Unsteady 3D nonlinear Kutta-Joukovski condition for thin lifting surfaces

Abstract

Considering unsteady potential flows, the fluid velocity regularity condition at a wake shedding edge – often referred to as the Kutta-Joukovski condition – has been clarified in the 2D nonlinear and 3D linearized cases only, and the usual nonlinear direct numerical approaches, as they are built on an already discretised view of the problem, skip over the theoretical question. A nonlinear Kutta-Joukovski condition is proposed here for unsteady 3D flows around thin airfoils, for which the body and its trailing-edge-shed wake are represented by double layers. The fluid velocity field, deriving from the potential function, is expressed using the two usual terms, relating to the edge of the sheet and its surface. Removing the singularity for the first term leads to the well-known condition for the potential jump across the shedding edge to be continuous from the body to the wake. For the second, works by Legras, for the steady case – and readily extendible to the unsteady one – exhibit its logarithmic singular behavior and allow to derive a regularity condition that deals with the surface gradient of the potential jump across the shedding edge. Lastly, implemented in a general non linearized situation, these conditions lead to an expression relating the geometric and doublets distribution characteristics of the two surfaces (wing and wake).