A comment on unsteady–periodic flow friction factor: An analysis on experimental data gathered in pulsatile pipe flows
1Department of Mechanical Engineering, Gaziantep University, Gaziantep, Turkey
J Ther Eng 2020; 1(6): 16-27 DOI: 10.18186/thermal.670986
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Abstract

In 1940’s, Schultz- Grunow proposed that time-average value of friction factor, λ u,ta was similar to its corresponding steady state value, λ for the presence of gradual and slow oscillations in pulsatile flows. A recent approach was available for low frequency pulsatile flows through narrow channels in transitional and turbulent regimes by Zhuang et al, in 2016 and 2017. In this analysis; extensive experimental data of 𝜆𝜆𝑢𝑢,𝑡𝑡𝑡𝑡 in fully laminar and turbulent sinusoidal flow are processed in the measured time-average Reynolds number range of 1390 ≤ 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡≤60000 disregarding the transitional regime. The ranges of dimensionless frequency-Womersley number, √𝜔𝜔′ and oscillation amplitude, 𝐴𝐴1 are 2.72≤√𝜔𝜔′≤28 and 0.05≤ 𝐴𝐴1≤0.96 respectively. A multiplication element is defined as 𝑀𝑀𝑀 𝑀𝑀= 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡× √𝜔𝜔′ . A modified friction multiplier, 𝜆𝜆𝑀𝑀𝑀𝑀𝑀𝑀 which is similar to the conceptual parameter of Zhuang et al’s friction factor ratio C ( 𝜆𝜆𝑀𝑀𝑀𝑀𝑀𝑀= 𝜆𝜆𝑢𝑢,𝑡𝑡𝑡𝑡𝜆𝜆 ) is also referred. The correlation of 𝜆𝜆𝑀𝑀𝑀𝑀𝑀𝑀= 𝜆𝜆𝑀𝑀𝑀𝑀𝑀𝑀(𝑀𝑀𝑀 𝑀𝑀) is dependent on flow regime and the magnitude of 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡 for the range of √𝜔𝜔′ >1.32. The proposal of Schultz-Grunow is verified irrespective of the oscillations in turbulent regime since the magnitude of 𝜆𝜆𝑀𝑀𝑀𝑀𝑀𝑀=1 is observed for turbulent flow cases with 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡 ≥ 35000. In laminar regime the magnitude of 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡 is governing the fact. The magnitude of 𝜆𝜆𝑀𝑀𝑀𝑀𝑀𝑀 varies in 0.589≤ 𝜆𝜆𝑀𝑀𝑀𝑀𝑀𝑀≤28.125 for 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡 ≤ 5000 while 𝜆𝜆𝑀𝑀𝑀𝑀𝑀𝑀=1 is obtained for 𝑅𝑅𝑅𝑅𝑡𝑡𝑡𝑡 >5000. The graphical representation of 𝜆𝜆𝑀𝑀𝑀𝑀𝑀𝑀= 𝜆𝜆𝑀𝑀𝑀𝑀𝑀𝑀(𝑀𝑀𝑀 𝑀𝑀) can be considered as a counterpart of Moody Diagram in pulsatile fields for a significant practice.