This effect is usually described in terms of the roughness function DU +. The classical treatment of rough wall turbulent boundary layers consists in determining the effect the roughness has on the mean velocity profile. There is nonetheless evidence that the small-scale structure over the three-dimensional mesh roughness conforms more closely with isotropy than that over the rod-roughened and smooth walls. The small-scale anisotropy and interniittency exhibit much smaller differences when the Taylor microscale Reynolds number and the Kolmogorov-normalized mean shear are nominally the same.
The Reynolds stress anisotropy is largest for a smooth wall. The differences are such that the Reynolds stress anisotropy is smaller over the mesh roughness than the rod roughness. Measurements over two different surface geometries (a mesh roughness and spanwise circular rods regularly spaced in the streamwise direction) with nominally the same delta U+ indicate significant differences in the Reynolds stresses, especially those involving the wall-normal velocity fluctuation, over the outer region. The general implication is that different roughness geometries with the same delta U+ will have similar turbulence characteristics, at least at a sufficient distance from the roughness elements. This effect is usually described in terms of the roughness function delta U+.