Tulin 



laminar turbulent' •' ■' 



or 



Bu .^ 1 u*2 



By (y ) ^ 2 ^"TT ' 



where u* is the usual friction velocity (a = pu*^). 



We may then argue that the local dissipation there due to the operation of 

 the laminar and turbulent stresses must also be about equal (this argument de- 

 pends upon the fact that turbulent production of energy there is approximately 

 equal to turbulent dissipation (Ref. 17, Fig. 9.13): 



(dissipation )l ^ (dissipation )j , y = y ; 



T- (y ) 



* 3 



This specification leads to a requirement on the Reynolds number based on 

 the viscous sublayer thickness: 



For Newtonian fluids, 



u y 



4K . 



12 , 



but the important thing to notice is that a twofold or threefold increase in the 

 coefficient of turbulent dissipation, to which K is proportional, will result in a 

 proportionate increase in this Reynolds number and thus in the sublayer thick- 

 ness: 



u y 



% 24-36 



L V J 



'- -■ max 



in polymer solutions. 



This maximum thickening will occur when c^/u* ^ 0(5), while larger and 

 smaller concentrations of polymers than correspond to this last condition will 

 result in a decrease in the polymer effectiveness. The sublayer thickening has 

 been observed (2) and so has the plateau in the drag reduction as a function of 

 polymer concentration (1-3). 



CONCLUSION 



Presented here is only a partial and, of course, hurried view of some of 

 our conclusions, the result of research supported by the Fluid Dynamics Branch 



16 



