Nonlinear evolution of interacting sinuous and varicose modes in plane wakes and jets: quasi-periodic structures
File(s)
Published version
Author(s)
Mikhaylov, Kirill
Wu, Xuesong
Type
Journal Article
Abstract
A plane wake or jet supports sinuous and varicose instability modes. The nonlinear interaction between them following their linear development was described previously by Leib and Goldstein [“Nonlinear interaction between the sinuous and varicose instability modes in a plane wake,” Phys. Fluids A 1, 513–521 (1989)] using the strongly nonlinear non-equilibrium critical-layer approach in the case of the Bickley jet for which the frequencies of the sinuous and varicose modes have an integer ratio of 2. This paper develops the theory for general profiles where the frequencies of the sinuous and varicose modes are non-commensurable. The disturbance is quasi-periodic in time and space and must be expressed as a function of two phase variables. Using matched asymptotic expansions simultaneously with the multi-scale method, we derived a set of coupled evolution equations governing the development of the amplitudes and critical-layer vorticities of these modes. The evolution system is solved for the base-flow profiles mimicking those in experiments. The sinuous mode suppresses the varicose mode but also causes the latter to saturate in a highly oscillatory manner. The varicose mode inhibits the sinuous mode initially. However, in the later stage, it lends the sinuous mode a significantly higher saturating amplitude. For a wide range of initial modal compositions and Reynolds numbers, the ratio of the varicose mode amplitude to that of the sinuous mode eventually tends to an almost constant value in the range of 0.4–0.6, in line with the experimental measurement. Due to the self and mutual interactions, the vorticities roll up to form vortices, which are non-symmetric in the transverse direction and quasi-periodic in the streamwise direction as well as in time. With such an increased complexity, the vortices resemble those observed in experiments. The nonlinear interactions of the sinuous and varicose modes in the critical layer generate all harmonics in the main layer, as a result of which the perturbation is non-periodic and may even appear “random-like.”
Date Issued
2020-06
Date Acceptance
2020-03-04
Citation
Physics of Fluids, 2020, 32 (6)
ISSN
1070-6631
Publisher
American Institute of Physics
Journal / Book Title
Physics of Fluids
Volume
32
Issue
6
Copyright Statement
© 2020 Author(s).
This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Kirill Mikhaylov, Xuesong Wu; Nonlinear evolution of interacting sinuous and varicose modes in plane wakes and jets: Quasi-periodic structures. Physics of Fluids 1 June 2020; 32 (6): 064104. and may be found at https://doi.org/10.1063/5.0007019
This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Kirill Mikhaylov, Xuesong Wu; Nonlinear evolution of interacting sinuous and varicose modes in plane wakes and jets: Quasi-periodic structures. Physics of Fluids 1 June 2020; 32 (6): 064104. and may be found at https://doi.org/10.1063/5.0007019
Identifier
https://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000544095500001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=a2bf6146997ec60c407a63945d4e92bb
Subjects
COMPRESSIBLE LAMINAR WAKE
DYNAMICS
FINITE-AMPLITUDE INSTABILITY
GLOBAL INSTABILITIES
Mechanics
NON-LINEAR MECHANICS
Physical Sciences
Physics
Physics, Fluids & Plasmas
PLATE PLACED PARALLEL
Science & Technology
SHEAR-FLOW
STABILITY
Technology
TRANSITION
WAVE DISTURBANCES
Publication Status
Published
Article Number
064104
Date Publish Online
2020-06-18