> Does the Navier-Stokes equation model (experimentally observed) turbulence accurately? I was under the impression that this is an open question.
No one I know working in turbulence regards the Navier-Stokes equations as a bad model for turbulence. They typically regard the equations as a source of truth about turbulence.
Focusing on turbulence in incompressible fluids [0], almost every time I've seen a comparison of "direct numerical simulations" using the Navier-Stokes equations against experiments measuring turbulence, the two matched well. Off the top of my head the only time I've encountered a discrepancy was deemed not due to the Navier-Stokes equations themselves but due to not modeling the initial or boundary conditions correctly [1]. Initial conditions refer to the flow field at the start of the simulation. Boundary conditions refer to conditions at a boundary, like a wall. Initial condition modeling is a big issue in certain flow instabilities like the Rayleigh-Taylor instability to my knowledge. The Navier-Stokes equations are chaotic, so they are sensitive to small changes in the initial conditions.
I'm not aware of anyone working on flow instabilities like the Rayleigh-Taylor instability who attributes a discrepancy between the experiments and simulations to an inadequacy of the Navier-Stokes equations. And to my knowledge I don't think any of the mathematicians working on the blow-up problem regard flow instabilities as clues indicating where to focus their attention to make the Euler/Navier-Stokes equations blow-up.
[0] You could, of course, pick a situation where the Navier-Stokes equations are known to not work well independent of turbulence considerations.
[1] Having worked a bit on initial and boundary condition modeling, though not in the context of the Rayleigh-Taylor instability, I can say that initial condition and boundary condition models are often bad, and they are understudied.