Someone wrote in today to ask whether stopper position is important.
When you bought your flute it probably came with a cleaning rod with a mark on it to indicate the manufacturer's recommended position for the stopper. The distance is usually around 17.3 mm (for modern Boehm concert flutes) from the centre of the embouchure hole to the stopper face.
Moving the cork effectively changes the length, and therefore pitch, of your flute. However, the effects are not even across the entire length of the instrument. If you think about the proportional change: if you move your cork 1mm outward, then you have increased the distance to C1 by very roughly half a percent, but the distance to D1 has only increased by a quarter of a percent. Uneven.
Is the position of my stopper/cork right?
The manufacturer of your flue will have designed the taper of the headjoint and other dimensions to their exacting standards. They will have then calculated or deduced the optimal stopper position for best intonation and sound quality, and that is what they have marked on the cleaning rod.
If you change cleaning rod, the mark may be in a different place. Not all headjoints will play best with the stopper in the same place.
Narrowing the distance between the stopper and the centre of the embouchure can raise the pitch of your instrument - but it is uneven across left hand vs. right hand notes as seen above, and also across registers: the increase is far greater in the 3rd register, less in the 2nd, and much less in the first. The effect of increasing the distance is the opposite.
The "correct" position of the stopper is one in which the flute is in tune with itself (not a very helpful answer I know). You don't normally adjust the stopper position to alter pitch when playing - you pull or push the headjoint instead.
In my Haynes flute, which has A at 442hz, the flute feels more in tune with itself, and harmonics and overtones are in tune, when I have the stopper a little less than 17.1mm from the centre of the embouchure, and pull out the headjoint a little more. I'm sure this due to my playing style, the headjoint taper, the bore size, and the fact that I'm most often playing with colleagues at A440.
Setting the correct position
This is really best only done by experienced players.
The rough rule of thumb is that the distance between stopper face and centre of the embouchure should be equal to the inside diameter of the bore at the centre of the embouchure. That's not a simple thing to measure without the right tools, but your local repair shop or engineer should be able to help. Or contact the manufacturer.
Begin with your manufacturer's mark!
If you consider experimenting, I'd suggest getting your teacher involved as they will be able to guide you and judge the effect of changes. Make tiny changes, and then compare octaves and harmonics (and listen carefully to overtones) to check the flute it in tune with itself - and pull-out or push-in the HJ from the body as necessary.
If you are using a tuner to help, make sure you set it to the pitch of your flute, e.g. 440, or 442. Note that, when overblowing to get harmonics, not all octaves are designed to be in tune - which is why this is an exercise for experienced players. For example, overblowing D2 to D3 should (correctly) result in a slightly flat D3, which is corrected when you when lift the RH fingers for the correct D3 fingering.
If you are thinking: "my stopper is in the right place, but my top octave is very sharp - so I'll just pull the stopper out 2mm to help", then I'd advise resisting the change and seeking the advice of a good teacher. Lengthening the tube can kill the quality of sound (as well as affecting general intonation), and it is likely that fixing another fault with your instrument, or adjusting technique will yield a far better long-term result.
And again, be careful when swabbing out. Don't hit the stopper with your cleaning rod, or you'll likely move it.