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TECHNICAL PAPERS

# Mass Flow Rate Controlled Fully Developed Laminar Pulsating Pipe Flows

[+] Author and Article Information
S. Ray1

Institute of Fluid Mechanics (LSTM), Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 4, D-91058 Erlangen, Germany

B. Ünsal

Institute of Fluid Mechanics (LSTM), Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 4, D-91058 Erlangen, Germanybuensal@lstm.uni-erlangen.de

F. Durst, Ö. Ertunc, O. A. Bayoumi

Institute of Fluid Mechanics (LSTM), Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 4, D-91058 Erlangen, Germany

According to the present definition, when, after the sudden increase (or decrease) in the axial pressure gradient, the resultant value reaches the mean of the overall waveform, $Δθ$ is equal to $π∕2$.

It may be recognized that the pressure gradient in Eq. 3 may also be expressed as

$−1ρ(dPdx)=P̂0[1+∑n=1∞{(P̂cn*)2+(P̂sn*)2}1∕2sin{2πnFτ+tan−1(P̂cn*∕P̂sn*)}]$
which is clearly a sine series, with a modified magnitude (the absolute value of the complex variable, $P̂en*$) and a corresponding phase. Thus, the foregoing analysis may also be applied for the general situation as well with minor modification in the expression for pressure gradient.

In Eq. 13, $ṁn*$ is the prescribed dimensionless amplitude of the $nth$ frequency of the mass flow rate oscillation. It will be further clarified when the analysis for triangularly pulsating mass flow rate controlled flow will be described.

It is important to note that Eq. 14 is the solution of the pressure gradient driven flows. In this equation, $ṁn*$ and $Δθm,n$ are obtained using Eqs. 15 and Eq. 16.

For example, consider power wave or sawtooth wave type pulsations, where a Fourier series cannot express the pulsation with reasonable accuracy without application of Lanczos’ Sigma factor.

If the pressure gradient is expressed in the form of a Fourier series, which is also not essential for the numerical solution. Also note that this definition is different (by a factor of 8) from the definition given in Eq. 17.

1

On leave from the Department of Mechanical Engineering, Jadavpur University Kolkata 700 032, India.

J. Fluids Eng 127(3), 405-418 (Mar 06, 2005) (14 pages) doi:10.1115/1.1906265 History: Received February 18, 2004; Revised March 06, 2005

## Abstract

Pressure gradient driven, laminar, fully developed pulsating pipe flows have been extensively studied by various researchers and the data for the resultant flow field are available in a number of publications. The present paper, however, concentrates on related flows that are mass flow driven, i.e., the flows where the mass flow rate is prescribed as $ṁ=ṁM+ṁAfm(t)$ and $fm(t)$ is periodically varying in time. Sinusoidal and triangular mass flow rate pulsations in time are analytically considered in detail. Results of experimental investigations are presented and are complemented by data deduced from corresponding analytical and numerical studies. Overall, the results provide a clear insight into mass flow rate driven, laminar, fully developed pulsating pipe flow. To the best of the authors’ knowledge, flows of this kind have not been studied before experimentally, analytically and numerically.

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## Figures

Figure 1

Schematic diagram of the experimental setup

Figure 2

Applied sinusoidal mass flow rate and measured pressure gradients for mA*=0.9473, for different pulsation frequencies

Figure 3

Variation of Δθ and P̂A* with f and ṁA* for sinusoidal pulsations

Figure 4

Applied triangular mass flow pulsations and measured pressure drop wave forms for different pulsation frequencies

Figure 5

Variation of Δθ and P̂A* with f for triangular pulsations

Figure 6

Comparison of experimental and analytical pressure gradients for sinusoidal mass flow rate pulsations for F=0.01, 0.1, 1 and 10

Figure 7

Variation of Δθ and P̂A* with F and ṁA* for sinusoidal pulsations

Figure 8

Applied triangular mass flow pulsations and comparison of experimental and analytical pressure gradient wave forms for F=0.01, 0.1, 1 and 10

Figure 9

Variation of Δθ and P̂A* with F for triangular pulsations

Figure 10

Power spectrum of pressure wave forms for F=0.01, 0.1, 1 and 10 triangular mass flow rate pulsations

Figure 11

Effect of number of terms used in Fourier series in analytic solution of triangular mass flow rate pulsations for F−10 (left one with 40 terms corresponding to 125Hz, right one with ten terms corresponding to 33Hz)

Figure 12

Results for flows with power pulsation for F=0.1 and 10 (top: the complete variation, bottom: zoomed view near the discontinuity)

Figure 13

Comparison of analytical solution with applied sigma factor and numerical solution for flows with power pulsation for F=0.1 and 10 (top: the complete variation, bottom: zoomed view near the discontinuity)

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