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Fundamental Theory of Panel Method. Mª Victoria Lapuerta González Ana Laverón Simavilla. Introduction. Allows one to solve the potential incompressible problem (or linearized compressible) for complex geometries. - PowerPoint PPT Presentation
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Fundamental Theory of Panel Method
Mª Victoria Lapuerta GonzálezAna Laverón Simavilla
Introduction
Allows one to solve the potential incompressible problem (or linearized compressible) for complex geometries.
Based on distributing singularities over the body’s surface and calculating their intensities so as to comply with the boundary conditions on the body.
Lowers the problem’s dimension 3D to 2D (the variables are on the wing’s surface) 2D to 1D (the variables are on the profile’s curvature
line)
Green’s Integral
1
2
, where is a class vectorial field and
is the inner normal to .
Taking:
, with and functions in
which comply 0, 0, we have:
D
C
D
F G G F F G C D
F
ds
G
A
N
A N
A
A
0F G G F ds
N
Apply Green’s integral, with:
: Velocity potential at the point of a source with unit strength located at the point P
2D:
3D:
: Velocity potential at the point
Basic Formulation
( , )m pG x x
1; log2m P P P Pf f x x x x x x x x
1;4m P P P
Pf f
x x x x x x
x x
x
( )F x x
0m mF G G F ds ds
N N
Basic Formulation Definition of the contour surfaces:
S : Infinity
surface
0m mF G G F ds ds
N N
B WS S S S
P SB : Body
surface
SW: Outflow
surface or discontinuity
N
N
N
N
N
SSphere
centered on P
Basic Formulation
S : Infinity
surface
0B W
m m m m
S S S S
ds ds
N N
P SB : Body
surface
SW: Outflow
surface
N
N
N
N
N
SSphere
centered on P
Basic Formulation
B W
m m m m
S S S S S
ds ds ds
N N N
P SB : Body
surface
SW: Outflow
surface
N
N
N
N
N
S : Infinity
surfaceSSphere
centered on P
→0
( )P x
Basic Formulation
P SB : Body
surface
SW: Outflow
surface
N
N
N
N
N
S : Infinity
surfaceSSphere
centered on P
( )B W
p m m
S S S
ds
x N
: Inner velocity potential at the point . The boundary condition is undetermined
Basic Formulation
( ), con 0 en i i iF D x
0B
i m m iSF G G F ds ds
N N
P SB : Body
surfaceN
x
Basic Formulation
P m m
S S SB W
dS
x N
0 m i i m
SB
dS N
SPSB
SW
N
N
N
N
N
S
SBN
Subtracting the second equation from the first we get:
0
( cos sin )
+
P m i i m
SB
m m m m
S SW
U x y
dS
dS dS
x N N
N N N N
Basic Formulation Green’s formula:
Potential in induced by asource distribution over the body bo
Potential in induced by adoublet distribution over the body boundaryundary
p
iP m i m
S SB B
p
dS dSn n
x x
x N
Potential in induced by adoublet distribution over the discontinuity surface
m
SW
p
dS
x
N
Source distribution
Doublet distribution
and i must satisfy the following equations:
The boundary condition for the “inner potential” is undertermined; it’s the degree of freedom that allows one to choose amoung different singularities.
Basic Formulation
0 in 0 in 0 in ? o ? in cos sin in
Kutta condition
i iB
ii B
DDS
nSU x y S n
Dirichlet’s Formulation
P n x
im i m
S SB B
m
SW
dS dSn
dS
N
N
The integral equation must be solved by making the point P tend toward the surface of the body
Potential of a doublet: d
im P i d P
SB
d P
SB
SW
P dS dSn
dS
x x x x
x
x x x
x
x
Source distribution
Doublet distribution
Potential of a doublet: d
Dirichlet’s Formulation
( )
im P i d P
S B
S
B S
d P
P
W
dS dSn
dS
x x x x x x
x x
x x
• Choosing the inner potential as zero:
Doublet distribution
S S
P d
W
d P P
B
dS dS x x xx x x
Neumann Formulation
The derivative of the equation is taken to calculate the perpendicular velocity on the body ( ) and then it is set to zero. The variables are: and . If the inner potential is zero, the only variable is
(doublet distribution).
Information about the constant part of the potential is lost
P n x
im i m
S SB B
S S
m
SW
m
B B
PP d P
SW
dS dSn
dS
dS dS
x x x xx
N
N
x x
x x
P V x x
