Mathematical Theory of Flight

We show by computational solution of the incompressible Navier-Stokes equations with friction force boundary conditions, that the classical inviscid circulation theory by Kutta-Zhukovsky for lift of a wing and laminar viscous boundary layer theory by Prandtl for drag, which have dominated 20th century flight mechanics, do not correctly describe the real turbulent airflow around a wing. We show that lift and drag essentially originate from a turbulent wake of counter-rotating rolls of low-pressure streamwise vorticity generated by a certain instability mechanism of potential flow at rear separation. The new theory opens the possibility of computational prediction of flight characteristics of an airplane using millions of meshpoints without resolving thin boundary layers, instead of the imposssible quadrillions required according to state-of-the-art for boundary layer resolution. 1 New and Old Theories of Flight As a corollary of the resolution of d’Alembert’s paradox of zero lift/drag of potential flow recently presented in this journal [26, 28], we outline in this article a mathematical theory for the generation of lift and drag of a wing in subsonic flight, which is fundamentally different from the classical theory by Kutta-Zhukovsky for lift in inviscid flow and by Prandtl for drag in viscous flow. We give evidence that the turbulent flow around a wing can be seen as a perturbation of zero-lift/drag potential flow resulting from a specific three-dimensional instability mechanism at separation generating a turbulent wake of counter-rotating low-pressure rolls of streamwise vorticity, a mechanism which changes the pressure distribution around the trailing edge so as to produce lift but also drag. By mathematical analysis and computation we thus identify the basic mechanism, seen as a modification of zero lift/drag potential flow, generating both lift and drag in real turbulent flow around a wing. On the other hand, we give evidence that the modification by Kutta-Zhukovsky consisting of large scale two-dimensional circulation around the section of the wing, which is the basic mechanism for lift according to classical theory representing state-of-the-art, is purely fictional without counterpart in real turbulent flow. We thus identify the true mechanism for both lift and drag of a wing, which is not captured by classical theory. The problem of explaining why it is possible to fly in the air using wings has haunted scientists since the birth of mathematical sciences. To fly, an upward force on the wing, referred to as lift L, has to be generated from the flow of air around the