Spatial-temporal transformation for primary and secondary instabilities in weakly non-parallel shear flows
File(s)
Accepted version
Author(s)
Xu, Jiakuan
Liu, Jianxin
Zhang, Zhongyu
Wu, Xuesong
Type
Journal Article
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.
Date Issued
2023-03-25
Date Acceptance
2023-03-01
Citation
Journal of Fluid Mechanics, 2023, 959
ISSN
0022-1120
Publisher
Cambridge University Press
Journal / Book Title
Journal of Fluid Mechanics
Volume
959
Copyright Statement
© The Author(s), 2023. Published by Cambridge University Press
This article has been published in a revised form in Journal of Fluid Mechanics https://doi.org/10.1017/jfm.2023.67. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works.
This article has been published in a revised form in Journal of Fluid Mechanics https://doi.org/10.1017/jfm.2023.67. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works.
Identifier
https://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000953051300001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=a2bf6146997ec60c407a63945d4e92bb
Subjects
ABSOLUTE
absolute/convective instability
boundary layer stability
BOUNDARY-LAYER
CONVECTIVE INSTABILITIES
DISTURBANCES
EVOLUTION
GROWTH
Mechanics
Physical Sciences
Physics
Physics, Fluids & Plasmas
Science & Technology
shear-flow instability
Technology
VORTICES
Publication Status
Published
Article Number
ARTN A21
Date Publish Online
2023-03-17