Assumptions About The Mixing Length
From the figure above, we can see that the size of a turbulent eddy near the surface is constrained by its proximity to the surface; turbulent eddies centered near the surface cannot be as large as those centered further from the surface. From this consideration, it is reasonable to assume that the mixing length, is proportional to the eddy's depth in the surface:
,
where is the depth and is known as the von Kármán constant. Thus the gradient can be integrated to solve for :
.
So we see that the mean flow in the surface layer has a logarithmic relationship with depth.
Read more about this topic: Surface Layer
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