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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.
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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. • 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)
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
P SB : Body surface Se: Sphere centered on P SW: Outflow surface or discontinuity S :Infinity surface Basic Formulation • Definition of the contour surfaces:
P SB : Body surface Se: Sphere centered on P SW: Outflow surface S :Infinity surface Basic Formulation
S :Infinity surface Basic Formulation P SB : Body surface Se: Sphere centered on P SW: Outflow surface e→0
S :Infinity surface Basic Formulation 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
P SB Se SW S Basic Formulation • Subtracting the second equation from the first we get: SB
Source distribution Doublet distribution Basic Formulation • Green’s formula:
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.
Source distribution Doublet distribution Dirichlet’s Formulation Potential of a doublet: Fd Potential of a doublet: Fd • The integral equation must be solved by making the point P tend toward the surface of the body
Doublet distribution Dirichlet’s Formulation • Choosing the inner potential as zero: =G
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