torsdag 15 november 2012

Lifting Line Theory Illposed

Circulation theory in the form of the Prandtl/Lanchester Lifting Line Theory relies on the following theorems by Helmholtz concerning "vortex filaments" or lines of concentrated vorticity:
  • Theorem 1: The strength of a vortex filament is constant along its length.
  • Theorem 2: A vortex filament cannot end in the fluid; it must extend to the boundaries of the fluid or form a closed path.
  • Theorem 3: In the absence of of rotational forces, a fluid that is initially irrotational remains irrotational. (Kelvin's theorem).
Here, Theorem is 3 is not wellposed and thus does not describe physics: An infinitesimal perturbation can give rise to substantial vorticity without rotational forcing, as shown in the Basic Principle of New Theory of Flight
Nevertheless, Lifting Line Theory is based on Theorem 3 with a closed lifting line of vorticity formed by (i) transversal vorticity along the wing, (ii) streamwise vorticity extending from the wing tips (iii) transversal starting vortex, formally adding to zero as illustrated in the above left picture. The right picture shows Prandtl's conception of streamwise vorticity from the trailing edge compensating variation in transversal vorticity again from an idea of closed circuits of vorticity summing to zero
The streamwise vorticity emerging from the airfoil compensating variation of transversal vorticity in Lifting Line Theory superficially connects to the counter-rotating vortex lines of 3d rotational slip separation of the New Theory attaching to the trailing edge, but the physics is entirely different:
In the New Theory "vortex filaments" are really created on the surface from opposing flow instability, while in the Lifting Line Theory they appear formally from variation of transversal vorticity as a consequence to Helmholtz's (illposed) Theorem 3.
Note that the "vortex filaments" of streamwise vorticity attaching to the trailing edge in the New Theory conforms to Helmholtz's Theorem 2 with "vortex filaments" allowed to start from the boundary, as a correct statement. Helmholtz Theorem 1 is questionable, Theorem 2 is ok and Theorem 3 is illposed. Lifting Line Theory is based on Theorem 3.
One may say that the New Theory is based on unstable physics, while Lifting Line Theory is based on illposed formal mathematics. Unstable physics is real, while unstable formal mathematics is unreal.

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