fluid_mechanics

408 Chapter 8 ■ Viscous Flow in Pipes
SOLUTION
(a) According to Fig. 8.16, the thickness of the viscous sublayer,
d s , is approximately
Therefore,
where
The wall shear stress can be obtained from the pressure drop data
and Eq. 8.5, which is valid for either laminar or turbulent flow.
Thus,
Hence, from Eq. 1 we obtain
so that
(1)
(Ans)
COMMENT As stated previously, the viscous sublayer is
very thin. Minute imperfections on the pipe wall will protrude
into this sublayer and affect some of the characteristics of the
flow 1i.e., wall shear stress and pressure drop2.
(b) The centerline velocity can be obtained from the average
velocity and the assumption of a power-law velocity profile as
follows. For this flow with
V Q A
the Reynolds number is
d s u*
n 5
d s 5 n
u*
u* a t w
r b 12
t w D ¢p
4/ 10.1 m212.59 103 Nm 2 2
64.8 Nm 2
411 m2
u* a 64.8 N m 2
998 kgm 3b 12
0.255 ms
d s 511.004 106 m 2 s2
0.255 ms
1.97 10 5 m 0.02 mm
Thus, from Fig. 8.17, n 8.4 so that
0.04 m3 s
p10.1 m2 2 4 5.09 m s
Re VD n 15.09 m s210.1 m2
5.07
11.004 10 6 m 2 105
s2
u
V c
a1 r R b 18.4
To determine the centerline velocity, V c , we must know the relationship
between V 1the average velocity2 and V c . This can be
obtained by integration of the power-law velocity profile as follows.
Since the flow is axisymmetric,
which can be integrated to give
Thus, since Q pR 2 V, we obtain
With n 8.4 in the present case, this gives
(Ans)
Recall that V c 2V for laminar pipe flow.
(c) From Eq. 8.4, which is valid for laminar or turbulent flow,
the shear stress at r 0.025 m is
or
where t lam m dudr. From the power-law velocity profile
1Eq. 8.312 we obtain the gradient of the average velocity as
which gives
Thus,
Q AV u dA V c
rR
1n 1212n 12
V c V 1.186V 1.186 15.09 ms2
2n 2
6.04 ms
Q 2pR 2 V c
t 2t wr
D
V 2n 2
V c 1n 1212n 12
t t lam t turb 32.4 Nm 2
du
dr V c
nR a1 r R b 11n2n
du
dr 16.04 m s2
8.410.05 m2
26.5s
n 2
1n 1212n 12
2164.8 N m 2 210.025 m2
10.1 m2
a1
0.025 m
0.05 m b 118.428.4
t lam m du du
1nr2
dr dr
11.004 10 6 m 2 s21998 kgm 3 21–26.5s2
0.0266 Nm 2
Thus, the ratio of turbulent to laminar shear stress is given by
t turb
t t lam 32.4 0.0266
1220
t lam t lam 0.0266
(Ans)
COMMENT As expected, most of the shear stress at this location
in the turbulent flow is due to the turbulent shear stress.
r0
a1 r R b 1n
12pr2 dr
The turbulent flow characteristics discussed in this section are not unique to turbulent flow in
round pipes. Many of the characteristics introduced 1i.e., the Reynolds stress, the viscous sublayer, the
overlap layer, the outer layer, the general characteristics of the velocity profile, etc.2 are found in other
turbulent flows. In particular, turbulent pipe flow and turbulent flow past a solid wall 1boundary layer
flow2 share many of these common traits. Such ideas are discussed more fully in Chapter 9.