The alternative is numerical approximation, as I mentioned. Like always in physics, any analytic (ie. "closed form") solutions would also necessarily have to be special-cases, approximations, and simplifications. The problem with turbulence is that it's so chaotic and complex that it doesn't seem to be reducible to simpler models while still retaining some predictive power. But that's not very surprising; we're talking about the chaotic motion of molecules at the Avogadro scale! After all, we can't even write closed-form solutions (or even approximations) to the motion of merely three bodies under gravity, never mind ten to the power of twenty-three.

But what is fascinating about turbulence, though, is the "edge of chaos" – the boundary conditions at which laminar (non-turbulent) flow suddenly turns turbulent. On one side of the boundary analytic treatment is possible, on the other it is not.