The low-dimensional dynamics of the structures in a turbulent wall flow are studied
by means of numerical simulations. These are made both ‘minimal’, in the sense
that they contain a single copy of each relevant structure, and ‘autonomous’ in
the sense that there is no outer turbulent flow with which they can interact. The
interaction is prevented by a numerical mask that damps the flow above a given
wall distance, and the flow behaviour is studied as a function of the mask height.
The simplest case found is a streamwise wave that propagates without change. It
takes the form of a single wavy low-velocity streak flanked by two counter-rotating
staggered quasi-streamwise vortices, and is found when the height of the numerical
masking function is less than δ+1 ≈ 50. As the mask height is increased, this solution
bifurcates into an almost-perfect limit cycle, a two-frequency torus, weak chaos, and
full-edged bursting turbulence. The transition is essentially complete when δ+1 ≈ 70,
even if the wall-parallel dimensions of the computational box are small enough for
bursting turbulence to be metastable, lasting only for a few bursting cycles. Similar
low-dimensional dynamics are found in somewhat larger boxes, containing two copies
of the basic structures, in which the bursting turbulence is self-sustaining.
Low-dimensional models of subcritical transition to turbulence
