A review of the use of boundary integral equations in aerodynamics is presented, with the objective of addressing what has been accomplished and, even more, what remains to be done. The paper is limited to aerodynamics of aeronautical type, with emphasis on unsteady flows (incompressible and compressible, potential and viscous). For potential flows, both incompressible and compressible flows are considered; the issue of the boundary conditions on the wake and on the trailing edge are addressed in some detail (in particular, some unresolved issues related to the impulsive start are pointed out). For incompressible viscous flows, the use of boundary integral equations in the non-primitive variable formulation are addressed: the Helmholtz decomposition and a decomposition recently introduced (and here referred to as the Poincare´ decomposition) are presented, along with their relationship. The latter is used to examine the relationship between potential and attached viscous flows (in particular, it is shown how the Poincare´ representation, for vortex layers of infinitesimal thickness, reduces to the potential-flow representation). The extension to compressible flows is also briefly outlined and the relative advantages of the two decompositions are discussed. Throughout the paper the emphasis is on the derivation and the interpretation of the boundary integral equations; issues related to the discretization (ie, panel methods, boundary element methods) are barely addressed. For numerical results, which are not included here, the reader is referred to the original references.

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