Generalized Kutta-Joukowski theorem for multi-vortex and multi-airfoil flow (a lumped vortex model)

For purpose of easy identification of the role of free vortices on the lift and drag and for purpose of fast or engineering evaluation of forces for each individual body, we will extend in this paper the Kutta-Joukowski (KJ) theorem to the case of inviscid flow with multiple free vortices and multiple airfoils. The major simplification used in this paper is that each airfoil is represented by a lumped vortex, which may hold true when the distances between vortices and bodies are large enough. It is found that the Kutta-Joukowski theorem still holds provided that the local freestream velocity and the circulation of the bound vortex are modified by the induced velocity due to the outside vortices and airfoils. We will demonstrate how to use the present result to identify the role of vortices on the forces according to their position, strength and rotation direction. Moreover, we will apply the present results to a two-cylinder example of Crowdy and the Wagner example to demonstrate how to perform fast force approximation for multi-body and multi-vortex problems. The lumped vortex assumption has the advantage of giving such kinds of approximate results which are very easy to use. The lack of accuracy for such a fast evaluation will be compensated by a rigorous extension, with the lumped vortex assumption removed and with vortex production included, in a forthcoming paper.