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

Article

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.
Article Title

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

Article CategoryArticle
Authors
AuthorSuslov, S. A.
Journal TitleDiscrete and Continuous Dynamical Systems Series S, Selected Topics
Journal Citation1 (1), pp. 165-176
Number of Pages12
Year2008
Place of PublicationUnited States
Web Address (URL)http://aimsciences.org/journals/dcdsS/index.htm
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.

Keywordsamplitude expansion, resonances, subcritical instability
ANZSRC Field of Research 2020401207. Fundamental and theoretical fluid dynamics
499999. Other mathematical sciences not elsewhere classified
490409. Ordinary differential equations, difference equations and dynamical systems
Public Notes

Invited paper for the Inaugural issue of DCDS-S.

Byline AffiliationsDepartment of Mathematics and Computing
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