Abstract
Transient simulation is very important to protect the water supply pipeline system from extreme pressures. In order to numerically simulate the transient response in a variable–property series pipe system, the water hammer model and matched boundary conditions are developed by introducing the MacCormack time marching scheme. Based on the proposed method, the transient pressure and flow velocity are numerically predicted for a variable–property series pipe, and then the results are compared to the classical method of characteristics (MOC). The improved method can yield a reasonable numerical solution for a closed pipe, and the solution agrees well with the MOC and existing experimental results. In the proposed model, the time step is no longer subjected to the length of the space step; consequently, it is more convenient in meshing and modifying the \(x-t\) grid. Especially, it is very advantageous in establishing the simultaneous calculation in water hammer simulation for a variable–property series pipes system or complicated distribution networks.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- A :
-
Area of section \((\hbox {m}^{2})\)
- a :
-
Wave speed of water hammer (m/s)
- C :
-
Constant
- \(Z_{\mathrm{u}} \) :
-
Upstream constant water level (m)
- \(C^{{*}} \) :
-
Vardy’s shear decay coefficient
- \(C_{\mathrm{d}} \) :
-
Discharge coefficient of valve
- \(C_{\mathrm{T}} \) :
-
Dimensionless time consumed
- \(C_{\mathrm{R}} \) :
-
Refining coefficient of time interval
- \(C_{\mathrm{TMO}} \) :
-
Dimensionless time consumed in MOC
- \(C_{\mathrm{TMT}} \) :
-
Dimensionless time consumed in MTMS
- \(C_{\mathrm{TR}} \) :
-
The ratio of the computing times
- \(C_{{x}} \) :
-
Dimensionless space step
- D :
-
Main pipe diameter (m)
- e :
-
Internal energy (J)
- f :
-
Darcy–Weisbach friction factor
- \(f_{\mathrm{q}} \) :
-
Quasi-steady friction factor
- g :
-
Acceleration of gravity \((\hbox {m/s}^{2})\)
- h :
-
Pressure head (m)
- \(h_{ns} \) :
-
The head of the valve (m)
- \(h_w \) :
-
The head loss between section (m)
- i :
-
Serial number of nodes (s)
- j :
-
Serial number of nodes (s)
- k :
-
Brunone friction coefficient
- n :
-
The number of elements in a single pipe
- \(p_{\mathrm{a}} \) :
-
Atmospheric pressure (Pa)
- p :
-
Pressure (Pa)
- Q :
-
Instantaneous discharge at section (m\(^{3}\)/s)
- s :
-
Section area at nodes (m\(^{2})\)
- S :
-
Square integral domain
- \(\mathbf{S} \) :
-
Vector surface area
- T :
-
Time (s)
- t :
-
Time, as subscript to denote time (s)
- \(T_{\mathrm{MO}} \) :
-
Total time consumed in MOC
- \(T_{\mathrm{MT}} \) :
-
Total time consumed in MTMS
- v :
-
Flow velocity (m/s)
- \(\mathbf{v} \) :
-
Velocity vector
- x :
-
Distance along pipe from the inlet (m)
- \(\theta \) :
-
Pipe slope
- \(\rho \) :
-
Fluid density \((\hbox {kg/m}^{3})\)
- \(\Delta x \) :
-
Length of element, space interval step (m)
- \(C_{\mathrm{xMO}} \) :
-
Dimensionless space step in MOC
- \(C_{\mathrm{xMT}} \) :
-
Dimensionless space step in MTMS
- \(\Delta t \) :
-
Time interval step (s)
- \(\Omega \) :
-
Volume integral domain
- \(\sim \) :
-
Superscript denotes estimated values
- \(- \) :
-
Superscript denotes average values
- MOC:
-
Method of characteristics
- MTMS:
-
MacCormack time marching scheme
- FW:
-
Forward wave
- RW:
-
Reverse wave
References
Wylie, E.B., Streeter, V.L., Suo, L.: Fluid Transients in Systems. Prentice Hall, Englewood Cliffs (1993)
Triki, A.: Water-hammer control in pressurized-pipe flow using an in-line polymeric short-section. Acta Mech. 227(3), 777–793 (2016). https://doi.org/10.1007/s00707-015-1493-1
Triki, A.: Dual-technique-based inline design strategy for water-hammer control in pressurized pipe flow. Acta Mech. (2017). https://doi.org/10.1007/s00707-017-2085-z
Rezghi, A., Riasi, A.: Sensitivity analysis of transient flow of two parallel pump-turbines operating at runaway. Renew. Energy 86, 611–622 (2016). https://doi.org/10.1016/j.renene.2015.08.059
Li, X., Zhu, M., Xie, J.: Numerical simulation of transient pressure control in a pumped water supply system using an improved bypass pipe. Strojniski Vestn. J. Mech. Eng. 62(10), 614–622 (2016). https://doi.org/10.5545/sv-jme.2016.3535
Rohani, M., Afshar, M.H.: Simulation of transient flow caused by pump failure: point-implicit method of characteristics. Ann. Nucl. Energy 37(12), 1742–1750 (2010). https://doi.org/10.1016/j.anucene.2010.07.004
Yu, X.D., Zhang, J., Miao, D.: Innovative closure law for pump-turbines and field test verification. J. Hydraul. Eng. (2015). https://doi.org/10.1061/(asce)hy.1943-7900.0000976
Riasi, A., Tazraei, P.: Numerical analysis of the hydraulic transient response in the presence of surge tanks and relief valves. Renew. Energy 107, 138–146 (2017). https://doi.org/10.1016/j.renene.2017.01.046
Ghidaoui, M.S., Zhao, M., McInnis, D.A., Axworthy, D.H.: A review of water hammer theory and practice. Appl. Mech. Rev. 58(1), 49–76 (2005)
Ghidaoui, M.S., Karney, B.W.: Equivalent differential-equations in fixed-grid characteristics method. J. Hydraul. Eng. ASCE 120(10), 1159–1175 (1994). https://doi.org/10.1061/(asce)0733-9429(1994)120:10(1159)
Afshar, M.H., Rohani, M.: Water hammer simulation by implicit method of characteristic. Int. J. Press. Vessels Pip. 85(12), 851–859 (2008). https://doi.org/10.1016/j.ijpvp.2008.08.006
Chaudhry, M., Hussaini, M.: Second-order accurate explicit finite-difference schemes for waterhammer analysis. J. Fluids Eng. 107(4), 523–529 (1985)
Guinot, V.: Riemann solvers for water hammer simulations by Godunov method. Int. J. Numer. Methods Eng. 49(7), 851–870 (2000). https://doi.org/10.1002/1097-0207(20001110)49:7%3c851::aid-nme978%3e3.0.co;2-%23
Wood, D.J.: Waterhammer analysis—essential and easy (and efficient). J. Environ. Eng. ASCE 131(8), 1123–1131 (2005). https://doi.org/10.1061/(asce)0733-9372(2005)131:8(1123)
Kim, S.H.: Impulse response method for pipeline systems equipped with water hammer protection devices. J. Hydraul. Eng. ASCE 134(7), 961–969 (2008). https://doi.org/10.1061/(asce)0733-9429(2008)134:7(961)
Kim, S.H.: Dynamic memory computation of impedance matrix method. J. Hydraul. Eng. ASCE 137(1), 122–128 (2011). https://doi.org/10.1061/(asce)hy.1943-7900.0000278
Niroomandi, A., Borghei, S.M., Bohluly, A.: Implementation of time splitting projection method in water hammer modeling in deformable pipes. Int. J. Press. Vessels Pip. 98, 30–42 (2012). https://doi.org/10.1016/j.ijpvp.2012.07.002
Alamian, R., Behbahani-Nejad, M., Ghanbarzadeh, A.: A state space model for transient flow simulation in natural gas pipelines. J. Nat. Gas Sci. Eng. 9, 51–59 (2012). https://doi.org/10.1016/j.jngse.2012.05.013
Hwang, Y.H.: Development of a characteristic particle method for water hammer simulation. J. Hydraul. Eng. 139(11), 1175–1192 (2013). https://doi.org/10.1061/(asce)hy.1943-7900.0000771
Bazargan-Lari, M.R., Kerachian, R., Afshar, H., Bashi-Azghadi, S.N.: Developing an optimal valve closing rule curve for real-time pressure control in pipes. J. Mech. Sci. Technol. 27(1), 215–225 (2013). https://doi.org/10.1007/s12206-012-1208-7
Karadzic, U., Bulatovic, V., Bergant, A.: Valve-induced water hammer and column separation in a pipeline apparatus. Strojniski Vestn. J. Mech. Eng. 60(11), 742–754 (2014). https://doi.org/10.5545/sv-jme.2014.1882
Holler, S., Jaberg, H.: A contribution to water hammer analysis in pumped-storage power plants. Wasserwirtschaft 103(1–2), 78–84 (2013)
Wan, W., Li, F.: Sensitivity analysis of operational time differences for a pump-valve system on a water hammer response. J. Press. Vessel Technol. 138(1), 011303 (2016). https://doi.org/10.1115/1.4031202
Vasconcelos, J.G., Klaver, P.R., Lautenbach, D.J.: Flow regime transition simulation incorporating entrapped air pocket effects. Urban Water J. 12(6), 488–501 (2015). https://doi.org/10.1080/1573062x.2014.881892
Travas, V., Basara, S.: A mixed MOC/FDM numerical formulation for hydraulic transients. Tech. Gaz. 22(5), 1141–1147 (2015)
MacCormack, R.: The effect of viscosity in hypervelocity impact cratering. J. Spacecr. Rockets 40(5), 757–763 (2003)
Gottlieb, D., Turkel, E.: Dissipative two–four methods for time-dependent problems. Math. Comput. 30(136), 703–723 (1976). https://doi.org/10.2307/2005392
Anderson, J.D.: Computational Fluid Dynamics: The Basics with Applications. McGrawhill Inc, New York (1995)
Triki, A.: Resonance of free-surface waves provoked by floodgate maneuvers. J. Hydrol. Eng. 19(6), 1124–1130 (2014). https://doi.org/10.1061/(asce)he.1943-5584.0000895
Triki, A.: Further investigation on the resonance of free-surface waves provoked by floodgate maneuvers: negative surge waves. Ocean Eng. 133, 133–141 (2017). https://doi.org/10.1016/j.oceaneng.2017.02.003
Amara, L., Berreksi, A., Achour, B.: Adapted MacCormack finite-differences scheme for water hammer simulation. J. Civ. Eng. Sci. 2(4), 226–233 (2013)
Bergant, A., Simpson, A.R.: Estimating unsteady friction in transient cavitating pipe flow. In: 2nd International Conference on Water Pipeline Systems, Edinburgh, Scotland (1994)
Bergant, A., Vitkovsky, J., Simpson, A.R., Lambert, M.: Valve induced transients influenced by unsteady pipe flow friction. In: 10th International Meeting of the Work Group on the Behaviour of Hydraulic Machinery under Steady Oscillatory Conditions, pp. 12–23 (2001)
Bergant, A., Simpson, A.R., Vitkovsky, J.: Developments in unsteady pipe flow friction modelling. J. Hydraul. Res. 39(3), 249–257 (2001). https://doi.org/10.1080/00221680109499828
Zielke, W.: Frequency-dependent friction in transient pipe flow. J. Basic Eng. 90(1), 109–115 (1968)
Hino, M., Sawamoto, M., Takasu, S.: Study on the transition to turbulence and frictional coefficient in an oscillatory pipe flow. Trans. JSCE 9, 282–284 (1977)
Brunone, B., Golia, U.M., Greco, M.: Some remarks on the momentum equation for fast transients. In: Proceedings of International Conference on Hydraulic Transients With Water Column Separation, pp. 201–209 (1991)
Pezzinga, G.: Discussion of ’Developments in unsteady pipe flow friction modelling’. J. Hydraul. Res. 40(5), 650–653 (2002)
Pezzinga, G.: Evaluation of unsteady flow resistances by quasi-2D or 1D models. J. Hydraul. Eng. ASCE 126(10), 778–785 (2000). https://doi.org/10.1061/(asce)0733-9429(2000)126:10(778)
Vardy, A.E., Brown, J.M.B.: Transient turbulent friction in smooth pipe flows. J. Sound Vib. 259(5), 1011–1036 (2003). https://doi.org/10.1006/jsvi.2002.5160
Brunone, B., Ferrante, M., Calabresi, F.: Discussion of “Evaluation of unsteady flow Resistances by Quasi-2D or 1D models” by Giuseppe Pezzinga. J. Hydraul. Eng. ASCE 128(6), 646–647 (2002). https://doi.org/10.1061/(asce)0733-9429(2002)128:6(646)
Urbanowicz, K., Zarzycki, Z.: Improved lumping friction model for liquid pipe flow. J. Theor. Appl. Mech. 53(2), 295–305 (2015)
Ioriatti, M., Dumbser, M., Iben, U.: A comparison of explicit and semi-implicit finite volume schemes for viscous compressible flows in elastic pipes in fast transient regime. ZAMM-Zeitschrift Für Angewandte Mathematik und Mechanik 97(11), 1358–1380 (2017). https://doi.org/10.1002/zamm.201700018
Ioriatti, M., Dumbser, M.: Semi-implicit staggered discontinuous Galerkin schemes for axially symmetric viscous compressible flows in elastic tubes. Comput. Fluids 167, 166–179 (2018). https://doi.org/10.1016/j.compfluid.2018.02.019
Vardy, A.E., Brown, J.M.B.: On turbulent, unsteady, smooth-pipe flow. In: International Conference on Pressure Surges and Fluid Transients, Harrogate, England, pp. 289–311. BHR Group (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The project was supported by National Natural Science Foundation of China (Grant Nos. 51779216, 51279175) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LZ16E090001).
Rights and permissions
About this article
Cite this article
Wan, W., Huang, W. Water hammer simulation of a series pipe system using the MacCormack time marching scheme. Acta Mech 229, 3143–3160 (2018). https://doi.org/10.1007/s00707-018-2179-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-018-2179-2