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Panel Methods
Source and Vortex
• Point Source
• Point Vortex
+
dz in Polar Form
x
ix
z
z+dzds
Panels
Consider a point source)(2
)(1zz
qzW
Imagine spreading the source along a line. We would then end up with a
certain strength per unit length q(s) that could vary with distance s along
the line.
sds
Example: The Source Panel (or Sheet)
z1
z
+ z1
Singularity distributed along a line
Constant Strength Source Panel
z
a
b
ds
z1vn vs
The panel is not a solid boundary to the flow. To make it behave like one you would
set the strength q so that the total vn (due to the panel and the flow) is zeroVortex?
• Break up the body surface into Nstraight panels.
• Write an expression for the normal component of velocity at the middle of the panel from the sum of all the velocities produced by the panels and the free stream. Gives N expressions.
• Given that each expression must be equal to zero, solve the N equations for the N strengths
panel
control
point
A Simple Source Panel MethodFor flow past an arbitrary body
Defining the N Panels
• We pick a control point very close to the center of the panel at
mth
control
point
za(2)
za(3)
za(N-1)
zb(2)
zb(N-1)
)(21
abbac zzizzz Center point Displacement <<1 (say 0.001)
• Number the panels anticlockwise, 1 to N
• Define N coordinates za that identify the start of every panel (going counter clockwise), and zb that identify the end of every panel.
• Each panel has a slopeab
abi
zz
zze
ds
dz
So, if W(z) is the velocity of the whole flow,
ds
dzzW )(Im is the component normal
to the panel
zc
za
zb
1
23
N
N-1
nth
panel
Completing the MethodVelocity produced by whole
flow is
N
n
n
n
b
n
ae
n
dz
ds
zz
zzqWzW
1
)(
1
)(
)(
)( log2
1)(
Velocity at the control point of
the mth panel zc(m) in panel
aligned components is
N
n
nmn
m
CqWds
dz
1
),()(
)(
1 }Im{Im
We want the normal velocity to be zero, so this is what we use to get the q’s
We write
So, velocity normal to
the mth panel is
N
n
nmn
m
CqWds
dz
1
),()(
)(
1 }Im{Im
Or 1xN result matrix
(known)
1xN matrix
of strengths
(unknown)
NxN matrix of
ceoffs (known)= x
Once we have solved
this for the q’s we can
use eqn. 2 to get the
velocities along the body
surface, or eqn. 1 to get
them anywhere else
1
2
)(
1
)(
11
)()(
)()()(
)(
1 log2
1mn
N
n
n
b
m
c
n
a
m
ce
n
m
ds
dz
dz
ds
zz
zzq
ds
dzW
),( nmC
Velocity parallel to
the mth panel is
N
n
nmn
m
CqWds
dz
1
),()(
)(
1 }Re{Re
Computational Steps
• Define coordinates of start and end of panels za and zb
• Compute the panel slopes
• Put the control points next to the panel centers
• Determine the component of W∞ normal to each panel
• Determine the influence coefficients
• Solve the matrix problem, i.e. matrix divide by
• Compute the flow velocities and pressures
ab
abi
zz
zze
ds
dz
)()( , n
b
n
a zz
)(21
abbac zzizzz
Wds
dzm)(
1Im}Im{ ),( nmC
Wds
dzm)(
1Im
)(
1
)(
1
)()(
)()(),( log
2
1m
n
n
b
m
c
n
a
m
ce
nm
ds
dz
dz
ds
zz
zzC
Matlab Code
)(
1
)()( ,,nn
b
n
a dsdzzz
)(n
cz
),( nmC
Wds
dzm)(
1Im
Matrix div.
ConstantSourcePanel.m
N
n
nmn
m
CqWds
dz
1
),()(
)(
1 }Re{Re
Result
matrix
Velocities
along body
surface
N
n
nmn
m
CqWds
dz
1
),()(
)(
1 }Im{Im
mth
control
point
1
23
N
N-1
nth
panel
)(
)(
)2(
)1(
),()1,(
),()1,(
)1,2(
),1(),1()2,1()1,1(
}Im{}Im{
}Im{}Im{
}Im{
}Im{}Im{}Im{}Im{
)(
)(
)2(
)1(
N
n
NNN
nmm
Nn
q
q
q
q
CC
CC
C
CCCC
Nres
mres
res
res
q(n)
Matlab Code Ideas:
1. Non-Uniform Free Stream
E.g. Suppose
free stream
includes, say a
doublet outside
the body at a
location x=5, so
2)5(
101
zW
ConstantSourcePanel.m
Matlab Code Ideas:
2. More than one body
E.g. Suppose
we have two
circles
ConstantSourcePanel.m
Matlab Code Ideas:
3. Use more sophisticated panels
E.g. Panels with
linearly varying
strength
ConstantSourcePanel.m
3. Linear Source Panel Method
E.g. Panels with
linearly varying
strength
LinearSourcePanel.m
3. Linear Source Panels
a
bqa
Constant
strength panels
Linear strength
panels
Influence of qb
depends on panel
ab and panel bc
Influence of qa
depends only
panel ab
a
bqa
qb
qc
c
3. Linear Source Panel Method
E.g. Panels with
linearly varying
strength
LinearSourcePanel.m
Matlab Code Ideas:
4.Change to vortex panel method
)(2)(
1zz
qzW
)(2)(
1zz
izW
LinearSourcePanel.m
N
n
nmn
m
CqWds
dz
1
),()(
)(
1 }Im{Im
mth
control
point
1
23
N
N-1
nth
panel
)(
)(
)2(
)1(
),()1,(
),()1,(
)1,2(
),1(),1()2,1()1,1(
}Im{}Im{
}Im{}Im{
}Im{
}Im{}Im{}Im{}Im{
)(
)(
)2(
)1(
N
n
NNN
nmm
Nn
q
q
q
q
CC
CC
C
CCCC
Nres
mres
res
res
q(n)
Matlab Code Ideas:
4. Vortex panel method
LinearVortexPanel.m
(see also
ConstantVortexPanel.m)
Matlab Code Ideas:
5. Set a Kutta ConditionKutta condition
requires that
surface vorticity
at trailing edge is
zero.
LinearVortexPanel.m
(see also
ConstantVortexPanel.m)
Matlab Code Ideas:
5. Kutta Condition CodeKutta condition
requires that
surface vorticity
at trailing edge is
zero.
LinearVortexPanelKutta.m
N
n
nmn
m
CqWds
dz
1
),()(
)(
1 }Im{Im
mth
control
point
1
23
N
N-1
nth
panel
)(
)(
)2(
)1(
),()1,(
),()1,(
)1,2(
),1(),1()2,1()1,1(
}Im{}Im{
}Im{}Im{
}Im{
}Im{}Im{}Im{}Im{
)(
)(
)2(
)1(
N
n
NNN
nmm
Nn
q
q
q
q
CC
CC
C
CCCC
Nres
mres
res
res
q(n)
Something to watch out for…
• The control-point equation
assumes that as you move from a to b you are progressing counter-clockwise around the body surface. (For clockwise you need to reverse the sign before i).
)(21
abbac zzizzz Center point Displacement
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