There has been endless confusion about how this type of whistle really works. If you investigate the flow pattern, you see the sort of vortex patterns in your picture, but that doesn't necessarily mean the vortices cause the sound. In fact the reverse is true - the sound causes the vortices!
The basic explanation (which also applies to organ pipes, and musical instruments like the tin whistle, and recorder) depends on Bernouilli's principle. It's simpler to think about making a sound by blowing over the open end of a bottle, because the "pipe" part of the whistle is really the same as the bottle bent round through 90 degrees. The important thing is not that you blow "into the pipe", but that you blow "across the hole at the end of the pipe".
When you blow across a bottle, the pressure in the moving air stream is decreased, and some air is "sucked out" of the bottle into the low pressure air stream.
However, the extra air merging into the airstream bends the airstream away from the mouth of the bottle, which reduces the amount of "suction."
The air inside the bottle has a natural frequency of vibration, which depends on the size and shape of the bottle. This vibration is excited by the sudden pressure change at the mouth of the bottle, and after half a cycle of vibration it acts to suck some air back into the bottle. That pulls the air stream you are blowing across the bottle back towards the mouth of the bottle, and the cycle repeats.
The critical parameter here is the time it takes the blown air stream to travel across the mouth of the bottle, compared with the time for one vibration cycle of the air inside the bottle. If the two time intervals have the correct relationship, the oscillations can build up in amplitude. This explains why if you blow gently (low velocity), you produce no sound at all, and if you gradually blow harder, suddenly the sound starts. Depending on the geometry of the whole system, if you blow very hard the sound may "jump" to a different higher frequency. In fact it is possible to adjust an organ pipe so that it will successively produce a sound at with 3 or 4 different pitches, depending of the wind pressure (and therefore the air velocity) used to blow it.
The vortices are simply by-products of the oscillating air stream hitting the edge of the bottle mouth.
The "wrong" explanations start from the correct observation that simply blowing a jet of air through a nozzle can produce a pattern of vortices, which may repeat at a definite frequency. But for the geometry and blowing pressure of a typical whistle, the frequency of that vortex pattern (if it exists at all) is very different from the frequency of the sound produced by the whistle, and it's hard to invent a good reason why the vortices should cause the sound.