588 



by some law of probability, or by a maximum- or miiiimum-pon- 



dition. Then the mean values are calculated for this assembly. 



Prof. VON Karman from Aix-la-Cliapelle pointed out to me that before 



trying to find a condition governing tlie weight of the different types 



of motions, it would be advisable at first to search for the H/a^imw?» 



value of S, or of a. In this way a higher limit for the resistance 



of turbulent flow would be found. 



That a maximum value exists may be shown thus: 



From (17) it is deduced that n may become great (i.e. especially : 



1 

 great as compared to -) only '\ï y. <^ R and if t becomes small. 



The value of r is determined by the distribution of the values of 

 uv over the interval 0<[//<^l. Only if uv assumes a constant 

 value throughout this interval, r can attain its minimum value 0. 

 However, uv cannot be a constant everywhere, as u and d decrease 

 to in the neighl)oiirhood of the walls. Hence we will obtain the 

 smallest possible value of t if uv has a constant value throughout 

 the whole region with the exception of two very thin layers along 



the walls, in which layers \uv\ decreases to zero. If the thickness 

 of these "boundary" layers is represented by e, t will be of the 

 same order of magnitude as f, hence with a numerical constante,: 



T = C, f . (19) 



In the boundary layers — and ? will be of the order of magnitude 



6—1, and so ?' will be proportional to e~^. Hence if this intensive 

 vorticity occurs in the boundary layers only : 



X = c, 8-1 (20) 



Now equation (17j gives: 



1 0. 



c,sR c, 6* R' 

 This expression attains a maximum value if: 



•=!• <-) 



The thickness of the boundary layer appears to be inversely 

 proportional to R. The value of ö becomes: 



1 



(T„„^ = (22) 



4 c, c. 



It appears that a takes a value which is independent of R; 



