Singular Value Decomposition–Based Collocation Spectral Method for Quasi-Two-Dimensional Laminar Water Hammer Problems
Publication: Journal of Hydraulic Engineering
Volume 143, Issue 7
Abstract
The radial distributions of velocity components need to be resolved in quasi-two-dimensional laminar water hammer problems. In a collocation spectra method, the radial distributions are approximated with Chebyshev expansions and the equations are assumed valid at the collocation points. The traditional collocation method requires an equal number of equations and unknown expansion coefficients, which is sometimes difficult to implement. The proposed model adopts extra collocation points to provide extra equations for expansion coefficients to construct an overdetermined system. Singular value decomposition is used to solve the overdetermined system. In the new method, the boundary conditions can be naturally incorporated into the system. However, the accuracy of the boundary condition equation is not acceptable because of least-squares approximation. Large multipliers are introduced to enhance the accuracy of the boundary condition equations. Spatial variation in the axial direction and time advancement are treated using the method of characteristics.
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Acknowledgments
This work is supported by the National Natural Science Foundation of China under Project No. 11472074.
References
Anderson, E., et al. (1999). LAPACK users’ guide, 3rd Ed., SIAM, Philadelphia.
Canuto, G., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. (1988). Spectral methods in fluid dynamics, Springer, Berlin.
Cheney, W., and Kincaid, D. (1994). Numerial mathematics and computing, 3rd Ed., Brooks/Cole, Pacific Grove, CA.
Holmboe, E. L., and Rouleau, W. T. (1967). “The effect of viscous shear on transients in liquid lines.” J. Basic Eng., 89(1), 174–180.
Horn, R. A., and Johnson, C. R. (2012). Matrix analysis, Cambridge University Press, Cambridge, U.K.
Korbar, R., Virag, Z., and Savar, M. (2014). “Truncated method of characteristics for quasi-two-dimensional water hammer model.” J. Hydraul. Eng., .
Lawson, C. L., and Hanson, R. J. (1974). Solving least squares problems, Prentice-Hall, Englewood Cliffs, NJ.
Lopez, J. M., Marques, F., and Shen, J. (2002). “An efficient spectral-projection method for the Navier-Stokes equations in cylindrical geometries. II: Three dimensional cases.” J. Comput. Phys., 176(2), 384–401.
Moin, P., and Kim, J. (1980). “On the numerical solution of time-dependent viscous incompressible fluid flows involving solid boundaries.” J. Comput. Phys., 35(3), 381–392.
Pezzinga, G. (1999). “Quasi-2D Model for unsteady flow in pipe networks.” J. Hydraul. Eng., 676–685.
Prado, R. A., and Larreteguy, A. E. (2002). “A transient shear stress model for the analysis of laminar water-hammer problems.” J. Hydraul. Res., 40(1), 45–53.
Riasi, A., Nourbakhsh, A., and Raisee, M. (2013). “Energy dissipation in unsteady turbulent pipe flows caused by water hammer.” Comput. Fluids, 73, 124–133.
Silva-Araya, W. F., and Chaudhry, M. H. (1997). “Computation of energy dissipation in transient flow.” J. Hydraul. Eng., 108–115.
Vardy, A. E., and Hwang, K. L. (1991). “A characteristic model of transient friction in pipes.” J.Hydraul. Res., 29(5), 669–684.
Wahba, E. M. (2006). “Runge-Kutta time stepping schemes with TVD central differencing for the water hammer equations.” Int. J. Numer. Methods Fluids, 52(5), 571–590.
Wahba, E. M. (2008). “Modelling the attenuation of laminar fluid transients in piping systems.” Appl. Math. Modell., 32(12), 2863–2871.
Wahba, E. M. (2009). “Turbulence modeling for two-dimensional water hammer simulations in the low Reynolds number range.” Comput. Fluid, 38(9), 1763–1770.
Wahba, E. M. (2011). “A computational study of viscous dissipation and entropy generation in unsteady pipe flow.” Acta Mech., 216(1–4), 75–86.
Zhang, X., Liu, X., Song, K., and Lu, M. (2001). “Least-squares collocation meshless method.” Int. J. Numer. Method Eng., 51(9), 1089–1100.
Zhao, M. (2016). “Numerical solutions of quasi-two-dimensional models for laminar water hammer problems.” J. Hydraul. Res., 54(3), 360–368.
Zhao, M., and Ghidaoui, M. S. (2003). “An efficient quasi-two-dimensional model for water hammer problems.” J. Hydraul. Eng., 1007–1013.
Zhao, M., Ghidaoui, M. S., and Kolyshkin, A. A. (2004). “Investigation of the mechanisms responsible for the breakdown of axis-symmetry in pipe transient.” J. Hydraul. Res., 42(6), 645–656.
Zhu, T., and Atluri, S. N. (1998). “A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method.” Comput. Mech., 21(3), 211–222.
Zielke, W. (1968). “Frequency-dependent friction in transient pipe flow.” J. Basic Eng., 90(1), 109–115.
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Published In
Journal of Hydraulic Engineering
Volume 143 • Issue 7 • July 2017
Copyright
©2017 American Society of Civil Engineers.
History
Received: Jul 27, 2016
Accepted: Nov 3, 2016
Published ahead of print: Feb 23, 2017
Published online: Feb 24, 2017
Published in print: Jul 1, 2017
Discussion open until: Jul 24, 2017
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