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 irrotattional. (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. 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, all together formally adding to zero, as illustrated in the above left picture. The right picture shows Prandtl’s conception of streamwise vorticity issued along the wing compensating for variation in transversal vorticity, again from an idea of closed circuits of vorticity all 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 created on the surface from opposing flow instability, while in the Lifting Line Theory they appear 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. Helmholz 1 is questionable, 2 is ok and 3 is illposed. Lifting Line Theory is based on 3.

One may say that the New Theory is based on unstable physics, while Lifting Line Theory is based on ill posed formal mathematics. Unstable physics is real, while unstable formal mathematics is unreal.

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