7
Flow Past a SphereUniform flow + opposing doublet
0 90 180 270 360
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
Cp
SphereCylinder
8
Flow past sphere Re=300
Re=15000
Re=30000
ON
ER
A ph
otog
raph
s, W
erle
198
0
-3 -2 -1 0 1 2 3-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Ideal Flow, AOE 5104
9
Lines/FilamentsSource, doublet or vortex distributed along a curved or straight line – source and doublet analysis analogous to panel
ds
r1
r
V
O
Filament path denoted by curve C
14 rr
qPoint source
10
Vortex FilamentMust obey Helmholtz’s Laws so filament strength must be constant, and filament must be looped or extend to infinity
ds
r1
r
V
O
Velocity field is determined from the Biot Savart Law:
The Biot Savart Law cannot be inferred from simple integration since there is no comparable point singularity. Instead it is determined from considering the general problem of determining a velocity field from a given vorticity field.
C
d3
1
11
||)()(
4)(
rrrsrrrV
Filament path denoted by curve C
11
Biot Savart Law1. How to invert
? VΩ
0. V2. For incompressible flow so we may write V as the curl of a vector potential AV
3. SoAAAAΩ 22).(
4. Which has the solution
spaceAll
d||
)()(41)(
1
11
rrrrrA
5. Which differentiates to
spaceAll
d3
1
111
||)()()(
41
rrrrrrAV
ds
r1
r
V
O
Curve C
6. Which, when applied to the singular vorticity field of a filament gives
C
d3
1
11
||)()(
4)(
rrrsrrrV
0. AChoose
12
Example: Velocity induced by a section of a straight filament
C
d3
1
11
||)()(
4)(
rrrsrrrV
r1
r
V
O
Curve C
dshs
2
1e
PanelsSource or doublet distributed over finite (often flat) sheet or ‘panel’
dS r1
r
V
O
Point doublet
31
1
||4).(
rrrrμ
n
Area SPerimeter C
ds
15
Constant Strength Doublet PanelSame flow as a vortex filament ring around the panel perimeter
r1 r
V
O
ds
C
d3
1
11
||)()(
4)(
rrrsrrrV
sw
nw
se
ne
=
Therefore for a quadrilateral panel,
),,(),,(
),,(),,()(
nwnefilnesefil
seswfilswnwfil
rrrfrrrf
rrrfrrrfrV
This makes doublet / vortex ring panels ideal for 3D panel methods, since their velocity fields are easy to compute. Since they contain a vortex element they can also be extended to situations (like wings) where vorticity is shed into the wake.