Word for "rush of air"

As far as I know, there is no distinction made between air motion caused by natural means, air motion caused by mechanical means, and air motion experienced due to the observer being in motion while the air stands still; they're all "wind" or "breeze" or "gale" or "gust" or "zephyr" or "scirocco" or whatever other word suits the velocity of the air.

I stuck my head out the window to feel the cool breeze.

I relaxed in the gentle breeze from the fan.

I hate riding my motorcycle with a helmet on because it means I can't feel the wind in my hair.

In aerodynamic and aeronautics, it is called the relative wind. I don't know if there exist a non-technical to describe it.

I'd prefer rush of wind...

I know you are looking for one word, but I don't see a perfect one-word alternative, and rush of wind (not air) fits perfectly in poetic/emotional terms.

Certainly with airplanes you have the term "slipstream" to refer to the air coming from the propeller(s) and following the craft.