The physical interpretation of the lift discrepancy in Lanchester–Prandtl lifting wing theory for Euler flow, leading to the proposal of an alternative model in Oseen flow

Abstract

Consider uniform, steady potential and incompressible flow past a fixed thin wing inclined at a small angle to the flow. An investigation is conducted into the physical interpretation and consequences of the revision by Chadwick (Chadwick 2005 Proc. R. Soc. A 461 , 1–18) of the Lanchester–Prandtl lifting wing theory in Euler flow. In the present paper, the lift is evaluated from the pressure distribution over the top and bottom surfaces together with a contribution across the trailing edge of the wing. It is shown that this contribution across the trailing edge has previously been erroneously omitted in the standard approach but confirms and provides a physical explanation for the discrepancy in the lift calculation found by Chadwick. This results in a reduction of the lift by a half, but this reduction in lift from the additional calculation is not the right answer, and instead arises from a mathematical discrepancy with the physically observed lift. The discrepancy is due to the pressure becoming singular at the trailing edge in the Euler model. The physical explanation is that in real flow the pressure is regularized by the action of viscosity and so is not singular at the trailing edge. So this lift force at the trailing edge is present in the Euler model but not in a real flow. In a real flow, the viscous effects prevent the pressure becoming singular and so there is no lift force, and consequently no large torque, concentrated at the trailing edge. That the lift force at the trailing edge has been ignored in the Lanchester–Prandtl theory in Euler flow has led to fortuitous agreement with the experimental results on real flows. This shows that the Euler model does not properly predict forces for this problem in which there are singularities (vorticity) within the flow field. We propose a revision to the Euler model by allowing a counterbalancing singular viscous velocity term to reside on the trailing vortex sheet, which is derived from the lift oseenlet. This viscous term ensures that the pressure and velocity are not singular in the flow field. The consequences for the flow due to the inclusion of this term for extending triple-deck and similar asymptotic theories to the case for flow past wings rather than aerofoils are discussed, as well as for the (ideal) high Reynolds number limit and for slender body lift.