All Issue

2017 Vol.50, Issue 5
Review on the divergence form for bed slope source term and correction of the volume/free-surface relationship

발산형 바닥 경사 생성항의 재검토와 체적-수위 관계의 수정

Seung-Yong Hwang*
황 승용*
Hydro Science and Engineering Research Institute, KICT
한국건설기술연구원 수자원․ 하천연구소
*교신저자. *Corresponding Author. 
31 May 2017. pp. 289-302
PDF XML Citation
Abstract

DFB (Divergence Form for Bed slope source term) was rigorously derived and the error of mDFB using mean water depth at the cell face in DFB was clearly demonstrated. In addition, DFB technique turned out to be an exact method to the bed slope source term. The existing volume/free-surface relationship to the PSC (Partially Submerged Cell) has been corrected. It was discussed that treatment for the partially submerged edge is required to satisfy the C-property in PSC. It is expected that this study will provides a more accurate means in analyzing the shallow water equations with the approximate Riemann solver.

Keywords
Divergence form for bed slope source term
Volume/free-surface relationship
Shallow water equations
Well-balancedness
Riemann solver

발산형 바닥 경사 생성항(DFB, Divergence Form for Bed slope source term)을 엄밀하게 유도하였으며, DFB 중에서 격자의 변에서 평균 수 심을 이용하는 mDFB의 오차를 명백하게 입증하였다. 또한, DFB 기법은 바닥 경사 생성항에 대해 정확한 방법임을 밝혔다. 완전히 잠기기 않은 격 자에 대한 기존의 체적-수심 관계의 오류를 수정하였으며, C-특성의 충족을 위해 완전히 잠기지 않은 변에 대한 처리가 필요함을 검토하였다. 이 연 구를 통해 근사 Riemann 해법으로 천수방정식을 해석할 때 보다 정확한 수단을 제공할 수 있을 것으로 기대한다.

키워드
발산형 바닥 경사 생성항
체적-수심 관계
천수방정식
선평형성
Riemann 해법
References
1
Aquaveo, LLC. (2016). Mesh editing. SMS 12.2 tutorials. www.aquaveo. com/software/sms-learning-tutorials. Accessed 20 Mar. 2017.
2
Aureli, F., Maranzoni, A., Mignosa, P., and Ziveri, C. (2008). “A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography.” Adv. Water Resour., Vol. 31, pp. 962-974.
3
Begnudelli, L., and Sanders, B. F. (2006). “Unstructured grid finite- volume algorithm for shallow-water flow and scalar transport with wetting and drying.” J. Hydraul. Eng., Vol. 132, No. 4, pp. 371-384.
4
Begnudelli, L., and Sanders, B. F. (2007). “Conservative wetting and drying methodology for quadrilateral grid finite-volume methods.” J. Hydraul. Eng., Vol. 133, No. 3, pp. 312-322.
5
Bermúdez, A., Dervieux, A., Desideri, J.-A., and Vázquez, M. E. (1998). “Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes.” Comput. Methods Appl. Mech. Engrg., Vol. 155, pp. 49-72.
6
Bermúdez, A., and Vázquez, M. E. (1994). “Upwind methods for hyperbolic conservation laws with source terms.” Computers Fluids, Vol. 23, No. 8, pp. 1049-1071.
7
Bradford, S. F., and Katopodes, N. D. (2001). “Finite volume model for nonlevel basin irrigation.” J. Irrig. Drain. Eng., Vol. 127, No. 4, pp. 216-223.
8
Bradford, S. F., and Sanders, B. F. (2002). “Finite-volume model for shallow-water flooding of arbitrary topography.” J. Hydraul. Eng., Vol. 128, No. 3, pp. 289-298.
9
Greco, M., Iervolino, M., and Leopardi, A. (2008). “Discussion of ‘Divergence form for bed slope source term in shallow water equations’ by Alessandro Valiani and Lorenzo Begnudelli.” J. Hydraul. Eng., Vol. 134, No. 5, pp. 676-678.
10
Hwang, S.-Y. (2013). “Finite-volume model for shallow-water flow over uneven bottom.” J. Korea Water Resour. Assoc., Vol. 46, No. 2, pp. 139-153 (in Korean).
11
Komaei, S., and Bechteler, W. (2004). “An improved, robust implicit solution for the two dimensional shallow water equations on unstructured grids.” Proc. 2nd Conf. on Fluvial Hydraulics, IAHR, Edited by Greco, M., Carravetta, A., and Della Morte, R., Balkema, Vol. 2, pp. 1065-1072.
12
Kuiry, S. N., Pramanik, K., and Sen, D. (2008). “Finite volume method for shallow water equations with improved treatment of source terms.” J. Hydraul. Eng., Vol. 134, No. 2, p. 231-242.
13
Kuiry, S. N., and Sen, D. (2009). “Closure to ‘Finite volume method for shallow water equations with improved treatment of source terms’ by Soumendra Nath Kuiry, Kiran Pramanik, and Dhrubajyoti Sen.” J. Hydraul. Eng., Vol. 135, No. 11, pp. 1017.
14
LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems. Cambridge University Press.
15
Liu, X., and García, M. H. (2008). “Discussion of ‘Divergence form for bed slope source term in shallow water equations’ by Alessandro Valiani and Lorenzo Begnudelli.” J. Hydraul. Eng., Vol. 134, No. 5, pp. 678-679.
16
Maurel, F. (1997). “Test case 1: momentum equation source terms calculation - 1D codes.” Proc. 2nd workshop on dam-break wave simulation, Edited by Goutal, N., and Maurel, F., IAHR, pp. 2-5.
17
Messerschmidt, B. C. (2012). A high order finite volume scheme for the 2D shallow water equations including topography. Ph. D. dissertation, University of Kassel.
18
Roe, P. L. (1987). Error estimates for cell-vertex solutions of the compressible Euler equations. ICASE Report No. 87-6, NASA.
19
Simões, F. J. M. (2011). “Finite volume model for two-dimensional shallow environmental flow.” J. Hydraul. Eng., Vol. 137, No. 2, pp. 173-182.
20
Song, L., Zhou, J., Li, Q., Yang, X., et al. (2011). “An unstructured finite volume model for dam-break floods with wet/dry fronts over complex topography.” Int. J. Numer. Meth. Fluids., Vol. 67, pp. 960-980.
21
Toro, E. F. (2001). Shock-capturing methods for free-surface shallow flows. John Wiley & Sons.
22
Valiani, A., and Begnudelli, L. (2006). “Divergence form for bed slope source term in shallow water equations.” J. Hydraul. Eng., Vol. 132, No. 7, pp. 652-665.
23
Valiani, A., and Begnudelli, L. (2008). “Closure to ‘Divergence form for bed slope source term in shallow water equations’ by Alessandro Valiani and Lorenzo Begnudelli.” J. Hydraul. Eng., Vol. 134, No. 5, pp. 680-682.
24
Valiani, A., and Begnudelli, L. (2009). “Discussion of ‘Finite volume method for shallow water equations with improved treatment of source terms’ by Soumendra Nath Kuiry, Kiran Pramanik, and Dhrubajyoti Sen.” J. Hydraul. Eng., Vol. 135, No. 11, pp. 1016-1017.
25
Valiani, A., Caleffi, V., and Zanni, A. (2002). “Case study: Malpasset dam-break simulation using a two-dimensional finite volume method.” J. Hydraul. Eng., Vo. 128, No. 5, pp. 460-472.
26
van Leer, B. (1979). “Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method.” J. Comput. Phys., Vol. 32, pp. 101-136.
27
Vázquez-Cendón, M. E. (1999). “Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry.” J. Comput. Phys., Vol. 148, pp. 497-526.
28
Weiyan, T. (1992). Shallow water hydrodynamics. Water & Power Press.
29
Zhao, D. H., Shen, H. W., Tabios III, G. Q., Lai, J. S., et al. (1994). “Finite-volume two-dimensional unsteady-flow model for river basins.” J. Hydraul. Eng., Vol. 120, No. 7, pp. 863-883.
30
Zhou, J. G., Causon, D. M., Mingham, C. G., and Ingram, D. M. (2001). “The surface gradient method for the treatment of source terms in the shallow-water equations.” J. Comput. Phys., Vol. 168, pp. 1-25.
Information
  • Publisher :KOREA WATER RESOURECES ASSOCIATION
  • Publisher(Ko) :한국수자원학회
  • Journal Title :Journal of Korea Water Resources Association
  • Journal Title(Ko) :한국수자원학회 논문집
  • Volume : 50
  • No :5
  • Pages :289-302
  • Received Date : 2017-02-14
  • Revised Date : 2017-03-07
  • Accepted Date : 2017-03-07
  • DOI :https://doi.org/10.3741/JKWRA.2017.50.5.289
Journal Informaiton Journal of Korea Water Resources Association Journal of Korea Water Resources Association
Online Submission
Archives
  • scopus
  • KCI-수자원
  • NRF
  • KOFST
  • KISTI Current Status
  • KISTI Cited-by
  • crosscheck
  • orcid
  • open access
  • ccl
Journal Informaiton Journal Informaiton - close