The inertia force per unit volume is equal to
ρ
u
(
∂
u
/
∂
x
). For a distance
x
, the gradient
∂
u
/
∂
x
is
proportional to
U
/
x
, where
U
is the velocity
outside the B.L. Hence the inertia force is of the
order
U
2
/
x
.
The friction force per unit volume is equal to
∂
τ
/
∂
y
, which for laminar flow is equal to
μ
∂
2
u
/
∂
y
2
.
The
velocity
gradient
∂
u
/
∂
y
perpendicular to the wall is of the order
U
/
δ
, so
the friction force is of the order
μ
U
/
δ
2
.
If the two forces are to balance each other, then:
x
U
U
2
2
~
ρ
δ
μ
and solving for the B.L. thickness,
U
x
⋅
ν
δ
~

89
The numerical factor is deduced from the exact
solution given by Blasius, which is equal to 5.
Therefore, for laminar boundary layer:
U
x
⋅
=
ν
δ
5
or
x
Re
x
5
=
δ
A more accurate factor of 5.48 is found to better
match experimental data, hence:
x
Re
x
48
.
5
=
δ
where
ν
x
U
Re
x
⋅
=
9.2. Estimation of wall shear stress
The wall shear stress may be expressed as,
τ
w
=
μ
∂
u
∂
y
y
=0
so that,
τ
w
=
0.365
ρ
U
2
Re
x
and the skin friction coefficient,
C
f
, is given by,
C
f
=
τ
w
1
2
ρ
U
2
=
0.730
Re
x

90
The boundary layer thickness,
δ
, increases with
x
1/2
, while the wall shear stress and the skin
friction coefficient vary as
x
-1/2
.
These are the characteristics of a laminar
boundary layer over a flat plate.
9.3. Turbulent boundary layer
A suitable velocity profile for turbulent B.L. is
the empirical power-law profile. An exponent of
1
/
7
is used to model the velocity profile,
u
/
U
= (
y
/
δ
)
1/7
=
η
1/7
This profile doesn't hold in the immediate
vicinity of the wall, since at the wall it predicts
d
u
/d
y
=
∞
. Hence, we cannot use this profile in
the definition of
τ
w
to obtain an expression in
terms of
δ
.
For turbulent boundary-layer flow we use the
expressions developed for pipe flow,
τ
w
= 0.03325
ρ
V
2
ν
RV
1/4
For a
1
/
7
-power profile in a pipe,
V
/
U
=0.8. So,
substituting
V
=0.8
U
and
R
=
δ
one gets,

91
τ
w
= 0.00225
ρ
U
2
(
ν
/
U
δ
)
1/4
or
C
f
= 0.045(
ν
/
U
δ
)
1/4
For turbulent boundary layer,
δ
x
=
0.382
(Re
x
)
1/5
and so,
C
f
=
τ
w
1
2
ρ
U
2
=
0.0594
Re
x
1/5
Experiment show that this eq'n predicts the
turbulent skin friction on a flat plate within
about 3% for 5
⋅
10
5
<
Re
x
<10
7
.

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- Fall '08
- ZOHAR
- Fluid Dynamics, Force, Aerodynamics