The high-Reynolds-number asymptotic development of nonlinear equilibrium states in plane Couette flow

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The relationship between nonlinear equilibrium solutions of the full Navier-Stokes equations and the high-Reynolds-number asymptotic vortex-wave interaction (VWI) theory developed for general shear flows by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641-666) is investigated. Using plane Couette flow as a prototype shear flow, we show that all solutions having O(1) wavenumbers converge to VWI states with increasing Reynolds number. The converged results here uncover an upper branch of VWI solutions missing from the calculations of Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178-205). For small values of the streamwise wavenumber, the converged lower-branch solutions take on the long-wavelength state of Deguchi, Hall & Walton (J. Fluid Mech., vol. 721, 2013, pp. 58-85) while the upper-branch solutions are found to be quite distinct, with new states associated with instabilities of jet-like structures playing the dominant role. Between these long-wavelength states, a complex 'snaking' behaviour of solution branches is observed. The snaking behaviour leads to complex 'entangled' states involving the long-wavelength states and the VWI states. The entangled states exhibit different-scale fluid motions typical of those found in shear flows.

Original languageEnglish
Pages (from-to)99-112
Number of pages14
JournalJournal of Fluid Mechanics
Publication statusPublished - 1 Jan 2014
Externally publishedYes


  • general fluid mechanics
  • mathematical foundations
  • nonlinear instability

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