On the displacement height in the logarithmic velocity profile

The displacement height appears in the logarithmic velocity profile for rough-wall boundary layers as a reference height for the vertical co-ordinate. It is shown that this height should be regarded as the level at which the mean drag on the surface appears to act. The equations of motion then show that this also coincides with the average displacement thickness for the shear stress. A simple analytical model, experimental results and dimensional analysis are all used to indicate how the displacement height depends upon the detailed geometry of the roughness elements.