fluid_mechanics

8.3 Fully Developed Turbulent Flow 407
11
10
9
n
8
7
6
5
10 4 10 5 10 6
Re = ____ ρ VD
μ
F I G U R E 8.17 Exponent, n, for power-law velocity profiles.
(Adapted from Ref. 1.)
1.0
n = 10
n = 6
Laminar
n = 8
r __
R
0.5
V8.9 Laminar/
turbulent velocity
profiles
Turbulent
0
0 0.5 1.0
_
__ u
Vc
F I G U R E 8.18
Typical laminar flow and
turbulent flow velocity
profiles.
reasonable approximate results are often obtained by using the inviscid Bernoulli equation and by
assuming a fictitious uniform velocity profile. Since most flows are turbulent and turbulent flows
tend to have nearly uniform velocity profiles, the usefulness of the Bernoulli equation and the uniform
profile assumption is not unexpected. Of course, many properties of the flow cannot be accounted
for without including viscous effects.
E XAMPLE 8.4
Turbulent Pipe Flow Properties
GIVEN Water at 20 °C 1r 998 kgm 3 and n 1.004
10 6 m 2 s2 flows through a horizontal pipe of 0.1-m diameter
with a flowrate of Q 4 10 2 m 3 s and a pressure gradient of
2.59 kPa m.
(b) Determine the approximate centerline velocity, V c .
(c) Determine the ratio of the turbulent to laminar shear stress,
t turbt lam , at a point midway between the centerline and the pipe
wall 1i.e., at r 0.025 m2.
FIND (a) Determine the approximate thickness of the viscous
sublayer.