Skip to main content
SpringerLink
Log in

Two-dimensional numerical modeling of dam-break flow using a new TVD finite-element scheme

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

A new numerical scheme based on the finite-element method with a total-variation-diminishing property is developed with the aim of studying the shock-capturing capability of the combination. The proposed model is formulated within the framework of the one-step Taylor–Galerkin scheme in conjunction with the Van Leer’s limiter function. The approach is applied to the two-dimensional shallow water equations by different test cases, i.e., the partial, circular, and one-dimensional dam-break flow problems. For the one-dimensional case, the sub- and supercritical flow regimes are considered. The results are compared with the analytical, finite-difference, and smoothed particle hydrodynamics solutions in the literature. The findings show that the proposed model can effectively mask the sources of errors in the abrupt changes of the flow conditions and is able to resolve the shock and rarefaction waves where other numerical models produce spurious oscillations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Abbreviations

B(r):

Van Leer’s limiter

d :

Constant coefficient

F :

Flux vectors’ matrix

g :

Gravity

G :

Flux vectors’ matrix

h :

Water depth

h e :

Element size

h L :

Water depth in the upstream channel

h R :

Water depth in the downstream channel

h i :

Numerical depth results in each node

i :

Node number

K :

Stiffness matrix

L :

Length of the computational domain

M :

Mass matrix

n :

Manning’s roughness coefficient

S :

Topographical and frictional source terms

S bx :

Depth gradients in the x-directions

S by :

Depth gradients in the y-directions

S fx :

Friction slopes along x-directions

S fy :

Friction slopes along y-directions

s :

Bore speed

t :

Time

U :

Matrix of variables

u :

Depth-integrated velocity in the x-directions

v :

Depth-integrated velocity in the y-directions

x :

x-direction space

y :

y-direction space

\(\varPsi_{i}\) :

Basis-function

References

  1. Lauritzen PH (2011) Numerical techniques for global atmospheric models. In: Lecture notes in computational science and engineering, vol 73. Springer, Berlin, p 572

  2. Akkerman I et al (2011) Isogeometric analysis of free-surface flow. J Comput Phys 230:4137–4152

    Article  MATH  MathSciNet  Google Scholar 

  3. Ouyang C, He S, Xu Q (2014) MacCormack-TVD finite difference solution for dam break hydraulics over erodible sediment beds. J Hydraul Eng 141:06014026

    Article  Google Scholar 

  4. Luo Z, Gao J (2015) The numerical simulations based on the NND finite difference scheme for shallow water wave equations including sediment concentration. Comput Methods Appl Mech Eng 294:245–258

    Article  MathSciNet  Google Scholar 

  5. Amini R, Maghsoodi R, Moghaddam N (2016) Simulating free surface problem using isogeometric analysis. J Braz Soc Mech Sci Eng 38:413–421

    Article  Google Scholar 

  6. Wang P, Zhang N (2014) A large-scale wave-current coupled module with wave diffraction effect on unstructured meshes. Sci China Phys Mech Astron 57:1331–1342

    Article  Google Scholar 

  7. Zhang M-L et al (2016) Numerical simulation of flow and bed morphology in the case of dam break floods with vegetation effect. J Hydrodyn Ser B 28:23–32

    Article  Google Scholar 

  8. Triki A (2013) A finite element solution of the unidimensional shallow-water equation. J Appl Mech 80:021001

    Article  Google Scholar 

  9. Ortiz P (2014) Shallow water flows over flooding areas by a flux-corrected finite element method. J Hydraul Res 52:241–252

    Article  Google Scholar 

  10. Isakson MJ, Chotiros NP, Piper J (2015) Finite element modeling of propagation and reverberation shallow water waveguide with a variable environment. J Acoust Soc Am 138:1898

    Article  Google Scholar 

  11. Yin J, Sun J-W, Jiao Z-F (2015) A TVD-WAF-based hybrid finite volume and finite difference scheme for nonlinearly dispersive wave equations. Water Sci Eng 8:239–247

    Article  Google Scholar 

  12. Crespo A, Gómez-Gesteira M, Dalrymple R (2008) Modeling dam break behavior over a wet bed by a sph technique. J Waterw Port Coast Ocean Eng 134:313–320

    Article  Google Scholar 

  13. Chang T-J et al (2011) Numerical simulation of shallow-water dam break flows in open channels using smoothed particle hydrodynamics. J Hydrol 408:78–90

    Article  Google Scholar 

  14. Zhang M et al (2014) Integrating 1D and 2D hydrodynamic, sediment transport model for dam-break flow using finite volume method. Sci China Phys Mech Astron 57:774–783

    Article  Google Scholar 

  15. Zarmehi F, Tavakoli A, Rahimpour M (2011) On numerical stabilization in the solution of Saint-Venant equations using the finite element method. Comput Math Appl 62:1957–1968

    Article  MATH  MathSciNet  Google Scholar 

  16. Atallah MH, Hazzab A (2013) A Petrov–Galerkin scheme for modeling 1D channel flow with varying width and topography. Acta Mech 224:707–725

    Article  MATH  MathSciNet  Google Scholar 

  17. Wang H-H et al (1972) Numerical solutions of the one-dimensional primitive equations using Galerkin approximations with localized basis functions. Mon Weather Rev 100:738–746

    Article  Google Scholar 

  18. Sheu TWH, Fang CC (2001) High resolution finite-element analysis of shallow water equations in two dimensions. Comput Methods Appl Mech Eng 190:2581–2601

    Article  MATH  Google Scholar 

  19. Harten A (1983) High resolution schemes for hyperbolic conservation laws. J Comput Phys 49:357–393

    Article  MATH  MathSciNet  Google Scholar 

  20. Wang J, Ni H, He Y (2000) Finite-difference TVD scheme for computation of dam-break problems. J Hydraul Eng 126:253–262

    Article  Google Scholar 

  21. Boulahia A, Abboudi S, Belkhiri M (2014) Simulation of viscous and reactive hypersonic flows behaviour in a shock tube facility: TVD schemes and flux limiters application. J Appl Fluid Mech 7:315–328

    Google Scholar 

  22. Ouyang C et al (2013) A MacCormack-TVD finite difference method to simulate the mass flow in mountainous terrain with variable computational domain. Comput Geosci 52:1–10

    Article  Google Scholar 

  23. Huang Z, Lee J-J (2014) Modeling the spatial evolution of roll waves with diffusive Saint Venant equations. J Hydraul Eng 141:06014022

    Article  Google Scholar 

  24. Vacondio R, Dal Palù A, Mignosa P (2014) GPU-enhanced finite volume shallow water solver for fast flood simulations. Environ Model Softw 57:60–75

    Article  Google Scholar 

  25. Stoker JJ (1992) Water waves; the mathematical theory with applications. Pure and applied mathematics. Interscience Publishers, New York

    Book  MATH  Google Scholar 

  26. Zachariah C (2000) A characteristics finite-element algorithm for computational open channel flow analysis. The University of Tennessee, Knoxville

    Google Scholar 

  27. Selmin V (1987) Finite element solution of hyperbolic equations II. Two dimensional case. Inria, Paris

    Google Scholar 

  28. Yee H, Warming R, Harten A (1983) Implicit total variation diminishing (TVD) schemes for steady-state calculations. In: 6th computational fluid dynamics conference danvers. American Institute of Aeronautics and Astronautics

  29. Toro EF (2001) Shock-capturing methods for free-surface shallow flows, vol xv. Wiley, Chichester, p 309

    MATH  Google Scholar 

  30. Zienkiewicz OC, Taylor RL (eds) (2000) The finite element method. Butterworth-Heinemann, Oxford

    MATH  Google Scholar 

  31. Falconer RA (1986) Water quality simulation study of a natural harbor. J Waterw Port Coast Ocean Eng 112:15–34

    Article  Google Scholar 

  32. Ransom O, Younis B (2015) Selective application of a total variation diminishing term to an implicit method for two-dimensional flow modelling. J Flood Risk Manag 8:52–61

    Article  Google Scholar 

  33. Alcrudo F, Garcia-Navarro P (1993) A high-resolution Godunov-type scheme in finite volumes for the 2D shallow-water equations. Int J Numer Meth Fluids 16:489–505

    Article  MATH  Google Scholar 

  34. Ying X, Wang S, Khan A (2003) Numerical simulation of flood inundation due to dam and levee breach. In: World water and environmental resources congress. American Society of Civil Engineers, pp 1–9

  35. Liang S-J, Tang J-H, Wu M-S (2008) Solution of shallow-water equations using least-squares finite-element method. Acta Mech Sinica 24:523–532

    Article  MATH  MathSciNet  Google Scholar 

  36. Chou C et al (2015) Extrapolated local radial basis function collocation method for shallow water problems. Eng Anal Bound Elem 50:275–290

    Article  MathSciNet  Google Scholar 

  37. Neumann P, Bungartz H-J (2015) Dynamically adaptive Lattice Boltzmann simulation of shallow water flows with the Peano framework. Appl Math Comput 267:795–804

    MathSciNet  Google Scholar 

  38. Touma R, Klingenberg C (2015) Well-balanced central finite volume methods for the Ripa system. Appl Numer Math 97:42–68

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Kermanshah University of Technology, 6715685420, Kermanshah, Iran

    Omid Seyedashraf

  2. Department of Civil Engineering, Razi University, 6714967346, Kermanshah, Iran

    Ali Akbar Akhtari

Authors
  1. Omid Seyedashraf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ali Akbar Akhtari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ali Akbar Akhtari.

Additional information

Technical Editor: Jader Barbosa Jr.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Seyedashraf, O., Akhtari, A.A. Two-dimensional numerical modeling of dam-break flow using a new TVD finite-element scheme. J Braz. Soc. Mech. Sci. Eng. 39, 4393–4401 (2017). https://doi.org/10.1007/s40430-017-0776-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40430-017-0776-y

Keywords

Search

Navigation