Two-equation model of mean flow resonances in subcritical flow systems

Suslov, S. A. (2008) Two-equation model of mean flow resonances in subcritical flow systems. Discrete and Continuous Dynamical Systems Series S, Selected Topics, 1 (1). pp. 165-176. ISSN 1937-1632

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Abstract

[Abstract]: Amplitude equations of Landau type, which describe the dynamics ofthe most amplified periodic disturbance waves in slightly supercritical flow systems, have been known to form reliable and sufficiently accurate low-dimensional models capable of predicting the asymptotic magnitude of saturated perturbations. However the derivation of similar models for estimating the threshold disturbance amplitude in subcritical systems faces multiple resonances which lead to the singularity of model coefficients. The observed resonances are traced back to the interaction between the mean flow distortion induced by the decaying fundamental disturbance harmonic and other decaying disturbance modes. Here we suggest a methodology of deriving a two-equation dynamical system of coupled cubic amplitude equations with non-singular coefficients which resolve the resonances and are capable of predicting the threshold amplitude for weakly nonlinear subcritical regimes. The suggested reduction procedure is based on the consistent use of an appropriate orthogonality condition which is different from a conventional solvability condition. As an example, a developed procedure is applied to a system of Navier-Stokes equations describing a subcritical plane Poiseuille flow. Predictions of the so-developed model are found to be in reasonable agreement with experimentally detected threshold amplitudes reported in literature.


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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Additional Information: Invited paper for the Inaugural issue of DCDS-S.
Depositing User: Dr Sergey Suslov
Faculty / Department / School: Historic - Faculty of Sciences - Department of Maths and Computing
Date Deposited: 04 Feb 2008 02:02
Last Modified: 05 Sep 2014 04:16
Uncontrolled Keywords: amplitude expansion, resonances, subcritical instability
Fields of Research (FOR2008): 02 Physical Sciences > 0203 Classical Physics > 020303 Fluid Physics
01 Mathematical Sciences > 0199 Other Mathematical Sciences > 019999 Mathematical Sciences not elsewhere classified
01 Mathematical Sciences > 0101 Pure Mathematics > 010109 Ordinary Differential Equations, Difference Equations and Dynamical Systems
Socio-Economic Objective (SEO2008): E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
URI: http://eprints.usq.edu.au/id/eprint/3824

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