  Turbulent flows are fundamental in engineering and the environment, but their
chaotic and three-dimensional (3-D) nature makes them computationally expensive
to simulate. In this work, a dimensionality reduction technique is investigated
to exploit flows presenting an homogeneous direction, such as wake flows of
extruded two-dimensional (2-D) geometries. First, we examine the effect of the
homogeneous direction span on the wake turbulence dynamics of incompressible
flow past a circular cylinder at $Re=10^4$. It is found that the presence of a
solid wall induces 3-D structures even in highly constricted domains. The 3-D
structures are rapidly two-dimensionalised by the large-scale K\'{a}rm\'{a}n
vortices if the cylinder span is 50\% of the diameter or less, as a result of
the span being shorter than the natural wake Mode B instability
wavelength[...]. With this physical understanding, a 2-D data-driven model that
incorporates 3-D effects, as found in the 3-D wake flow, is presented. The 2-D
model is derived from a novel flow decomposition based on a local spanwise
average of the flow, yielding the spanwise-averaged Navier-Stokes (SANS)
equations[...]. A machine-learning (ML) model is employed to provide closure to
the SANS equations. In the a-priori framework, the ML model yields accurate
predictions of the SSR terms, in contrast to a standard eddy-viscosity model
which completely fails to capture the closure term structures[...]. In the
a-posteriori analysis, while we find evidence of known stability issues with
long-time ML predictions for dynamical systems, the closed SANS equations are
still capable of predicting wake metrics and induced forces with errors from
1-10%. This results in approximately an order of magnitude improvement over
standard 2-D simulations while reducing the computational cost of 3-D
simulations by 99.5%.
