B,23 • EFFECT OF ROUGHNESS 



then the rule does hold true for the entire layer. This is true when k is 

 proportional to 5, and at the same time Vrk/v is sufficiently large to make 

 viscous effects negligible. If we fully grasp the foregoing facts, it does not 

 seem so strange that a small quantity like k should be associated with a 

 much larger quantity like 5 and furthermore occupy a position of equal 

 importance. 



An important characteristic of the roughness effect, first pointed out 

 by Nikuradse [95], is a downward shift of the velocity near the wall from 

 that corresponding to the smooth wall condition at a given value of Ut. 

 This is understandable in view of the fact that the mixing action of the 

 roughness elements increases the rate of momentum transfer, and a lower 

 velocity near the wall is required to keep Ut the same. In connection with 

 this downward shift it is necessary to recall that we now have two wall 

 laws: 



Smooth wall jj = -^ In ( ^=^^ ) + const 



h^i'f) 



Fully rough wall jj- = -^ln(|j + const 



Both are dependent on conditions near the wall and both are independent 

 of stream conditions, such as boundary layer thickness and pressure gradi- 

 ent. If we subtract the second equation from the first and call the differ- 

 ence AU/Ut, the downward shift in velocity is found to be 



-JJ- ^ X \ I "^ ^° (23-4) 



This equation applied only for values of Urk/v for which the surface is 

 fully rough. 



The behavior of AU/Ur over a wide range of values of Urk/v has been 

 determined by a number of investigators. A representative summary of 

 results given by Clauser [72] is reproduced in Fig. B,23a. This figure is 

 very instructive. It shows the behavior of different kinds of roughness 

 through the range smooth, partially rough, and fully rough conditions. 

 The limits of such ranges can be judged from this figure. Where the 

 roughness elements are of uniform size, as for example uniform sand, the 

 limit below which the wall is smooth is reasonably definite. It appears to 

 be Urk/v = 4. However, when the roughness consists of a mixture of 

 sizes or is not densely packed and a fictitious k is chosen to bring the 

 curves into coincidence in the fully rough regime, then the lower limit 

 cannot be specified. The lower limit for the fully rough condition is seen 

 to be somewhere between 50 and 100. 



It is interesting to interpret these limits in terms of k/di^^, where di^^ is 

 the thickness of the laminar sublayer on a smooth wall. The sublayer is 



< 149 ) 



