Abstract
We analyse a low-dimensional model for turbulent shear flows. The model is based on Fourier modes and describes sinusoidal shear flow, in which the fluid between two free-slip walls experiences a sinusoidal body force. The model contains a total of nine modes, including modes describing the basic mean velocity profile and its modification, downstream vortices, streaks, and instabilities of streaks, with other modes being a consequence of the non-linear interactions. The transition to turbulence for the model is subcritical and intermittent, and the distributions of turbulent lifetimes are exponential, in agreement with observations in many shear flows.
Highlights
The understanding of the transition to turbulence in many flows has benefited from the analysis of simplified models
Describing the transition to turbulence in shear flows such as plane Couette flow or pipe flow has proven much more difficult: there is no linear instability around which centre manifold and other reduction techniques can be applied; numerical simulations and experiments indicate that the dynamics are spatially and temporally complicated right above the onset of turbulence [6]–[10]
What makes the development of such low-dimensional models complicated is the fact that the peculiar form of the non-linearity of the Navier–Stokes equations constrains the possible interactions among modes, so that a poor choice of modes in a model can lead to trivial dynamics
Summary
The understanding of the transition to turbulence in many flows has benefited from the analysis of simplified models. Visual inspection of direct numerical simulations [11] and experiments in the transition region [10, 12], proper orthogonal decomposition of the flow field [5, 13], and results from linear stability analysis [7, 14, 15] all suggest that downstream vortices should play a major role These modes drive streaks, and, as discussed perhaps most strongly by Waleffe [14, 16], the unstable modes of these streaks are recycled as downstream vortices.
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