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Analytical solution for laminar entrance flow in circular pipes

Published online by Cambridge University Press:  25 January 2024

Taig Young Kim Open the ORCID record for Taig Young Kim [Opens in a new window]
Taig Young Kim*
Affiliation:
Department of Mechanical Eng., Tech University of Korea, 237 Sangideahak-ro, Siheung-si, Gyeonggi-do 15073, South Korea
*
†Email address for correspondence: tagikim@tukorea.ac.kr
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Abstract

This study introduces an analytical solution for the laminar entrance flow in circular pipes, aiming to confirm the occurrence of velocity overshoot. Velocity overshoot is characterised by the maximum axial velocity appearing near the pipe wall instead of the central axis. Similar to the previous studies, the analytical solution is derived from the parabolised Navier–Stokes equation; however, the specific approach used in linearising the momentum equation has not been attempted before. The accuracy of this analytical solution has been verified through a comprehensive comparison with various published experimental data. The existence of velocity overshoot at a short distance from the inlet, which is evident in numerous numerical calculations based on the full Navier–Stokes equations and corroborated by recent magnetic resonance (MR) velocimetry experiments, is identified analytically for the first time. The parabolised Navier–Stokes equation has inherent self-similarity with respect to the Reynolds number, implying that $Re$ is incorporated into the dimensionless variables rather than serving as an independent flow parameter. According to both MR velocimetry measurements and the present analytical solution, the self-similarity is not valid immediately following the pipe inlet, and this becomes more evident as $Re$ decreases; hence, the analytical solution derived from the parabolised Navier–Stokes equation cannot accurately predict the evolution of the velocity profile within this region near the pipe inlet.

JFM classification

Boundary Layers: Pipe flow boundary layer Mathematical Foundations: General fluid mechanics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 979 , 25 January 2024 , A51
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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