Panel.Method

Panel.Method - Chapter IV. HandOut #1 Panel Methods Panel...

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Chapter IV. HandOut #1 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 + = 0 (1) and, irrotationality: u y v x - = 0 (2) One can define a velocity potential Φ such that
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∂φ x u y v = = ; (3) This equation satisfies the irrotationality. Continuity equation becomes: 2 2 2 2 0 x y + = (4) One can also define a stream function ψ such that ∂ψ y u y v = = ; - (5) which yields the following relation: 2 2 2 2 0 x y + = (6) Equations (3) and (6) are each called Laplace’ equation. In subsonic compressible flow, (see AE 4001) the potential flow equation
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Panel.Method - Chapter IV. HandOut #1 Panel Methods Panel...

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