Singularities in BIEs for the Laplace equation; Joukowski trailing-edge conjecture revisited

Abstract

The paper deals with trailing-edge issues connected with the analysis of three-dimensional incompressible quasi-potential flows (i.e. flows that are potential everywhere, except for a zero-thickness vortex layer, called the wake). Specifically, following the Joukowski conjecture of smooth flow at the trailing edge, all the trailing-edge conditions that are required to avoid singularities in the boundary integral representation for the velocity, in a quasi-potential incompressible flow around a wing, are identified. In particular, these include the Kondrat'ev and Oleinik singularity as well as the vortex-line and the edge-jet singularities of Epton. Also, following Mangler and Smith, the behavior of the wake geometry at the trailing edge is determined, using the Kutta condition of no pressure discontinuity at the trailing edge. Specific theoretical issues are addressed which include (1) the relationship between Joukowski conjecture and Kutta condition, and (2) identification of those trailing-edge conditions that are necessary to assure the uniqueness of the solution (as opposite to relationships that are automatically satisfied by the solution). Regarding the first issue, in the main body of the paper, the Joukowski conjecture and the Kutta condition are used as if they were independent assumptions; then, in Appendix A, it is shown that the Kutta condition need not be invoked as a separate assumption since it may be obtained as a consequence of the governing equations and of the Joukowski conjecture. In order to clarify the second issue, the theoretical analysis is coupled with a numerical one. In particular, the conditions necessary to insure uniqueness are inferred (not proven) through numerical experimentation: only the no-vortex-line condition appears to be necessary to insure uniqueness. This is accomplished by using a piecewise-cubic boundary-element method for quasi-potential flows that is an extension of a high-order formulation introduced by the authors and their collaborators (the order of the formulation is adequate to address all the theoretical trailing-edge conditions uncovered). The emphasis is on steady flows in simply connected regions; however, some issues related to unsteady flows in multiply connected regions are also examined. Finally, several open problems that require additional work are identified.