Maximally, a closed form solution would remove the need for Computational Fluid Dynamics. Any property could be derived from a (presumably expensive) analytic function.

Minimally. It could say that no such analytic function can ever exist. Which would be rather boring.

Turbulent fluids look awfully predictable with their spirals….

That would be surprising IMO. We have closed form solutions to Newton’s Laws plus gravity (albeit not very many of them), we have several closed form solutions to Einstein’s equation in GR, and we have a whole lot of closed form solutions to Maxwell’s equations. But we still use numerical methods to solve interesting problems in all of these fields.