ME 309 Fall 2008
Section 2
Homework # 32
Due Wednesday 19 November 2008
Problem 321
a)
Show that the log law of the boundary layer:
5
.
5
ln
5
.
2
+
=
+
+
y
u
can
be expressed in terms of skin friction and the Reynolds number based on the
boundary layer thickness as:
5
.
5
2
Re
ln
5
.
2
2
+
=
f
f
C
C
δ
.
To do this, evaluate the
log law at
y
=
δ
where the velocity equals the free stream value.
b.
This equation between the Reynolds number and the skin friction coefficient can be
approximately fit to the more accessible relation,
6
/
1
Re
02
.
0

=
δ
f
C
.
Using this skin
friction relation in the momentum integral relation for a flat plate (zero pressure
gradient) along with a oneseventh power law velocity profile, find relations for
δ
,
θ,
and C
f
as function of Re
x
.
Also find a relation between the local skin friction
coefficient at the end of the plate, C
fL
and the drag coefficient for the entire plate
length, C
D
.
Solution:
Given Loglaw profile:
5
.
5
ln
5
.
2
+
=
+
+
y
u
Evaluate loglaw at boundary layer edge,
δ
where the velocity equals its free stream value
5
.
5
ln
5
.
2
+
=
∞
ν
δ
τ
τ
u
u
u
but:
5
.
5
ln
5
.
2
2
2
+
=
=
∞
∞
∞
∞
u
u
u
u
u
w
w
ρ
τ
ν
δ
τ
ρ
τ
Thus, log law immediately give:
5
.
5
2
Re
ln
5
.
2
2
+
=
f
f
C
C
δ
Curve fit of C_f vs Re_delta:
6
/
1
Re
02
.
0

=
δ
f
C
Combine with Karman momentum (for zero pressure gradient)
dx
d
C
f
θ
=
2
:
Use power law to describe turbulent velocity profile:
7
/
1
=
∞
δ
y
u
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 Spring '08
 MERKLE
 Force, Aerodynamics

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