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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

A numerical study of effects of the axial stress on unsteady liquid pipeline flows  
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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 one-dimensional 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 Darcy-Weisbach 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. 777-788, 2004
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