Universality of the logarithmic law for streamwise turbulence intensity
In plain words
A picture of wall turbulence as a hierarchy of eddies attached to the wall predicts that the strength of streamwise fluctuations falls with the logarithm of height in the middle of the layer. Whether this law holds with the same slope in pipes, channels, boundary layers and the atmosphere is debated.
Precise statement
In the logarithmic region of smooth-wall turbulence, Townsend's attached-eddy hypothesis gives $\langle u'^2\rangle+ = B1 - A1 \ln(y/\delta)$, with $\delta$ the outer length scale. Determine whether A1 is the same for pipe, channel, zero-pressure-gradient boundary layer and atmospheric surface layer as $\mathrm{Re}_{\tau} \to \infty$, its value (data give $A1$ of about 1.25, approximately), and its relation to $\kappa$. Answer: $A1$ per flow with uncertainty below 0.05, or evidence that the variance has no logarithmic region.
What would settle it
Spatially resolved streamwise-variance profiles at $\mathrm{Re}_{\tau} > 10^{5}$ in at least three geometries with corrections for probe spatial averaging.
Status in the literature
Boundary-layer, pipe and atmospheric data for $\mathrm{Re}_{\tau}$ from 2 x $10^{4}$ to 6 x $10^{5}$ support a universal logarithmic region (Marusic, Monty, Hultmark, Smits, J. Fluid Mech. 2013); universality across geometries and the value of $A1$ are still debated.