  Data-driven dimensionality reduction methods such as proper orthogonal
decomposition (POD) and dynamic mode decomposition (DMD) have proven to be
useful for exploring complex phenomena within fluid dynamics and beyond. A
well-known challenge for these techniques is posed by the continuous
symmetries, e.g. translations and rotations, of the system under consideration
as drifts in the data dominate the modal expansions without providing an
insight into the dynamics of the problem. In the present study, we address this
issue for fluid flows in rectangular channels by formulating a continuous
symmetry reduction method that eliminates the translations in the streamwise
and spanwise directions simultaneously. We demonstrate our method by computing
the symmetry-reduced dynamic mode decomposition (SRDMD) of sliding windows of
data obtained from the transitional plane-Couette and turbulent
plane-Poiseuille flow simulations. In the former setting, SRDMD captures the
dynamics in the vicinity of the invariant solutions with translation
symmetries, i.e. travelling waves and relative periodic orbits, whereas in the
latter, our calculations reveal episodes of turbulent time evolution that can
be approximated by a low-dimensional linear expansion.
