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Spatial–temporal transformation for primary and secondary instabilities in weakly non-parallel shear flows

Published online by Cambridge University Press:  17 March 2023

Jiakuan Xu Open the ORCID record for Jiakuan Xu [Opens in a new window] ,
Jianxin Liu Open the ORCID record for Jianxin Liu [Opens in a new window] ,
Zhongyu Zhang Open the ORCID record for Zhongyu Zhang [Opens in a new window]  and
Xuesong Wu Open the ORCID record for Xuesong Wu [Opens in a new window]
Jiakuan Xu
Affiliation:
School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, PR China
Jianxin Liu*
Affiliation:
Laboratory for High-Speed Aerodynamics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, PR China
Zhongyu Zhang
Affiliation:
Laboratory for High-Speed Aerodynamics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, PR China
Xuesong Wu*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK
*
†Email addresses for correspondence: shookware@tju.edu.cn, x.wu@ic.ac.uk
†Email addresses for correspondence: shookware@tju.edu.cn, x.wu@ic.ac.uk
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Abstract

When studying instability of weakly non-parallel flows, it is often desirable to convert temporal growth rates of unstable modes, which can readily be computed, to physically more relevant spatial growth rates. This has been performed using the well-known Gaster's transformation for primary instability and Herbert's transformation for the secondary instability of a saturated primary mode. The issue of temporal–spatial transformation is revisited in the present paper to clarify/rectify the ambiguity/misunderstanding that appears to exist in the literature. A temporal mode and its spatial counterpart may be related by sharing either the real frequency or wavenumber, and the respective transformations between their growth rates are obtained by a simpler consistent derivation than the original one. These transformations, which consist of first- and second-order versions, are valid under conditions less restrictive than those for Gaster's and Herbert's transformations, and reduce to the latter under additional conditions, which are not always satisfied in practice. The transformations are applied to inviscid Rayleigh instability of a mixing layer and a jet, secondary instability of a streaky flow as well as general detuned secondary instability (including subharmonic and fundamental resonances) of primary Mack modes in a supersonic boundary layer. Comparison of the transformed growth rates with the directly calculated spatial growth rates shows that the transformations derived in this paper outperform Gaster's and Herbert's transformations consistently. The first-order transformation is accurate when the growth rates are small or moderate, while the second-order transformations are sufficiently accurate across the entire instability bands, and thus stand as a useful tool for obtaining spatial instability characteristics via temporal stability analysis.

JFM classification

Boundary Layers: Boundary layer stability Instability: Absolute/convective instability Instability: Shear-flow instability

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JFM Papers
Information
Journal of Fluid Mechanics , Volume 959 , 25 March 2023 , A21
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© The Author(s), 2023. Published by Cambridge University Press

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