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

机译：BIE中Laplace方程的奇点；约科夫斯基后缘猜想

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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 f 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.
机译：该论文涉及与三维不可压缩的准势流（即零厚度涡流层（称为尾流）除外，到处都有势能的流）分析相关的前沿问题。具体来说，遵循Joukowski猜想在后缘处进行平滑流动，确定了避免边界积分表示中的奇异性或速度（在机翼周围的准潜在不可压缩流动中）所需的所有后缘条件。特别是，这些特征包括Kondrat'ev和Oleinik的奇点，以及Epton的涡旋线和边缘喷射的奇点。此外，跟随芒格勒和史密斯。使用后缘无压力不连续的Kutta条件，确定后缘尾流几何形状的行为。解决了特定的理论问题，其中包括（1）Joukowski猜想与Kutta条件之间的关系，以及（2）识别确保解决方案唯一性所必需的那些后沿条件（与自动满足的关系相反）解决方案）。关于第一个问题，在本文的主体中，使用Joukowski猜想和Kutta条件，就好像它们是独立的假设一样。然后，在附录A中，表明无需调用Kutta条件作为单独的假设，因为可以通过控制方程式和Joukowski猜想来获得它。为了阐明第二个问题，将理论分析与数值分析相结合。特别是，通过数值实验推论（未证明）确保唯一性所需的条件，只有为确保唯一性而似乎无涡流条件才是必需的。这是通过对准势流使用分段三次边界元方法实现的，这是作者及其合作者引入的高阶公式的扩展（公式的顺序足以解决所有理论上的尾随边缘条件）。重点是在简单相连的区域中的稳定流量。但是，还研究了与多重连通区域中的非稳定流动有关的一些问题。最后，确定了一些需要额外工作的未解决问题。

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