This is a pretty broad, vague question and may not be appropriate (apologies, feel free to close if so.)
I'm going into computational fluid mechanics, but coming from a pure math and physics background.
One of the things that keeps striking me is the similarity of several issues to cohomology.
For instance, in finite volume methods, we discretize the domain into cells, compute averages around boundaries, or in finite differences, we compare boundary values, etc.
Meanwhile, changes from steady to quasisteady to turbulent flows come with global bifurcations of the associated vector fields. We're really looking at global structures of vector fields on complicated geometric domains.
Most of the recent developments in fluids concerns global structures revealed by Dynamic Mode Decomposition (i.e. Koopman operator theory) and Lagrangian Coherent Structures (which arose out of a mathematical study of vortices.)
All of this smacks of cohomology to me. A quick google turns things like these up:
https://arxiv.org/ftp/arxiv/papers/1412/1412.3059.pdf https://www.ima.umn.edu/2013-2014/W10.28-11.1.13
which seems to support that there is a connection here.
I was wondering if there are any systematic overviews of (co)homology in fluids, computational or otherwise, either in the form of books or articles.