Mathematical Theory of Sailing

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 and laminar viscous boundary layer theory by Prandtl for drag, which have dominated 20th century fluid dynamics, do not correctly describe the real turbulent airflow around a sail under tacking. 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 ab initio computational prediction of characteristics of a sailing boat using les than a million meshpoints without resolving thin boundary layers, instead of the imposssible quadrillions required according to state-of-the-art for boundary layer resolution. 1 New Theory of Sailing As a corollary of the resolution of d’Alembert’s paradox of zero lift/drag of potential flow [32, 37] and the related mathematical theory of flight [34, 35, 36], we outline in this article a mathematical theory for the generation of forward drive force and sideway heeling force from the combined action of the sail and keel of a sailing boat under tacking against the wind, which is fundamentally different from the classical theory by Kutta-Zhukovsky for lift in inviscid flow and by Prandtl for drag in viscous flow. A keel moving through water acts like a symmetric wing generating lift which balances the heeling from the sail. A sail in a flow of air also acts like a wing with the drive coming from a forward component of lift and the heeling from the sideway component of lift. But there is an important difference in the action of a sail and a keel, with the purpose of the sail to give forward drive at the price of heeling, and the purpose of the keel to give lift at the price of drag. A sail requires a relatively large angle of attack α = 15 − 25 degrees to give sufficient drive to overcome the the total drag from the sail, keel and hull, while for a keel the angle of attack is smaller with α = 5− 10. In the gliding flight of birds and airplanes with fixed wings at subsonic speeds, the lift/drag ratio L D with L the lift and D the drag, is typically between 10 and 20, which means that a good glider can glide up to 20 meters upon loosing 1 meter in altitude, or that Charles Lindberg could cross the Atlantic in 1927 at a speed of 50 m/s in his 2000 kg Spirit of St Louis at an effective engine thrust of 150 kp (with L D = 2000/150 ≈ 13) from 100 horse powers.