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Scaling and mechanism of the propagation speed of the upstream turbulent front in pipe flow

Published online by Cambridge University Press:  21 December 2023

Haoyang Wu  and
Baofang Song Open the ORCID record for Baofang Song [Opens in a new window]
Haoyang Wu
Affiliation:
Center for Applied Mathematics, Tianjin University, Tianjin 300072, PR China
Baofang Song*
Affiliation:
State Key Laboratory of Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, PR China
*
†Email address for correspondence: baofang.song@pku.edu.cn
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Abstract

The scaling and mechanism of the propagation speed of turbulent fronts in pipe flow with the Reynolds number has been a long-standing problem in the past decades. Here, we derive an explicit scaling law for the upstream front speed, which approaches a power-law scaling at high Reynolds numbers, and we explain the underlying mechanism. Our data show that the average wall distance of low-speed streaks at the tip of the upstream front, where transition occurs, appears to be constant in local wall units in the wide bulk-Reynolds-number range investigated, between 5000 and 60 000. By further assuming that the axial propagation of velocity fluctuations at the front tip, resulting from streak instabilities, is dominated by the advection of the local mean flow, the front speed can be derived as an explicit function of the Reynolds number. The derived formula agrees well with the speed measured by front tracking. Our finding reveals a relationship between the structure and speed of a front, which enables a close approximation to be obtained of the front speed based on a single velocity field without having to track the front over time.

JFM classification

Instability: Shear-flow instability Instability: Transition to turbulence Turbulent Flows: Pipe flow
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 977 , 25 December 2023 , R4
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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Supplementary material: File

Wu and Song supplementary movie

Contours of the transverse velocity fluctuations on a crosssection along the pipe axis in a frame of reference co-moving with the upstream front of a pipe flow at Re = 25000. It is upstream on the left and downstreamm on the right. The movie shows that transition to turbulence continuously occurs at the tip of the front (the most upstream part of the front) and turbulence spreads toward the pipe center while going downstream.
Download Wu and Song supplementary movie(File)
File 4.2 MB