Chapter IV.
HandOut #1
Panel Methods
Panel methods are techniques for solving potential flow over 2D and 3D
geometries.
The governing equation (Laplace’s equation, or the linearized form
in compressible flow) is recast into an integral equation. This integral equation
involves quantities such as velocity, only on the surface, whereas the original
equation involved the velocity potential
Φ
all over the flow field. The surface is
divided into panels or “boundary elements”, and the integral is approximated by
an algebraic expression on each of these panels. A system of linear algebraic
equations result for the unknowns at the solid surface, which may be solved
using techniques such as Gaussian elimination to determine the unknowns at the
body surface.
In some publications, especially from Europe, panel methods are referred
to as
boundary element
methods. Panel methods have been the workhorse of
the aircraft industry since the 1960s until today because:
i) They can handle complex configurations such as a complete aircraft, or even
a 747 aircraft + space shuttle configuration.
ii) They are fast compared to “field” methods or finite difference methods that
compute the flow properties in the entire field surrounding the aircraft.
iii) They are the only techniques that can quickly predict interference effects
between the various aircraft components such as stores, pylons, nacelle, jet
exhaust, etc. The speed and reliability is appreciated by designers who have
to parametrically analyze a number of configurations.
In this chapter we will restrict ourselves to 2D flows, and in particular to
potential flow over single and multielement airfoils. Some papers will be given
(or cited) where 3D panel methods are discussed. Later on, we will address how
these analyses can be turned into design tools, and study how to model viscous
effects on airfoil/aircraft performance.
Governing Equations
The equations governing 2D, incompressible, irrotational flow are:
Continuity:
∂
u
x
v
y
+ =
0
(1)
and, irrotationality:
u
y
v
x
 =
0
(2)
One can define a velocity potential
Φ
such that
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View Full Document∂φ
∂
x
u
y
v
= =
;
(3)
This equation satisfies the irrotationality. Continuity equation becomes:
2
2
2
2
0
x y
+ =
(4)
One can also define a stream function
ψ
such that
∂ψ
y
u
y
v
= =
;

(5)
which yields the following relation:
2
2
2
2
0
x y
+ =
(6)
Equations (3) and (6) are each called
Laplace’ equation.
In subsonic compressible flow, (see AE 4001) the potential flow equation
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 Fall '08
 Yeung
 Fluid Dynamics, stream function, Vθ

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