Travelling waves in two-dimensional plane Poiseuille flow

Warren Smith, JG Wissink

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)
82 Downloads (Pure)

Abstract

The asymptotic structure of laminar modulated travelling waves in two-dimensional high-Reynolds-number plane Poiseuille flow is investigated on the upper-energy branch. A finite set of independent slowly varying parameters are identified which parameterize the solution of the Navier--Stokes equations in this subset of the phase space. Our parameterization of the weakly stable modes describes an attracting manifold of maximum-entropy configurations. The complementary modes, which have been neglected in this parameterization, are strongly damped. In order to seek a closure, a countably infinite number of modulation equations are derived on the long viscous time scale: a single equation for averaged kinetic energy and momentum; and the remaining equations for averaged powers of vorticity. Only a finite number of these vorticity modulation equations are required to determine the finite number of unknowns. The new results show that the evolution of the slowly varying amplitude parameters is determined by the vorticity field and that the phase velocity responds to these changes in the amplitude in accordance with the kinetic energy and momentum. The new results also show that the most crucial physical mechanism in the production of vorticity is the interaction between vorticity and kinetic energy, this interaction being responsible for the existence of the attractor.
Original languageEnglish
Pages (from-to)2147–2169
JournalSIAM Journal on Applied Mathematics
Volume75
Issue number5
Early online date30 Sep 2015
DOIs
Publication statusE-pub ahead of print - 30 Sep 2015

Keywords

  • strongly nonlinear analysis
  • travelling waves
  • Navier--Stokes equations

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