Abstract
The fix-grid method of characteristic (MOC) has been the main numerical scheme for modeling the transient pipe flows with an entrapped air pocket, where the Courant number usually equals one (i.e., ) to ensure its accuracy and stability. However, cannot always be guaranteed in each pipe of real pipe systems; thus, the MOC needs to be approximated by interpolation or wavespeed adjustment. This could lead to large accumulated numerical errors and serious shape distortion of simulated pressure curves. To address this problem, an alternative coupled scheme, which combines the second-order Godunov-type scheme (GTS) and the MOC, is developed. Specifically, the conservation equations with unsteady friction of the water column are numerically solved by the GTS, and the moving air-water interface is modeled and captured by the coupled GTS-MOC scheme. The simulated pressure curves by the GTS-MOC scheme are compared with both MOC results and laboratory experiments. The proposed scheme with unsteady friction can better reproduce the experimental pressure oscillations, and is more robust and efficient than the MOC. The MOC scheme with and coarse grids causes more obvious numerical dissipation during an intensive transient induced by relatively high inlet pressure, in which more high-frequency waves occur.
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Data Availability Statement
The data during the study are available from the corresponding author by request.
Acknowledgments
The writers gratefully acknowledge the financial support for this research from the National Natural Science Foundation of China (Grant Nos. 51839008, 51679066, and 52209084), Fok Ying Tong Education Foundation (Grant No. 161068).
References
Alexander, J., Z. Li, P. J. Lee, M. Davidson, and H. F. Duan. 2020. “Experimental investigation of the effects of air pocket configuration on fluid transients in a pipeline.” J. Hydraul. Eng. 146 (12): 04020081. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001823.
Bergant, A., A. Ross Simpson, and J. Vìtkovsk. 2001. “Developments in unsteady pipe flow friction modelling.” J. Hydraul. Res. 39 (3): 249–257. https://doi.org/10.1080/00221680109499828.
Brunone, B., U. M. Golia, and M. Greco. 1995. “The effects of two dimensionality on pipe transients modeling.” J. Hydraul. Eng. 121 (12): 906–912. https://doi.org/10.1061/(ASCE)0733-9429(1995)121:12(906).
Che, T. C., H. F. Duan, and P. J. Lee. 2021. “Transient wave-based methods for anomaly detection in fluid pipes: A review.” Mech. Syst. Signal Process. 160 (Nov): 107874. https://doi.org/10.1016/j.ymssp.2021.107874.
Coronado-Hernández, O. E., V. S. Fuertes-Miquel, P. L. Iglesias Rey, and F. J. Martínez-Solano. 2018. “Rigid water column model for simulating the emptying process in a pipeline using pressurized air.” J. Hydraul. Eng. 144 (4): 06018004. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001446.
Duan, H. F., P. J. Lee, M. S. Ghidaoui, and Y. K. Tung. 2011. “Leak detection in complex series pipelines by using the system frequency response method.” J. Hydraul. Res. 49 (2): 213–221. https://doi.org/10.1080/00221686.2011.553486.
Guinot, V. 2000. “Riemann solvers for water hammer simulations by Godunov method.” Int. J. Numer. Methods Eng. 49 (7): 851–870. https://doi.org/10.1002/1097-0207(20001110)49:7%3C851::AID-NME978%3E3.0.CO;2-.
Hou, Q., S. Li, A. S. Tijsseling, and J. Laanearu. 2020. “Discussion of ‘Rigid water column model for simulating the emptying process in a pipeline using pressurized air’ by Oscar E. Coronado-Hernandez, Vicente S.Fuertes-Miquel, Pedro L. Iglesias-Rey, and Francisco J. Martinez-Solano.” J. Hydraul. Eng. 146 (3): 07020001. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001682.
Lee, N., and C. Martin. 1999. “Experimental and analytical investigation of entrapped air in a horizontal pipe.” In Proc., 3rd ASME/JSME Joint Fluids Engineering Conf., 1–8. New York: ASME.
Lee, P. J., M. F. Lambert, A. R. Simpson, J. P. Vítkovský, and J. Liggett. 2006. “Experimental verification of the frequency response method for pipeline leak detection.” J. Hydraul. Res. 44 (5): 693–707. https://doi.org/10.1080/00221686.2006.9521718.
Liu, D., L. Zhou, B. Karney, Q. Zhang, and C. Ou. 2011. “Rigid-plug elastic-water model for transient pipe flow with entrapped air pocket.” J. Hydraul. Res. 49 (6): 799–803. https://doi.org/10.1080/00221686.2011.621740.
Martin, C. S. 1976. “Entrapped air in pipelines.” In Proc., 2nd Int. Conf. on Pressure Surges, 15–27. Bedford, UK: British Hydromechanics Research Association.
Martins, N. M. C., J. N. Delgado, and H. M. Ramos. 2017. “Maximum transient pressures in a rapidly filling pipeline with entrapped air using a CFD model.” J. Hydraul. Res. 55 (4): 506–519. https://doi.org/10.1080/00221686.2016.1275046.
Pan, T., L. Zhou, C. Ou, P. Wang, and D. Liu. 2022. “Smoothed particle hydrodynamics with unsteady friction model for water hammer pipe flow.” J. Hydraul. Eng. 148 (2): 04021057. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001966.
Pozos-Estrada, O., O. A. Fuentes-Mariles, and A. Pozos-Estrada. 2012. “Gas pockets in a wastewater rising main: A case study.” Water Sci. Technol. 66 (10): 2265–2274. https://doi.org/10.2166/wst.2012.462.
Ramezani, L., B. Karney, and A. Malekpour. 2016. “Encouraging effective air management in water pipelines: A critical review.” J. Water Resour. Plann. Manage. 142 (12). https://doi.org/10.1061/(ASCE)WR.1943-5452.0000695.
Tijsseling, A. S., Q. Hou, and Z. Bozkus. 2019. “Rapid liquid filling of a pipe with venting entrapped gas: Analytical and numerical solutions.” J. Pressure Vessel Technol. 141 (4): 041301. https://doi.org/10.1115/1.4043321.
Tijsseling, A. S., Q. Hou, Z. Bozkus, and J. Laanearu. 2016. “Improved one-dimensional models for rapid emptying and filling of pipelines.” J. Pressure Vessel Technol. 138 (3): 031301. https://doi.org/10.1115/1.4031508.
Toro, E. F. 1997. Riemann solvers and numerical methods for fluid dynamics: A practical introduction. New York: Springer.
Toro, E. F. 2009. Riemann solvers and numerical methods for fluid dynamics: A practical introduction. 3rd ed. New York: Springer.
Vardy, A. E., and J. M. B. Brown. 1995. “Transient, turbulent, smooth pipe friction.” J. Hydraul. Res. 33 (4): 435–456. https://doi.org/10.1080/00221689509498654.
Vítkovský, J. P., A. Bergant, A. R. Simpson, and M. F. Lambert. 2006. “Systematic evaluation of one-dimensional unsteady friction models in simple pipelines.” J. Hydraul. Eng. 132 (7): 696–708. https://doi.org/10.1061/(ASCE)0733-9429(2006)132:7(696).
Wiggert, D. C., and M. J. Sundquist. 1977. “Fixed-grid characteristics for pipeline transients.” J. Hydraul. Div. 103 (12): 1403–1416. https://doi.org/10.1061/JYCEAJ.0004888.
Wylie, E. B., V. L. Streeter, and L. Suo. 1993. Fluid transient in systems. Hoboken, NY: Prentice Hall.
Xue, Z., L. Zhou, B. Karney, D. Liu, and P. Wang. 2020. “Primitive form Godunov-type scheme for two-phase homogeneous water hammer flows.” J. Hydraul. Eng. 146 (4): 04020018. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001725.
Zhao, M., and M. S. Ghidaoui. 2004. “Godunov-type solutions for water hammer flows.” J. Hydraul. Eng. 130 (4): 341–348. https://doi.org/10.1061/(ASCE)0733-9429(2004)130:4(341).
Zhou, F., F. Hicks, and P. Steffler. 2002. “Observations of air–water interaction in a rapidly filling horizontal pipe.” J. Hydraul. Eng. 128 (6): 635–639. https://doi.org/10.1061/(ASCE)0733-9429(2002)128:6(635).
Zhou, F., F. Hicks, and P. Steffler. 2004. “Analysis of effects of air pocket on hydraulic failure of urban drainage infrastructure.” Can. J. Civ. Eng. 31 (1): 86–94. https://doi.org/10.1139/l03-077.
Zhou, L., Y. Cao, B. Karney, A. Bergant, A. S. Tijsseling, D. Liu, and P. Wang. 2020. “Expulsion of entrapped air in a rapidly filling horizontal pipe.” J. Hydraul. Eng. 146 (7): 04020047. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001773.
Zhou, L., Y. Li, B. Karney, Y. Cheng, and D. Liu. 2021a. “Godunov-type solutions for transient pipe flow implicitly incorporating Brunone unsteady friction.” J. Hydraul. Eng. 147 (7): 04021021. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001895.
Zhou, L., Y. Li, Y. Zhao, C. Ou, and Y. Zhao. 2021b. “An accurate and efficient scheme involving unsteady friction for transient pipe flow.” J. Hydroinf. 23 (4): 879–896. https://doi.org/10.2166/hydro.2021.160.
Zhou, L., D. Liu, B. Karney, and Q. Zhang. 2011. “Influence of entrapped air pockets on hydraulic transients in water pipelines.” J. Hydraul. Eng. 137 (12): 1686–1692. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000460.
Zhou, L., H. Wang, A. Bergant, A. S. Tijsseling, D. Liu, and S. Guo. 2018a. “Godunov-type solutions with discrete gas cavity model for transient cavitating pipe flow.” J. Hydraul. Eng. 144 (5): 04018017. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001463.
Zhou, L., H. Wang, B. Karney, D. Liu, P. Wang, and S. Guo. 2018b. “Dynamic behavior of entrapped air pocket in a water filling pipeline.” J. Hydraul. Eng. 144 (8): 04018045. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001491.
Zhou, L., H. Wang, D. Liu, J. Ma, P. Wang, and L. Xia. 2017. “A second-order finite volume method for pipe flow with water column separation.” J. Hyro-Environ. Res. 17 (Dec): 47–55. https://doi.org/10.1016/j.jher.2016.11.004.
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© 2023 American Society of Civil Engineers.
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Received: Aug 21, 2022
Accepted: Apr 19, 2023
Published online: Jun 21, 2023
Published in print: Sep 1, 2023
Discussion open until: Nov 21, 2023
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