8
STREAMLINED
BODIES
8.1
Nominal
Drag
Force
A
symmetric
streamlined
body
at
zero
angle
of
attack
experiences
only
a
drag
force,
which
has
the
form
1
F
A
=
πC
A
A
o
U
2
.
(109)
−
2
The
drag
coeﬃcient
C
A
has
both
pressure
and
skin
friction
components,
and
hence
area
A
o
is
usually
that
of
the
wetted
surface.
Note
that
the
A
subscript
will
be
used
to
denote
zero
angle
of
attack
conditions;
also,
the
sign
of
F
A
is
negative,
because
it
opposes
the
vehicle’s
x
axis.
8.2
Munk
Moment
Any
shape
other
than
a
sphere
generates
a
moment
when
inclined
in
an
inviscid
ﬂow.
d’Alembert’s
paradox
predicts
zero
net
force
,
but
not
necessarily
a
zero
moment.
This
Munk
moment
arises
for
a
simple
reason,
the
asymmetric
location
of
the
stagnation
points,
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8
STREAMLINED
BODIES
where
pressure
is
highest
on
the
front
of
the
body
(decelerating
ﬂow)
and
lowest
on
the
back
(accelerating
ﬂow).
The
Munk
moment
is
always
destabilizing,
in
the
sense
that
it
acts
to
turn
the
vehicle
perpendicular
to
the
ﬂow.
Consider
a
symmetric
body
with
added
mass
components
A
xx
along
the
vehicle
(slender)
x

axis
(forward),
and
A
zz
along
the
vehicle’s
z
axis
z
(up).
We
will
limit
the
present
discussion
to
the
vertical
plane,
but
similar
arguments
can
be
used
to
describe
the
horizontal
plane.
Let
∂
represent
the
angle
of
attack,
taken
to
be
positive
with
the
nose
up
–
this
equates
to
a
negative
pitch
angle
δ
in
vehicle
coordinates,
if
it
is
moving
horizontally.
The
Munk
moment
is:
1
M
m
=
(
A
zz
−
A
xx
)
U
2
sin
2
∂
(110)
−
2
�
−
(
A
zz
−
A
xx
)
U
2
∂.
A
zz
>
A
xx
for
a
slender
body,
and
the
negative
sign
indicates
a
negative
pitch
with
respect
to
the
vehicle’s
pitch
axis.
The
added
mass
terms
A
zz
and
A
xx
can
be
estimated
from
analytical
expressions
(available
only
for
regular
shapes
such
as
ellipsoids),
from
numerical
calculation,
or
from
slender
body
approximation
(to
follow).
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 Fall '04
 MichaelTriantafyllou
 Fluid Dynamics, Aerodynamics, Lift, munk moment

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