Document Type: Research Paper


1 Civil Engineering Department, Shahrood University of Technology, Shahrood, 009823, IRAN

2 Civil Engineering Department, Jundi-Shapur University of Technology, Dezful, 009861, IRAN


Four explicit finite difference schemes, including Lax-Friedrichs, Nessyahu-Tadmor, Lax-Wendroff and Lax-Wendroff with a nonlinear filter are applied to solve water hammer equations. The schemes solve the equations in a reservoir-pipe-valve with an instantaneous and gradual closure of the valve boundary. The computational results are compared with those of the method of characteristics (MOC), and with the results of Godunov''s scheme to verify the proposed numerical solution. The computations reveal that the proposed Lax-Friedrichs and Nessyahu-Tadmor schemes can predict the discontinuities in fluid pressure with an acceptable order of accuracy in cases of instantaneous and gradual closure. However, Lax-Wendroff and Lax-Wendroff with nonlinear filter schemes fail to predict the pressure discontinuities in instantaneous closure. The independency of time and space steps in these schemes are allowed to set different spatial grid size with a unique time step, thus increasing the efficiency with respect to the conventional MOC. In these schemes, no Riemann problems are solved; hence field-by-field decompositions are avoided. As provided in the results, this leads to reduced run times compared to the Godunov scheme.

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Main Subjects

[1]     A. S. Tijsseling, “Fluid-structure interaction in liquid-filled pipe systems: a review”, Journal of Fluids and Structures, Vol. 10, No. 2, pp. 109-146, (1996).

[2]     H. J. Kwon and J. J Lee, “Computer and experimental models of transient flow in pipe involving backflow preventers”, Journal of Hydraulic Engineering, Vol. 134, No. 4, pp. 426-434, (2008).

[3]     M. H. Afshar and M. Rohani, “Exploring the Versatility of the implicit method of characteristic (MOC) for Transient simulation of pipeline systems”, Twelfth International Water Technology Conference, Alexandria, Egypt, (2008).

[4]     S. R. Sabbagh-yazdi, A. Abbasi and N. Mastorakis, “Water hammer modeling using 2nd order Godunov finite volume method”. Proceeding of European Computing Conference. Vol. 2, pp. 215-223, (2009).

[5]     M. Zhao, M. S. Ghidaoui, “Godunov-Type Solutions for Water Hammer Flows”, Journal ofHydraulic Engineering, Vol. 130, No. 4, pp. 341-348, (2004).

[6]     M. H. Chaudhry and M. Y. Hussaini, “Second-order accurate explicit finite-difference schemes for water hammer analysis”, Journal of Fluids Engineering, Vol. 107, No. 4, pp. 523-529, (1985).

[7]     A. S. Tijsseling and A. Bergant, “Meshless computation of water hammer”, 2nd IAHR International Meeting of the Work groupon Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Timisoara, Vol. 52, No. 66, pp. 65-76 (2007).

[8]     H. Hou, A. C. H. Kruisbrink, A. S. Tijsseling and A. Keramat,  “Simulating water hammer with corrective smoothed particle method”, Eindhoven University of Technology, Eindhoven, (2012).

[9]     H. Chaudhry, Applied hydraulic transients, Van Nostrand Reinhold Company, New York, (1979).

[10]   E. Wylie and V. Streeter, Applied hydraulic transients, Fluid Transient in Systems. Prentice-Hall, New York, (1993).

[11]    A. Bergant, A. S. TIJSSELING and J. P. VÍTKOVSKÝ, “Parameters affecting water-hammer wave attenuation, shape and timing-part1: Mathematical tools”, Journal of Hydraulic Research, Vol. 46, No. 3, pp. 373-381, (2003).

[12]    K. A. Hoffmann and S. T. Chiang, “Computational fluid dynamics for engineers”. Engineering Education Systems, Austin, Texas, (1993).

[13]   A. V. Chikitkin, B. V. Rogov and S. V. Utyuzhnikov, “High-order accurate monotone compact running scheme for multidimensional hyperbolic equations”. Applied Numerical Mathematics, Vol. 93, No. 3, pp. 150-163, (2015).

[14]   L. F. Shampine, “Two-step Lax-Friedrichs method”, Applied Mathematics Letters, Vol. 18, No. 10, pp. 1134-1136, (2004).

[15]   L. F. Shampine, “Solving hyperbolic PDEs in MATLAB”, Applied  Numerical




Analysis & Computational Mathematics, Vol. 2, No. 3, pp. 346-358, (2005).

[16]   A. S. Tijsseling and C. S. W. Lavooij, “Water hammer with fluid-structure interaction”, Applied Scientific Research, Vol. 47, No. 3, pp. 273-285, (1990).

[17]    A. S. Tijsseling and A. Bergant, “Meshless computation of water hammer”, 2nd IAHR International Meeting of the Work groupon Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Timisoara, Vol. 52, No. 6, pp. 65-76, (2007).