On the Structure of Wind near the Ground, IV

Abstract

The vertical structure of wind near the ground is assumed to be composed of the mean flow and many ranks of vertical turbulons. At a certain height z there are the largest vertical turbulons which are coupled with the mean flow. They take the energy from the mean flow, and give it to turbulons of smaller size at that height. This rate of energy transfer is represented by the relations K (dU/dz)²_??_W³/_??__z, K_??_W_??__z. where _??__z denotes the diameter of the largest vertical turbulon and is thought to be proportional to the height z. Moreover, the vertical velocity W of the largest vertical turbulon is deduced to be independent of height. Thus the logarithmic law for the vertical distribution of wind velocity is obtained.
The largest turbulon is thought obviously nonisotropic, but turbulons of smaller scale may be conceived to be locally isotropic. Then the spectral function of vertical turbulon energy is also deduced to obey the negative five-thirds power law.
The thickness of the laminar sub-layer is obtained as the limiting length of application of this turbulent boundary layer theory, and is given by z₀_??_ν/V_*, where V_* means the friction velocity and is proportinal to W. Thus z₀ is known to correspond to the particular case of the diameter of the smallest turbulon.