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Research Papers: Fundamental Issues and Canonical Flows

Alternative Convolution Approach to Friction in Unsteady Pipe Flow

[+] Author and Article Information
Romuald Szymkiewicz

Faculty of Civil and Environmental Engineering,
Gdansk University of Technology,
ul. Narutowicza 11/12,
80-233 Gdansk, Poland
e-mail: rszym@pg.gda.pl

Marek Mitosek

Faculty of Environmental Engineering,
Warsaw University of Technology,
ul. Nowowiejska 20,
00-653 Warsaw, Poland
e-mail: marek.mitosek@is.pw.edu.pl

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received February 18, 2013; final manuscript received August 28, 2013; published online October 15, 2013. Assoc. Editor: D. Keith Walters.

J. Fluids Eng 136(1), 011202 (Oct 15, 2013) (9 pages) Paper No: FE-13-1096; doi: 10.1115/1.4025509 History: Received February 18, 2013; Revised August 28, 2013

In this paper some aspects of the unsteady friction in pipe flow expressed by the convolution are analyzed. This additional term introduced into the motion equation involves the accelerations of fluid occurring in the past and a weighting function. The essence of such approach is to assume the appropriate form of weighting function. However, until now, no fully reliable formula for this function has been found. To avoid some inconveniences typical for the commonly used weighting functions, an alternative form of the convolution is proposed. Instead of a weighting function an impulse response function in a general form is introduced. This function, defined in the real time domain, having clear physical interpretation and some useful properties is not related to the usually assumed viscosity distribution over the pipe's cross section. The proposed approach involves two parameters. The convergence of the impulse response function, characterized by the flow memory, is determined by a parameter which can be related to the pressure wave frequency. The second parameter determines the magnitude of the unsteady friction force. The proposed alternative convolution approach was tested basing on the laboratory measurements for a water hammer event initiated by turbulent flow in pipes made of steel. Although the alternative convolution approach causes a very good damping of the pressure wave amplitude, it appears to be unable to ensure appropriate smoothing of the pressure heads. This is because it acts in the dynamic equation as a source/sink term. To ensure the required smoothing of the pressure wave the diffusive term was included into the dynamic equation.

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References

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Figures

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Fig. 1

The weighting function W(τ) assumed by Zielke [1]

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Fig. 2

The weighting function W(τ) proposed by Vardy and Brown [7]

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Fig. 3

Observed and calculated head oscillations at the end of steel pipe of length L = 177.40 m using the MOC with unsteady friction proposed by Vardy and Brown [7]

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Fig. 4

Observed and calculated head oscillations at the end of steel pipe of length L = 72 m using the MOC with unsteady friction proposed by Vardy and Brown [7]

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Fig. 5

Flow velocity V(t) (a) and its time derivative ∂V/∂t (b) calculated with the steady friction only for the steel pipeline of length L = 177.40 m in the cross section next to the reservoir

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Fig. 6

Illustration of the flow memory

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Fig. 7

The impulse response function (20) obtained for various values of K

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Fig. 8

Observed and calculated head oscillations at the end of steel pipeline of L = 72.0 m length using the MOC and the unsteady friction according to the proposed approach

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Fig. 9

Steady and unsteady friction force according to the proposed approach computed next to reservoir in steel pipe of L = 72 m length

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Fig. 10

Observed and calculated pressure heads at the end of steel pipe of length L = 72 m with the steady and unsteady friction as well as with the diffusive term

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Fig. 11

Observed and calculated pressure heads at the end of steel pipe of length L = 177.40 m with the steady and unsteady friction as well as with the diffusive term

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