since continuity requires that V(y) and v(y) start out parabolically from y = 0. 

 Further, since 



u(y)~y + , (49) 



w(y) ~y-\ , 



v(y) ~ y 2 H , 



we see that the turbulent shear stress can rise no faster than a cubic: 



— uv >~ y 3 H . 



(50) 

 (51) 



(52) 



In fact, there is some evidence that 



hV 



sometimes decreases toward the wall [24], 



in which case — uv may build up even more slowly. 



Some of these conditions have been exploited, for example, in a report by 

 Phillips [25]. 



A different but qualitative look at the wall region can be gotten by dye injection 

 methods. The "leaching" of colored water from a glass tube by flowing clear water 

 through it in a turbulent state leaves dye only near the wall after a short time. In 

 figure 14 is such a situation as photographed by Beatty, Ferrell and Richardson.* The 

 Reynolds number is about 7000. The significant property seems to be the strong orienta- 

 tion into stream-wise filaments of the residual dye. Presumably this indicates a pre- 

 dominance of axial vorticity near the wall, "sweeping" the (dyed) wall fluid into these 

 long narrow strips. Of course the very high mean velocity gradient in this zone is 

 expected to rotate and strain all fluid lines axially, but the parabolic decrease of v' 



Figure 14. Flushing of dye by a turbulent pip flow, R 7000. [Courtesy of Chemical Engineering 

 Department, North Carolina State College] 



toward the wall would seem to prevent the dominance of this vorticity component in 

 the laminar sublayer itself. 



Figure 15 is a somewhat analogous picture taken by Hama, with dye injected 

 from a flush cross-stream slit into the wall layer of a turbulent boundary layer. In this 

 case the interpretation is made more difficult by the apparent non-uniformity of dye 



* I should like to thank Drs. Beatty, Ferrell and Richardson of the Chemical Engineer- 

 ing Dept. at North Carolina State College for the loan of their manuscript, and also Dr. Hama 

 of the University of Maryland for supplying figure 15. 



393 



