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TECHNICAL PAPERS

Two-Dimensional Pipe Model for Laminar Flow

[+] Author and Article Information
M. C. P. Brunelli

 Siemens VDO Automotive AG, Siemensstrasse 12, D-93055 Regensburg, Germanymarco.brunelli@siemens.com

J. Fluids Eng 127(3), 431-437 (Mar 16, 2005) (7 pages) doi:10.1115/1.1905645 History: Received November 19, 2004; Revised March 16, 2005

The one-dimensional Zielke model of the energy loss in laminar pipe flow is exact but gives no information about the velocity profile. Here a two-dimensional pipe model is presented which gives the two-dimensional velocity profile in the time domain for an unstationary pipe flow of a compressible fluid that follows an equation of state. The continuity and the motion equations are projected over two sets of functions accounting for the radial and the axial dependence. A set of ordinary differential equations for the time-dependent coefficients is obtained, which is numerically integrated according to the boundary conditions at the pipe ends needed in practical applications. The model reproduces the experimental results of a water hammer and the analytical transfer functions over a wide range of frequencies.

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Copyright © 2005 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Setup of the water hammer experiment as in the article of Zielke (2)

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Figure 2

Pressure at the pipe downstream end. The Chebyshev and Jacobi polynomials have been expanded, respectively, up to the order m=0,1,…,15 and k=0,1,…,20.

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Figure 3

Velocity profile over the pipe length and pipe cross section at t=14c∕L. The units in the figure are: 1∕L for the pipe length, 1∕R for the pipe section, and 1∕V0 for the velocity.

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Figure 4

Transfer function Pa for a pipe with velocity–velocity boundary condition

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Figure 5

Transfer function Pb for a pipe with velocity–velocity boundary condition

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Figure 6

Transfer function Pa for a pipe with pressure–velocity boundary condition

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Figure 7

Transfer function Vb for a pipe with pressure–velocity boundary condition

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Figure 8

Transfer function Va for a pipe with pressure–pressure boundary condition

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Figure 9

Transfer function Vb for a pipe with pressure–pressure boundary condition

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Figure 10

Transfer function Pa for a pipe with pressure–velocity boundary condition for three different expansion orders of the Jacobi polynomials

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