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Subcritical transition in channel flows. (English) Zbl 1037.76023
From the summary: Certain laminar flows are known to be linearly stable at all Reynolds numbers, \(R\), although in practice they always become turbulent for sufficiently large \(R\). Other flows typically become turbulent well before the critical Reynolds number of linear instability. One resolution of these paradoxes is that the domain of attraction for the laminar state shrinks for large \(R\) (as \(R^\gamma\) say, with \(\gamma< 0)\), so that small but finite perturbations lead to transition. Here, through a formal asymptotic analysis of Navier-Stokes equations, it is found that for streamwise initial perturbations \(\gamma=-1\) and \(-3/2\) for plane Couette and plane Poiseuille flow respectively (factoring out the unstable modes for plane Poiseuille flow), while for oblique initial perturbations \(\gamma=-1\) and \(-5/4\). Furthermore, it is shown why the numerically determined threshold exponents are not the true asymptotic values.

MSC:
76F06 Transition to turbulence
76E05 Parallel shear flows in hydrodynamic stability
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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