Predicting rough-wall drag from surface topography alone
In plain words
Rough surfaces such as fouled ship hulls or corroded pipes raise turbulent friction, and engineers summarize the effect by one number, an equivalent sand-grain roughness height. Nobody can yet compute that number from a measured map of a surface without testing the surface in a flow.
Precise statement
In fully rough wall turbulence the mean-velocity shift is $\Delta U+ = (1/\kappa)\ln k_s+ + \mathrm{const}$, which defines the equivalent sand-grain height $k_s$. Determine whether $k_s/k$ ($k$ a roughness height) is a function of a finite set of topographic statistics (rms height, effective slope, skewness, correlation lengths) for generic random surfaces, give that function, and predict the transitionally rough curve $\Delta U+(k_s+)$. Answer: a relation with error well below 10 percent on surfaces not used for calibration.
What would settle it
Blind predictions of $k_s$ for new random surfaces confirmed by roughness-resolved DNS or experiments in the fully rough regime.
Status in the literature
Prediction uncertainty for full-scale drag is at least 10 percent (Chung, Hutchins, Schultz, Flack, Annual Review of Fluid Mechanics 2021).