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arXiv · 2004.12418

Bifurcations of a plane parallel flow with Kolmogorov forcing

Kannabiran Seshasayanan·Vassilios Dallas·Stephan Fauve
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Abstract

We study the primary bifurcations of a two-dimensional Kolmogorov flow in a channel subject to boundary conditions chosen to mimic a parallel flow, i.e. periodic and free-slip boundary conditions in the streamwise and spanwise directions, respectively. The control parameter is the Reynolds number based on the friction coefficient, denoted as $Rh$. We find that as we increase $Rh$ the laminar steady flow goes through a degenerate Hopf bifurcation with both the oscillation frequency and the amplitude of the growing mode being zero at the threshold. A reduced four-mode model captures the scalings that are obtained from the numerical simulations. As we increase $Rh$ further we observe a secondary instability which excites the largest mode in the domain. The saturated amplitude of the largest mode is found to scale as a $3/2$ power-law of the distance to the threshold which is also explained using a low-dimensional model.

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Kannabiran Seshasayanan, Vassilios Dallas, Stephan Fauve. 2020-04-26. Bifurcations of a plane parallel flow with Kolmogorov forcing. https://arxiv.org/abs/2004.12418

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