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Keywords
(8)
Continuity Equation
High Pressure
Laminar Flow
Numerical Model
Temporal Change
Unsteady Flow
Water Hammer
Shear Stress
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A numerical study of effects of the axial stress on unsteady liquid pipeline flows
A numerical study of effects of the axial stress on unsteady liquid pipeline flows,10.1155/S0161171204211024,International Journal of Mathematics and
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A numerical study of effects of the axial stress on unsteady liquid pipeline flows
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Masaji Watanabe
,
Yukio Kono
,
Hiroshi Suito
We study the effects of the axial component of the
shear stress
on unsteady pipeline flows. We show that the axial component of the
shear stress
should be introduced in the modeling of unsteady flows, and as a numerical model, we propose a onedimensional momentum equation in which a term containing the second derivative of the velocity with respect to space is introduced. The momentum equation and the
continuity equation
are converted into a system suitable for the application of upstream difference approximations. Numerical results are presented, and their correspondence with experimental results is examined to see how our model captures phenomena observed experimentally. 2000 Mathematics Subject Classification: 65M06, 76F99. 1. Introduction. A small disturbance in a liquid pipeline flow may result in an un expectedly rapid increase of the pressure. Such a burst of the pressure is called water hammer, and it is one of phenomena peculiar to unsteady pipeline flows. On the other hand, the generation of the
high pressure
due to waterhammer becomes a pressure wave, and it propagates with the sound speed. The rapid transmission of a pressure wave is another phenomenon characteristic of unsteady pipeline flows. In this paper, we show how these phenomena can be treated mathematically. In particular, we intro duce a new model to capture phenomena observed experimentally and show how it can be analyzed numerically. In Section 2, we introduce some experimental results to illustrate unsteady pipeline flows. A pipeline was connected to a water tank at one end and a valve was set at the other end. The valve was set open initially to allow a uniform flow to form. Then the valve was closed suddenly, and waterhammer was generated. We present the
temporal change
of the pressure which was measured in the transition. In Section 3, we propose a new model to analyze unsteady liquid pipeline flows numerically. We show that an expression derived from the DarcyWeisbach equation corresponds to the radial com ponent of the
shear stress
in a steady radially symmetric laminar flow, and that the axial component should be retained in the modeling of unsteady flows. In Section 4 ,w e turn to a numerical aspect of the problem. We show how upstream difference approxima tions can be applied to the momentum equation and the continuity equation. We also present some numerical results and examine their correspondence with experimental results introduced in Section 2.
Journal:
International Journal of Mathematics and Mathematical Sciences
, vol. 2004, no. 15, pp. 777788, 2004
DOI:
10.1155/S0161171204211024
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References
(2)
Transient Cavitating Pipe Flow
(
Citations: 6
)
Victor L. Streeter
Journal:
Journal of Hydraulic Engineeringasce  J HYDRAUL ENGASCE
, vol. 109, no. 11, 1983
Physical Fluid Dynamics
(
Citations: 560
)
D. J. Tritton
Journal:
American Journal of Physics  AMER J PHYS
, vol. 46, no. 4, 1978