Singular partial integro-differential equations arising in thin aerofoil theory

Abstract

A study is undertaken of three physical problems in which singular partial integro-differential equations arise. The equations result from applying thin aerofoil theory to two-dimensional potential flow over a free boundary. In all cases the work builds on existing steady models, for which ordinary singular integro-differential equations have been derived, to obtain time-dependent equations. The first of the physical problems examined is the sail equation, which describes the behaviour of a thin, flexible, inextensible sail in a high-Reynolds-number cross-flow at a small, but time-dependent, angle of incidence to the sail. A review of literature on the sail problem is presented. The unsteady equation is then solved numerically for various angles of incidence and sail masses. Variations of the unsteady sail equation are also derived to describe the sail as the sail mass becomes very small or very large, and to describe the flapping of a flag. An analytic expression is found for the low-mass sail given a particular set of functions for the angle of attack, for which the tension in the sail, and hence the lift, are zero.

The second problem examined is that of slot injection into a high-Reynolds-number cross-flow driven by an excess pressure in the slot. A literature review of slot injection to, and suction from, a free stream is presented, and then the unsteady injection problem is considered. Three different regimes are found for slot injection, according to the time scale of the pressure variations. Of these, the most important is the 'interactive' time scale, for which a third order singular partial integro-differential equation is obtained for the height of the shear layer separating the injected fluid from the free stream. For the other two cases, analytic solutions are found in terms of the known steady solutions.

More information

Identifiers

Catalogue record

Export record

ASCII CitationAtomBibTeXData Cite XMLDublin CoreDublin CoreEP3 XMLEndNoteHTML CitationHTML CitationHTML ListJSONMETSMODSMPEG-21 DIDLOpenURL ContextObjectOpenURL ContextObject in SpanRDF+N-TriplesRDF+N3RDF+XMLRIOXX2 XMLReferReference ManagerSimple Metadata