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. 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

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

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

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

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

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

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

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

Basic Formulation Green’s formula: Source distribution Doublet distribution

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.

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

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

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