Flows with Drag Reduction (Veloetty and Frictton) 
uy? Sa + u (19) 
The first part, describing the law of the wall, is given by the solution 
of Eq. (26) with 7rt= 1-y¥YR* , namely 
dps Di (aiek y PREY (20) 
dy" 14V 14 4k? y" 2 - exp(-y'/A")]2( -y'/R’) 
Jap 
It is easy to see that the limit of Eq. (20) for small values of R /A 
1s 
+ + 
du," / Gg, eae Te eu (21) 
which describes the parabolic velocity distribution (18). This result 
implies that a ,» which is zero near the wall, has to vanish iden- 
tically for small values of RAL. In other words, the deviation from 
the law of the wall has to decrease as the region where the damping is 
effective increases. This condition is satisfied by the following equa- 
tion proposed for a 
+ + +, + 
us = Se [1 - cos(ry'/R')] [1 - exp (-2R'/A')] (22) 
where II = 0.67 isa universal constant for pipe flows. The value of 
Il has been determined so that the Newtonian friction factor at 
Re = 5.10° would satisfy Eq. (5) with a= 0, a, = 4.0 and b, = 0.4. 
Note that for large values of BVA, which is always the case if 
A* = 26, the exponential term in Eq. (22) vanishes and igs becomes 
identical to Coles' Wake Function. 
Undoubtedly, many other schemes can be used to describe the 
deviation of the velocity a ae near the center of the pipe from me 
and its dependence on A’. As we shall see later the relative contri- 
bution of uz is very small and thus any consistent model which 
complies with the boundary conditions would be satisfactory. The choice 
of Coles'Wake Function is justified mainly for convenience in future 
applications of the model to boundary layer flows. 
DISCUSSION AND COMPARISON WITH EXPERIMENTAL DATA 
A clear distinction between the new model and the constant 
shear approximation used by van Driest, is the dependence of the 
velocity profile on R*. Both oes and ce are functions of R* and 
ewe 
