Kutta-Joukowski Lift Theorem

Two early aerodynamicists, Kutta in Germany and Joukowski in Russia, worked to quantify the lift achieved by an airflow over a spinning cylinder. The lift relationship is

两位早期空气动力学家，德国的库塔和俄罗斯的茹可夫斯基，致力于量化气流经过旋转圆柱体时所产生的升力。升力关系为

where ρ is the air density, V is the velocity of air flow relative to the cylinder, and G is called the "vortex strength". The vortex strength is given by

其中ρ是空气密度，V是空气相对于圆柱体的流速，G称为‘涡旋强度’。涡旋强度由下式给出

where ω is the angular velocity of spin of the cylinder. This can also be described as the surface speed (speed V_r = ωr of the surface associated with the rotation) times the circumference of the cylinder.

其中ω是圆柱体的自旋角速度。这也可以描述为表面速度（与旋转相关的表面速度V = ωr）乘以圆柱体的周长。

Like all aerodynamic lift, this seems a bit mysterious, but it can be looked at in terms of a redirection of the air motion. If the cylinder traps some air in a boundary layer at the cylinder surface and carries it around with it, shedding it downward, then it has given some of the air a downward momentum. That can act to give the cylinder an upward momentum in accordance with the principle of conservation of momentum. Another approach is to say that you have exerted a downward component of force on the air and by Newton's 3rd law there must be an upward force on the cylinder. Yet another approach is to say that the top of the cylinder is assisting the airstream, speeding up the flow on the top of the cylinder. Then by the Bernoulli equation, the pressure on the top of the cylinder is diminished, giving an effective lift.

如同所有空气动力学升力一样，这似乎有些神秘，但可以考虑为气流的重新定向。如果圆柱体在表面形成一个边界层并携带它一起运动，然后将它向下甩出，那么它就给部分空气赋予了向下的动量。根据动量守恒原理，这可以给圆柱体一个向上的动量。另一种说法是，你对空气施加了一个向下的力分量，根据牛顿第三定律，空气对圆柱体必须有一个向上的力。还有一种说法是，圆柱体的顶部协助了气流，加快了顶部的流速。然后根据伯努利方程，圆柱体顶部的压强降低，从而产生有效的升力。