The boundary behaves as hydraulically smooth even with two and three- 

 dimensional roughness provided that the parameter B e where e is the rough- 

 ness diameter is smaller than a certain value; B of course is equal to 



/— The range of B- for the three classifications determined from the 



V2v ° e 



experiments is the following. 



Two-Dimensional Three-Dimensional 

 Roughness Roughness 



Smooth boundary B e < .678 



Transition .678 < B e < .1.77 .153 < 8 e < .249 



Rough boundary 1.77 < B e .249 < B e 



(i) For the hydraulically smooth boundary it was found that the 

 critical value of the Reynolds number defined as 



I 

 ura 



was constant and equal to 400. 



(ii) In the transition region the Reynolds number was of the form 



U o S 

 N R = — 



with constant critical value for the two-dimensional and three-dimensional 

 roughness ; namely 



N = 640 two-dimensional 

 R 

 cr 



N = 104 three-dimensional 

 cr 



(iii) In the rough region the governing parameter was not a Reynolds 

 number but the dimensional quantity 



u e 



v 



which was found to have the critical values of 



0.2 

 u e _ 4 



- — ~ 2.52 x 10 two-dimensional 



0.2 

 u e _ 4 

 — — - 1.78 x 10 three-dimensional 



A-3 



