KSCE Journal of Civil and Environmental Engineering Research (대한토목학회논문집)

Korean Society of Civil Engeneers (대한토목학회)
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Dam-Break and Transcritical Flow Simulation of 1D Shallow Water Equations with Discontinuous Galerkin Finite Element Method

불연속 갤러킨 유한요소법을 이용한 1차원 천수방정식의 댐 붕괴류 및 천이류 해석

  • 윤광희 (단국대학교(천안) 토목환경공학과) ;
  • 이해균 (단국대학교(천안) 토목환경공학과) ;
  • 이남주 (경성대학교 토목공학과)
  • Received : 2014.01.29
  • Accepted : 2014.06.30
  • Published : 2014.10.01
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Abstract

Recently, with rapid improvement in computer hardware and theoretical development in the field of computational fluid dynamics, high-order accurate schemes also have been applied in the realm of computational hydraulics. In this study, numerical solutions of 1D shallow water equations are presented with TVD Runge-Kutta discontinuous Galerkin (RKDG) finite element method. The transcritical flows such as dam-break flows due to instant dam failure and transcritical flow with bottom elevation change were studied. As a formulation of approximate Riemann solver, the local Lax-Friedrichs (LLF), Roe, HLL flux schemes were employed and MUSCL slope limiter was used to eliminate unnecessary numerical oscillations. The developed model was applied to 1D dam break and transcritical flow. The results were compared to the exact solutions and experimental data.

최근, 급속한 컴퓨터 하드웨어의 성능 향상과 전산유체역학 분야의 이론적 발전으로, 고차 정확도의 수치기법들이 계산수리학 분야에 적용되어 왔다. 본 연구에서는 1차원 천수방정식에 대한 수치 해법으로 TVD Runge-Kutta 불연속 갤러킨(RKDG) 유한요소법을 적용하였다. 대표적인 천이류(transcritical flow)의 예로 순간적인 댐 붕괴에 의한 댐 붕괴류(dam-break flow) 흐름과 지형변화에 의한 천이류를 모의하였다. 리만(Riemann) 근사해법으로 로컬 Lax-Friedrichs (LLF), Roe, HLL 흐름률(flux) 기법을 사용하였고, 불필요한 진동을 제거하기 위하여, 기울기 제한자로서 MUSCL 제한자를 사용하였다. 개발된 모델은 1차원 댐 붕괴류와 천이류에 적용하였다. 수치해석 결과는 해석해, 수리실험 결과와 비교하였다.

Keywords

References

  1. Chavent, G. and Salzano, G. (1982). "A finite element method for the 1D water flooding problem with gravity." J. Comp. Phys., Vol. 45, pp. 307-344. https://doi.org/10.1016/0021-9991(82)90107-3
  2. Cockburn, B. (1999). "Discontinuous galerkin methods for convection dominated problems." Lecture Notes in Computational Science and Engineering, Springer, Vol. 9, pp. 69-224.
  3. Cunge, J. A., Holly, F. M. and Verwey, A. (1980). Practical aspects of computational river hydraulics, Pitman, London.
  4. Harten, A., Lax. P. D., van Leer, B. (1983). "On upstream differencing and Godunov-type schemes for hyperbolic conservation laws." SIAM Rev. Vol. 25, No. 1, pp. 35-61. https://doi.org/10.1137/1025002
  5. Hesthaven, J. S. and Warburton, T. (2007). Nodal discontinuous galerkin methods: Algorithms, Analysis, and Applications, Springer, New York.
  6. Hughes, T. J. R. and Brooks, A. N. (1982). "A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions:Application to the Streamline-Upwind Procedure." Finite Elements in Fluids, R. H. Gallagher et al., eds., Wiley, Chichester, U.K., Vol. 4, pp. 46-65.
  7. Jain, S. C. (2001). Open-channel flow, John Wiley & Sons, New York.
  8. Kim, J. S. and Han, K. Y. (2009). "One-dimensional hydraulic modeling of open channel flow using the Riemann approximate solver - Application for natural river." J. Korea Water Resources Association, Vol. 42, No. 4, pp. 271-279 (in Korean). https://doi.org/10.3741/JKWRA.2009.42.4.271
  9. Lai, W. and Khan, A. A. (2012). "Discontinuous galerkin method for 1D shallow water flow in nonrectangular and nonprismatic channels." J. Hydraul. Engrg., Vol. 138, No. 3, pp. 285-296. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000501
  10. Lax, P. D. (1954). "Weak solutions of nonlinear hyperbolic equations and their numerical computation." Comm. Pure Appl. Math., Vol. 7, pp. 159-193. https://doi.org/10.1002/cpa.3160070112
  11. Lee, H. and Lee, N. J. (2013). "Simulation of shallow water flow by discontinuous galerkin finite element method." Proc. of 2013 IAHR World Congress, Chengdu, China.
  12. LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems, Cambridge University Press, Cambridge.
  13. Meselhe, E. A., Sotiropoulos, F. and Holly, F. M. (1997). "Numerical simulation of transcritical flow in open channels." J. Hydraul. Engrg., Vol. 23, No. 9, pp. 774-782.
  14. Mohapatra, P. K. and Bhallamudi, S. M. (1996). "Computation of a dam-break flood wave in channel transitions." Adv. Water Resour., Vol. 19, No. 3, pp. 181-187. https://doi.org/10.1016/0309-1708(95)00036-4
  15. Reed, W. H. and Hill, T. R. (1973). Triangular mesh methods for the neutron transport equation, Scientific Laboratory Report, Los Alamos, LA-UR-73-479.
  16. Roe, P. (1981). "Approximate riemann solvers, parameter vectors, and difference schemes." J. Comput. Phys., Vol. 43, No. 2, pp. 357-372. https://doi.org/10.1016/0021-9991(81)90128-5
  17. Schwanenberg, D. and Harms, M. (2004). "Discontinuous galerkin finite-element method for transcritical two-dimensional shallow water flows." J. Hydraul. Engrg., Vol. 130, No. 5, pp. 412-421. https://doi.org/10.1061/(ASCE)0733-9429(2004)130:5(412)
  18. Townson, J. M. and Al-Salihi, A. H. (1989). "Models of dam-break flow in R-T space." J. Hydraul. Engrg., Vol. 115, No. 5. pp. 561-575. https://doi.org/10.1061/(ASCE)0733-9429(1989)115:5(561)
  19. Zienkiewicz, O. C. and Codina, R. (1995). "A general algorithm for compressible and incompressible flow, Part I: The Split Characteristic Based Scheme." Int. J. Numer. Meth. Fluids, Vol. 20, pp. 869-885. https://doi.org/10.1002/fld.1650200812