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Chapter IV. HandOut # Panel Methods
Panel methods are techniques for solving potential flow over 2-D and 3-D 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 2-D flows, and in particular to potential flow over single and multi-element airfoils. Some papers will be given (or cited) where 3-D 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 2-D, incompressible, irrotational flow are:
Continuity: ∂ ∂
∂ ∂
u x
v y
and, irrotationality:
u y
v x
One can define a velocity potential Φ such that ∂φ ∂
∂φ x ∂
u y
= ; = v
(3) This equation satisfies the irrotationality. Continuity equation becomes:
2 2
2
x^ +^ y^2 =^0 (4) One can also define a stream function ψ such that
y ∂
u y
= ; - = v
(5) which yields the following relation:
2 2
2
x^ +^ y^2 =^0 (6) Equations (3) and (6) are each called Laplace’ equation.
In subsonic compressible flow, (see AE 4001) the potential flow equation is modified to give the following, approximate equation:
( 1 2 ) 0
2 2
2 − M (^) ∞ x + (^) y 2 =
Assuming one can solve for either the velocity potential or the stream function and its derivatives (which yield the flow velocities u and v), the pressure can be computed for incompressible flows, from Bernoulli’equation as:
p + ( u + v )= p (^) ∞ + ∞ V ∞
ρ^2 2 ρ^2
From pressure, a non-dimensional quantity called the pressure coefficient may be computed:
C
p p
V
u v V
V
p (^) V
∞ ∞ ∞^ ∞
2
2 2 2
2 2 ρ
(9) where u and v are Cartesian components of velocity V. This pressure coefficient may be then integrated to compute loads on the airfoil or other components such as flaps and slats.
θ
ln r
v
r
v
r r
r
v
v r Circulation v rd
r =
=
θ (^2)
θ
Pay attention to the signs. A counterclockwise vortex is considered “positive” in the twisted world of fluid dynamics! Here ‘r’ represents the distance between the position of the vortex, and the point where the velocity is evaluated. If the vortex had been placed at a location r 0 (where r 0 represents a point in the
x-y plane with coordinates x 0 and y 0 ), then the stream function at a point r (with coordinates x and y) would be
( ) (^ )^ (^ )
0
2 0
2
ln r r
where
r r x x y y
o
o
As in the case of sources and sinks, a number of vortices can be superposed to produce any flow field. At the center of the vortex, the velocities are singular.
Because these solutions are singular, they should be made as weak as possible, to produce a smooth flow field. In the panel method we use here, we
therefore make these vortices to be of infinitesimal strength γ 0 ds 0 , where ds 0 is
the length of a small segment of the airfoil, and γ 0 is the vortex strength per unit length. Then, the stream function due to all such infinitesimal vortices at a point
in space may be written as: ( )
0
∫ 2 ln^ r^ − r 0^ ds 0 , where the integral is done over all
the vortex elements on the airfoil surface.
The freestream itself has a stream function associated with it, given by u (^) ∞ y − v x ∞ , where u ∞ and v ∞ are the x- and y- components of the freestream
velocity. Adding all these effects, the stream function at any point r in space is given by
= u ∞ y − v ∞ x − ∫γ r − r dso
ln
(13) If we consider only those points that are on the airfoil, then due to the fact that the body itself is a streamline, the stream function value at all these points will be a constant value C. Then, for all points r on the airfoil surface, equation (13) becomes
u y v x r r ds C
or
u y v x r r ds C
o
o
∞ ∞
∞ ∞
0 0
0 0
ln
ln
(14) Physical meaning of γ:
Before we go too far, we need to associate a physical interpretation with the abstract quantity γ 0. Consider a small segment ds 0 on the airfoil surface.
Then γ 0 ds 0 represents the vortex strength associated with this segment, or more
precisely its circulation. This circulation is caused by velocity field surrounding this element:
Ouside the airfoil
Inside the airfoil If we think of the interior of the airfoil as “dead” air, and if we allow the height of the box shown above to shrink to zero, then only the top part will contribute to the circulation surrounding this box. That is,
We also assume that the vortex strength γ 0 is a piecewise constant on each of these panels. Then, we can approximate the line integral over the entire airfoil as several line integrals over the N panels, over each of which γ 0 is a constant. Then, equation 17 becomes:
u yi v x i ( r r ) ds C
j i o j N j
∞ ∞
0 0 1 2
..
ln
(18) Notice that we use two indices ‘i’ and ‘j’. The index ‘I’ refers to the control point where equation (14) is applied. The index ‘j’ refers to the panel over which the line integral is evaluated. The integrals over the individual panels depends only on the panel shape (straight line segment), its end points and the control point í’. Therefore this integral may be computed analytically. We refer to the resulting quantity as
Ai j , = Influence of Panel j on index i = ∫ln ( ri − r ) ds
Equation (18) then becomes:
u y (^) i v xi Ai j (^) j C j
N ∞ ∞ =
1
One such equation may be written at each of the N control points. The ‘N+1’th equation is the Kutta Condition :
γ 01 + γ 0 N = 0
Equation (20) and (21) represent N+1 equations for the unknowns: the constant C which represents the stream function value on the airfoil surface, and the N values of the vortex strength γ 0. This system may be easily inverted and solved.
Extension to Multi-element Airfoils:
Multi-element airfoils can be analyzed in a similar manner. For example, for a two element airfoil-flap system, there will be N+2 equations, since there are N panels and (corresponding vortex strength) , and two values of streamfunction C 1 and C 2 for the two airfoils. We will have two Kutta conditions, one for the main airfoil and the second for the second element (flap or slat).
