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Lecture
12
 Marine
Hydrodynamics
Lecture
12
3.14
Lifting
Surfaces
3.14.1
2D
Symmetric
Streamlined
Body
No
separation,
even
for
large
Reynolds
numbers.
U
stream line
•
Viscous
eﬀects
only
in
a
thin
boundary
layer.
•
Small
Drag
(only
skin
friction).
•
No
Lift.
1
2.20  Marine Hydrodynamics, Spring 2005
2.20
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View Full Document 3.14.2
Asymmetric
Body
(a)
Angle
of
attack
α
,
chord line
U
α
(b)
or
camber
η
(
x
),
chord line
mean camber line
U
(c)
or
both
amount of camber
U
α
angle of attack
mean camber line
chord line
Lift
⊥
to
U
±
and
Drag
±
to
U
±
2
3.15
Potential
Flow
and
Kutta
Condition
From
the
PFlow
solution
for
ﬂow
past
a
body
we
obtain
PFlow
solution,
inﬁnite
velocity
at
trailing
edge.
Note
that
(a)
the
solution
is
not
unique
 we
can
always
superimpose
a
circulatory
ﬂow
without
violating
the
boundary
conditions,
and
(b)
the
velocity
at
the
trailing
edge
→
∞
.
We
must
therefore,
impose
the
Kutta
condition,
which
states
that
the
‘ﬂow
leaves
tangentially
the
trailing
edge,
i.e.,
the
velocity
at
the
trailing
edge
is
ﬁnite’
.
To
satisfy
the
Kutta
condition
we
need
to
add
circulation.
Circulatory
ﬂow
only.
Superimposing
the
PFlow
solution
plus
circulatory
ﬂow,
we
obtain
Figure
1:
PFlow
solution
plus
circulatory
ﬂow.
3
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View Full Document 3.15.1
Why
Kutta
condition
?
Consider
a
control
volume
as
illustrated
below.
At
t
=
0,
the
foil
is
at
rest
(top
control
volume).
It
starts
moving
impulsively
with
speed
U
(middle
control
volume).
At
t
=0
+
,
a
starting
vortex
is
created
due
to
ﬂow
separation
at
the
trailing
edge.
As
the
foil
moves,
viscous
eﬀects
streamline
the
ﬂow
at
the
trailing
edge
(no
separation
for
later
t
),
and
the
starting
vortex
is
left
in
the
wake
(bottom
control
volume).
t = 0
+
t = 0
for later
t
Kelvin’s
theorem:
After
a
while
the
Γ
S
Γ
Γ
Γ
Γ
S
S
S
S
U
U
no
Γ
starting vortex left in wake
Γ = 0
starting vortex
due to separation
(a real fluid effect,
no infinite vel of
potetial flow)
d
Γ
→
Γ = 0
for
t
≥
0i
fΓ(
t
=0)=0
dt
in
the
wake
is
far
behind
and
we
recover
Figure
1.
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.20 taught by Professor Dickk.p.yue during the Spring '05 term at MIT.
 Spring '05
 DickK.P.Yue

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