THE NONLINEAR STABILITY OF PLANE PARALLEL SHEAR FLOWS WITH RESPECT TO TILTED PERTURBATIONS

doi:10.1007/s10473-024-0315-8

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (3): 1036-1045.doi: 10.1007/s10473-024-0315-8

THE NONLINEAR STABILITY OF PLANE PARALLEL SHEAR FLOWS WITH RESPECT TO TILTED PERTURBATIONS

Lanxi XU^*, Fangfang GUAN

College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

收稿日期:2022-10-11 修回日期:2023-03-23 出版日期:2024-06-25 发布日期:2024-05-21

THE NONLINEAR STABILITY OF PLANE PARALLEL SHEAR FLOWS WITH RESPECT TO TILTED PERTURBATIONS

Lanxi XU^*, Fangfang GUAN

College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

Received:2022-10-11 Revised:2023-03-23 Online:2024-06-25 Published:2024-05-21
Contact: *Lanxi XU, E-mail:xulx@mail.buct.edu.cn
About author:Fangfang GUAN, erytar@outlook.com
Supported by:
Xu's research was supported by the National Natural Science Foundation of China (21627813).

摘要/Abstract

摘要： The nonlinear stability of plane parallel shear flows with respect to tilted perturbations is studied by energy methods. Tilted perturbation refers to the fact that perturbations form an angle $\theta\in(0,\frac{\pi}{2})$ with the direction of the basic flows. By defining an energy functional, it is proven that plane parallel shear flows are unconditionally nonlinearly exponentially stable for tilted streamwise perturbation when the Reynolds number is below a certain critical value and the boundary conditions are either rigid or stress-free. In the case of stress-free boundaries, by taking advantage of the poloidal-toroidal decomposition of a solenoidal field to define energy functionals, it can be even shown that plane parallel shear flows are unconditionally nonlinearly exponentially stable for all Reynolds numbers, where the tilted perturbation can be either spanwise or streamwise.

关键词: plane parallel shear flows, energy method, energy functional, nonlinear stability, Reynolds number

Abstract: The nonlinear stability of plane parallel shear flows with respect to tilted perturbations is studied by energy methods. Tilted perturbation refers to the fact that perturbations form an angle $\theta\in(0,\frac{\pi}{2})$ with the direction of the basic flows. By defining an energy functional, it is proven that plane parallel shear flows are unconditionally nonlinearly exponentially stable for tilted streamwise perturbation when the Reynolds number is below a certain critical value and the boundary conditions are either rigid or stress-free. In the case of stress-free boundaries, by taking advantage of the poloidal-toroidal decomposition of a solenoidal field to define energy functionals, it can be even shown that plane parallel shear flows are unconditionally nonlinearly exponentially stable for all Reynolds numbers, where the tilted perturbation can be either spanwise or streamwise.

Key words: plane parallel shear flows, energy method, energy functional, nonlinear stability, Reynolds number

中图分类号:

76E05

Lanxi XU, Fangfang GUAN. THE NONLINEAR STABILITY OF PLANE PARALLEL SHEAR FLOWS WITH RESPECT TO TILTED PERTURBATIONS[J]. 数学物理学报(英文版), 2024, 44(3): 1036-1045.

Lanxi XU, Fangfang GUAN. THE NONLINEAR STABILITY OF PLANE PARALLEL SHEAR FLOWS WITH RESPECT TO TILTED PERTURBATIONS[J]. Acta mathematica scientia,Series B, 2024, 44(3): 1036-1045.

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THE NONLINEAR STABILITY OF PLANE PARALLEL SHEAR FLOWS WITH RESPECT TO TILTED PERTURBATIONS

THE NONLINEAR STABILITY OF PLANE PARALLEL SHEAR FLOWS WITH RESPECT TO TILTED PERTURBATIONS

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