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1
BoundaryLayer Thickness,
δ
This is a very thin layer in the immediate neighborhood of the body where the fluid
viscosity exerts an essential influence, in the sense that the shearing stresses are important
here.
Displacement Thickness,
δ
*
This is a more meaningful measure for boundarylayer thickness.
It is the distance by
which the external flow
streamlines are displaced outwards as a consequence of the
formation of the boundary layer, i.e., due to the decrease in velocity in the boundary
layer.
Decrease in volume flow rate is U
o
δ
* which is also equal to and replaced by:
Momentum Thickness,
δ
m
The loss of momentum in the boundary layer is expressed as:
Note:
The upper limit in the above integrals can be replaced by
∞
, since the contribution
to the integrand, and hence to the resulting integral, from the portion beyond
δ
is
negligible.
e
U
99
.
o
u

y
=
=
δ
∫∫
∫
δδ
δ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
δ
⇒
−
=
δ
⇒
−
=
δ
00
o
o
o
0
o
o
dy
U
u
1
*
dy
U
)
u
U
(
*
dy
)
u
U
(
*
U
∫
∫
δ
δ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
δ
∴
−
ρ
=
δ
ρ
0
2
o
2
o
m
0
o
m
2
o
dy
U
u
U
u
dy
)
u
U
(
u
U
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Given
: Velocity profiles for laminar boundary layer.
Find
:
*
δ
δ
for each profile
Definition
:
00
uu
*1d
y1d
y
UU
∞δ
⎛⎞
δ=
−
−
⎜⎟
⎝⎠
∫∫
±
Solution
:
()
1
*1
u
1d
y
1
u
d
y
U
δ
δ
=−=
−
δδ
²²
or
1
0
*
1ud
δ
=−
η
δ
∫
²
where
yu
y, u
U
η
δ
³±
³
(1) Laminar: u
y
==
η
1
2
1
0
0
1
d
0.500
22
⎡⎤
δη
η
η
=
η
−
=
=
⎢⎥
δ
⎣⎦
∫
(2) Parabolic:
2
u2
=η
−η
²
1
3
1
0
0
12
d
0
.
3
3
3
33
∴
η
+
η
η
=
η
−
η
+
=
=
δ
∫
(3) Cubic:
3
31
u
²
1
1
32
4
0
0
*3
1
3
1
3
1
d
0.375
4
8
8
δ
⎡
⎤
∴
η
+
η
η
=
η
−
η
+
η
=
=
δ
⎣
⎦
∫
(4) Sinusoidal:
us
i
n
2
π
²
1
1
0
0
*2
2
1
sin
d
cos
1
0.363
δπ
η
π
η
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This note was uploaded on 10/13/2010 for the course MECH 414 taught by Professor Ghia during the Winter '06 term at University of Cincinnati.
 Winter '06
 Ghia
 Shear, Stress

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