These things usually work by feedback that causes pressure changes, that then causes the original flow to be re-directed, which then has the opposite effect on pressure, etc.

This is simpler to think about when considering a common ball whistle, like athletic coaches and refs usually have.  The part you blow in has a thin slot, which makes a laminar sheet of air blow accross a opening, but which then continues on into the round part.  However, there is no other opening in the round part, so eventually the pressure builds up and this pressure "breaks through" the laminar sheet that sort of "covered up" the opening.  This allows the pressure to release through the opening, which also re-directs the air from the nozzle to not go into the round part.  The pressure in the whistle then drops, the laminar sheet can re-form or resume its original un-redirected flow, which cause the pressure to build up again, etc.

The ball inside the whistle isn't necessary for making sound.  It periodically breaks up the process above, effectively modulating the whistle frequency at a much lower frequency.  I'm not sure why exactly this is done, but I think the sound with the ball is more interesting or attention getting or easier for humans to localize.