60 HIGHER-ORDER ELEMENTS
torsional rigidity J obtaining
Γ4-560 for finite elements
J = 2
νάΩ =
Λ
Ω
[4-487 for boundary elements
The approximate solutions are about 10 % of the analytical values.
Notice that better agreement is obtained for the problem variable
than for the torsional rigidity, which is computed approximately from
the approximate values of the problem variables.
Table
3.1 VALUE OF TORSIONAL RIGIDITY AND V FUNCTION FOR INTERNAL POINTS
r
Finite elements Boundary elements
i
iii
x
act
/f
. /i.
(linear) (linear)
0
035
0
0-75
0
0-75
0
0-350
0-414
0-638
0-566
0-782
0-638
0-800
0-341
0-392
0-627
0-561
0-790
0-665
0-793
0-334
0-401
0-626
0-557
0-772
0-629
0-791
Rigidity 5026 4-560 4-487
Example 3.2
Figure 3.4 describes the aerofoil shape called NACA 0018 which can
be studied using boundary elements. The surface of the aerofoil was
* 100
Figure 3.4 NACA 0018 aerofoil
divided into a series of elements. Because of symmetry only half the
aerofoil needs to be considered. The problem can be expressed in
terms of a stream function u for a perfect fluid, such that the velocities
are
HIGHER-ORDER ELEMENTS 61
du du
V
*
=
Ty>
V
>
=
-Tx
(a)
The velocities far from the aerofoil are,
v
x
= V, v
y
= 0 (b)
The solution is considered to be the superposition of two solutions,
Mj for the flow without the aerofoil and u
2
for the disturbed flow
around the aerofoil,
u = u
l
+u
2
(c)
where u
1
produces the velocity field defined by (b). Considering
u
= 0
on the aerofoil equation (c) gives
u
l
=
—
u
2
for all the points on the surface. The results for the tangential velocity
are shown in Table 3.2 where they are compared against those
presented by NACA. The agreement is satisfactory.
Table
3.2
COMPARISON
OF
BOUNDARY
ELEMENTS
AND
ANALYTICAL
SOLUTIONS FOR THE TANGENTIAL VELOCITIES
NACA
0018 AEROFOIL
Solution
for (vjV)
X
00
1
25
2-5
50
7-5
10.0
150
200
250
300
400
500
60-0
700
800
900
950
1000
Analytical
0000
0-926
1103
1-228
1-267
1-276
1-278
1-275
1-262
1-247
1-205
1157
1116
1074
1025
0-966
0-914
0000
Boundary element
0000
0-931
1093
1-212
1-253
1-265
1-273
1-269
1-254
1-236
1198
1158
1118
1079
1036
0-975
0-941
0000
v is the tangential velocity.
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