Geoscience Reference
In-Depth Information
where
r
is the radial coordinate,
R
=
D/
2 is the pipe radius, and
u
max
is the
maximum (centerline) velocity. The velocity averaged over the cross section is
πR
2
R
1
u
max
2
u
ave
=
u(r)
2
πr dr
=
.
(1.2)
0
The wall shear stress is
r
=
R
=
μ
∂u
∂r
8
μ
u
ave
D
τ
wall
=−
,
(1.3)
with
μ
the dynamic viscosity of the fluid. Since
∂p/∂x
does not depend on
x
(Problem 1.1)
, we can write the axial force balance on a slug of fluid of length
L
and diameter
D
as
∂x
L
πD
2
∂P
∂P
∂x
D
τ
wall
πDL
=−
,
so that
−
=
4
τ
wall
.
(1.4)
4
The mean pressure gradient nondimensionalized with
ρ(u
ave
)
2
/
2and
D
is called
the
Darcy friction factor
,
†
∂P
−
∂x
D
4
τ
wall
f
≡
ρ(u
ave
)
2
/
2
=
ρ(u
ave
)
2
/
2
.
(1.5)
Thus
f
is, from
Eq. (1.3)
,
64
μu
ave
Dρ (u
ave
)
2
64
Re
.
f
lam
=
=
(1.6)
Figure 1.2
shows this inverse-
Re
dependence of
f
in the laminar-flow regime.
we'll discuss in detail in
Chapter 2
) we wo
rk
with their me
an
values
u
ave
and
τ
wall
.
In the turbulent regime
Eq. (1.5)
implies
τ
wall
f
turb
ρ(u
ave
)
2
/
8. Therefore the
ratio of the mean wall stress in turbulent pipe flow and the wall stress in laminar
flow at the same average velocity is
=
τ
wall
f
turb
f
turb
Re
64
τ
wall
(
laminar flow
)
=
f
lam
=
.
(1.7)
The
Fanning friction factor
is the wall stress nondimensionalized with
ρ(u
ave
)
2
/
2. The Darcy friction factor,
Eq. (1.5)
, is larger by a factor of four.
†
