Characteristic Based Split Finite Element for Unsteady Dam-Break Problem

Document Type: Research Paper

Author

Water Engineering Department, Faculty of Agriculture, University of Tabriz, Tabriz, Iran

Abstract

In this paper, an efficient numerical model for solution of the two-dimensional unsteady dam-break problem is described. The model solves the shallow water equations through Characteristic-Based Split (CBS) finite element method. The formulation of the model is based upon the fractional time step technique primarily used in the finite difference method for the incompressible Navier-Stokes equations. In addition to well-known advantages of the finite element discretization in introducing complex geometries and making accurate results near the boundaries, the CBS utilizes interesting advantages. These include the ability of the method to simulate both compressible and incompressible flows using the same formulation. Improved stability of the CBS algorithm along with its capability to simulate both sub- and super-critical flows are other main advantages of the method. These useful advantages of the algorithm introduce the CBS as a unique procedure to solve fluid dynamics problems under various conditions. Since dam-break problem has principally a high non-linear nature, the model is verified firstly by modeling one-dimensional problems of dam-break and bore formation problems. Furthermore, application of the model to a two-dimensional hypothetical dam-break problem shows the robustness and efficiency of the procedure. Despite the high non-linearity nature of the solved problems, the computational results, compared with the analytical solutions and reported results of other numerical models, indicate the favorable performance of the used procedure in modeling the dam-break problems.

Keywords


  1. Wu G,Yang Zh, Zhang K, Dong P, Lin Y, (2018). A Non-Equilibrium Sediment Transport Model for Dam Break Flow over Moveable Bed Based on Non-Uniform Rectangular Mesh. Water, pp:10(5): 1-22.
  2. Kocaman S, Ozmen-Cagatay H, (2015). Investigation of dam-break induced shock waves impact on a vertical wall. Journal of Hydrology, pp:525: 1–12.
  3. Zheng XG, Pu JH, Chen RD, Liu XN, Shao SD, (2016). A Novel Explicit-Implicit Coupled Solution Method of SWE for Long-term River Meandering Process Induced by Dam break. Journal of Applied Fluid Mechanics, pp:9(6): 2647-2660.
  4. Seyedashraf O, Mehrabi M, Akhtari AA, (2018). Novel approach for dam break flow modeling using computational intelligence. Journal of Hydrology, pp:559: 1028–1038.
  5. Fang Q, Tang Ch, Chen Zh, Wang Sh, Yang T, (2019). A calculation method for predicting the run out volume of dam-break and non-dam-break debris flows in the Wenchuan earthquake area. Geomorphology, pp:327: 201–214.
  6. Issakhov A, Zhandaulet Y, Nogaeva A, (2018). Numerical simulation of dam break flow for various forms of the obstacle by VOF method. International Journal of Multiphase Flow, pp:109 : 191–206.
  7. Erpicum S, Dewals BJ, Archambeau P, Pirotton M, (2010). Dam break flow computation based on an efficient flux vector splitting. Journal of Computational and Applied Mathematics, pp:234: 2143–2151.
  8. Sun X, Zhang J, Ren X, (2012). Characteristic-Based Split (CBS) Finite Element Method for Incompressible Viscous Flow with Moving Boundaries. Engineering Applications of Computational Fluid Mechanics, pp:6(3): 461-474.
  9. Zoppou C, Roberts S, (2000). Numerical solution of the two-dimensional unsteady dam-break. Applied Mathematical Modeling, pp:24 (7):457-475.
  10. Nithiarasu P, Zienkiewicz OC, (2000). On stabilization of the CBS algorithm: Internal and external time steps. International Journal for Numerical Methods in Engineering, pp:48:875-880.
  11. Baggio, G. A. P., Silva, J. B. C. (2016). Tridimensional flow simulation with finite element stabilized by CBS scheme. Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering, Brasília, DF, Brazil, November 6-9.
  12. Douglas J, Russell TF, (1982). Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM Journal of Numerical Analysis, pp:19:871-885.
  13. Jiang C, Zhang Z, Han X, Liu G, Lin T, (2018). A cell-based smoothed finite element method with semi-implicit CBS procedures for incompressible laminar viscous flows. International Journal for Numerical Methods in Fluids, pp:86:20-45.
  14. Morandi-Cecchi M, Venturin M, (2006). Characteristic-based split (CBS) algorithm finite element modeling for shallow waters in the Venice lagoon. International Journal for Numerical Methods in Engineering, pp:66:1641-1657.
  15. Ortiz P, (2012). Non-oscillatory continuous FEM for transport and shallow water flows. Computer Methods in Applied Mechanics and Engineering, pp:223:55–69.
  16. Ortiz P, Zienkiewicz OC, Szmelter J, (2006). Hydrodynamics and transport in estuaries and rivers by the CBS finite element method. International Journal for Numerical Methods in Engineering, pp:66:1569-1586.
  17. Ortiz, P., Zienkiewicz, O. C., Szmelter, J. (2004). CBS finite element modeling of shallow water and transport problems. European Congress on Computational Methods in Applied Sciences and Engineering, pp. 1–14.
  18. Parsa, J., Afshar, M. H. (2006). CBS finite element model for shallow water problems. 7th International Conference on Coasts, Ports and Marine Structures, Book of abstract, pp. 32, Tehran, Iran, November 27-29.
  19. Wang D, Tham LG, Shui Q, (2013). Dam-break model with Characteristic-Based Operator-Splitting Finite Element Method. Computer Modeling in Engineering and Science, pp:91(5):355-376.
  20. Zienkiewicz OC, Nithiarasu P, Codina R, Vazquez M, Ortiz P, (1999). An efficient and accurate algorithm for fluid mechanics problems: The characteristic based split procedure. International Journal of Numerical Methods in Fluids, pp:31:359–392.
  21. Zienkiewicz OC, Ortiz P, (1995). A split characteristic based finite element model for the shallow water equations. International Journal for Numerical Methods in Fluids, pp:20:1061–1080.
  22. Zienkiewicz O.C., Taylor R.L., The finite element method: Volume 3: Fluid dynamics, Butterworth Heinemann, pp.218–228, 2000.
  23. Chorin A, (1968). Numerical solution of the Navier Stokes equations. Mathematics of Computation, pp:22:745–762.
  24. Daubert A, Graffe O, (1967) Quelques aspects des ecoulements presque horizontaux a deux dimensions en plan et nonpermanents aplication aux estuaries. La Houille Blanche, pp:8:847-860.
  25. Verboom G., Stelling G., Officier M., Boundary conditions for the shallow water equations: Engineering Applications of Computational Hydraulics, Vol.1, Pitman, London, 1982.
  26. Ortiz P, (2004). Finite elements using a plane wave basis for scattering of surface water waves. Philosophical Transactions of the Royal Society of London, Series A, pp:362:1-16.
  27. Ortiz P, Sanchez E, (2001). Improved partition of unity finite element model for diffraction problems. International Journal for Numerical Methods in Engineering, pp:50:2727-2740.
  28. Labadie G., Dalsecco S., Latteaux B., Resolution des equations de Saint Venant par une methode de elements finis, EDF Report HE/41/82, 1982.
  29. Stoker J.J., Water waves: The mathematical theory with application, Interscience Publication, John Wiley & Sons, Inc., New York, 1957.
  30. Alcrudo F, Garcia-Navarro P, (1993). A high-resolution Gudnov-type scheme in finite volumes for the 2-D shallow water equations. International Journal for Numerical Methods in Fluids, pp:16(6):489-505.
  31. Hervouet J.M., Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method, Wiley, London, 2007.
  32. Fagherazzi S, Rasetarinera P, Hussaini MY, Furbish DJ, (2004). Numerical solution of the dam-break problem with a discontinuous Galerkin method. Journal of Hydraulic Engineering, pp:130(6):532-539.
  33. Fennema RJ, Chaudhry MH, (1990) Explicit methods for 2D transient free-surface flows. Journal of Hydraulic Engineering, pp:116(8):1013-1034.
  34. Biscarini C, Francesco SDi, Manciola P, (2010). CFD modelling approach for dam break flow studies. Hydrology and Earth System Sciences, pp:14: 705–718.