What is the law of the wall?
Close to a wall, turbulent velocity profiles can be expressed with the same inner variables. The profile is scaled by the wall shear stress rather than by pipe size or mean velocity.
The dark curve is a continuous teaching model. The orange line marks the selected mean velocity; it is a control, not a wall-law equation.
Wall units
u+ = u / u*
y+ = y u* / ν
ks+ = ks u* / ν
The bottom and left axes show physical velocity and distance. The top and right axes show the same point in wall units.
What does the brown band mean?
The brown band runs from the wall to ks.
Equivalent sand-grain roughness is a hydraulic scale. It is the sand-grain size that would produce comparable drag; it is not necessarily the actual height of the surface texture.
The profile inside this band should therefore be read as a smooth continuation to zero, not as a resolved velocity field between real roughness elements.
The fractional-log scale
The current setting is close to logarithmic.
The slider changes the mathematical mapping continuously from y at the bottom toward ln y at the top. Moving it upward expands small y-values and the near-wall decades; moving it downward restores physical linear spacing. It changes only the axis, never the profile or any calculated quantity.