fluid_mechanics

406 Chapter 8 ■ Viscous Flow in Pipes
25
Experimental data
20
u ___
u*
15
10
Eq. 8.29
Eq. 8.30
Turbulent effects
Pipe
centerline
5
Viscous
sublayer
Viscous and
turbulent effects
0
1 10 10
2
yu* ____
v
10 3 10 4
F I G U R E 8.16
Typical structure of the
turbulent velocity profile in
a pipe.
A turbulent flow velocity
profile can be
divided into various
regions.
V8.8 Laminar to
turbulent flow from
a pipe
where y R r is the distance measured from the wall, u is the time-averaged x component of velocity,
and u* 1t wr2 1 2
is termed the friction velocity. Note that u* is not an actual velocity of the
fluid—it is merely a quantity that has dimensions of velocity. As is indicated in Fig. 8.16, Eq. 8.29
1commonly called the law of the wall2 is valid very near the smooth wall, for
Dimensional analysis arguments indicate that in the overlap region the velocity should vary
as the logarithm of y. Thus, the following expression has been proposed:
u
2.5 ln ayu*
u*
(8.30)
where the constants 2.5 and 5.0 have been determined experimentally. As is indicated in Fig. 8.16,
for regions not too close to the smooth wall, but not all the way out to the pipe center, Eq. 8.30
gives a reasonable correlation with the experimental data. Note that the horizontal scale is a logarithmic
scale. This tends to exaggerate the size of the viscous sublayer relative to the remainder of
the flow. As is shown in Example 8.4, the viscous sublayer is usually quite thin. Similar results
can be obtained for turbulent flow past rough walls 1Ref. 172.
A number of other correlations exist for the velocity profile in turbulent pipe flow. In the central
region 1the outer turbulent layer2 the expression 1V c u2u* 2.5 ln1Ry2, where V c is the centerline
velocity, is often suggested as a good correlation with experimental data. Another often-used
1and relatively easy to use2 correlation is the empirical power-law velocity profile
u
V c
a1 r R b 1n
n b 5.0 0 yu*n f 5.
(8.31)
In this representation, the value of n is a function of the Reynolds number, as is indicated in
Fig. 8.17. The one-seventh power-law velocity profile 1n 72 is often used as a reasonable approximation
for many practical flows. Typical turbulent velocity profiles based on this power-law
representation are shown in Fig. 8.18.
A closer examination of Eq. 8.31 shows that the power-law profile cannot be valid near the
wall, since according to this equation the velocity gradient is infinite there. In addition, Eq. 8.31
cannot be precisely valid near the centerline because it does not give dudr 0 at r 0. However,
it does provide a reasonable approximation to the measured velocity profiles across most of
the pipe.
Note from Fig. 8.18 that the turbulent profiles are much “flatter” than the laminar profile
and that this flatness increases with Reynolds number 1i.e., with n2. Recall from Chapter 3 that