Multiple scales analysis of water hammer attenuation
Authors:
S. Y. Han, D. Hansen and G. Kember
Journal:
Quart. Appl. Math. 69 (2011), 677-690
MSC (2000):
Primary 76M99; Secondary 76M45
DOI:
https://doi.org/10.1090/S0033-569X-2011-01258-9
Published electronically:
July 7, 2011
MathSciNet review:
2893995
Full-text PDF Free Access
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Additional Information
Abstract: A multiple scales analysis is used to construct a uniformly accurate approximation to water hammer pressure wave attenuation that is initiated by a sudden valve opening. The method of analysis is well suited to the study of a water hammer that possesses several time scales and is applied to a mild generalization of the classical equations. It should prove useful for finding attenuation curves when effects such as unsteady friction and fluid-structure interaction are added. The analytical results are numerically verified using the method of characteristics.
References
- Fathi M. Allan and Kamel Al-Khaled, An approximation of the analytic solution of the shock wave equation, J. Comput. Appl. Math. 192 (2006), no. 2, 301–309. MR 2228815, DOI https://doi.org/10.1016/j.cam.2005.05.009
- A. S. Tijsseling and A. E. Vardy, Time scales and fsi in unsteady liquid-filled pipe flow, BHR Group, Proc. of the 9th Int. Conf. on Pressure Surges (Editor S. J. Murray), 2004, pp. 135–150.
- ---, Time scales and fsi in oscillatory liquid-filled pipe flow, BHR Group, Proc. of the 10th Int. Conf. on Pressure Surges (Editor S. Hunt), 2008, pp. 553–568.
- A. Bergant, A. S. Tijsseling, J. P. Vitkovsky, D. I. C. Covas, A. R. Simpson, and M. F. Lambert, Parameters affecting water-hammer wave attenuation, shape, and timing part i: Mathematical tools, IAHR J. of Hydraulic Research 46(3) (2008), 373–381.
- ---, Parameters affecting water-hammer wave attenuation, shape, and timing part ii: Case studies, IAHR J. of Hydraulic Research 46(3) (2008), 382–391.
- G. C. F. M. R. de Prony Baron, Recherches physico-mathematiques sur la theorie du mouvement des eaux courantes (1804).
- M. S. Ghidaoui and A. A. Kolyshkin, Stability analysis of velocity profiles in water hammer flows, ASCE J. of Hydraulic Engineering 127(6) (2001), 499–511.
- G. O. Brown, The history of the Darcy-Weisbach equation for pipe flow resistance, Proceedings of the 150th Anniversary Conference of ASCE (Editors J. Rogers and A. Fredrich), 2002, pp. 34–43.
- D. Hansen, V. K. Garga, and D. R. Townsend, Selection and application of a 1-d non-Darcy flow equation for 2-d flow through rockfill embankments, Can. Geotechnical J. 32(2) (1995), 223–232.
- D. J. Leslie and A. S. Tijsseling, Travelling discontinuities in waterhammer theory: attenuation due to friction, BHR Group, Proc. of the 8th Int. Conf. on Pressure Surges (Editor A. Anderson), 2000, pp. 323–335.
- A. S. Leon, M. S. Ghidaoui, A. R. Schmidt, and M. H. Garcia, Efficient second-order-accurate shock-capturing scheme for modelling one and two-phase water hammer flows, ASCE J. of Hydraulic Engineering 134(7) (2008), 970–983.
- L. F. Moody, Friction factors for pipe flow, Transactions of American Society of Mechanical Engineers 66(8) (1944), 671–678.
- A. M. A. Sattar, A. R. Dickerson, and M. H. Chaudhry, Wavelet-Galerkin solution to the water hammer equations, ASCE J. of Hydraulic Engineering 134(4) (2009), 283–295.
- A. S. Tijsseling, Fluid-structure interaction in liquid-filled pipe systems, J. of Fluids and Structures 10 (1996), 109–146.
- W. Driedger, Controlling centrifugal pumps, Hydrocarbon Processing 74(7) (1995), 43–49.
- X. Q. Wang, J. G. Sun, and W. T. Sha, Transient flows and pressure waves in pipes, Journal of Hydrodynamics 7(2) (1995), 51–59.
- D. H. Wilkinson and E. M. Curtis, Water hammer in a thin-walled pipe, Proc. $3$rd Int. Conf. on Pressure Surges, 1980, pp. 221–240.
- E. B. Wylie and V. L. Streeter, Fluid transients in systems, Prentice-Hall, Englewood Cliffs, NJ, 1993.
References
- M. F. Allan and K. Al-Khaled, An approximation of the analytic solution of the shock wave equation, Journal of Computational and Applied Mathematics 192(2) (2006), 301–309. MR 2228815
- A. S. Tijsseling and A. E. Vardy, Time scales and fsi in unsteady liquid-filled pipe flow, BHR Group, Proc. of the 9th Int. Conf. on Pressure Surges (Editor S. J. Murray), 2004, pp. 135–150.
- ---, Time scales and fsi in oscillatory liquid-filled pipe flow, BHR Group, Proc. of the 10th Int. Conf. on Pressure Surges (Editor S. Hunt), 2008, pp. 553–568.
- A. Bergant, A. S. Tijsseling, J. P. Vitkovsky, D. I. C. Covas, A. R. Simpson, and M. F. Lambert, Parameters affecting water-hammer wave attenuation, shape, and timing part i: Mathematical tools, IAHR J. of Hydraulic Research 46(3) (2008), 373–381.
- ---, Parameters affecting water-hammer wave attenuation, shape, and timing part ii: Case studies, IAHR J. of Hydraulic Research 46(3) (2008), 382–391.
- G. C. F. M. R. de Prony Baron, Recherches physico-mathematiques sur la theorie du mouvement des eaux courantes (1804).
- M. S. Ghidaoui and A. A. Kolyshkin, Stability analysis of velocity profiles in water hammer flows, ASCE J. of Hydraulic Engineering 127(6) (2001), 499–511.
- G. O. Brown, The history of the Darcy-Weisbach equation for pipe flow resistance, Proceedings of the 150th Anniversary Conference of ASCE (Editors J. Rogers and A. Fredrich), 2002, pp. 34–43.
- D. Hansen, V. K. Garga, and D. R. Townsend, Selection and application of a 1-d non-Darcy flow equation for 2-d flow through rockfill embankments, Can. Geotechnical J. 32(2) (1995), 223–232.
- D. J. Leslie and A. S. Tijsseling, Travelling discontinuities in waterhammer theory: attenuation due to friction, BHR Group, Proc. of the 8th Int. Conf. on Pressure Surges (Editor A. Anderson), 2000, pp. 323–335.
- A. S. Leon, M. S. Ghidaoui, A. R. Schmidt, and M. H. Garcia, Efficient second-order-accurate shock-capturing scheme for modelling one and two-phase water hammer flows, ASCE J. of Hydraulic Engineering 134(7) (2008), 970–983.
- L. F. Moody, Friction factors for pipe flow, Transactions of American Society of Mechanical Engineers 66(8) (1944), 671–678.
- A. M. A. Sattar, A. R. Dickerson, and M. H. Chaudhry, Wavelet-Galerkin solution to the water hammer equations, ASCE J. of Hydraulic Engineering 134(4) (2009), 283–295.
- A. S. Tijsseling, Fluid-structure interaction in liquid-filled pipe systems, J. of Fluids and Structures 10 (1996), 109–146.
- W. Driedger, Controlling centrifugal pumps, Hydrocarbon Processing 74(7) (1995), 43–49.
- X. Q. Wang, J. G. Sun, and W. T. Sha, Transient flows and pressure waves in pipes, Journal of Hydrodynamics 7(2) (1995), 51–59.
- D. H. Wilkinson and E. M. Curtis, Water hammer in a thin-walled pipe, Proc. $3$rd Int. Conf. on Pressure Surges, 1980, pp. 221–240.
- E. B. Wylie and V. L. Streeter, Fluid transients in systems, Prentice-Hall, Englewood Cliffs, NJ, 1993.
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Additional Information
S. Y. Han
Affiliation:
Worley-Parsons Canada, 400-10201 Southport Rd. SW, Calgary, Alberta, Canada, T2W 4X9
Email:
Sang-Yoon.Han@WorleyParsons.com
D. Hansen
Affiliation:
Department of Civil and Resource Engineering, Dalhousie University, 1360 Barrington St., Halifax, NS, Canada, B3J 1Z1
Email:
David.Hansen@dal.ca
G. Kember
Affiliation:
Department of Engineering Mathematics, Dalhousie University, 1340 Barrington St., Halifax, NS, Canada, B3J 1Y9
Email:
Guy.Kember@dal.ca
Received by editor(s):
March 2, 2010
Published electronically:
July 7, 2011
Article copyright:
© Copyright 2011
Brown University
The copyright for this article reverts to public domain 28 years after publication.
