Affiliations: Weierstraß‐Institut für Angewandte Analysis und Stochastik im Forschungsverbund Berlin e.V. (WIAS), Mohrenstraße 39, 10117 Berlin, Germany E‐mail: hinder@wias‐berlin.de | Laboratory of Math. Modelling of Wave Phenomena, Institute of Mechanical Engineering Problems, V.O. Bol’shoy pr., 61, Saint Petersburg, 199178, Russia E‐mail: [email protected]
Note: [] Corresponding author.
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 (\Delta + k^{ 2}) \varPhi =0 in \mathbb{R}^{ 3}\backslash \overline{S}, where \varPhi denotes the perturbation velocity potential, induced by the presence of the wings and \overline{S} := \overline{L} \cup \overline{W} with the projection L of the wings onto the (y_{ 1},y_{ 2})‐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 \varGamma . \varGamma has to be fixed by the Kutta–Joukovskii condition: \nabla \varPhi should be finite near the trailing edge x_{\rm t} of L. We reduce here this screen problem to an equivalent mixed boundary value problem in \mathbb{R}^{ 3}_{ +}. The main problem is in both cases the calculation of \varGamma . In order to find \varPhi we use the method of matched asymptotics for some small geometrical parameter \varepsilon and the ansatz \varGamma = \varGamma _{ 0} + \varepsilon \varGamma _{ 1} + \cdots which makes it possible to split the problem into a sequence of problems for \varGamma _{ 0},\varGamma _{ 1}, \ldots\,. Concretely, we calculate \varGamma _{ 0} and \varGamma _{ 1} explicitly by the demand of vanishing intensity factors of the solutions of the corresponding mixed problems at the borderline between L and W. Especially, we point out that \varGamma _{ 0} can be obtained by solving a two‐dimensional problem for every cross‐section of L while \varGamma _{ 1} indicates the interaction of these cross‐sections.
