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On the subcritical transitional threshold in two-dimensional plane Poiseuille flow

Published online by Cambridge University Press:  15 December 2025

Yuanyuan Gao ,
Tongsheng Wang ,
Jianan Chen ,
Guang Xi  and
Zhu Huang Open the ORCID record for Zhu Huang [Opens in a new window]
Yuanyuan Gao
Affiliation:
Department of Fluid Machinery and Engineering, Xi’an Jiaotong University, Shaanxi 710049, PR China
Tongsheng Wang
Affiliation:
Department of Fluid Machinery and Engineering, Xi’an Jiaotong University, Shaanxi 710049, PR China
Jianan Chen
Affiliation:
Department of Fluid Machinery and Engineering, Xi’an Jiaotong University, Shaanxi 710049, PR China
Guang Xi
Affiliation:
Department of Fluid Machinery and Engineering, Xi’an Jiaotong University, Shaanxi 710049, PR China
Zhu Huang*
Affiliation:
Department of Fluid Machinery and Engineering, Xi’an Jiaotong University, Shaanxi 710049, PR China
*
Corresponding author: Zhu Huang, zhuhuang@xjtu.edu.cn
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Abstract

The nonlinear Tollmien–Schlichting waves mechanism of subcritical transitional flow in quasi-two-dimensional flow and two-dimensional (2-D) plane Poiseuille flow have been investigated (Camobreco et al. 2023 J. Fluid Mech., vol. 963, p. R2; Huang et al. 2024 J. Fluid Mech., vol. 994, p. A6). However, the subcritical transitional flow threshold has remained unsolved for 2-D shear flows since the problem was proposed in Trefethen et al. (1993 Science vol. 261, no. 5121, pp. 578–584). In this study, we proposed a theoretical analysis based on the nonlinear non-modal analysis and asymptotic analysis to quantify the scaling law for subcritical transitional flow of 2-D plane Poiseuille flow. The subcritical transitional flow induced by the critical disturbance experiences the nonlinear edge state with invariant disturbance kinetic energy (Huang et al. 2024 J. Fluid Mech. vol. 994, p. A6). Consequently, the required magnitude along with the edge state is predicted by asymptotic analysis, and the a priori threshold is achieved theoretically. All stages are validated by the numerical minimal seeds of different channels. The proposed theory predicts that the scaling laws are $O(Re^{-11/3})$ and $O(\textit{Re}^{-7/3})$ for the critical disturbances and their edge state, respectively. While the numerical thresholds of the subcritical transitional flow are $ \textit{Re}^{-11/3 \pm 0.06}$ and $ \textit{Re}^{-7/3 \pm 0.05}$, respectively.

JFM classification

Instability: Nonlinear instability Instability: Transition to turbulence Instability: Shear-flow instability

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JFM Papers
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Journal of Fluid Mechanics , Volume 1025 , 25 December 2025 , A25
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© The Author(s), 2025. Published by Cambridge University Press

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