caa a22 4500 397499760 CHVBK 20180308164530.0 cr unu---uuuuu 161202e199505 xx s 000 0 eng 10.1093/qjmam/48.2.159 doi (NATIONALLICENCE)oxford-10.1093/qjmam/48.2.159 ON THE NONLINEAR EVOLUTION OF INSTABILITY MODES IN UNSTEADY SHEAR LAYERS: THE STOKES LAYER AS A PARADIGM [Elektronische Daten] [XUESONG WU, STEPHEN J. COWLEY] The Stokes layer generated by a flat plate oscillating sinusoidally in an infinite fluid is an important prototype of unsteady flows. It is known to be susceptible to high-frequency instabilities at large Reynolds numbers. Linear theory shows that small disturbances can grow exponentially over part of a period, before becoming neutral at a later time. We show that at times when a disturbance is almost neutral, it can evolve through a weakly nonlinear stage. Our approach is a simple extension of previous non-equilibrium critical-layer theories developed for quasi-parallel flows. For any inviscidly unstable, unsteady shear layer, we present a unified explanation of the asymptotic scalings required to describe three distinct types of small disturbance. We then specialize to weakly nonlinear two-dimensional disturbances, and show that when the critical layer is ‘singular' the amplitude equation is of Hickernell's (1) integro-differential type. Solutions to this equation develop finitetime singularities if viscous effects are not too large. For the particular case of a Stokes layer, we find that for a class of disturbances with normalized wavenumbers in the range [0, 0-6] or [1-29, 1-43], the singularity can be eliminated by sufficiently large viscous effects; the instability wave then evolves to an equilibrium state. However, for wavenumbers in the range [0-6, 1-29], a finite-time singularity occurs no matter how large the scaled viscosity parameter is. This nonlinear explosive amplification may be related to the rapid growth of high-frequency disturbances observed as bursts in experiments. © Oxford University Press Articles nationallicence WU XUESONG Department of Mathematics, Imperial College 180 Queens Gate, London SW7 2BZ aut COWLEY STEPHEN J. DAMTP, University of Cambridge Silver Street, Cambridge CB3 9EW aut The Quarterly Journal of Mechanics and Applied Mathematics Oxford University Press 48/2(1995-05), 159-188 0033-5614 48:2<159 1995 48 qjmamj https://doi.org/10.1093/qjmam/48.2.159 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1093/qjmam/48.2.159 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- WU XUESONG Department of Mathematics, Imperial College 180 Queens Gate, London SW7 2BZ aut NATIONALLICENCE 700 1- COWLEY STEPHEN J. DAMTP, University of Cambridge Silver Street, Cambridge CB3 9EW aut NATIONALLICENCE 773 0- The Quarterly Journal of Mechanics and Applied Mathematics Oxford University Press 48/2(1995-05), 159-188 0033-5614 48:2<159 1995 48 qjmamj Metadata rights reserved CC BY-NC-4.0 http://creativecommons.org/licenses/by-nc/4.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-oxford