Controlled closure calculation of the Kolmogorov constant
In plain words
The energy spectrum of turbulence has a known shape with one dimensionless prefactor, the Kolmogorov constant, measured near 1.6. No calculation from the equations of motion produces this number with an error that the method itself controls.
Precise statement
In the inertial range $E(k) = C_K \epsilon^{2/3} k^{-5/3}$ up to small intermittency corrections, with measured $C_K \sim 1.5 \text{ to } 1.7$ (approximately). Compute $C_K$ from the Navier-Stokes equations by a closure (Lagrangian closures of Kraichnan type, eddy-damped quasi-normal Markovian closure, functional renormalization group) in which the neglected terms are bounded or shown to be small. Answer: a value with an error estimate derived within the method.
What would settle it
A closure or renormalization-group calculation with a demonstrated small parameter or convergent truncation giving $C_K$ within the experimental range.
Status in the literature
Existing closures and renormalization-group calculations give values near the measured range but rely on uncontrolled truncations or expansion parameters set to physical values.