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the Prandtl hyper-singular integral operatorTheboundary layer circulation modelling challenge is caused by the Euler fluid model restriction allowing only potential flows. The lift of an airfoil in inviscid flow requires circulation in the flow around the airfoil, but a single potential function that is continuous throughout the domain around the airfoil cannot represent a flow with non-zero circulation. Therefore potential flow theory requires special treatment and an additional assumption which was formulated first by L. Prandtl. The Prandtl operator with H(1/2) domain enables a dynamic fluid accompanied by frictional forces, which is applicable as mathematical tool for exterior space problems in physics. Note: For a closed connected surface in the three-dimension space the Noetherian Prandtl operator as a mapping from H(r) to H(r-1) is defined by the double layer potential providing the unique solution of the Neumann boundary value problem for values r which are greater or equal than 1/2, and less than 1, (LiI).
From (GlH) we summarize (not quote) the following (1.2): "The development of an airfoil lift theory goes back to Kutta, Joukowski (n=2) and Lancester, Prandtl (n=3). The theory delivers results, which matches quite good with experimental results. However it remains the difficulty to explain the origin of the circulation, based on the assumption that the "fluid" (air) is considered as homogeneous medium, whereby in its equilibrium state no shear stresses can be transmitted. The viscosity of air can be neglected in many cases. Considering air as an incompressible, frictonless medium then a body, which moves through this ideal "liquid" does not experience any resulting forces (Helmholtz, Kirchhoff). This miss match between theoretical assumptions and experimental results manifests in the question of the root cause of the circulation."
From Trefftz R., §1, we quote: "(man) erhält einen Auftrieb, wenn man in die Fläche einen Wirbelfaden legt, der eine Zirkulation um den Flügel hervorruft, ... mathematisch gesprochen, wenn man eine Mehrdeutigkeit des Potentials in den die Tragfläche umgebenden zweifach zusammenhängenden Raum zulässt. ...."Gamma" (ist) die Zirkulations-/Wirbelstärke ("vortex force") dieses Wirbelfadens";
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