  New reduced-order models (ROMs) are derived for sinusoidal shear flow (also
known as Waleffe flow) and plane Couette flow in small periodic domains. A
first derivation for Waleffe flow exploits Fourier modes that form a natural
orthonormal basis for the problem. A ROM for such basis is obtained by a
Galerkin projection of the Navier-Stokes equation. A large basis was reduced to
12 modes that contribute significantly in maintaining chaotic, turbulent
dynamics. A key difference from earlier ROMs is the inclusion of two
roll-streak structures, with spanwise wavelengths equal to $L_z$ and $L_z/2$,
where $L_z$ is the spanwise length of the computational box. The resulting
system was adapted to Couette flow by rewriting the Galerkin system for the
same 12 modes, modified so as to satisfy no-slip conditions on the walls. The
resulting dynamical systems lead to turbulence with finite lifetimes, in
agreement with earlier ROMs and simulations in small domains. However, the
present models display lifetimes that are much longer than in earlier ROMs,
with differences of more than an order of magnitude. The Couette-flow model is
compared to results of direct numerical simulation (DNS), with statistics
displaying fair agreement. The inclusion of the $L_z$ and $L_z/2$ lengthscales
is seen to be a key feature for longer turbulence lifetimes: neglecting any of
the roll modes, or their non-linear interaction, leads to drastic reductions of
turbulence lifetimes. The present ROMs thus highlight some of the dominant
nonlinear interactions that are relevant in maintaining turbulence for long
lifetimes.
