Poreh and Dtmant 
it is not possible to describe ut asa function of ¥ and At alone. 
We have plotted in Fig. 1 numerically computed distributions of u,t : 
a wand ud for At =26 and At = 300* We see that the various 
velocity distribution curves for Rt = 10.000 are practically the same 
at small values of y?/R+. The maximum difference between u+ and 
u,;t is about 5% for A = 26 at the center of the pipe and only half of 
it for At = 300. This indicates of course that the contribution of the 
u+ is relatively small. Another interesting observation is that the 
differences at this value of Rt , between ut and as can be hardly 
noticed. They are bette distinguishedin Fig. 2 where velocity defects 
ut 35 + 
max 72 > Umax7u, and ea tae ae are plotted. Note that at center 
of the pipe dut/dy+ = 0 whereas du,+/dy* + 0. 
in We have also shown in Fig. 1 the distribution of a for 
R = 1000 and Rt= 100. We see that the differences between the 
velocity profiles for R+ = 10.000 and Rt = 1000 are small. Prac- 
tically the same profile is also obtained in the Newtonian case for 
Rt = 100; however, the velocity distribution for R* = 100 and 
At = 300 does not coincide any more with the other profiles which 
have larger values of Ry A, The velocity distributions according to 
the various models for Rt = 100 are plotted separately in Fig.3. We 
see from this figure that the difference between ut and u? for 
Ass = B00. ds large. Note that the velocity ut near the wall merges 
with the parabolic equation ut = yt(1- y?/2R+) whereas uj} is 
tangent to the ut= yt curve and goes above the parabolic profile. 
We have also plotted in this figure Virk's ultimate profile (Eq. 9). 
Virk's profile is quite close to u‘* but it also gives at one region 
slightly larger velocities thanin a laminar pipe flow. 
Measured velocity distributions are compared with the calcul- 
ated profiles of ut in Figs. 4 - 7. The values of At were chosen 
arbitrarily (The data is taken from Virk (1971), Fig.3, using the same 
symbols to denote the various entries.) The agreement with the data 
is very good, In particular the velocity profiles in the maximum drag 
reduction regime, Figs. 6 and 7, describe the measurements much 
better than the velocity profiles proposed by Virk's elastic sublayer 
model, 
: - : : 
* A computer program for the calculations of u and f is available 
on request from the authors. 
ave 
