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A conjecture on the least stable mode for the energy stability of plane parallel flows

Published online by Cambridge University Press:  29 October 2019

Xiangming Xiong Open the ORCID record for Xiangming Xiong [Opens in a new window]  and
Zhi-Min Chen Open the ORCID record for Zhi-Min Chen [Opens in a new window]
Xiangming Xiong
Affiliation:
School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China The Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
Zhi-Min Chen*
Affiliation:
School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China
*
†Email address for correspondence: [email protected]
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Abstract

In the energy stability theory, the critical Reynolds number is usually defined as the minimum of the first positive eigenvalue $R_{1}$ of an eigenvalue equation for all wavenumber pairs $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are the streamwise and spanwise wavenumbers of the normal mode. We prove that $(\cos \unicode[STIX]{x1D703}\pm 1)R_{1}$ are decreasing functions of $\unicode[STIX]{x1D703}=\arctan (\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D6FC})$ for the parallel flows between no-slip or slip parallel plates with or without variations in temperature. Numerical results inspire us to conjecture that $R_{1}$ is also a decreasing function of $\unicode[STIX]{x1D703}$ for the parallel shear flows under the no-slip boundary condition and without variations in temperature. If the conjecture is correct, the least stable normal modes for the energy stability will be streamwise vortices for these base flows.

JFM classification

Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 881 , 25 December 2019 , pp. 794 - 814
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Copyright
© 2019 Cambridge University Press 

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