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Fundamental Theory of Panel Method

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

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

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

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

  8. S :Infinity surface Basic Formulation 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. P SB Se SW S Basic Formulation • Subtracting the second equation from the first we get: SB

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

  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. 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

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

  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

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