & Samani, J. M. V. (2009). A 2D numerical depth-averaged model for unsteady flow in open channel bends. Journal of Agricultural Science and Technology, 11, 457–468.Google Scholar
  16. 16.
    Duan, J. G., & Nanda, S. K. (2006). Two-dimensional depth-averaged model simulation of suspended sediment concentration distribution in a Groyne field. Journal of Hydrology, 327, 426–437.CrossRefGoogle Scholar
  17. 17.
    Bradford, S. F., & Sanders, B. F. (2002). Finite volume model for shallow water flooding of arbitrary topography. Journal of Hydraulic Engineering, 128, 289–298.CrossRefGoogle Scholar
  18. 18.
    Paulo, G. S., Tom, M. F., & McKee, S. (2007). A marker-and-cell approach to viscoelastic free surface flows using the PTT model. Journal Non-Newtonian Fluid Mechanic, 147, 149–174.CrossRefGoogle Scholar
  19. 19.
    Chow, V. T., & Zvi, A. B. (1973). Hydrodynamic modeling of two-dimensional watershed flow. Journal of the Hydraulics Division, 99(11), 2023–2040.Google Scholar
  20. 20.
    Bellos, C., Hrissanthou, V. (2003). Numerical simulation of morphological changes in rivers and reservoirs. Computers & Mathematics with Applications, 45, 453-467.Google Scholar
  21. 21.
    McKee, S., Tom, M. F., Cuminato, J. A., Castelo, A., & Ferreira, V. G. (2004). Recent advances in the marker and cell method. Archives of Computational Methods Engineering, 11–2, 107–142.CrossRefGoogle Scholar
  22. 22.
    Ferreira, V. G., Tomé, M. F., Mangiavacchi, N., Castelo, A., Cuminato, J. A., Fortuna, A. O., & McKee, S. (2002). High-order upwinding and the hydraulic jump. International Journal of Numerical Methods in Fluids, 39, 549–583.CrossRefGoogle Scholar
  23. 23.
    Dang, T. A. (2013). Development of two dimensional riverbed variation model in mountainous rivers. South Korea: Dissertation Gangneung-Wonju National University.Google Scholar
  24. 24.
    Dang, T. A., & Park, S. D. (2015). Numerical simulation of bed level variation in open channels under steady flow conditions. Journal of Civil & Environmental Engineering, 5, 204. doi: 10.4172/2165-784X.1000204.Google Scholar
  25. 25.
    Hoffmann, K. A., Chiang, S. T., (2000). Computational fluid dynamics. Volume 1, fourth edition.
  26. 26.
    Matyka, M. (2004). Solution to 2D incompressible Navier–Stokes equations with SIMPLE, SIMPLER and Vorticity stream function approaches driven lid cavity problem: solution and visualization. Poland: University of Wrocław.Google Scholar
  27. 27.
    Nash, J. E., & Sutcliffe, J. V. (1970). River flow forecasting through conceptual models part I—A discussion of principles. Journal of Hydrology, 10(3), 282–290.CrossRefGoogle Scholar
  28. 28.
    Tomé, M. F., Mangiavacchia, N., Cuminatoa, J. A., Casteloa, A., & McKeeb, S. (2002). A finite difference technique for simulating unsteady viscoelastic free surface flows. Journal Non-Newtonian Fluid Mechanics, 106, 61–106.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Sustainable Management of Natural Resources and Environment Research Group, Faculty of Environment and Labour SafetyTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Department of Civil EngineeringGangneung-Wonju National UniversityGangneungSouth Korea

Personalised recommendations