Impact of Interpolation on Numerical Properties of the Method of Characteristics Used for Solution of the Transient Pipe Flow Equations
Publication: Journal of Hydraulic Engineering
Volume 150, Issue 5
Abstract
This paper presents a comparison of numerical properties of the fixed-grid method of characteristics resulting from space and time linear interpolation used for the solution of transient flow in pipe equations. Modified equation analysis method is applied, in which the modified equations derived for the simplified linear version of the transient pipe flow equations provide explicitly the coefficients of numerical diffusion and dispersion generated by the method of characteristics. Through this approach the value of the numerical diffusion coefficient generated in the solution can be precisely estimated. Consequently, the effects resulting from the interpolation in time and space can be compared. The conclusions presented were confirmed by the results of numerical calculations.
Practical Applications
If a valve is suddenly closed at the end of a pipeline in which water flows, a pressure wave will appear in it. This phenomenon, known as the water hammer, commonly occurs in water supply networks and in various industrial installations. The wave propagation process is described by a system of differential equations, which are solved using numerical methods. Numerical methods are approximate; they allow to solve the mathematical problems in which both the data and the results of calculations are given in the form of numbers. Numerical methods enable the calculation of approximate pressure and velocity values at selected cross-sections of the pipeline and at selected moments of time. The calculated values contain an error that depends on the solution method. This error determines the quality of the water hammer simulation results. There are a number of methods for numerical solution of water hammer equations. In this paper, the method of characteristics is used, which requires interpolating pressure and velocity values between selected points. The article shows that the generated numerical error depends on the applied technique of interpolation. Two techniques, interpolation in space and interpolation in time, are considered.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request. The available data are experimental results for pressure at the end of a 170-m-long steel pipeline measured with a water hammer.
References
Abbott, M. B., and D. R. Basco. 1989. Computational fluid dynamics. New York: Longman Scientific and Technical.
Almeida, A. B., and E. Koelle. 1992. Fluid transients in pipe networks. London: Elsevier.
Fletcher, C. A. J. 1991. Computational techniques for fluid dynamics. Berlin: Springer.
Ghidaoui, M. S., and B. W. Karney. 1994. “Equivalent differential equations in fixed grid characteristics method.” J. Hydraul. Eng. 120 (Apr): 1159–1175. https://doi.org/10.1061/(ASCE)0733-9429(1994)120:10(1159).
Ghidaoui, M. S., B. W. Karney, and D. A. McInnis. 1998. “Energy estimates for discretization errors in water hammer problems.” J. Hydraul. Eng. 124 (Jun): 384–393. https://doi.org/10.1061/(ASCE)0733-9429(1998)124:4(384).
Goldberg, D. E., and E. B. Wylie. 1983. “Characteristics method using time-line interpolations.” J Hydraul. Eng. 109 (5): 670–683. https://doi.org/10.1061/(ASCE)0733-9429(1983)109:5(670).
Karney, B. W., and M. S. Ghidaoui. 1997. “Flexible discretization algorithm for fixed-grid MOC in pipelines.” J. Hydraul. Eng. 123 (Jun): 1004–1011. https://doi.org/10.1061/(ASCE)0733-9429(1997)123:11(1004).
Leendertse, J. J. 1967. Aspect of a computational model for long-period water wave propagation. Santa Monica, CA: RAND Corporation.
Streeter, V. L., and C. Lai. 1962. “Water hammer analysis including fluid friction.” J. Hydraul. Div. 88 (3): 79–112. https://doi.org/10.1061/JYCEAJ.0000736.
Streeter, V. L., and E. B. Wylie. 1967. Hydraulic transients. New York: McGraw-Hill.
Szymkiewicz, R. 2010. Numerical modeling in open channel hydraulics. New York: Springer.
Warming, R. F., and B. J. Hyett. 1974. “The modified equation approach to the stability and accuracy analysis of finite difference methods.” J. Comput. Phys. 14 (2): 159–179. https://doi.org/10.1016/0021-9991(74)90011-4.
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., and V. L. Streeter. 1993. Fluid transients in systems. New York: Prentice-Hall.
Information & Authors
Information
Published In
Journal of Hydraulic Engineering
Volume 150 • Issue 5 • September 2024
Copyright
© 2024 American Society of Civil Engineers.
History
Received: Jul 5, 2023
Accepted: Mar 7, 2024
Published online: May 23, 2024
Published in print: Sep 1, 2024
Discussion open until: Oct 23, 2024
ASCE Technical Topics:
- Comparative studies
- Diffusion
- Engineering fundamentals
- Engineering mechanics
- Flow (fluid dynamics)
- Fluid dynamics
- Fluid mechanics
- Hydrologic engineering
- Infrastructure
- Linear analysis
- Methodology (by type)
- Numerical methods
- Pipe flow
- Pipeline systems
- Pipes
- Research methods (by type)
- Structural analysis
- Structural engineering
- Thermodynamics
- Transient flow
- Transport phenomena
- Water and water resources
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.