American Institute of Mathematical Sciences

Advanced
March  2008, 1(1): 165-176. doi: 10.3934/dcdss.2008.1.165

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

Sergey A. Suslov 1,

1. 

Department of Mathematics and Computing and Computational Engineering and Science Research Centre, University of Southern Queensland, Toowoomba, Queensland 4350, Australia

Received  September 2006 Revised  August 2007 Published  December 2007

Amplitude equations of Landau type, which describe the dynamics of the 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.
Keywords: Subcritical shear flows, resonance mode interaction..
Mathematics Subject Classification: Primary: 76E05, 76E30; Secondary: 35Q3.
Citation: Sergey A. Suslov. Two-equation model of mean flow resonances in subcritical flow systems. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 165-176. doi: 10.3934/dcdss.2008.1.165
[1]

Eric S. Wright. Macrotransport in nonlinear, reactive, shear flows. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2125-2146. doi: 10.3934/cpaa.2012.11.2125
[2]

Anna Geyer, Ronald Quirchmayr. Weakly nonlinear waves in stratified shear flows. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2309-2325. doi: 10.3934/cpaa.2022061
[3]

Zhenlu Cui, Qi Wang. Permeation flows in cholesteric liquid crystal polymers under oscillatory shear. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 45-60. doi: 10.3934/dcdsb.2011.15.45
[4]

Alex Mahalov, Mohamed Moustaoui, Basil Nicolaenko. Three-dimensional instabilities in non-parallel shear stratified flows. Kinetic and Related Models, 2009, 2 (1) : 215-229. doi: 10.3934/krm.2009.2.215
[5]

Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods for incompressible flows. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 817-840. doi: 10.3934/dcdsb.2005.5.817
[6]

M. Bulíček, P. Kaplický. Incompressible fluids with shear rate and pressure dependent viscosity: Regularity of steady planar flows. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 41-50. doi: 10.3934/dcdss.2008.1.41
[7]

James Nolen, Jack Xin. Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1217-1234. doi: 10.3934/dcds.2005.13.1217
[8]

Xavier Perrot, Xavier Carton. Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 971-995. doi: 10.3934/dcdsb.2009.11.971
[9]

Thierry Gallay. Stability and interaction of vortices in two-dimensional viscous flows. Discrete and Continuous Dynamical Systems - S, 2012, 5 (6) : 1091-1131. doi: 10.3934/dcdss.2012.5.1091
[10]

Hong Zhou, M. Gregory Forest, Qi Wang. Anchoring-induced texture & shear banding of nematic polymers in shear cells. Discrete and Continuous Dynamical Systems - B, 2007, 8 (3) : 707-733. doi: 10.3934/dcdsb.2007.8.707
[11]

Siwei Yu, Jianwei Ma, Stanley Osher. Geometric mode decomposition. Inverse Problems and Imaging, 2018, 12 (4) : 831-852. doi: 10.3934/ipi.2018035
[12]

Raphael Stuhlmeier. Effects of shear flow on KdV balance - applications to tsunami. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1549-1561. doi: 10.3934/cpaa.2012.11.1549
[13]

Vicenţiu D. Rădulescu. Noncoercive elliptic equations with subcritical growth. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 857-864. doi: 10.3934/dcdss.2012.5.857
[14]

Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, J. Nathan Kutz. On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics, 2014, 1 (2) : 391-421. doi: 10.3934/jcd.2014.1.391
[15]

Steven L. Brunton, Joshua L. Proctor, Jonathan H. Tu, J. Nathan Kutz. Compressed sensing and dynamic mode decomposition. Journal of Computational Dynamics, 2015, 2 (2) : 165-191. doi: 10.3934/jcd.2015002
[16]

Hao Zhang, Scott T. M. Dawson, Clarence W. Rowley, Eric A. Deem, Louis N. Cattafesta. Evaluating the accuracy of the dynamic mode decomposition. Journal of Computational Dynamics, 2020, 7 (1) : 35-56. doi: 10.3934/jcd.2020002
[17]

Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139
[18]

Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu. Nonlinear Dirichlet problems with double resonance. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1147-1168. doi: 10.3934/cpaa.2017056
[19]

Leszek Gasiński, Nikolaos S. Papageorgiou. Dirichlet $(p,q)$-equations at resonance. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2037-2060. doi: 10.3934/dcds.2014.34.2037
[20]

D. Bonheure, C. Fabry. A variational approach to resonance for asymmetric oscillators. Communications on Pure and Applied Analysis, 2007, 6 (1) : 163-181. doi: 10.3934/cpaa.2007.6.163

2021 Impact Factor: 1.865

Tools

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy