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도수의 수치 모의

Numerical Simulation of Hydraulic Jump

  • 황승용 (한국건설기술연구원 수자원환경연구본부)
  • 투고 : 2023.09.22
  • 심사 : 2023.10.11
  • 발행 : 2023.12.01

초록

천수 방정식 흐름률 계산에 근사 Riemann 해법을 채택한 수심 적분 모형을 도수 실험에 적용하였다. 도수 때문에 단일 수로에서 서로 다른 흐름 양상이 동시에 나타나므로 흐름 저항에 대해 수심이나 유속에 무관한 Manning 조도 계수보다는 흐름 조건을 반영할 수 있는 Weisbach 저항 계수를 채택하였다. 모의 결과는 실험 결과에 잘 부합되었으며, Weisbach 계수로부터 환산한 Manning 계수는 사류 구간과 상류 구간에서 각 각 적절하게 설정되고 있음을 확인하였다. 도수 실험과 비교에서 정수압 가정에 기반한 천수 방정식의 한계가 드러나 비정수압 천수 흐름 모형 도입의 필요성이 높아졌다.

A depth-integrated model with an approximate Riemann solver for flux computation of the shallow water equations was applied to hydraulic jump experiments. Due to the hydraulic jump, different flow regimes occur simultaneously in a single channel. Therefore, the Weisbach resistance coefficient, which reflects flow conditions rather than the Manning roughness coefficient that is independent of depth or flow, has been employed for flow resistance. Simulation results were in good agreement with experimental results, and it was confirmed that Manning coefficients converted from Weisbach coefficients were appropriately set in the supercritical and subcritical flow reaches, respectively. Limitations of the shallow water equations that rely on hydrostatic assumptions have been revealed in comparison with hydraulic jump experiments, highlighting the need for the introduction of a non-hydrostatic shallow-water flow model.

키워드

과제정보

This study is a sequel of Oh et al. (1998). The author would like to thank Oh, Seong-Taek for reviewing an early version of this article. This work was supported by the Korea Environment Industry & Technology Institute (KEITI) through the R&D Project: Water Management Program for Drought (Grant # 2022003610003; Korea Institute of Civil Engineering and Building Technology (KICT) Project # 20230076), funded by the Korea Ministry of Environment (MOE).

참고문헌

  1. Abbott, M. B. and Rodenhuis, G. S. (1972). "A numerical simulation of the undular hydraulic jump." Journal of Hydraulic Research, IAHR, Vol. 10, No. 3, pp. 239-257, https://doi.org/10.1080/00221687209500160. 
  2. Abdo, K., Riahi-Nezhad, C. K. and Imran, J. (2019). "Steady supercritical flow in a straight-wall open-channel contraction." Journal of Hydraulic Research, IAHR, Vol. 57, No. 5, pp. 647-661, https://doi.org/10.1080/00221686.2018.1504126. 
  3. Batten, P., Lambert, C. and Causon, D. M. (1996). "Positively conservative high-resolution convection schemes for unstructured elements." International Journal for Numerical Methods in Engineering, John Wiley & Sons, Vol. 39, No. 11, pp. 1821-1838, https://doi.org/10.1002/(SICI)1097-0207(19960615)39:11<1821::AID-NME929>3.0.CO;2-E. 
  4. Belanger, J.-B.-C.-J. (1849). Notes sur le cours d'hydraulique : Session 1849-1850, Ecole nationale des ponts et chaussees, unpublished (in French). 
  5. Bidone, G. (1820). "Experiences sur le remou et sur la propagation des ondes." Memorie Della Reale Accademia Delle Scienze Di Torino, Torino Dalla Stamperia Reale, Vol. 25, pp. 21-112 (in French). 
  6. Bristeau, M.-O., Mangeney, A., Sainte-Marie, J. and Seguin, N. (2015). "An energy-consistent depth-averaged Euler system: Derivation and properties." Discrete and Continuous Dynamical Systems - Series B, AIMS, Vol. 20, No. 4, pp. 961-988, https://doi.org/10.3934/dcdsb.2015.20.961. 
  7. Chanson, H. (2009). "Development of the Belanger equation and backwater equation by Jean-Baptiste Belanger (1828)." Journal of Hydraulic Engineering, ASCE, Vol. 135, No. 3, pp. 159-163, https://doi.org/10.1061/(ASCE)0733-9429(2009)135:3(159). 
  8. Cheng, C.-K., Tai, Y.-C. and Jin, Y.-C. (2017). "Particle image velocity measurement and mesh-free method modeling study of forced hydraulic jumps." Journal of Hydraulic Engineering, ASCE, Vol. 143, No. 9, 04017028, https://doi.org/10.1061/(ASCE)HY.1943-7900.0001325. 
  9. Chippada, S., Ramaswamy, B. and Wheeler, M. F. (1994). "Numerical simulation of hydraulic jump." International Journal for Numerical Methods in Engineering, John Wiley & Sons, Vol. 37, No. 8, pp. 1381-1397, https://doi.org/10.1002/nme.1620370807. 
  10. Chow, V. T. (1959). Open-channel hydraulics, McGraw-Hill Book Company. 
  11. Courant, R. and Friedrichs, K. (1948). Supersonic flow and shock waves, Interscience Publishers Inc. 
  12. Cueto-Felgueroso, L., Santillan, D., Garcia-Palacios, J. H. and Garrote, L. (2019). "Comparison between 2D shallow-water simulations and energy-momentum computations for transcritical flow past channel contractions." Water, MDPI, Vol. 11, No. 7, 1476, https://doi.org/10.3390/w11071476. 
  13. Echeverribar, I., Brufau, P. and Garcia-Navarro, P. (2023). "Extension of a Roe-type Riemann solver scheme to model non-hydrostatic pressure shallow flows." Applied Mathematics and Computation, Elsevier, Vol. 440, 127642, https://doi.org/10.1016/j.amc.2022.127642. 
  14. Escalante, C., Dumbser, M. and Castro, M. J. (2019). "An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes." Journal of Computational Physics, Elsevier, Vol. 394, pp. 385-416, https://doi.org/10.1016/j.jcp.2019.05.035. 
  15. Fennema, R. J. and Chaudhry, M. H. (1986). "Explicit numerical schemes for unsteady free-surface flows with shocks." Water Resources Research, AGU, Vol. 22, No. 13, pp. 1923-1930, https://doi.org/10.1029/WR022i013p01923. 
  16. Forster, J. W. and Skrinde, R. A. (1950). "Control of the hydraulic jump by sills." Transactions of the American Society of Civil Engineers, ASCE, Vol. 115, No. 1, pp. 973-1022, https://doi.org/10.1061/TACEAT.0006341. 
  17. Gharangik, A. M. and Chaudhry, M. H. (1991). "Numerical simulation of hydraulic jump." Journal of Hydraulic Engineering, ASCE, Vol. 117, No. 9, pp. 1195-1211, https://doi.org/10.1061/(ASCE)0733-9429(1991)117:9(1195). 
  18. Gottlieb, D. and Turkei, E. (1976). "Dissipative two-four methods for time-dependent problems." Mathematics of Computation, AMS, Vol. 30, No. 136, pp. 703-723, https://doi.org/10.2307/2005392. 
  19. Green, A. E. and Naghdi, P. M. (1976). "A derivation of equations for wave propagation in water of variable depth." Journal of Fluid Mechanics, Cambridge, Vol. 78, No. 2, pp. 237-246, https://doi.org/10.1017/S0022112076002425. 
  20. Hager, W. H. (1992). Energy dissipators and hydraulic jump, Springer, Netherlands, Dordrecht. 
  21. Henderson, F. (1966). Open channel flow, Macmillan Publishing Co., Inc., New York. 
  22. Hirt, C. W. and Nichols, B. D. (1981). "Volume of fluid (VOF) method for the dynamics of free boundaries." Journal of Computational Physics, Elsevier, Vol. 39, No. 1, pp. 201-225, https://doi.org/10.1016/0021-9991(81)90145-5. 
  23. Hwang, S.-Y. (2015). "A novel scheme to depth-averaged model for analyzing Shallow-water flows over discontinuous topography." KSCE Journal of Civil and Environmental Engineering Research, KSCE, Vol. 35, No. 6, pp. 1237-1246, https://doi.org/10.12652/Ksce.2015.35.6.1237 (in Korean). 
  24. Hwang, S.-Y. (2022). "Numerical analysis of shallow-water flow over the square-edged broad-crested weir." Journal of Korea Water Resources Association, KWRA, Vol. 55, No. 10, pp. 811-821, https://doi.org/10.3741/JKWRA.2022.55.10.811 (in Korean). 
  25. Hwang, S.-Y. (2023). "Numerical simulation of shallow-water flow over perpendicular broad-crested weir." Proceedings of 2023 Conference of the Korea Water Resources Association, KWRA, pp. 503 (in Korean). 
  26. Hwang, S.-Y. and Lee, S. H. (2012). "An application of the HLLL approximate Riemann solver to the shallow water equations." KSCE Journal of Civil and Environmental Engineering Research, KSCE, Vol. 32, No. 1B, pp. 21-27, https://doi.org/10.12652/Ksce.2012.32.1B.021 (in Korean). 
  27. Idelchik, I. E. (2008). Handbook of hydraulic resistance, 4th ed., Begell House Inc., New York. 
  28. Jimenez, O. F. and Chaudhry, M. H. (1988). "Computation of supercritical free-surface flows." Journal of Hydraulic Engineering, ASCE, Vol. 114, No. 4, pp. 377-395, https://doi.org/10.1061/(ASCE)0733-9429(1988)114:4(377). 
  29. Katopodes, N. D. (1984). "A dissipative Galerkin scheme for open-channel flow." Journal of Hydraulic Engineering, ASCE, Vol. 110, No. 4, pp. 450-466, https://doi.org/10.1061/(ASCE)0733-9429(1984)110:4(450). 
  30. Ketcheson, D. I. and de Luna, M. Q. (2022). "Numerical simulation and entropy dissipative cure of the carbuncle instability for the shallow water circular hydraulic jump." International Journal for Numerical Methods in Fluids, Vol. 94, No. 6, pp. 655-677, https://doi.org/10.1002/fld.5070. 
  31. Khan, A. A. and Steffler, P. M. (1996). "Physically based hydraulic jump model for depth-averaged computations." Journal of Hydraulic Engineering, ASCE, Vol. 122, No. 10, pp. 540-548, https://doi.org/10.1061/(ASCE)0733-9429(1996)122:10(540). 
  32. Kupka, F. and Muthsam, H. J. (2017). "Modelling of stellar convection." Living Reviews in Computational Astrophysics, Springer, Vol. 3, No. 1, https://doi.org/10.1007/s41115-017-0001-9. 
  33. Lee, K. S. and Lee, S.-T. (1998). "Two-dimensional finite-volume unsteady-flow model for shocks." Journal of Korea Water Resources Association, KWRA, Vol. 31, No. 3, pp. 279-290 (in Korean). 
  34. LeVeque, R. (2002). Finite volume methods for hyperbolic problems, Cambridge University Press. 
  35. Linde, T. (2002). "A practical, general-purpose, two-state HLL Riemann solver for hyperbolic conservation laws." International Journal for Numerical Methods in Fluids, Jhon Wiley & Sons, Vol. 40, Nos. 3-4, pp. 391-402, https://doi.org/10.1002/fld.312. 
  36. Liu, Q. and Drewes, U. (1994). "Turbulence characteristics in free and forced hydraulic jumps." Journal of Hydraulic Research, IAHR, Vol. 32, No. 6, pp. 877-898, https://doi.org/10.1080/00221689409498696. 
  37. Long, D., Steffler, P. M. and Rajaratnam, N. (1991). "A numerical study of submerged hydraulic jumps." Journal of Hydraulic Research, IAHR, Vol. 29, No. 3, pp. 293-308, https://doi.org/10.1080/00221689109498435. 
  38. Madsen, P. A. and Svendsen, I. A. (1983). "Turbulent bores and hydraulic jumps." Journal of Fluid Mechanics, Cambridge University Press, Vol. 129, pp. 1-25, https://doi.org/10.1017/S0022112083000622. 
  39. McCorquodale, J. A. and Khalifa, A. (1983). "Internal flow in hydraulic jumps." Journal of Hydraulic Engineering, ASCE, Vol. 109, No. 5, pp. 684-701, https://doi.org/10.1061/(ASCE)0733-9429(1983)109:5(684). 
  40. Molls, T. and Chaudhry, M. H. (1995). "Depth-averaged open-channel flow model." Journal of Hydraulic Engineering, ASCE, Vol. 121, No. 6, pp. 453-465, https://doi.org/10.1061/(ASCE)0733-9429(1995)121:6(453). 
  41. Mossa, M. and Petrillo, A. (2003). "A brief history of the jump of Bidone." Proceedings of 30th International Association for Hydraulic Engineering and Research World Congress, IAHR, Thessaloniki, Greece, pp. 57-64. 
  42. Navas-Montilla, A. and Murillo, J. (2019). "Improved Riemann solvers for an accurate resolution of 1D and 2D shock profiles with application to hydraulic jumps." Journal of Computational Physics, Elsevier, Vol. 378, pp. 445-476, https://doi.org/10.1016/j.jcp.2018.11.023. 
  43. Oh, S.-T., Hwang, S.-Y. and Lee, K. S. (1998). "Numerical analysis of hydraulic jump by the flux splitting method." KSCE Journal of Civil and Environmental Engineering Research, KSCE, Vol. 18, No. II-3, pp. 215-221 (in Korean). 
  44. Pandolfi, M. (1973). "Numerical computation of one-dimensional unsteady flow in channels." Meccanica, Vol. 8, No. 4, pp. 236-242, https://doi.org/10.1007/BF02342409. 
  45. Pandolfi, M. (1975). "Numerical experiments on free surface water motion with bores." Proceedings of the 4th International Conference on Numerical Methods in Fluid Dynamics, pp. 304-312, https://doi.org/10.1007/BFb0019766. 
  46. Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T. (1992). Numerical recipes in C - the art of scientific computing, 2nd ed., Cambridge University Press, New York. 
  47. Rahman, M. and Chaudry, M. H. (1995). "Simulation of hydraulic jump with grid adaptation." Journal of Hydraulic Research, IAHR, Vol. 33, No. 4, pp. 555-569, https://doi.org/10.1080/00221689509498660. 
  48. Stoker, J. J. (1948). "The formation of breakers and bores the theory of nonlinear wave propagation in shallow water and open channels." Communications on Pure and Applied Mathematics, Wiley, Vol. 1, No. 1, pp. 1-87, https://doi.org/10.1002/cpa.3160010101. 
  49. Strutt, J. W. (1914). "On the theory of long waves and bores." Proceedings of the Royal Society of London - Series A, Royal Society, Vol. 90, No. 619, pp. 324-328, https://doi.org/10.1098/rspa.1914.0055. 
  50. Suzuki, Y. and van Leer, B. (2005). "Application of the 10-moment model to mems flows." Proceedings of 43rd American Institute of Aeronautics and Astronautics Aerospace Sciences Meeting and Exhibit, AIAA, pp. 1-13, https://doi.org/10.2514/ 6.2005-1398. 
  51. Ting, W. K., Puay, H. T. and Zakaria, N. A. (2022). "Numerical simulation of hydraulic jump with the inclusion of Boussinesq term." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, Vol. 93, No. 1, pp. 186-199, https://doi.org/10.37934/arfmts.93.1.186199. 
  52. Toro, E. F. (2001). Shock-capturing methods for free-surface shallow flows, Wiley, Chichester, England. 
  53. van Leer, B. (1976). "MUSCL, a new approach to numerical gas dynamics." Proceedings of 2nd European Conference on Computational Physics, EPS, Garching, Germany, pp. 1-4. 
  54. van Leer, B. (2003). "Upwind and high-resolution methods for compressible flow - from donor cell to residual-distribution schemes." Proceedings of 16th American Institute of Aeronautics and Astronautics Computational Fluid Dynamics Conference, AIAA, Orlando, USA, pp. 1-8, https://doi.org/10.2514/6.2003-3559. 
  55. Viti, N., Valero, D. and Gualtieri, C. (2019). "Numerical simulation of hydraulic jumps. part 2: Recent results and future outlook." Water, MDPI, Vol. 11, No. 1, https://doi.org/10.3390/w11010028. 
  56. Weiyan, T. (1992). Shallow water hydrodynamics, 1st edition, Elsevier Science Publishers, Amsterdam, The Netherland. 
  57. White, F. M. (2011). Fluid mechanics, 7th edition, McGraw-Hill. 
  58. Wilsey, E. F. (1923). The history and mathematics of the hydraulic jump, Master's thesis, State University of Iowa, Iowa, USA, https://doi.org/10.17077/etd.005638. 
  59. Yen, B. C. (2002). "Open channel flow resistance." Journal of Hydraulic Engineering, ASCE, Vol. 128, No. 1, pp. 20-39, https://doi.org/10.1061/(ASCE)0733-9429(2002)128:1(20). 
  60. Younus, M. and Chaudhry, M. H. (1994). "A depth-averaged turbulence model for the computation of free-surface flow." Journal of Hydraulic Research, IAHR, Vol. 32, No. 3, pp. 415-444, https://doi.org/10.1080/00221689409498744. 
  61. 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." Journal of Computational Physics, Elsevier, Vol. 168, No. 1, pp. 1-25, https://doi.org/10.1006/jcph.2000.6670