B • TURBULENT FLOW 



Tw depends on the velocity U at some small distance y from the wall and 

 on k and p. By dimensional reasoning similar to that leading to Eq. 16-1 

 we find 



As we have already noted, the velocity-defect law is unaffected by rough- 

 ness. Since it again develops that there exists a region of overlap where 

 both laws are valid, a logarithmic function is indicated in Eq. 23-1, and 

 the law may be written 



| = S'°(l) + "™^* ^23-2) 



where K is the same as that appearing in the smooth wall law and in the 

 velocity-defect law. 



Just as in the case of the smooth wall law there is a Hnear relation- 

 ship between U/Uj and In {y/k) only for the region of the wall, not 

 throughout the whole boundary layer. Obviously there is some question 

 about a suitable reference point from which to measure y. If y is not 

 expressed correctly, the region that should be linear becomes curved. 

 Experimentally this is used to find the origin of y. No cases are known 

 where the origin did not lie somewhere between the top and bottom of 

 the roughness elements. 



The well-known skin friction law for fully rough walls is obtained by 

 adding Eq. 23-2 and the defect law (Eq. 17-4) and using the relationship 

 Ue/Ur = VVcf. The result is 



V| = 5'°(l) + --* (23-3) 



Since the defect law is affected by the pressure gradient, Eq. 23-3 applies 

 only to cases where the effect of the pressure gradient is negligible. The 

 effect of the free stream conditions is also present, but this effect is small 

 and may be absorbed in the constant. 



The effect of roughness is seen to depend on its height compared to 

 the boundary layer thickness. The effect is independent of Reynolds 

 number. These two circumstances illustrate in a very direct way an in- 

 herent characteristic of turbulent diffusion in shear flow, namely that the 

 length scale in eddy diffusion processes tends to remain proportional to 

 the thickness of the shear layer. In other words, mixing tends to take 

 place on a scale of coarseness proportional to the boundary layer thick- 

 ness, or the radius of a pipe. Ordinarily this rule cannot hold true in the 

 immediate neighborhood of a wall where the turbulent motions are influ- 

 enced by the presence of the wall; but if flow about roughness elements 

 introduces a scale of mixing proportional to the scale of the shear layer, 



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