1. Fundamental Theory of Panel Method
Mª Victoria Lapuerta González
Ana Laverón Simavilla

2. 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)

3. Green’s Integral

4. Basic Formulation 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

5. Basic Formulation Definition of the contour surfaces:P
SB : Body surface
SW: Outflow
surface or discontinuity
Se: Sphere centered on P
S : Infinity surface

6. Se: Sphere centered on P
Basic Formulation P
SB : Body surface
SW: Outflow surface
Se: Sphere centered on P
S : Infinity surface

7. Se: Sphere centered on P
Basic Formulation
S : Infinity surface P
SB : Body surface
Se: Sphere centered on P
SW: Outflow surface
e→0

8. Se: Sphere centered on P
Basic Formulation
S : Infinity surface P
SB : Body surface
Se: Sphere centered on P
SW: Outflow surface

9. Basic Formulation P
SB : Body surface
: Inner velocity potential at the point The boundary condition is undetermined

10. Basic Formulation S P SB SW Se SB
Subtracting the second equation from the first we get:

11. Basic Formulation Green’s formula: Source distribution
Doublet distribution

12. Basic Formulation F and Fi 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.

13. Dirichlet’s Formulation
Potential of a doublet: Fd
Potential of a doublet: Fd
Source distribution
Doublet distribution
The integral equation must be solved by making the point P tend toward the surface of the body

14. Dirichlet’s Formulation
Choosing the inner potential as zero: =G
Doublet distribution

15. 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