Kavli Affiliate: Michael P. Brenner
| First 5 Authors: Jacob Page, Joe Holey, Michael P. Brenner, Rich R. Kerswell,
| Summary:
Convolutional autoencoders are used to deconstruct the changing dynamics of
two-dimensional Kolmogorov flow as $Re$ is increased from weakly chaotic flow
at $Re=40$ to a chaotic state dominated by a domain-filling vortex pair at
$Re=400$. The highly accurate embeddings allow us to visualise the evolving
structure of state space and are interpretable using `latent Fourier analysis’
(Page {em et. al.}, emph{Phys. Rev. Fluids} textbf{6}, 2021). Individual
latent Fourier modes decode into vortical structures with a streamwise
lengthscale controlled by the latent wavenumber, $l$, with only a small number
$l lesssim 8$ required to accurately represent the flow. Latent Fourier
projections reveal a detached class of bursting events at $Re=40$ which merge
with the low-dissipation dynamics as $Re$ is increased to $100$. We use doubly-
($l=2$) or triply- ($l=3$) periodic latent Fourier modes to generate guesses
for UPOs (unstable periodic orbits) associated with high-dissipation events.
While the doubly-periodic UPOs are representative of the high-dissipation
dynamics at $Re=40$, the same class of UPOs move away from the attractor at
$Re=100$ — where the associated bursting events typically involve larger-scale
($l=1$) structure too. At $Re=400$ an entirely different embedding structure is
formed within the network in which no distinct representations of small-scale
vortices are observed; instead the network embeds all snapshots based around a
large-scale template for the condensate. We use latent Fourier projections to
find an associated `large-scale’ UPO which we believe to be a finite-$Re$
continuation of a solution to the Euler equations.
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