The evolution of finite-amplitude wavetrains in plane channel flow

R. E. Hewitt, P. Hall

    Research output: Contribution to journalArticlepeer-review

    Abstract

    We consider a viscous incompressible fluid flow driven between two parallel plates by a constant pressure gradient. The flow is at a finite Reynolds number, with an O(1) disturbance in the form of a travelling wave. A phase equation approach is used to discuss the evolution of slowly varying fully nonlinear two-dimensional wavetrains. We consider uniform wavetrains in detail, showing that the development of a wavenumber perturbation is governed by the Burgers equation in most cases. The wavenumber perturbation theory, constructed by using the phase equation approach for a uniform wavetrain, is shown to be distinct from an amplitude perturbation expansion about the periodic flow. In fact, we show that the amplitude equation contains only linear terms and is simply the heat equation. We review, briefly, the well-known dynamics of the Burgers equation, which imply that both shock structures and finite-time singularities of the wavenumber perturbation can occur with respect to the slow scales. Numerical computations have been performed to identify areas of the {wavenumber, Reynolds number, energy} neutral surface for which each of these possibilities can occur. We note that the evolution equations will break down under certain circumstances, in particular for a weakly nonlinear secondary flow. Finally, we extend the theory to three dimensions and discuss the limit of a weak spanwise dependence for uniform wavetrains, showing that two functions are required to describe the evolution. These unknowns are a phase and a pressure function which satisfy a pair of linearly coupled partial differential equations. The results obtained from applying the same analysis to the fully three-dimensional problem are included as an appendix.
    Original languageEnglish
    Pages (from-to)2413-2446
    Number of pages33
    JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
    Volume356
    Issue number1746
    DOIs
    Publication statusPublished - 15 Oct 1998

    Keywords

    • Nonlinear stability
    • Parallel shear flow
    • Phase equation methods

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