Regarding Some Problems of the Kutta — Joukovskii Condition in Lifting Surface Theory

Abstract

We are interested in finding the velocity distribution at the wings of an aeroplane.Within the scope of a three-dimensional linear theory we analyse a model which is formulated as a mixed screen boundary value problem for the Helmholtz equation ( A + k 2 ) Qi = 0 in R3\S where Q, denotes the perturbation velocity potential, induced by the presence of the wings and s := L U w with the projection L of the wings onto the (x,y)-plane and the wake W .Not all Cauchy data are given explicitly on L , respectively W . These missing Cauchy data depend on the wing circulation r. r has to be fixed by the Kutta-Joukovskii condition: V@ should be finite near the trailing edge xt of L. To fulfil this condition in a way that all appearing terms can be defined mathematically exactly and belong to spaces which are physically meaningful, we propose to fix r by the condition of vanishing stress intensity factors of Qi near xt up to a certain order such thatIn the two-dimensional case, and if L is the left half-plane in IRz , we have an explicit formula to E W'2(2t) c L2(xt), > 0.calculate r and we can control the regularity of r and a.

Asymptotics of solutions to jou kovskii—kutta-type problems at infinity

“…Especially Meister considered a time-harmonically oscillating wing with reduced wavenumber k under the assumption of a weakly damping gas, i.e., Re k > 0, Im k > 0. The resulting model has been extended 1988 by Hebeker [5] under Meister's assumption of a weakly damping gas to the three-dimensional situation and has been investigated in [6], [12], 1131 in more detail. The full problem is: Find the perturbation velocity potential Q, as the sc+ lution of the problem…”

About singularities at angular points of a trailing edge under the Joukowskii-Kutta Condition

“…This problem appears in the linear theory of lifting surfaces and its solution implies a perturbation by a slightly curved wing of the velocity potential for a plane-parallel flow in the space 1 (see [27,10,11] etc.). The distinguishing features of the problem are an incomplete prescription of data on the wake behind the wing and the formulation of the Joukowskii-Kutta condition [12,14] which requires an additional regularity of solutions on a part of the polygonal line.…”

“…That is why we formulate the problem for v in the half-space 1 > "+x:z'0, only. According to References 10, 11 Since we are not going to prove existence of the solution v, there is no restriction on k3". The right-hand side g is supposed to be smooth on and M stands for the Machnumber.…”