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NPTEL IIT Kharagpur: Prof. K.P. Sinhamahapatra, Dept. of
Aerospace Engineering
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Module6:Bodies of Revolution
Lecture 29:Slender Body Theory
(Contd.)
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NPTEL IIT Kharagpur: Prof. K.P. Sinhamahapatra, Dept. of
Aerospace Engineering
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Hence
( )2 2 2
2
2 2 2 2
1 11 0M
x r r r r x
+ + + =
Boundary conditions
2D flow,( )
( )0022
1
2
uU
u
uU
u
x
dx
bodybody +
+=
( )
U
u 0~ 2
For a body of revolution, aligned in cylindrical coordinates,
the component of velocity w is
automatically tangent to the surface. Hence, only the boundary
condition in the meridian plane is to be
considered. In the meridian plane the body contour is ( )r R x=
. The exact condition is
RuUdx
dR
+=
Approximation similar to 2D or planar case is not applicable
Consider longitudinal section of either 2D/planar body or a body
of revolution and radial velocity near
the surface
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NPTEL IIT Kharagpur: Prof. K.P. Sinhamahapatra, Dept. of
Aerospace Engineering
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x2=
2D/planar Body of revolution
r R=
x2= 0
0r=
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NPTEL IIT Kharagpur: Prof. K.P. Sinhamahapatra, Dept. of
Aerospace Engineering
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This velocity field may be obtained by a suitable source
distribution on 02 =x or 0r= . In 2D/planer
flow, the velocity near the axis is nearly the same as at the
boundary. In axially symmetric flow the
radial velocity at the axis must be infinite if it is to be
finite on the boundary.
Using power series expansion, for 2D case
( ) ( ) ...0,, 222211211 ++++= xaxaxuUxxu
( ) ( ) ...0,, 2222112212 +++= xbxbxuxxu
Only the first term is retained for approximate boundary
condition.
Power series expansion is not possible in the axially symmetric
case since velocity gradients near the
axis are singular; because of the term ( )1
rr r
. However, this term must be of the same order as the
other terms
( )1
~ u
rr r x
or ( ) ~ u
r rr x
( ) ~ 0rr
as ( )00r r a x =
Hence, near the axis is of the order of1
r
01 2
...a
a a rr
= + + +
The correct form for an approximate boundary condition on the
axis in case of an elongated body is
( )0
R
rdR rR
dx U u U
=
+
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NPTEL IIT Kharagpur: Prof. K.P. Sinhamahapatra, Dept. of
Aerospace Engineering
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Using irrotationality,
u
r x
=
01 ...
aa
r
+ +
0 1log ...u a r a r = + +
The pressure coefficient
( )
+++
=
12
2
11
21
2
2
2
2
2
22
2
U
w
Uu
u
u
uM
MC
p
For small perturbation, using series expansion, to second
order
( )
+++=
2
22
2
22
12U
w
U
uM
U
uCp
For 2-D case, it is possible to neglect all but the first term
in the first order theory. But for elongated
axi-symmetric body, the radial component is of different order
form , near the axis, as shown
above. Since is very large near 0r= , the pressure coefficient
for axially symmetric case to the first
order accuracy is
2
2
= UU
uCp
